[{"data":1,"prerenderedAt":13629},["ShallowReactive",2],{"navigation_docs":3,"learnalog_locale_counterpart__zh_time-series_03-prediction":1782,"-en-time-series-03-prediction":1783,"-en-time-series-03-prediction-surround":13624},[4,1038],{"title":5,"path":6,"stem":7,"children":8},"En","\u002Fen","en",[9,12,58,250,406,477,534,715,798,824,943],{"title":10,"path":6,"stem":11},"","en\u002Findex",{"title":13,"path":14,"stem":15,"children":16},"Research Skills and Academic Writing","\u002Fen\u002Facademic-writing","en\u002Facademic-writing\u002Findex",[17,18,22,26,30,34,38,42,46,50,54],{"title":13,"path":14,"stem":15},{"title":19,"path":20,"stem":21},"1. Research Questions, Scope, and Feasibility","\u002Fen\u002Facademic-writing\u002F01-research-questions-and-planning","en\u002Facademic-writing\u002F01-research-questions-and-planning",{"title":23,"path":24,"stem":25},"2. Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js Visualizations","\u002Fen\u002Fplayground\u002F05-chartjs","en\u002Fplayground\u002F05-chartjs",{"title":825,"path":826,"stem":827,"children":828,"page":249},"Statistics For Insurance","\u002Fen\u002Fstatistics-for-insurance","en\u002Fstatistics-for-insurance",[829,835,857,883,905,927,937],{"title":830,"path":831,"stem":832,"children":833},"Statistics for General Insurance","\u002Fen\u002Fstatistics-for-insurance\u002F01-intro","en\u002Fstatistics-for-insurance\u002F01-intro\u002Findex",[834],{"title":830,"path":831,"stem":832},{"title":836,"path":837,"stem":838,"children":839},"Claims Development and Reserving","\u002Fen\u002Fstatistics-for-insurance\u002F02-chain-ladder","en\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002Findex",[840,841,845,849,853],{"title":836,"path":837,"stem":838},{"title":842,"path":843,"stem":844},"Basic Chain Ladder","\u002Fen\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F01-basic-chain","en\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F01-basic-chain",{"title":846,"path":847,"stem":848},"Frequency–Severity Reserving","\u002Fen\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F02-average-per-claim","en\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F02-average-per-claim",{"title":850,"path":851,"stem":852},"Bornhuetter–Ferguson Method","\u002Fen\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F03-b-f-method","en\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F03-b-f-method",{"title":854,"path":855,"stem":856},"Uncertainty and Diagnostics","\u002Fen\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F04-uncertainty-diagnostics","en\u002Fstatistics-for-insurance\u002F02-chain-ladder\u002F04-uncertainty-diagnostics",{"title":858,"path":859,"stem":860,"children":861},"Frequency and Severity Models","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002Findex",[862,863,867,871,875,879],{"title":858,"path":859,"stem":860},{"title":864,"path":865,"stem":866},"Core Severity Models","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist\u002F01-loss-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002F01-loss-dist",{"title":868,"path":869,"stem":870},"Tail Models and Extreme Values","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist\u002F02-more-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002F02-more-dist",{"title":872,"path":873,"stem":874},"Claim Count Models","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist\u002F03-case-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002F03-case-dist",{"title":876,"path":877,"stem":878},"Fitting and Validating Loss Models","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist\u002F04-fit-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002F04-fit-dist",{"title":880,"path":881,"stem":882},"Mixtures and Heterogeneity","\u002Fen\u002Fstatistics-for-insurance\u002F03-common-dist\u002F05-mix-dist","en\u002Fstatistics-for-insurance\u002F03-common-dist\u002F05-mix-dist",{"title":884,"path":885,"stem":886,"children":887},"Reinsurance as a Loss Transformation","\u002Fen\u002Fstatistics-for-insurance\u002F04-reinsurance","en\u002Fstatistics-for-insurance\u002F04-reinsurance\u002Findex",[888,889,893,897,901],{"title":884,"path":885,"stem":886},{"title":890,"path":891,"stem":892},"Proportional Reinsurance","\u002Fen\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F01-proportional","en\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F01-proportional",{"title":894,"path":895,"stem":896},"Excess-of-Loss Reinsurance","\u002Fen\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F02-excess","en\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F02-excess",{"title":898,"path":899,"stem":900},"Inflation and Layer Erosion","\u002Fen\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F03-inflation","en\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F03-inflation",{"title":902,"path":903,"stem":904},"Reinsurance Decision Lab","\u002Fen\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F04-examples","en\u002Fstatistics-for-insurance\u002F04-reinsurance\u002F04-examples",{"title":906,"path":907,"stem":908,"children":909},"Aggregate Risk and Capital","\u002Fen\u002Fstatistics-for-insurance\u002F05-risk-theory","en\u002Fstatistics-for-insurance\u002F05-risk-theory\u002Findex",[910,911,915,919,923],{"title":906,"path":907,"stem":908},{"title":912,"path":913,"stem":914},"Collective Risk Model","\u002Fen\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F01-collective","en\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F01-collective",{"title":916,"path":917,"stem":918},"Individual Risk Model","\u002Fen\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F02-individual","en\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F02-individual",{"title":920,"path":921,"stem":922},"Aggregate Risk Computation Lab","\u002Fen\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F03-examples","en\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F03-examples",{"title":924,"path":925,"stem":926},"Tail Risk, Dependence, and Capital","\u002Fen\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F04-tail-capital","en\u002Fstatistics-for-insurance\u002F05-risk-theory\u002F04-tail-capital",{"title":928,"path":929,"stem":930,"children":931},"Surplus and Ruin Theory","\u002Fen\u002Fstatistics-for-insurance\u002F06-ruin-theory","en\u002Fstatistics-for-insurance\u002F06-ruin-theory\u002Findex",[932,933],{"title":928,"path":929,"stem":930},{"title":934,"path":935,"stem":936},"Finite-Time Ruin Simulation","\u002Fen\u002Fstatistics-for-insurance\u002F06-ruin-theory\u002F01-finite-time-simulation","en\u002Fstatistics-for-insurance\u002F06-ruin-theory\u002F01-finite-time-simulation",{"title":938,"path":939,"stem":940,"children":941},"Portfolio Risk Capstone","\u002Fen\u002Fstatistics-for-insurance\u002F07-capstone","en\u002Fstatistics-for-insurance\u002F07-capstone\u002Findex",[942],{"title":938,"path":939,"stem":940},{"title":944,"path":945,"stem":946,"children":947,"page":249},"Time Series","\u002Fen\u002Ftime-series","en\u002Ftime-series",[948,958,964,970,976,982,988,994,1016],{"title":949,"path":950,"stem":951,"children":952},"Classical Time Series — Course Guide","\u002Fen\u002Ftime-series\u002F00-intro","en\u002Ftime-series\u002F00-intro\u002Findex",[953,954],{"title":949,"path":950,"stem":951},{"title":955,"path":956,"stem":957},"Preparation — Stationarity in 30 Minutes","\u002Fen\u002Ftime-series\u002F00-intro\u002F01-stationary","en\u002Ftime-series\u002F00-intro\u002F01-stationary",{"title":959,"path":960,"stem":961,"children":962},"Module 1 — Processes, Dependence, and Stationarity","\u002Fen\u002Ftime-series\u002F01-stochastic-process","en\u002Ftime-series\u002F01-stochastic-process\u002Findex",[963],{"title":959,"path":960,"stem":961},{"title":965,"path":966,"stem":967,"children":968},"Module 2 — ARMA, ARIMA, and Model Identification","\u002Fen\u002Ftime-series\u002F02-arma","en\u002Ftime-series\u002F02-arma\u002Findex",[969],{"title":965,"path":966,"stem":967},{"title":971,"path":972,"stem":973,"children":974},"Module 3 — Linear Prediction and State-Space Recursions","\u002Fen\u002Ftime-series\u002F03-prediction","en\u002Ftime-series\u002F03-prediction\u002Findex",[975],{"title":971,"path":972,"stem":973},{"title":977,"path":978,"stem":979,"children":980},"Module 4 — Estimation, Likelihood, and Inference","\u002Fen\u002Ftime-series\u002F04-estimation","en\u002Ftime-series\u002F04-estimation\u002Findex",[981],{"title":977,"path":978,"stem":979},{"title":983,"path":984,"stem":985,"children":986},"Module 5 — Systems, Seasonality, and Cointegration","\u002Fen\u002Ftime-series\u002F05-multi-ar","en\u002Ftime-series\u002F05-multi-ar\u002Findex",[987],{"title":983,"path":984,"stem":985},{"title":989,"path":990,"stem":991,"children":992},"Module 6 — Spectral Analysis, Cycles, and Filters","\u002Fen\u002Ftime-series\u002F06-spectral-analysis","en\u002Ftime-series\u002F06-spectral-analysis\u002Findex",[993],{"title":989,"path":990,"stem":991},{"title":995,"path":996,"stem":997,"children":998},"R Matrix Laboratory","\u002Fen\u002Ftime-series\u002F07-r-implementation","en\u002Ftime-series\u002F07-r-implementation\u002Findex",[999,1000,1004,1008,1012],{"title":995,"path":996,"stem":997},{"title":1001,"path":1002,"stem":1003},"R Matrix Lab 1 — Covariance Geometry","\u002Fen\u002Ftime-series\u002F07-r-implementation\u002F01-covariance-matrices","en\u002Ftime-series\u002F07-r-implementation\u002F01-covariance-matrices",{"title":1005,"path":1006,"stem":1007},"R Matrix Lab 2 — AR Recursions and Yule–Walker Equations","\u002Fen\u002Ftime-series\u002F07-r-implementation\u002F02-ar-recursions","en\u002Ftime-series\u002F07-r-implementation\u002F02-ar-recursions",{"title":1009,"path":1010,"stem":1011},"R Matrix Lab 3 — Prediction and Gaussian Likelihood","\u002Fen\u002Ftime-series\u002F07-r-implementation\u002F03-prediction-likelihood","en\u002Ftime-series\u002F07-r-implementation\u002F03-prediction-likelihood",{"title":1013,"path":1014,"stem":1015},"R Matrix Lab 4 — State Space and Kalman Filtering","\u002Fen\u002Ftime-series\u002F07-r-implementation\u002F04-state-space","en\u002Ftime-series\u002F07-r-implementation\u002F04-state-space",{"title":1017,"path":1018,"stem":1019,"children":1020},"Optional Python Appendix — Empirical Forecasting","\u002Fen\u002Ftime-series\u002F08-python-implementation","en\u002Ftime-series\u002F08-python-implementation\u002Findex",[1021,1022,1026,1030,1034],{"title":1017,"path":1018,"stem":1019},{"title":1023,"path":1024,"stem":1025},"Optional Python Lab 1 — Explore Before Modeling","\u002Fen\u002Ftime-series\u002F08-python-implementation\u002F01-exploration","en\u002Ftime-series\u002F08-python-implementation\u002F01-exploration",{"title":1027,"path":1028,"stem":1029},"Optional Python Lab 2 — Diagnose Stationarity","\u002Fen\u002Ftime-series\u002F08-python-implementation\u002F02-stationarity","en\u002Ftime-series\u002F08-python-implementation\u002F02-stationarity",{"title":1031,"path":1032,"stem":1033},"Optional Python Lab 3 — Fit and Audit ARMA Errors","\u002Fen\u002Ftime-series\u002F08-python-implementation\u002F03-arma-fitting","en\u002Ftime-series\u002F08-python-implementation\u002F03-arma-fitting",{"title":1035,"path":1036,"stem":1037},"Optional Python Lab 4 — Forecast, Backtest, and Monitor","\u002Fen\u002Ftime-series\u002F08-python-implementation\u002F04-forecasting","en\u002Ftime-series\u002F08-python-implementation\u002F04-forecasting",{"title":1039,"path":1040,"stem":1041,"children":1042},"Zh","\u002Fzh","zh",[1043,1045,1063,1247,1323,1365,1421,1467,1543,1625,1633,1731],{"title":10,"path":1040,"stem":1044},"zh\u002Findex",{"title":1046,"path":1047,"stem":1048,"children":1049},"学术写作与文献综述","\u002Fzh\u002Facademic-writing","zh\u002Facademic-writing\u002Findex",[1050,1051,1055,1059],{"title":1046,"path":1047,"stem":1048},{"title":1052,"path":1053,"stem":1054},"4. 文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 从分块结构到问题链：文献综述示例","\u002Fzh\u002Facademic-writing\u002F10-从分块结构到问题链-文献综述示例","zh\u002Facademic-writing\u002F10-从分块结构到问题链-文献综述示例",{"title":59,"path":1064,"stem":1065,"children":1066,"page":249},"\u002Fzh\u002Faccounting","zh\u002Faccounting",[1067,1073,1087,1191],{"title":1068,"path":1069,"stem":1070,"children":1071},"会计学学习路线图","\u002Fzh\u002Faccounting\u002F00-index","zh\u002Faccounting\u002F00-index",[1072],{"title":1068,"path":1069,"stem":1070},{"title":1074,"path":1075,"stem":1076,"children":1077},"附录","\u002Fzh\u002Faccounting\u002Fappendix","zh\u002Faccounting\u002Fappendix\u002Findex",[1078,1079,1083],{"title":1074,"path":1075,"stem":1076},{"title":1080,"path":1081,"stem":1082},"综合示例与常见陷阱","\u002Fzh\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","zh\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":1084,"path":1085,"stem":1086},"会计术语速查表","\u002Fzh\u002Faccounting\u002Fappendix\u002F26-glossary","zh\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":1088,"path":1089,"stem":1090,"children":1091},"金融会计","\u002Fzh\u002Faccounting\u002Ffinancial-accounting","zh\u002Faccounting\u002Ffinancial-accounting\u002Findex",[1092,1093,1111,1125,1159,1173],{"title":1088,"path":1089,"stem":1090},{"title":1094,"path":1095,"stem":1096,"children":1097},"1. 基础","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","zh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[1098,1099,1103,1107],{"title":1094,"path":1095,"stem":1096},{"title":1100,"path":1101,"stem":1102},"会计信息目标与质量特征","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","zh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":1104,"path":1105,"stem":1106},"会计等式与要素","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","zh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":1108,"path":1109,"stem":1110},"记账基础与原则","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","zh\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":1112,"path":1113,"stem":1114,"children":1115},"2. 交易记录","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","zh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[1116,1117,1121],{"title":1112,"path":1113,"stem":1114},{"title":1118,"path":1119,"stem":1120},"复式记账与借贷规则","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","zh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":1122,"path":1123,"stem":1124},"会计循环","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","zh\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":1126,"path":1127,"stem":1128,"children":1129},"3. 计量与调整","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[1130,1131,1135,1139,1143,1147,1151,1155],{"title":1126,"path":1127,"stem":1128},{"title":1132,"path":1133,"stem":1134},"收入确认","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":1136,"path":1137,"stem":1138},"存货","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":1140,"path":1141,"stem":1142},"应收账款","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":1144,"path":1145,"stem":1146},"固定资产","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":1148,"path":1149,"stem":1150},"无形资产","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":1152,"path":1153,"stem":1154},"租赁","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":1156,"path":1157,"stem":1158},"所得税","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","zh\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":1160,"path":1161,"stem":1162,"children":1163},"4. 报表与现金","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","zh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[1164,1165,1169],{"title":1160,"path":1161,"stem":1162},{"title":1166,"path":1167,"stem":1168},"财务报表","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","zh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":1170,"path":1171,"stem":1172},"现金控制","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","zh\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":1174,"path":1175,"stem":1176,"children":1177},"5. 分析与比较","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","zh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[1178,1179,1183,1187],{"title":1174,"path":1175,"stem":1176},{"title":1180,"path":1181,"stem":1182},"财务比率分析","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","zh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":1184,"path":1185,"stem":1186},"IFRS 与 US GAAP 对比","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","zh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":1188,"path":1189,"stem":1190},"2026 准则更新与报告案例","\u002Fzh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F27-current-standards-2026","zh\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F27-current-standards-2026",{"title":1192,"path":1193,"stem":1194,"children":1195},"管理会计","\u002Fzh\u002Faccounting\u002Fmanagement-accounting","zh\u002Faccounting\u002Fmanagement-accounting\u002Findex",[1196,1197,1215,1233],{"title":1192,"path":1193,"stem":1194},{"title":1198,"path":1199,"stem":1200,"children":1201},"1. 成本基础","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","zh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[1202,1203,1207,1211],{"title":1198,"path":1199,"stem":1200},{"title":1204,"path":1205,"stem":1206},"成本概念","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","zh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":1208,"path":1209,"stem":1210},"成本核算系统","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","zh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":1212,"path":1213,"stem":1214},"变动成本法 vs. 吸收成本法","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","zh\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":1216,"path":1217,"stem":1218,"children":1219},"2. 计划与控制","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","zh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[1220,1221,1225,1229],{"title":1216,"path":1217,"stem":1218},{"title":1222,"path":1223,"stem":1224},"本量利分析","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","zh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":1226,"path":1227,"stem":1228},"预算与差异分析","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","zh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":1230,"path":1231,"stem":1232},"绩效评价","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","zh\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":1234,"path":1235,"stem":1236,"children":1237},"3. 决策与投资","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","zh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[1238,1239,1243],{"title":1234,"path":1235,"stem":1236},{"title":1240,"path":1241,"stem":1242},"短期决策","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","zh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":1244,"path":1245,"stem":1246},"资本预算","\u002Fzh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","zh\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",{"title":1248,"path":1249,"stem":1250,"children":1251},"资产定价理论","\u002Fzh\u002Fasset-pricing","zh\u002Fasset-pricing\u002Findex",[1252,1253,1259,1265,1271,1277,1283,1289,1295,1301,1307,1313,1319],{"title":1248,"path":1249,"stem":1250},{"title":1254,"path":1255,"stem":1256,"children":1257},"第一章：引言与基础","\u002Fzh\u002Fasset-pricing\u002F01-intro","zh\u002Fasset-pricing\u002F01-intro\u002Findex",[1258],{"title":1254,"path":1255,"stem":1256},{"title":1260,"path":1261,"stem":1262,"children":1263},"第二章：效用理论与风险偏好","\u002Fzh\u002Fasset-pricing\u002F02-utility","zh\u002Fasset-pricing\u002F02-utility\u002Findex",[1264],{"title":1260,"path":1261,"stem":1262},{"title":1266,"path":1267,"stem":1268,"children":1269},"第三章：均值-方差分析","\u002Fzh\u002Fasset-pricing\u002F03-mean-variance","zh\u002Fasset-pricing\u002F03-mean-variance\u002Findex",[1270],{"title":1266,"path":1267,"stem":1268},{"title":1272,"path":1273,"stem":1274,"children":1275},"第四章：资本资产定价模型(CAPM)","\u002Fzh\u002Fasset-pricing\u002F04-capm","zh\u002Fasset-pricing\u002F04-capm\u002Findex",[1276],{"title":1272,"path":1273,"stem":1274},{"title":1278,"path":1279,"stem":1280,"children":1281},"第五章：因子模型","\u002Fzh\u002Fasset-pricing\u002F05-factor-models","zh\u002Fasset-pricing\u002F05-factor-models\u002Findex",[1282],{"title":1278,"path":1279,"stem":1280},{"title":1284,"path":1285,"stem":1286,"children":1287},"第六章：跨期资产定价","\u002Fzh\u002Fasset-pricing\u002F06-intertemporal","zh\u002Fasset-pricing\u002F06-intertemporal\u002Findex",[1288],{"title":1284,"path":1285,"stem":1286},{"title":1290,"path":1291,"stem":1292,"children":1293},"第七章：期权定价理论","\u002Fzh\u002Fasset-pricing\u002F07-options","zh\u002Fasset-pricing\u002F07-options\u002Findex",[1294],{"title":1290,"path":1291,"stem":1292},{"title":1296,"path":1297,"stem":1298,"children":1299},"第八章：固定收益证券","\u002Fzh\u002Fasset-pricing\u002F08-fixed-income","zh\u002Fasset-pricing\u002F08-fixed-income\u002Findex",[1300],{"title":1296,"path":1297,"stem":1298},{"title":1302,"path":1303,"stem":1304,"children":1305},"第九章：市场有效性与异象","\u002Fzh\u002Fasset-pricing\u002F09-efficiency","zh\u002Fasset-pricing\u002F09-efficiency\u002Findex",[1306],{"title":1302,"path":1303,"stem":1304},{"title":1308,"path":1309,"stem":1310,"children":1311},"第十章：数值方法与实证应用","\u002Fzh\u002Fasset-pricing\u002F10-empirical","zh\u002Fasset-pricing\u002F10-empirical\u002Findex",[1312],{"title":1308,"path":1309,"stem":1310},{"title":1314,"path":1315,"stem":1316,"children":1317},"第十一章：资产定价浏览器交互实验","\u002Fzh\u002Fasset-pricing\u002F11-interactive-labs","zh\u002Fasset-pricing\u002F11-interactive-labs\u002Findex",[1318],{"title":1314,"path":1315,"stem":1316},{"title":1320,"path":1321,"stem":1322},"第十二章：前沿文献与现代资产定价案例（2023—2026）","\u002Fzh\u002Fasset-pricing\u002F12-frontier-literature-2026","zh\u002Fasset-pricing\u002F12-frontier-literature-2026",{"title":1324,"path":1325,"stem":1326,"children":1327},"计量经济学","\u002Fzh\u002Feconometrics","zh\u002Feconometrics\u002Findex",[1328,1329,1333,1337,1341,1345,1349,1353,1357,1361],{"title":1324,"path":1325,"stem":1326},{"title":1330,"path":1331,"stem":1332},"第一章：数据、概率与回归对象","\u002Fzh\u002Feconometrics\u002F01-data-and-regression","zh\u002Feconometrics\u002F01-data-and-regression",{"title":1334,"path":1335,"stem":1336},"第二章：OLS、矩阵与几何解释","\u002Fzh\u002Feconometrics\u002F02-ols-and-geometry","zh\u002Feconometrics\u002F02-ols-and-geometry",{"title":1338,"path":1339,"stem":1340},"第三章：统计推断与稳健标准误","\u002Fzh\u002Feconometrics\u002F03-inference-and-robustness","zh\u002Feconometrics\u002F03-inference-and-robustness",{"title":1342,"path":1343,"stem":1344},"第四章：内生性、工具变量与两阶段最小二乘","\u002Fzh\u002Feconometrics\u002F04-endogeneity-and-iv","zh\u002Feconometrics\u002F04-endogeneity-and-iv",{"title":1346,"path":1347,"stem":1348},"第五章：面板数据与时间序列","\u002Fzh\u002Feconometrics\u002F05-panel-and-time-series","zh\u002Feconometrics\u002F05-panel-and-time-series",{"title":1350,"path":1351,"stem":1352},"第六章：估计方法与因果设计的共同基础","\u002Fzh\u002Feconometrics\u002F06-estimation-and-causal-design","zh\u002Feconometrics\u002F06-estimation-and-causal-design",{"title":1354,"path":1355,"stem":1356},"第七章：可重复计量实证项目","\u002Fzh\u002Feconometrics\u002F07-reproducible-project","zh\u002Feconometrics\u002F07-reproducible-project",{"title":1358,"path":1359,"stem":1360},"第八章：计量经济学浏览器回归实验","\u002Fzh\u002Feconometrics\u002F08-interactive-regression-labs","zh\u002Feconometrics\u002F08-interactive-regression-labs",{"title":1362,"path":1363,"stem":1364},"第九章：前沿文献与现代计量案例（2023—2026）","\u002Fzh\u002Feconometrics\u002F09-frontier-literature-2026","zh\u002Feconometrics\u002F09-frontier-literature-2026",{"title":478,"path":1366,"stem":1367,"children":1368,"page":249},"\u002Fzh\u002Fintro-to-economics","zh\u002Fintro-to-economics",[1369,1373,1377,1381,1385,1389,1393,1397,1401,1405,1409,1413,1417],{"title":1370,"path":1371,"stem":1372},"经济学导论 (微观与宏观)","\u002Fzh\u002Fintro-to-economics\u002F00-intro","zh\u002Fintro-to-economics\u002F00-intro",{"title":1374,"path":1375,"stem":1376},"第1章：经济学基础原理","\u002Fzh\u002Fintro-to-economics\u002F01-foundations","zh\u002Fintro-to-economics\u002F01-foundations",{"title":1378,"path":1379,"stem":1380},"第2章：需求与供给","\u002Fzh\u002Fintro-to-economics\u002F02-demand-and-supply","zh\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":1382,"path":1383,"stem":1384},"第3章：弹性","\u002Fzh\u002Fintro-to-economics\u002F03-elasticity","zh\u002Fintro-to-economics\u002F03-elasticity",{"title":1386,"path":1387,"stem":1388},"第4章：市场结构","\u002Fzh\u002Fintro-to-economics\u002F04-market-structures","zh\u002Fintro-to-economics\u002F04-market-structures",{"title":1390,"path":1391,"stem":1392},"第5章：GDP 与财富","\u002Fzh\u002Fintro-to-economics\u002F05-gdp-and-wealth","zh\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":1394,"path":1395,"stem":1396},"第6章：通货膨胀与失业","\u002Fzh\u002Fintro-to-economics\u002F06-inflation-and-unemployment","zh\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":1398,"path":1399,"stem":1400},"第7章：经济增长","\u002Fzh\u002Fintro-to-economics\u002F07-economic-growth","zh\u002Fintro-to-economics\u002F07-economic-growth",{"title":1402,"path":1403,"stem":1404},"第8章：货币与银行","\u002Fzh\u002Fintro-to-economics\u002F08-money-and-banking","zh\u002Fintro-to-economics\u002F08-money-and-banking",{"title":1406,"path":1407,"stem":1408},"第9章：货币政策与 AD-AS 模型","\u002Fzh\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","zh\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":1410,"path":1411,"stem":1412},"第10章：财政政策","\u002Fzh\u002Fintro-to-economics\u002F10-fiscal-policy","zh\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":1414,"path":1415,"stem":1416},"第11章：开放经济与汇率","\u002Fzh\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","zh\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":1418,"path":1419,"stem":1420},"第12章：综合复习与案例分析","\u002Fzh\u002Fintro-to-economics\u002F12-review-and-case-studies","zh\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":1422,"path":1423,"stem":1424,"children":1425},"宏观经济学","\u002Fzh\u002Fmacroeconomics","zh\u002Fmacroeconomics\u002Findex",[1426,1427,1431,1435,1439,1443,1447,1451,1455,1459,1463],{"title":1422,"path":1423,"stem":1424},{"title":1428,"path":1429,"stem":1430},"第一章：国民账户与宏观指标","\u002Fzh\u002Fmacroeconomics\u002F01-national-accounts","zh\u002Fmacroeconomics\u002F01-national-accounts",{"title":1432,"path":1433,"stem":1434},"第二章：消费、投资与凯恩斯交叉","\u002Fzh\u002Fmacroeconomics\u002F02-consumption-investment","zh\u002Fmacroeconomics\u002F02-consumption-investment",{"title":1436,"path":1437,"stem":1438},"第三章：货币、银行与货币政策","\u002Fzh\u002Fmacroeconomics\u002F03-money-and-monetary-policy","zh\u002Fmacroeconomics\u002F03-money-and-monetary-policy",{"title":1440,"path":1441,"stem":1442},"第四章：AD–AS、通货膨胀与失业","\u002Fzh\u002Fmacroeconomics\u002F04-ad-as-inflation-unemployment","zh\u002Fmacroeconomics\u002F04-ad-as-inflation-unemployment",{"title":1444,"path":1445,"stem":1446},"第五章：财政政策、债务与稳定化","\u002Fzh\u002Fmacroeconomics\u002F05-fiscal-policy-and-debt","zh\u002Fmacroeconomics\u002F05-fiscal-policy-and-debt",{"title":1448,"path":1449,"stem":1450},"第六章：经济增长、生产率与发展","\u002Fzh\u002Fmacroeconomics\u002F06-growth-productivity","zh\u002Fmacroeconomics\u002F06-growth-productivity",{"title":1452,"path":1453,"stem":1454},"第七章：开放经济、汇率与国际收支","\u002Fzh\u002Fmacroeconomics\u002F07-open-economy","zh\u002Fmacroeconomics\u002F07-open-economy",{"title":1456,"path":1457,"stem":1458},"第八章：宏观研究项目与政策分析","\u002Fzh\u002Fmacroeconomics\u002F08-macro-research-project","zh\u002Fmacroeconomics\u002F08-macro-research-project",{"title":1460,"path":1461,"stem":1462},"第九章：宏观经济学浏览器交互实验","\u002Fzh\u002Fmacroeconomics\u002F09-interactive-policy-labs","zh\u002Fmacroeconomics\u002F09-interactive-policy-labs",{"title":1464,"path":1465,"stem":1466},"第十章：前沿文献与当代宏观案例（2023—2026）","\u002Fzh\u002Fmacroeconomics\u002F10-frontier-literature-2026","zh\u002Fmacroeconomics\u002F10-frontier-literature-2026",{"title":1468,"path":1469,"stem":1470,"children":1471},"微观计量经济学","\u002Fzh\u002Fmicroeconometrics","zh\u002Fmicroeconometrics\u002Findex",[1472,1473,1479,1485,1491,1497,1503,1509,1515,1521,1527,1533,1539],{"title":1468,"path":1469,"stem":1470},{"title":1474,"path":1475,"stem":1476,"children":1477},"第一章：微观计量与因果推断导论","\u002Fzh\u002Fmicroeconometrics\u002F01-intro","zh\u002Fmicroeconometrics\u002F01-intro\u002Findex",[1478],{"title":1474,"path":1475,"stem":1476},{"title":1480,"path":1481,"stem":1482,"children":1483},"第二章：线性回归与 OLS","\u002Fzh\u002Fmicroeconometrics\u002F02-ols","zh\u002Fmicroeconometrics\u002F02-ols\u002Findex",[1484],{"title":1480,"path":1481,"stem":1482},{"title":1486,"path":1487,"stem":1488,"children":1489},"第三章：工具变量法","\u002Fzh\u002Fmicroeconometrics\u002F03-iv","zh\u002Fmicroeconometrics\u002F03-iv\u002Findex",[1490],{"title":1486,"path":1487,"stem":1488},{"title":1492,"path":1493,"stem":1494,"children":1495},"第四章：面板数据方法","\u002Fzh\u002Fmicroeconometrics\u002F04-panel","zh\u002Fmicroeconometrics\u002F04-panel\u002Findex",[1496],{"title":1492,"path":1493,"stem":1494},{"title":1498,"path":1499,"stem":1500,"children":1501},"第五章：双重差分法","\u002Fzh\u002Fmicroeconometrics\u002F05-did","zh\u002Fmicroeconometrics\u002F05-did\u002Findex",[1502],{"title":1498,"path":1499,"stem":1500},{"title":1504,"path":1505,"stem":1506,"children":1507},"第六章：断点回归设计","\u002Fzh\u002Fmicroeconometrics\u002F06-rdd","zh\u002Fmicroeconometrics\u002F06-rdd\u002Findex",[1508],{"title":1504,"path":1505,"stem":1506},{"title":1510,"path":1511,"stem":1512,"children":1513},"第七章：匹配、倾向得分与加权","\u002Fzh\u002Fmicroeconometrics\u002F07-matching","zh\u002Fmicroeconometrics\u002F07-matching\u002Findex",[1514],{"title":1510,"path":1511,"stem":1512},{"title":1516,"path":1517,"stem":1518,"children":1519},"第八章：离散选择模型","\u002Fzh\u002Fmicroeconometrics\u002F08-discrete-choice","zh\u002Fmicroeconometrics\u002F08-discrete-choice\u002Findex",[1520],{"title":1516,"path":1517,"stem":1518},{"title":1522,"path":1523,"stem":1524,"children":1525},"第九章：计数数据与受限因变量","\u002Fzh\u002Fmicroeconometrics\u002F09-count-limited","zh\u002Fmicroeconometrics\u002F09-count-limited\u002Findex",[1526],{"title":1522,"path":1523,"stem":1524},{"title":1528,"path":1529,"stem":1530,"children":1531},"第十章：合成控制法","\u002Fzh\u002Fmicroeconometrics\u002F10-synthetic-control","zh\u002Fmicroeconometrics\u002F10-synthetic-control\u002Findex",[1532],{"title":1528,"path":1529,"stem":1530},{"title":1534,"path":1535,"stem":1536,"children":1537},"第十一章：机器学习与因果推断","\u002Fzh\u002Fmicroeconometrics\u002F11-ml-causal","zh\u002Fmicroeconometrics\u002F11-ml-causal\u002Findex",[1538],{"title":1534,"path":1535,"stem":1536},{"title":1540,"path":1541,"stem":1542},"第十二章：前沿文献与现代微观案例（2024—2026）","\u002Fzh\u002Fmicroeconometrics\u002F12-frontier-literature-2026","zh\u002Fmicroeconometrics\u002F12-frontier-literature-2026",{"title":716,"path":1544,"stem":1545,"children":1546,"page":249},"\u002Fzh\u002Fmicroeconomics","zh\u002Fmicroeconomics",[1547,1553,1559,1565,1571,1577,1583,1589,1595,1601,1607,1613,1619],{"title":1548,"path":1549,"stem":1550,"children":1551},"微观经济学 III","\u002Fzh\u002Fmicroeconomics\u002F00-intro","zh\u002Fmicroeconomics\u002F00-intro\u002Findex",[1552],{"title":1548,"path":1549,"stem":1550},{"title":1554,"path":1555,"stem":1556,"children":1557},"消费者理论与分析基础","\u002Fzh\u002Fmicroeconomics\u002F01-fundations","zh\u002Fmicroeconomics\u002F01-fundations\u002Findex",[1558],{"title":1554,"path":1555,"stem":1556},{"title":1560,"path":1561,"stem":1562,"children":1563},"比较静态分析与福利测度","\u002Fzh\u002Fmicroeconomics\u002F02-comparative-statics","zh\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[1564],{"title":1560,"path":1561,"stem":1562},{"title":1566,"path":1567,"stem":1568,"children":1569},"不确定性下的决策","\u002Fzh\u002Fmicroeconomics\u002F03-uncertainty","zh\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[1570],{"title":1566,"path":1567,"stem":1568},{"title":1572,"path":1573,"stem":1574,"children":1575},"一般均衡与福利经济学","\u002Fzh\u002Fmicroeconomics\u002F04-general-equilibrium","zh\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[1576],{"title":1572,"path":1573,"stem":1574},{"title":1578,"path":1579,"stem":1580,"children":1581},"博弈论：静态与动态博弈","\u002Fzh\u002Fmicroeconomics\u002F05-game-theory","zh\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[1582],{"title":1578,"path":1579,"stem":1580},{"title":1584,"path":1585,"stem":1586,"children":1587},"寡头垄断与策略性市场行为","\u002Fzh\u002Fmicroeconomics\u002F06-oligopoly","zh\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[1588],{"title":1584,"path":1585,"stem":1586},{"title":1590,"path":1591,"stem":1592,"children":1593},"信息经济学：逆向选择与道德风险","\u002Fzh\u002Fmicroeconomics\u002F07-information-economics","zh\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[1594],{"title":1590,"path":1591,"stem":1592},{"title":1596,"path":1597,"stem":1598,"children":1599},"机制设计与拍卖理论","\u002Fzh\u002Fmicroeconomics\u002F08-mechanism-design","zh\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[1600],{"title":1596,"path":1597,"stem":1598},{"title":1602,"path":1603,"stem":1604,"children":1605},"行为与实验微观经济学","\u002Fzh\u002Fmicroeconomics\u002F09-behavioural-economics","zh\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[1606],{"title":1602,"path":1603,"stem":1604},{"title":1608,"path":1609,"stem":1610,"children":1611},"外部性、公共物品与机制","\u002Fzh\u002Fmicroeconomics\u002F10-externalities-public-goods","zh\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[1612],{"title":1608,"path":1609,"stem":1610},{"title":1614,"path":1615,"stem":1616,"children":1617},"市场设计与匹配理论","\u002Fzh\u002Fmicroeconomics\u002F11-market-design","zh\u002Fmicroeconomics\u002F11-market-design\u002Findex",[1618],{"title":1614,"path":1615,"stem":1616},{"title":1620,"path":1621,"stem":1622,"children":1623},"第十二章：微观经济学：回顾与前沿应用","\u002Fzh\u002Fmicroeconomics\u002F12-review","zh\u002Fmicroeconomics\u002F12-review\u002Findex",[1624],{"title":1620,"path":1621,"stem":1622},{"title":799,"path":1626,"stem":1627,"children":1628,"page":249},"\u002Fzh\u002Fplayground","zh\u002Fplayground",[1629],{"title":1630,"path":1631,"stem":1632},"Chart.js 可视化","\u002Fzh\u002Fplayground\u002F05-chartjs","zh\u002Fplayground\u002F05-chartjs",{"title":1634,"path":1635,"stem":1636,"children":1637},"概率论与数理统计","\u002Fzh\u002Fprob-and-stats","zh\u002Fprob-and-stats\u002Findex",[1638,1639,1645,1680,1727],{"title":1634,"path":1635,"stem":1636},{"title":1640,"path":1641,"stem":1642,"children":1643},"第零章：概率统计的对象与学习方法","\u002Fzh\u002Fprob-and-stats\u002F00-intro","zh\u002Fprob-and-stats\u002F00-intro\u002Findex",[1644],{"title":1640,"path":1641,"stem":1642},{"title":1646,"path":1647,"stem":1648,"children":1649,"page":249},"01 Probability","\u002Fzh\u002Fprob-and-stats\u002F01-probability","zh\u002Fprob-and-stats\u002F01-probability",[1650,1656,1662,1668,1674],{"title":1651,"path":1652,"stem":1653,"children":1654},"第一章：概率论基础","\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F01-prob-theory","zh\u002Fprob-and-stats\u002F01-probability\u002F01-prob-theory\u002Findex",[1655],{"title":1651,"path":1652,"stem":1653},{"title":1657,"path":1658,"stem":1659,"children":1660},"第二章：随机变量与分布","\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F02-random-variables","zh\u002Fprob-and-stats\u002F01-probability\u002F02-random-variables\u002Findex",[1661],{"title":1657,"path":1658,"stem":1659},{"title":1663,"path":1664,"stem":1665,"children":1666},"第三章：期望、方差与条件矩","\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F03-moment","zh\u002Fprob-and-stats\u002F01-probability\u002F03-moment\u002Findex",[1667],{"title":1663,"path":1664,"stem":1665},{"title":1669,"path":1670,"stem":1671,"children":1672},"第四章：常见分布族与建模机制","\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F04-families","zh\u002Fprob-and-stats\u002F01-probability\u002F04-families\u002Findex",[1673],{"title":1669,"path":1670,"stem":1671},{"title":1675,"path":1676,"stem":1677,"children":1678},"第五章：收敛与渐近理论","\u002Fzh\u002Fprob-and-stats\u002F01-probability\u002F05-asymptotics","zh\u002Fprob-and-stats\u002F01-probability\u002F05-asymptotics\u002Findex",[1679],{"title":1675,"path":1676,"stem":1677},{"title":1681,"path":1682,"stem":1683,"children":1684,"page":249},"02 Statistics","\u002Fzh\u002Fprob-and-stats\u002F02-statistics","zh\u002Fprob-and-stats\u002F02-statistics",[1685,1691,1697,1703,1709,1715,1721],{"title":1686,"path":1687,"stem":1688,"children":1689},"第六章：抽样分布","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F01-sampling","zh\u002Fprob-and-stats\u002F02-statistics\u002F01-sampling\u002Findex",[1690],{"title":1686,"path":1687,"stem":1688},{"title":1692,"path":1693,"stem":1694,"children":1695},"第七章：区间估计","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F02-interval-estimation","zh\u002Fprob-and-stats\u002F02-statistics\u002F02-interval-estimation\u002Findex",[1696],{"title":1692,"path":1693,"stem":1694},{"title":1698,"path":1699,"stem":1700,"children":1701},"第八章：点估计理论","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F03-point-estimation","zh\u002Fprob-and-stats\u002F02-statistics\u002F03-point-estimation\u002Findex",[1702],{"title":1698,"path":1699,"stem":1700},{"title":1704,"path":1705,"stem":1706,"children":1707},"第九章：点估计方法","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F04-pe-method","zh\u002Fprob-and-stats\u002F02-statistics\u002F04-pe-method\u002Findex",[1708],{"title":1704,"path":1705,"stem":1706},{"title":1710,"path":1711,"stem":1712,"children":1713},"第十章：假设检验原理","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F05-hypothesis","zh\u002Fprob-and-stats\u002F02-statistics\u002F05-hypothesis\u002Findex",[1714],{"title":1710,"path":1711,"stem":1712},{"title":1716,"path":1717,"stem":1718,"children":1719},"第十一章：常用检验方法与选择","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F06-hypothesis-method","zh\u002Fprob-and-stats\u002F02-statistics\u002F06-hypothesis-method\u002Findex",[1720],{"title":1716,"path":1717,"stem":1718},{"title":1722,"path":1723,"stem":1724,"children":1725},"第十二章：Bootstrap 与重抽样","\u002Fzh\u002Fprob-and-stats\u002F02-statistics\u002F07-bootstrap","zh\u002Fprob-and-stats\u002F02-statistics\u002F07-bootstrap\u002Findex",[1726],{"title":1722,"path":1723,"stem":1724},{"title":1728,"path":1729,"stem":1730},"第十三章：前沿文献与现代统计案例（2023—2026）","\u002Fzh\u002Fprob-and-stats\u002F03-frontier-literature-2026","zh\u002Fprob-and-stats\u002F03-frontier-literature-2026",{"title":1732,"path":1733,"stem":1734,"children":1735,"page":249},"Quant","\u002Fzh\u002Fquant","zh\u002Fquant",[1736,1742,1746,1750,1754,1758,1762,1766,1770,1774,1778],{"title":1737,"path":1738,"stem":1739,"children":1740},"量化投资——从可检验信号到可执行组合","\u002Fzh\u002Fquant\u002F00-index","zh\u002Fquant\u002F00-index",[1741],{"title":1737,"path":1738,"stem":1739},{"title":1743,"path":1744,"stem":1745},"数据获取与预处理","\u002Fzh\u002Fquant\u002F01-research-data","zh\u002Fquant\u002F01-research-data",{"title":1747,"path":1748,"stem":1749},"因子与交易信号","\u002Fzh\u002Fquant\u002F02-factor-and-signals","zh\u002Fquant\u002F02-factor-and-signals",{"title":1751,"path":1752,"stem":1753},"策略建模与回测","\u002Fzh\u002Fquant\u002F03-modeling-and-backtest","zh\u002Fquant\u002F03-modeling-and-backtest",{"title":1755,"path":1756,"stem":1757},"组合构建与风险建模（资产定价视角）","\u002Fzh\u002Fquant\u002F04-portfolio-and-risk","zh\u002Fquant\u002F04-portfolio-and-risk",{"title":1759,"path":1760,"stem":1761},"执行策略与市场微结构概览","\u002Fzh\u002Fquant\u002F05-execution-and-microstructure","zh\u002Fquant\u002F05-execution-and-microstructure",{"title":1763,"path":1764,"stem":1765},"策略上线、监控与迭代","\u002Fzh\u002Fquant\u002F06-production-and-monitoring","zh\u002Fquant\u002F06-production-and-monitoring",{"title":1767,"path":1768,"stem":1769},"量化工程与工具链概览","\u002Fzh\u002Fquant\u002F07-engineering-stack","zh\u002Fquant\u002F07-engineering-stack",{"title":1771,"path":1772,"stem":1773},"计量方法与实证检验","\u002Fzh\u002Fquant\u002F08-econometric-methods","zh\u002Fquant\u002F08-econometric-methods",{"title":1775,"path":1776,"stem":1777},"案例研究：多因子股票 Alpha 策略全流程","\u002Fzh\u002Fquant\u002F09-case-study","zh\u002Fquant\u002F09-case-study",{"title":1779,"path":1780,"stem":1781},"量化投资文献与软件图谱","\u002Fzh\u002Fquant\u002F10-reading-software-map","zh\u002Fquant\u002F10-reading-software-map",null,{"id":1784,"title":971,"body":1785,"description":13617,"extension":13618,"features":1782,"hero":1782,"layout":1782,"locale":1782,"meta":13619,"navigation":1782,"path":972,"published":13621,"seo":13622,"stem":973,"__hash__":13623},"docs\u002Fen\u002Ftime-series\u002F03-prediction\u002Findex.md",{"type":1786,"value":1787,"toc":13600},"minimark",[1788,1792,1797,1801,1805,1808,1830,1834,1885,2290,2432,2755,2758,2885,2918,3192,3200,3204,3207,3405,3408,3525,3702,3710,3714,3970,4180,4183,4616,4882,4885,5227,5231,5234,5620,5682,5862,5894,5899,6236,6242,6381,6385,6418,7123,7301,7601,7870,8352,8355,8359,8491,8899,8902,9430,9433,9437,9440,9899,9902,10131,10299,10753,10756,11201,11204,11477,12069,12221,12294,12298,12708,13589,13593],[1789,1790,971],"h1",{"id":1791},"module-3-linear-prediction-and-state-space-recursions",[1793,1794,1796],"h2",{"id":1795},"core-question","Core question",[1798,1799,1800],"p",{},"Given a covariance structure and a vector of observations, which linear combination predicts an unobserved value with minimum mean-squared error?",[1793,1802,1804],{"id":1803},"learning-outcomes","Learning outcomes",[1798,1806,1807],{},"You will be able to:",[1809,1810,1811,1815,1818,1821,1824,1827],"ul",{},[1812,1813,1814],"li",{},"derive the normal equations for a best linear predictor;",[1812,1816,1817],{},"compute prediction-error variance from a Schur complement;",[1812,1819,1820],{},"explain when a linear predictor equals a conditional expectation;",[1812,1822,1823],{},"obtain multi-step AR predictions from powers of a companion matrix;",[1812,1825,1826],{},"derive the Kalman prediction and correction equations;",[1812,1828,1829],{},"distinguish filtering, prediction, and smoothing.",[1793,1831,1833],{"id":1832},"_1-prediction-is-a-projection-problem","1. Prediction is a projection problem",[1798,1835,1836,1837,1884],{},"Let ",[1838,1839,1842,1864],"span",{"className":1840},[1841],"katex",[1838,1843,1846],{"className":1844},[1845],"katex-mathml",[1847,1848,1850],"math",{"xmlns":1849},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1851,1852,1853,1860],"semantics",{},[1854,1855,1856],"mrow",{},[1857,1858,1859],"mi",{},"Y",[1861,1862,1859],"annotation",{"encoding":1863},"application\u002Fx-tex",[1838,1865,1869],{"className":1866,"ariaHidden":1868},[1867],"katex-html","true",[1838,1870,1873,1878],{"className":1871},[1872],"base",[1838,1874],{"className":1875,"style":1877},[1876],"strut","height:0.6833em;",[1838,1879,1859],{"className":1880,"style":1883},[1881,1882],"mord","mathnormal","margin-right:0.2222em;"," be a centred target and let",[1838,1886,1889],{"className":1887},[1888],"katex-display",[1838,1890,1892,2001],{"className":1891},[1841],[1838,1893,1895],{"className":1894},[1845],[1847,1896,1898],{"xmlns":1849,"display":1897},"block",[1851,1899,1900,1998],{},[1854,1901,1902,1906,1910,1914,1923,1926,1929,1931,1938,1948,1950,1954,1957,1959,1962,1965,1967,1969,1971,1973,1975,1978,1980,1983,1985,1987,1989,1991,1993,1995],{},[1857,1903,1905],{"mathvariant":1904},"bold","X",[1907,1908,1909],"mo",{},"=",[1907,1911,1913],{"stretchy":1912},"false","(",[1915,1916,1917,1919],"msub",{},[1857,1918,1905],{},[1920,1921,1922],"mn",{},"1",[1907,1924,1925],{"separator":1868},",",[1907,1927,1928],{},"…",[1907,1930,1925],{"separator":1868},[1915,1932,1933,1935],{},[1857,1934,1905],{},[1857,1936,1937],{},"m",[1939,1940,1941,1944],"msup",{},[1907,1942,1943],{"stretchy":1912},")",[1857,1945,1947],{"mathvariant":1946},"normal","⊤",[1907,1949,1925],{"separator":1868},[1951,1952],"mspace",{"width":1953},"2em",[1857,1955,1956],{"mathvariant":1946},"Γ",[1907,1958,1909],{},[1857,1960,1961],{"mathvariant":1946},"Var",[1907,1963,1964],{},"⁡",[1907,1966,1913],{"stretchy":1912},[1857,1968,1905],{"mathvariant":1904},[1907,1970,1943],{"stretchy":1912},[1907,1972,1925],{"separator":1868},[1951,1974],{"width":1953},[1857,1976,1977],{"mathvariant":1904},"g",[1907,1979,1909],{},[1857,1981,1982],{"mathvariant":1946},"Cov",[1907,1984,1964],{},[1907,1986,1913],{"stretchy":1912},[1857,1988,1905],{"mathvariant":1904},[1907,1990,1925],{"separator":1868},[1857,1992,1859],{},[1907,1994,1943],{"stretchy":1912},[1857,1996,1997],{"mathvariant":1946},".",[1861,1999,2000],{"encoding":1863},"\\mathbf X=(X_1,\\ldots,X_m)^\\top,\\qquad\n\\Gamma=\\operatorname{Var}(\\mathbf X),\\qquad\n\\mathbf g=\\operatorname{Cov}(\\mathbf X,Y).",[1838,2002,2004,2026,2210,2256],{"className":2003,"ariaHidden":1868},[1867],[1838,2005,2007,2011,2015,2019,2023],{"className":2006},[1872],[1838,2008],{"className":2009,"style":2010},[1876],"height:0.6861em;",[1838,2012,1905],{"className":2013},[1881,2014],"mathbf",[1838,2016],{"className":2017,"style":2018},[1951],"margin-right:0.2778em;",[1838,2020,1909],{"className":2021},[2022],"mrel",[1838,2024],{"className":2025,"style":2018},[1951],[1838,2027,2029,2033,2037,2094,2098,2102,2106,2109,2112,2115,2156,2188,2191,2195,2198,2201,2204,2207],{"className":2028},[1872],[1838,2030],{"className":2031,"style":2032},[1876],"height:1.1491em;vertical-align:-0.25em;",[1838,2034,1913],{"className":2035},[2036],"mopen",[1838,2038,2040,2044],{"className":2039},[1881],[1838,2041,1905],{"className":2042,"style":2043},[1881,1882],"margin-right:0.0785em;",[1838,2045,2048],{"className":2046},[2047],"msupsub",[1838,2049,2053,2085],{"className":2050},[2051,2052],"vlist-t","vlist-t2",[1838,2054,2057,2080],{"className":2055},[2056],"vlist-r",[1838,2058,2062],{"className":2059,"style":2061},[2060],"vlist","height:0.3011em;",[1838,2063,2065,2070],{"style":2064},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1838,2066],{"className":2067,"style":2069},[2068],"pstrut","height:2.7em;",[1838,2071,2077],{"className":2072},[2073,2074,2075,2076],"sizing","reset-size6","size3","mtight",[1838,2078,1922],{"className":2079},[1881,2076],[1838,2081,2084],{"className":2082},[2083],"vlist-s","​",[1838,2086,2088],{"className":2087},[2056],[1838,2089,2092],{"className":2090,"style":2091},[2060],"height:0.15em;",[1838,2093],{},[1838,2095,1925],{"className":2096},[2097],"mpunct",[1838,2099],{"className":2100,"style":2101},[1951],"margin-right:0.1667em;",[1838,2103,1928],{"className":2104},[2105],"minner",[1838,2107],{"className":2108,"style":2101},[1951],[1838,2110,1925],{"className":2111},[2097],[1838,2113],{"className":2114,"style":2101},[1951],[1838,2116,2118,2121],{"className":2117},[1881],[1838,2119,1905],{"className":2120,"style":2043},[1881,1882],[1838,2122,2124],{"className":2123},[2047],[1838,2125,2127,2148],{"className":2126},[2051,2052],[1838,2128,2130,2145],{"className":2129},[2056],[1838,2131,2134],{"className":2132,"style":2133},[2060],"height:0.1514em;",[1838,2135,2136,2139],{"style":2064},[1838,2137],{"className":2138,"style":2069},[2068],[1838,2140,2142],{"className":2141},[2073,2074,2075,2076],[1838,2143,1937],{"className":2144},[1881,1882,2076],[1838,2146,2084],{"className":2147},[2083],[1838,2149,2151],{"className":2150},[2056],[1838,2152,2154],{"className":2153,"style":2091},[2060],[1838,2155],{},[1838,2157,2160,2163],{"className":2158},[2159],"mclose",[1838,2161,1943],{"className":2162},[2159],[1838,2164,2166],{"className":2165},[2047],[1838,2167,2169],{"className":2168},[2051],[1838,2170,2172],{"className":2171},[2056],[1838,2173,2176],{"className":2174,"style":2175},[2060],"height:0.8991em;",[1838,2177,2179,2182],{"style":2178},"top:-3.113em;margin-right:0.05em;",[1838,2180],{"className":2181,"style":2069},[2068],[1838,2183,2185],{"className":2184},[2073,2074,2075,2076],[1838,2186,1947],{"className":2187},[1881,2076],[1838,2189,1925],{"className":2190},[2097],[1838,2192],{"className":2193,"style":2194},[1951],"margin-right:2em;",[1838,2196],{"className":2197,"style":2101},[1951],[1838,2199,1956],{"className":2200},[1881],[1838,2202],{"className":2203,"style":2018},[1951],[1838,2205,1909],{"className":2206},[2022],[1838,2208],{"className":2209,"style":2018},[1951],[1838,2211,2213,2217,2225,2228,2231,2234,2237,2240,2243,2247,2250,2253],{"className":2212},[1872],[1838,2214],{"className":2215,"style":2216},[1876],"height:1em;vertical-align:-0.25em;",[1838,2218,2221],{"className":2219},[2220],"mop",[1838,2222,1961],{"className":2223},[1881,2224],"mathrm",[1838,2226,1913],{"className":2227},[2036],[1838,2229,1905],{"className":2230},[1881,2014],[1838,2232,1943],{"className":2233},[2159],[1838,2235,1925],{"className":2236},[2097],[1838,2238],{"className":2239,"style":2194},[1951],[1838,2241],{"className":2242,"style":2101},[1951],[1838,2244,1977],{"className":2245,"style":2246},[1881,2014],"margin-right:0.016em;",[1838,2248],{"className":2249,"style":2018},[1951],[1838,2251,1909],{"className":2252},[2022],[1838,2254],{"className":2255,"style":2018},[1951],[1838,2257,2259,2262,2269,2272,2275,2278,2281,2284,2287],{"className":2258},[1872],[1838,2260],{"className":2261,"style":2216},[1876],[1838,2263,2265],{"className":2264},[2220],[1838,2266,1982],{"className":2267,"style":2268},[1881,2224],"margin-right:0.0139em;",[1838,2270,1913],{"className":2271},[2036],[1838,2273,1905],{"className":2274},[1881,2014],[1838,2276,1925],{"className":2277},[2097],[1838,2279],{"className":2280,"style":2101},[1951],[1838,2282,1859],{"className":2283,"style":1883},[1881,1882],[1838,2285,1943],{"className":2286},[2159],[1838,2288,1997],{"className":2289},[1881],[1798,2291,2292,2293,1925],{},"For a linear predictor ",[1838,2294,2296,2327],{"className":2295},[1841],[1838,2297,2299],{"className":2298},[1845],[1847,2300,2301],{"xmlns":1849},[1851,2302,2303,2324],{},[1854,2304,2305,2313,2315,2322],{},[2306,2307,2308,2310],"mover",{"accent":1868},[1857,2309,1859],{},[1907,2311,2312],{"stretchy":1868},"^",[1907,2314,1909],{},[1939,2316,2317,2320],{},[1857,2318,2319],{"mathvariant":1904},"a",[1857,2321,1947],{"mathvariant":1946},[1857,2323,1905],{"mathvariant":1904},[1861,2325,2326],{"encoding":1863},"\\widehat Y=\\mathbf a^\\top\\mathbf X",[1838,2328,2330,2392],{"className":2329,"ariaHidden":1868},[1867],[1838,2331,2333,2337,2383,2386,2389],{"className":2332},[1872],[1838,2334],{"className":2335,"style":2336},[1876],"height:0.9233em;",[1838,2338,2341],{"className":2339},[1881,2340],"accent",[1838,2342,2344],{"className":2343},[2051],[1838,2345,2347],{"className":2346},[2056],[1838,2348,2350,2360],{"className":2349,"style":2336},[2060],[1838,2351,2353,2357],{"style":2352},"top:-3em;",[1838,2354],{"className":2355,"style":2356},[2068],"height:3em;",[1838,2358,1859],{"className":2359,"style":1883},[1881,1882],[1838,2361,2365,2368],{"className":2362,"style":2364},[2363],"svg-align","top:-3.6833em;",[1838,2366],{"className":2367,"style":2356},[2068],[1838,2369,2371],{"style":2370},"height:0.24em;",[2372,2373,2379],"svg",{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","100%","0.24em","0 0 1062 239","none",[2380,2381],"path",{"d":2382},"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z",[1838,2384],{"className":2385,"style":2018},[1951],[1838,2387,1909],{"className":2388},[2022],[1838,2390],{"className":2391,"style":2018},[1951],[1838,2393,2395,2399,2429],{"className":2394},[1872],[1838,2396],{"className":2397,"style":2398},[1876],"height:0.8491em;",[1838,2400,2402,2405],{"className":2401},[1881],[1838,2403,2319],{"className":2404},[1881,2014],[1838,2406,2408],{"className":2407},[2047],[1838,2409,2411],{"className":2410},[2051],[1838,2412,2414],{"className":2413},[2056],[1838,2415,2417],{"className":2416,"style":2398},[2060],[1838,2418,2420,2423],{"style":2419},"top:-3.063em;margin-right:0.05em;",[1838,2421],{"className":2422,"style":2069},[2068],[1838,2424,2426],{"className":2425},[2073,2074,2075,2076],[1838,2427,1947],{"className":2428},[1881,2076],[1838,2430,1905],{"className":2431},[1881,2014],[1838,2433,2435],{"className":2434},[1888],[1838,2436,2438,2520],{"className":2437},[1841],[1838,2439,2441],{"className":2440},[1845],[1847,2442,2443],{"xmlns":1849,"display":1897},[1851,2444,2445,2517],{},[1854,2446,2447,2450,2453,2455,2457,2460,2466,2468,2475,2478,2480,2482,2484,2486,2488,2490,2492,2494,2500,2502,2505,2511,2513,2515],{},[1857,2448,2449],{},"E",[1907,2451,2452],{"stretchy":1912},"[",[1907,2454,1913],{"stretchy":1912},[1857,2456,1859],{},[1907,2458,2459],{},"−",[1939,2461,2462,2464],{},[1857,2463,2319],{"mathvariant":1904},[1857,2465,1947],{"mathvariant":1946},[1857,2467,1905],{"mathvariant":1904},[1939,2469,2470,2472],{},[1907,2471,1943],{"stretchy":1912},[1920,2473,2474],{},"2",[1907,2476,2477],{"stretchy":1912},"]",[1907,2479,1909],{},[1857,2481,1961],{"mathvariant":1946},[1907,2483,1964],{},[1907,2485,1913],{"stretchy":1912},[1857,2487,1859],{},[1907,2489,1943],{"stretchy":1912},[1907,2491,2459],{},[1920,2493,2474],{},[1939,2495,2496,2498],{},[1857,2497,2319],{"mathvariant":1904},[1857,2499,1947],{"mathvariant":1946},[1857,2501,1977],{"mathvariant":1904},[1907,2503,2504],{},"+",[1939,2506,2507,2509],{},[1857,2508,2319],{"mathvariant":1904},[1857,2510,1947],{"mathvariant":1946},[1857,2512,1956],{"mathvariant":1946},[1857,2514,2319],{"mathvariant":1904},[1857,2516,1997],{"mathvariant":1946},[1861,2518,2519],{"encoding":1863},"E[(Y-\\mathbf a^\\top\\mathbf X)^2]\n=\\operatorname{Var}(Y)-2\\mathbf a^\\top\\mathbf g+\\mathbf a^\\top\\Gamma\\mathbf a.",[1838,2521,2523,2550,2630,2660,2711],{"className":2522,"ariaHidden":1868},[1867],[1838,2524,2526,2529,2533,2537,2540,2543,2547],{"className":2525},[1872],[1838,2527],{"className":2528,"style":2216},[1876],[1838,2530,2449],{"className":2531,"style":2532},[1881,1882],"margin-right:0.0576em;",[1838,2534,2536],{"className":2535},[2036],"[(",[1838,2538,1859],{"className":2539,"style":1883},[1881,1882],[1838,2541],{"className":2542,"style":1883},[1951],[1838,2544,2459],{"className":2545},[2546],"mbin",[1838,2548],{"className":2549,"style":1883},[1951],[1838,2551,2553,2556,2585,2588,2618,2621,2624,2627],{"className":2552},[1872],[1838,2554],{"className":2555,"style":2032},[1876],[1838,2557,2559,2562],{"className":2558},[1881],[1838,2560,2319],{"className":2561},[1881,2014],[1838,2563,2565],{"className":2564},[2047],[1838,2566,2568],{"className":2567},[2051],[1838,2569,2571],{"className":2570},[2056],[1838,2572,2574],{"className":2573,"style":2175},[2060],[1838,2575,2576,2579],{"style":2178},[1838,2577],{"className":2578,"style":2069},[2068],[1838,2580,2582],{"className":2581},[2073,2074,2075,2076],[1838,2583,1947],{"className":2584},[1881,2076],[1838,2586,1905],{"className":2587},[1881,2014],[1838,2589,2591,2594],{"className":2590},[2159],[1838,2592,1943],{"className":2593},[2159],[1838,2595,2597],{"className":2596},[2047],[1838,2598,2600],{"className":2599},[2051],[1838,2601,2603],{"className":2602},[2056],[1838,2604,2607],{"className":2605,"style":2606},[2060],"height:0.8641em;",[1838,2608,2609,2612],{"style":2178},[1838,2610],{"className":2611,"style":2069},[2068],[1838,2613,2615],{"className":2614},[2073,2074,2075,2076],[1838,2616,2474],{"className":2617},[1881,2076],[1838,2619,2477],{"className":2620},[2159],[1838,2622],{"className":2623,"style":2018},[1951],[1838,2625,1909],{"className":2626},[2022],[1838,2628],{"className":2629,"style":2018},[1951],[1838,2631,2633,2636,2642,2645,2648,2651,2654,2657],{"className":2632},[1872],[1838,2634],{"className":2635,"style":2216},[1876],[1838,2637,2639],{"className":2638},[2220],[1838,2640,1961],{"className":2641},[1881,2224],[1838,2643,1913],{"className":2644},[2036],[1838,2646,1859],{"className":2647,"style":1883},[1881,1882],[1838,2649,1943],{"className":2650},[2159],[1838,2652],{"className":2653,"style":1883},[1951],[1838,2655,2459],{"className":2656},[2546],[1838,2658],{"className":2659,"style":1883},[1951],[1838,2661,2663,2667,2670,2699,2702,2705,2708],{"className":2662},[1872],[1838,2664],{"className":2665,"style":2666},[1876],"height:1.0935em;vertical-align:-0.1944em;",[1838,2668,2474],{"className":2669},[1881],[1838,2671,2673,2676],{"className":2672},[1881],[1838,2674,2319],{"className":2675},[1881,2014],[1838,2677,2679],{"className":2678},[2047],[1838,2680,2682],{"className":2681},[2051],[1838,2683,2685],{"className":2684},[2056],[1838,2686,2688],{"className":2687,"style":2175},[2060],[1838,2689,2690,2693],{"style":2178},[1838,2691],{"className":2692,"style":2069},[2068],[1838,2694,2696],{"className":2695},[2073,2074,2075,2076],[1838,2697,1947],{"className":2698},[1881,2076],[1838,2700,1977],{"className":2701,"style":2246},[1881,2014],[1838,2703],{"className":2704,"style":1883},[1951],[1838,2706,2504],{"className":2707},[2546],[1838,2709],{"className":2710,"style":1883},[1951],[1838,2712,2714,2717,2746,2749,2752],{"className":2713},[1872],[1838,2715],{"className":2716,"style":2175},[1876],[1838,2718,2720,2723],{"className":2719},[1881],[1838,2721,2319],{"className":2722},[1881,2014],[1838,2724,2726],{"className":2725},[2047],[1838,2727,2729],{"className":2728},[2051],[1838,2730,2732],{"className":2731},[2056],[1838,2733,2735],{"className":2734,"style":2175},[2060],[1838,2736,2737,2740],{"style":2178},[1838,2738],{"className":2739,"style":2069},[2068],[1838,2741,2743],{"className":2742},[2073,2074,2075,2076],[1838,2744,1947],{"className":2745},[1881,2076],[1838,2747,1956],{"className":2748},[1881],[1838,2750,2319],{"className":2751},[1881,2014],[1838,2753,1997],{"className":2754},[1881],[1798,2756,2757],{},"Differentiation gives the normal equations",[1838,2759,2761],{"className":2760},[1888],[1838,2762,2764,2800],{"className":2763},[1841],[1838,2765,2767],{"className":2766},[1845],[1847,2768,2769],{"xmlns":1849,"display":1897},[1851,2770,2771,2797],{},[1854,2772,2773,2795],{},[2774,2775,2777],"menclose",{"notation":2776},"box",[2778,2779,2781],"mstyle",{"scriptlevel":2780,"displaystyle":1912},"0",[2778,2782,2783],{"scriptlevel":2780,"displaystyle":1912},[2778,2784,2785],{"scriptlevel":2780,"displaystyle":1868},[1854,2786,2787,2789,2791,2793],{},[1857,2788,1956],{"mathvariant":1946},[1857,2790,2319],{"mathvariant":1904},[1907,2792,1909],{},[1857,2794,1977],{"mathvariant":1904},[1857,2796,1997],{"mathvariant":1946},[1861,2798,2799],{"encoding":1863},"\\boxed{\\Gamma\\mathbf a=\\mathbf g}.",[1838,2801,2803],{"className":2802,"ariaHidden":1868},[1867],[1838,2804,2806,2810,2882],{"className":2805},[1872],[1838,2807],{"className":2808,"style":2809},[1876],"height:1.5578em;vertical-align:-0.5344em;",[1838,2811,2813],{"className":2812},[1881],[1838,2814,2816,2873],{"className":2815},[2051,2052],[1838,2817,2819,2870],{"className":2818},[2056],[1838,2820,2823,2858],{"className":2821,"style":2822},[2060],"height:1.0233em;",[1838,2824,2826,2830],{"style":2825},"top:-3.5578em;",[1838,2827],{"className":2828,"style":2829},[2068],"height:3.5578em;",[1838,2831,2834],{"className":2832},[2833],"boxpad",[1838,2835,2837],{"className":2836},[1881],[1838,2838,2840,2843,2846,2849,2852,2855],{"className":2839},[1881],[1838,2841,1956],{"className":2842},[1881],[1838,2844,2319],{"className":2845},[1881,2014],[1838,2847],{"className":2848,"style":2018},[1951],[1838,2850,1909],{"className":2851},[2022],[1838,2853],{"className":2854,"style":2018},[1951],[1838,2856,1977],{"className":2857,"style":2246},[1881,2014],[1838,2859,2861,2864],{"style":2860},"top:-3.0233em;",[1838,2862],{"className":2863,"style":2829},[2068],[1838,2865],{"className":2866,"style":2869},[2867,2868],"stretchy","fbox","height:1.5578em;border-style:solid;border-width:0.04em;",[1838,2871,2084],{"className":2872},[2083],[1838,2874,2876],{"className":2875},[2056],[1838,2877,2880],{"className":2878,"style":2879},[2060],"height:0.5344em;",[1838,2881],{},[1838,2883,1997],{"className":2884},[1881],[1798,2886,2887,2888,2917],{},"When ",[1838,2889,2891,2905],{"className":2890},[1841],[1838,2892,2894],{"className":2893},[1845],[1847,2895,2896],{"xmlns":1849},[1851,2897,2898,2902],{},[1854,2899,2900],{},[1857,2901,1956],{"mathvariant":1946},[1861,2903,2904],{"encoding":1863},"\\Gamma",[1838,2906,2908],{"className":2907,"ariaHidden":1868},[1867],[1838,2909,2911,2914],{"className":2910},[1872],[1838,2912],{"className":2913,"style":1877},[1876],[1838,2915,1956],{"className":2916},[1881]," is positive definite,",[1838,2919,2921],{"className":2920},[1888],[1838,2922,2924,2995],{"className":2923},[1841],[1838,2925,2927],{"className":2926},[1845],[1847,2928,2929],{"xmlns":1849,"display":1897},[1851,2930,2931,2992],{},[1854,2932,2933,2935,2937,2947,2949,2951,2953,2956,2958,2960,2962,2964,2966,2968,2970,2972,2978,2988,2990],{},[1857,2934,2319],{"mathvariant":1904},[1907,2936,1909],{},[1939,2938,2939,2941],{},[1857,2940,1956],{"mathvariant":1946},[1854,2942,2943,2945],{},[1907,2944,2459],{},[1920,2946,1922],{},[1857,2948,1977],{"mathvariant":1904},[1907,2950,1925],{"separator":1868},[1951,2952],{"width":1953},[1857,2954,2955],{"mathvariant":1946},"MSPE",[1907,2957,1964],{},[1907,2959,1909],{},[1857,2961,1961],{"mathvariant":1946},[1907,2963,1964],{},[1907,2965,1913],{"stretchy":1912},[1857,2967,1859],{},[1907,2969,1943],{"stretchy":1912},[1907,2971,2459],{},[1939,2973,2974,2976],{},[1857,2975,1977],{"mathvariant":1904},[1857,2977,1947],{"mathvariant":1946},[1939,2979,2980,2982],{},[1857,2981,1956],{"mathvariant":1946},[1854,2983,2984,2986],{},[1907,2985,2459],{},[1920,2987,1922],{},[1857,2989,1977],{"mathvariant":1904},[1857,2991,1997],{"mathvariant":1946},[1861,2993,2994],{"encoding":1863},"\\mathbf a=\\Gamma^{-1}\\mathbf g,\\qquad\n\\operatorname{MSPE}=\\operatorname{Var}(Y)-\\mathbf g^\\top\\Gamma^{-1}\\mathbf g.",[1838,2996,2998,3017,3086,3116],{"className":2997,"ariaHidden":1868},[1867],[1838,2999,3001,3005,3008,3011,3014],{"className":3000},[1872],[1838,3002],{"className":3003,"style":3004},[1876],"height:0.4444em;",[1838,3006,2319],{"className":3007},[1881,2014],[1838,3009],{"className":3010,"style":2018},[1951],[1838,3012,1909],{"className":3013},[2022],[1838,3015],{"className":3016,"style":2018},[1951],[1838,3018,3020,3024,3059,3062,3065,3068,3071,3077,3080,3083],{"className":3019},[1872],[1838,3021],{"className":3022,"style":3023},[1876],"height:1.0585em;vertical-align:-0.1944em;",[1838,3025,3027,3030],{"className":3026},[1881],[1838,3028,1956],{"className":3029},[1881],[1838,3031,3033],{"className":3032},[2047],[1838,3034,3036],{"className":3035},[2051],[1838,3037,3039],{"className":3038},[2056],[1838,3040,3042],{"className":3041,"style":2606},[2060],[1838,3043,3044,3047],{"style":2178},[1838,3045],{"className":3046,"style":2069},[2068],[1838,3048,3050],{"className":3049},[2073,2074,2075,2076],[1838,3051,3053,3056],{"className":3052},[1881,2076],[1838,3054,2459],{"className":3055},[1881,2076],[1838,3057,1922],{"className":3058},[1881,2076],[1838,3060,1977],{"className":3061,"style":2246},[1881,2014],[1838,3063,1925],{"className":3064},[2097],[1838,3066],{"className":3067,"style":2194},[1951],[1838,3069],{"className":3070,"style":2101},[1951],[1838,3072,3074],{"className":3073},[2220],[1838,3075,2955],{"className":3076},[1881,2224],[1838,3078],{"className":3079,"style":2018},[1951],[1838,3081,1909],{"className":3082},[2022],[1838,3084],{"className":3085,"style":2018},[1951],[1838,3087,3089,3092,3098,3101,3104,3107,3110,3113],{"className":3088},[1872],[1838,3090],{"className":3091,"style":2216},[1876],[1838,3093,3095],{"className":3094},[2220],[1838,3096,1961],{"className":3097},[1881,2224],[1838,3099,1913],{"className":3100},[2036],[1838,3102,1859],{"className":3103,"style":1883},[1881,1882],[1838,3105,1943],{"className":3106},[2159],[1838,3108],{"className":3109,"style":1883},[1951],[1838,3111,2459],{"className":3112},[2546],[1838,3114],{"className":3115,"style":1883},[1951],[1838,3117,3119,3122,3151,3186,3189],{"className":3118},[1872],[1838,3120],{"className":3121,"style":2666},[1876],[1838,3123,3125,3128],{"className":3124},[1881],[1838,3126,1977],{"className":3127,"style":2246},[1881,2014],[1838,3129,3131],{"className":3130},[2047],[1838,3132,3134],{"className":3133},[2051],[1838,3135,3137],{"className":3136},[2056],[1838,3138,3140],{"className":3139,"style":2175},[2060],[1838,3141,3142,3145],{"style":2178},[1838,3143],{"className":3144,"style":2069},[2068],[1838,3146,3148],{"className":3147},[2073,2074,2075,2076],[1838,3149,1947],{"className":3150},[1881,2076],[1838,3152,3154,3157],{"className":3153},[1881],[1838,3155,1956],{"className":3156},[1881],[1838,3158,3160],{"className":3159},[2047],[1838,3161,3163],{"className":3162},[2051],[1838,3164,3166],{"className":3165},[2056],[1838,3167,3169],{"className":3168,"style":2606},[2060],[1838,3170,3171,3174],{"style":2178},[1838,3172],{"className":3173,"style":2069},[2068],[1838,3175,3177],{"className":3176},[2073,2074,2075,2076],[1838,3178,3180,3183],{"className":3179},[1881,2076],[1838,3181,2459],{"className":3182},[1881,2076],[1838,3184,1922],{"className":3185},[1881,2076],[1838,3187,1977],{"className":3188,"style":2246},[1881,2014],[1838,3190,1997],{"className":3191},[1881],[1798,3193,3194,3195,3199],{},"In computation, use ",[3196,3197,3198],"code",{},"solve(Gamma, g)",". The inverse notation explains the mathematics; explicitly forming an inverse is usually unnecessary.",[1793,3201,3203],{"id":3202},"_2-orthogonality-is-the-defining-check","2. Orthogonality is the defining check",[1798,3205,3206],{},"At the optimum,",[1838,3208,3210],{"className":3209},[1888],[1838,3211,3213,3262],{"className":3212},[1841],[1838,3214,3216],{"className":3215},[1845],[1847,3217,3218],{"xmlns":1849,"display":1897},[1851,3219,3220,3259],{},[1854,3221,3222,3224,3226,3228,3230,3232,3234,3236,3242,3244,3246,3248,3250,3252,3254,3256],{},[1857,3223,1982],{"mathvariant":1946},[1907,3225,1964],{},[1907,3227,1913],{"stretchy":1912},[1857,3229,1905],{"mathvariant":1904},[1907,3231,1925],{"separator":1868},[1857,3233,1859],{},[1907,3235,2459],{},[2306,3237,3238,3240],{"accent":1868},[1857,3239,1859],{},[1907,3241,2312],{"stretchy":1868},[1907,3243,1943],{"stretchy":1912},[1907,3245,1909],{},[1857,3247,1977],{"mathvariant":1904},[1907,3249,2459],{},[1857,3251,1956],{"mathvariant":1946},[1857,3253,2319],{"mathvariant":1904},[1907,3255,1909],{},[1920,3257,3258],{"mathvariant":1904},"0.",[1861,3260,3261],{"encoding":1863},"\\operatorname{Cov}(\\mathbf X,Y-\\widehat Y)\n=\\mathbf g-\\Gamma\\mathbf a=\\mathbf 0.",[1838,3263,3265,3301,3352,3371,3392],{"className":3264,"ariaHidden":1868},[1867],[1838,3266,3268,3271,3277,3280,3283,3286,3289,3292,3295,3298],{"className":3267},[1872],[1838,3269],{"className":3270,"style":2216},[1876],[1838,3272,3274],{"className":3273},[2220],[1838,3275,1982],{"className":3276,"style":2268},[1881,2224],[1838,3278,1913],{"className":3279},[2036],[1838,3281,1905],{"className":3282},[1881,2014],[1838,3284,1925],{"className":3285},[2097],[1838,3287],{"className":3288,"style":2101},[1951],[1838,3290,1859],{"className":3291,"style":1883},[1881,1882],[1838,3293],{"className":3294,"style":1883},[1951],[1838,3296,2459],{"className":3297},[2546],[1838,3299],{"className":3300,"style":1883},[1951],[1838,3302,3304,3308,3340,3343,3346,3349],{"className":3303},[1872],[1838,3305],{"className":3306,"style":3307},[1876],"height:1.1733em;vertical-align:-0.25em;",[1838,3309,3311],{"className":3310},[1881,2340],[1838,3312,3314],{"className":3313},[2051],[1838,3315,3317],{"className":3316},[2056],[1838,3318,3320,3328],{"className":3319,"style":2336},[2060],[1838,3321,3322,3325],{"style":2352},[1838,3323],{"className":3324,"style":2356},[2068],[1838,3326,1859],{"className":3327,"style":1883},[1881,1882],[1838,3329,3331,3334],{"className":3330,"style":2364},[2363],[1838,3332],{"className":3333,"style":2356},[2068],[1838,3335,3336],{"style":2370},[2372,3337,3338],{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},[2380,3339],{"d":2382},[1838,3341,1943],{"className":3342},[2159],[1838,3344],{"className":3345,"style":2018},[1951],[1838,3347,1909],{"className":3348},[2022],[1838,3350],{"className":3351,"style":2018},[1951],[1838,3353,3355,3359,3362,3365,3368],{"className":3354},[1872],[1838,3356],{"className":3357,"style":3358},[1876],"height:0.7778em;vertical-align:-0.1944em;",[1838,3360,1977],{"className":3361,"style":2246},[1881,2014],[1838,3363],{"className":3364,"style":1883},[1951],[1838,3366,2459],{"className":3367},[2546],[1838,3369],{"className":3370,"style":1883},[1951],[1838,3372,3374,3377,3380,3383,3386,3389],{"className":3373},[1872],[1838,3375],{"className":3376,"style":1877},[1876],[1838,3378,1956],{"className":3379},[1881],[1838,3381,2319],{"className":3382},[1881,2014],[1838,3384],{"className":3385,"style":2018},[1951],[1838,3387,1909],{"className":3388},[2022],[1838,3390],{"className":3391,"style":2018},[1951],[1838,3393,3395,3399,3402],{"className":3394},[1872],[1838,3396],{"className":3397,"style":3398},[1876],"height:0.6444em;",[1838,3400,2780],{"className":3401},[1881,2014],[1838,3403,1997],{"className":3404},[1881],[1798,3406,3407],{},"The error is therefore orthogonal to every predictor. This statement needs finite second moments, not Gaussianity.",[1798,3409,3410,3411,3524],{},"If ",[1838,3412,3414,3444],{"className":3413},[1841],[1838,3415,3417],{"className":3416},[1845],[1847,3418,3419],{"xmlns":1849},[1851,3420,3421,3441],{},[1854,3422,3423,3425,3427,3429,3435],{},[1907,3424,1913],{"stretchy":1912},[1857,3426,1859],{},[1907,3428,1925],{"separator":1868},[1939,3430,3431,3433],{},[1857,3432,1905],{"mathvariant":1904},[1857,3434,1947],{"mathvariant":1946},[1939,3436,3437,3439],{},[1907,3438,1943],{"stretchy":1912},[1857,3440,1947],{"mathvariant":1946},[1861,3442,3443],{"encoding":1863},"(Y,\\mathbf X^\\top)^\\top",[1838,3445,3447],{"className":3446,"ariaHidden":1868},[1867],[1838,3448,3450,3454,3457,3460,3463,3466,3495],{"className":3449},[1872],[1838,3451],{"className":3452,"style":3453},[1876],"height:1.0991em;vertical-align:-0.25em;",[1838,3455,1913],{"className":3456},[2036],[1838,3458,1859],{"className":3459,"style":1883},[1881,1882],[1838,3461,1925],{"className":3462},[2097],[1838,3464],{"className":3465,"style":2101},[1951],[1838,3467,3469,3472],{"className":3468},[1881],[1838,3470,1905],{"className":3471},[1881,2014],[1838,3473,3475],{"className":3474},[2047],[1838,3476,3478],{"className":3477},[2051],[1838,3479,3481],{"className":3480},[2056],[1838,3482,3484],{"className":3483,"style":2398},[2060],[1838,3485,3486,3489],{"style":2419},[1838,3487],{"className":3488,"style":2069},[2068],[1838,3490,3492],{"className":3491},[2073,2074,2075,2076],[1838,3493,1947],{"className":3494},[1881,2076],[1838,3496,3498,3501],{"className":3497},[2159],[1838,3499,1943],{"className":3500},[2159],[1838,3502,3504],{"className":3503},[2047],[1838,3505,3507],{"className":3506},[2051],[1838,3508,3510],{"className":3509},[2056],[1838,3511,3513],{"className":3512,"style":2398},[2060],[1838,3514,3515,3518],{"style":2419},[1838,3516],{"className":3517,"style":2069},[2068],[1838,3519,3521],{"className":3520},[2073,2074,2075,2076],[1838,3522,1947],{"className":3523},[1881,2076]," is jointly Gaussian, the conditional mean is linear:",[1838,3526,3528],{"className":3527},[1888],[1838,3529,3531,3578],{"className":3530},[1841],[1838,3532,3534],{"className":3533},[1845],[1847,3535,3536],{"xmlns":1849,"display":1897},[1851,3537,3538,3575],{},[1854,3539,3540,3542,3544,3546,3549,3551,3553,3555,3561,3571,3573],{},[1857,3541,2449],{},[1907,3543,1913],{"stretchy":1912},[1857,3545,1859],{},[1907,3547,3548],{},"∣",[1857,3550,1905],{"mathvariant":1904},[1907,3552,1943],{"stretchy":1912},[1907,3554,1909],{},[1939,3556,3557,3559],{},[1857,3558,1977],{"mathvariant":1904},[1857,3560,1947],{"mathvariant":1946},[1939,3562,3563,3565],{},[1857,3564,1956],{"mathvariant":1946},[1854,3566,3567,3569],{},[1907,3568,2459],{},[1920,3570,1922],{},[1857,3572,1905],{"mathvariant":1904},[1857,3574,1997],{"mathvariant":1946},[1861,3576,3577],{"encoding":1863},"E(Y\\mid\\mathbf X)=\\mathbf g^\\top\\Gamma^{-1}\\mathbf X.",[1838,3579,3581,3605,3626],{"className":3580,"ariaHidden":1868},[1867],[1838,3582,3584,3587,3590,3593,3596,3599,3602],{"className":3583},[1872],[1838,3585],{"className":3586,"style":2216},[1876],[1838,3588,2449],{"className":3589,"style":2532},[1881,1882],[1838,3591,1913],{"className":3592},[2036],[1838,3594,1859],{"className":3595,"style":1883},[1881,1882],[1838,3597],{"className":3598,"style":2018},[1951],[1838,3600,3548],{"className":3601},[2022],[1838,3603],{"className":3604,"style":2018},[1951],[1838,3606,3608,3611,3614,3617,3620,3623],{"className":3607},[1872],[1838,3609],{"className":3610,"style":2216},[1876],[1838,3612,1905],{"className":3613},[1881,2014],[1838,3615,1943],{"className":3616},[2159],[1838,3618],{"className":3619,"style":2018},[1951],[1838,3621,1909],{"className":3622},[2022],[1838,3624],{"className":3625,"style":2018},[1951],[1838,3627,3629,3632,3661,3696,3699],{"className":3628},[1872],[1838,3630],{"className":3631,"style":2666},[1876],[1838,3633,3635,3638],{"className":3634},[1881],[1838,3636,1977],{"className":3637,"style":2246},[1881,2014],[1838,3639,3641],{"className":3640},[2047],[1838,3642,3644],{"className":3643},[2051],[1838,3645,3647],{"className":3646},[2056],[1838,3648,3650],{"className":3649,"style":2175},[2060],[1838,3651,3652,3655],{"style":2178},[1838,3653],{"className":3654,"style":2069},[2068],[1838,3656,3658],{"className":3657},[2073,2074,2075,2076],[1838,3659,1947],{"className":3660},[1881,2076],[1838,3662,3664,3667],{"className":3663},[1881],[1838,3665,1956],{"className":3666},[1881],[1838,3668,3670],{"className":3669},[2047],[1838,3671,3673],{"className":3672},[2051],[1838,3674,3676],{"className":3675},[2056],[1838,3677,3679],{"className":3678,"style":2606},[2060],[1838,3680,3681,3684],{"style":2178},[1838,3682],{"className":3683,"style":2069},[2068],[1838,3685,3687],{"className":3686},[2073,2074,2075,2076],[1838,3688,3690,3693],{"className":3689},[1881,2076],[1838,3691,2459],{"className":3692},[1881,2076],[1838,3694,1922],{"className":3695},[1881,2076],[1838,3697,1905],{"className":3698},[1881,2014],[1838,3700,1997],{"className":3701},[1881],[1798,3703,3704,3705,3709],{},"Without joint Gaussianity, this remains the ",[3706,3707,3708],"strong",{},"best linear"," predictor but need not be the best among all nonlinear functions.",[1793,3711,3713],{"id":3712},"_3-worked-ar1-projection","3. Worked AR(1) projection",[1798,3715,3716,3717,3763,3764,3880,3881,3969],{},"For a stationary AR(1) with variance ",[1838,3718,3720,3741],{"className":3719},[1841],[1838,3721,3723],{"className":3722},[1845],[1847,3724,3725],{"xmlns":1849},[1851,3726,3727,3738],{},[1854,3728,3729,3732,3734,3736],{},[1857,3730,3731],{},"γ",[1907,3733,1913],{"stretchy":1912},[1920,3735,2780],{},[1907,3737,1943],{"stretchy":1912},[1861,3739,3740],{"encoding":1863},"\\gamma(0)",[1838,3742,3744],{"className":3743,"ariaHidden":1868},[1867],[1838,3745,3747,3750,3754,3757,3760],{"className":3746},[1872],[1838,3748],{"className":3749,"style":2216},[1876],[1838,3751,3731],{"className":3752,"style":3753},[1881,1882],"margin-right:0.0556em;",[1838,3755,1913],{"className":3756},[2036],[1838,3758,2780],{"className":3759},[1881],[1838,3761,1943],{"className":3762},[2159]," and correlation ",[1838,3765,3767,3804],{"className":3766},[1841],[1838,3768,3770],{"className":3769},[1845],[1847,3771,3772],{"xmlns":1849},[1851,3773,3774,3801],{},[1854,3775,3776,3779,3781,3784,3786,3788],{},[1857,3777,3778],{},"ρ",[1907,3780,1913],{"stretchy":1912},[1857,3782,3783],{},"h",[1907,3785,1943],{"stretchy":1912},[1907,3787,1909],{},[1939,3789,3790,3793],{},[1857,3791,3792],{},"ϕ",[1854,3794,3795,3797,3799],{},[1857,3796,3548],{"mathvariant":1946},[1857,3798,3783],{},[1857,3800,3548],{"mathvariant":1946},[1861,3802,3803],{"encoding":1863},"\\rho(h)=\\phi^{|h|}",[1838,3805,3807,3834],{"className":3806,"ariaHidden":1868},[1867],[1838,3808,3810,3813,3816,3819,3822,3825,3828,3831],{"className":3809},[1872],[1838,3811],{"className":3812,"style":2216},[1876],[1838,3814,3778],{"className":3815},[1881,1882],[1838,3817,1913],{"className":3818},[2036],[1838,3820,3783],{"className":3821},[1881,1882],[1838,3823,1943],{"className":3824},[2159],[1838,3826],{"className":3827,"style":2018},[1951],[1838,3829,1909],{"className":3830},[2022],[1838,3832],{"className":3833,"style":2018},[1951],[1838,3835,3837,3841],{"className":3836},[1872],[1838,3838],{"className":3839,"style":3840},[1876],"height:1.0824em;vertical-align:-0.1944em;",[1838,3842,3844,3847],{"className":3843},[1881],[1838,3845,3792],{"className":3846},[1881,1882],[1838,3848,3850],{"className":3849},[2047],[1838,3851,3853],{"className":3852},[2051],[1838,3854,3856],{"className":3855},[2056],[1838,3857,3860],{"className":3858,"style":3859},[2060],"height:0.888em;",[1838,3861,3862,3865],{"style":2419},[1838,3863],{"className":3864,"style":2069},[2068],[1838,3866,3868],{"className":3867},[2073,2074,2075,2076],[1838,3869,3871,3874,3877],{"className":3870},[1881,2076],[1838,3872,3548],{"className":3873},[1881,2076],[1838,3875,3783],{"className":3876},[1881,1882,2076],[1838,3878,3548],{"className":3879},[1881,2076],", predict ",[1838,3882,3884,3909],{"className":3883},[1841],[1838,3885,3887],{"className":3886},[1845],[1847,3888,3889],{"xmlns":1849},[1851,3890,3891,3906],{},[1854,3892,3893],{},[1915,3894,3895,3897],{},[1857,3896,1905],{},[1854,3898,3899,3902,3904],{},[1857,3900,3901],{},"t",[1907,3903,2504],{},[1920,3905,1922],{},[1861,3907,3908],{"encoding":1863},"X_{t+1}",[1838,3910,3912],{"className":3911,"ariaHidden":1868},[1867],[1838,3913,3915,3919],{"className":3914},[1872],[1838,3916],{"className":3917,"style":3918},[1876],"height:0.8917em;vertical-align:-0.2083em;",[1838,3920,3922,3925],{"className":3921},[1881],[1838,3923,1905],{"className":3924,"style":2043},[1881,1882],[1838,3926,3928],{"className":3927},[2047],[1838,3929,3931,3960],{"className":3930},[2051,2052],[1838,3932,3934,3957],{"className":3933},[2056],[1838,3935,3937],{"className":3936,"style":2061},[2060],[1838,3938,3939,3942],{"style":2064},[1838,3940],{"className":3941,"style":2069},[2068],[1838,3943,3945],{"className":3944},[2073,2074,2075,2076],[1838,3946,3948,3951,3954],{"className":3947},[1881,2076],[1838,3949,3901],{"className":3950},[1881,1882,2076],[1838,3952,2504],{"className":3953},[2546,2076],[1838,3955,1922],{"className":3956},[1881,2076],[1838,3958,2084],{"className":3959},[2083],[1838,3961,3963],{"className":3962},[2056],[1838,3964,3967],{"className":3965,"style":3966},[2060],"height:0.2083em;",[1838,3968],{}," from",[1838,3971,3973],{"className":3972},[1888],[1838,3974,3976,4022],{"className":3975},[1841],[1838,3977,3979],{"className":3978},[1845],[1847,3980,3981],{"xmlns":1849,"display":1897},[1851,3982,3983,4019],{},[1854,3984,3985,3987,3989,3991,4003,4005,4011,4017],{},[1857,3986,1905],{"mathvariant":1904},[1907,3988,1909],{},[1907,3990,1913],{"stretchy":1912},[1915,3992,3993,3995],{},[1857,3994,1905],{},[1854,3996,3997,3999,4001],{},[1857,3998,3901],{},[1907,4000,2459],{},[1920,4002,1922],{},[1907,4004,1925],{"separator":1868},[1915,4006,4007,4009],{},[1857,4008,1905],{},[1857,4010,3901],{},[1939,4012,4013,4015],{},[1907,4014,1943],{"stretchy":1912},[1857,4016,1947],{"mathvariant":1946},[1857,4018,1997],{"mathvariant":1946},[1861,4020,4021],{"encoding":1863},"\\mathbf X=(X_{t-1},X_t)^\\top.",[1838,4023,4025,4043],{"className":4024,"ariaHidden":1868},[1867],[1838,4026,4028,4031,4034,4037,4040],{"className":4027},[1872],[1838,4029],{"className":4030,"style":2010},[1876],[1838,4032,1905],{"className":4033},[1881,2014],[1838,4035],{"className":4036,"style":2018},[1951],[1838,4038,1909],{"className":4039},[2022],[1838,4041],{"className":4042,"style":2018},[1951],[1838,4044,4046,4049,4052,4101,4104,4107,4148,4177],{"className":4045},[1872],[1838,4047],{"className":4048,"style":2032},[1876],[1838,4050,1913],{"className":4051},[2036],[1838,4053,4055,4058],{"className":4054},[1881],[1838,4056,1905],{"className":4057,"style":2043},[1881,1882],[1838,4059,4061],{"className":4060},[2047],[1838,4062,4064,4093],{"className":4063},[2051,2052],[1838,4065,4067,4090],{"className":4066},[2056],[1838,4068,4070],{"className":4069,"style":2061},[2060],[1838,4071,4072,4075],{"style":2064},[1838,4073],{"className":4074,"style":2069},[2068],[1838,4076,4078],{"className":4077},[2073,2074,2075,2076],[1838,4079,4081,4084,4087],{"className":4080},[1881,2076],[1838,4082,3901],{"className":4083},[1881,1882,2076],[1838,4085,2459],{"className":4086},[2546,2076],[1838,4088,1922],{"className":4089},[1881,2076],[1838,4091,2084],{"className":4092},[2083],[1838,4094,4096],{"className":4095},[2056],[1838,4097,4099],{"className":4098,"style":3966},[2060],[1838,4100],{},[1838,4102,1925],{"className":4103},[2097],[1838,4105],{"className":4106,"style":2101},[1951],[1838,4108,4110,4113],{"className":4109},[1881],[1838,4111,1905],{"className":4112,"style":2043},[1881,1882],[1838,4114,4116],{"className":4115},[2047],[1838,4117,4119,4140],{"className":4118},[2051,2052],[1838,4120,4122,4137],{"className":4121},[2056],[1838,4123,4126],{"className":4124,"style":4125},[2060],"height:0.2806em;",[1838,4127,4128,4131],{"style":2064},[1838,4129],{"className":4130,"style":2069},[2068],[1838,4132,4134],{"className":4133},[2073,2074,2075,2076],[1838,4135,3901],{"className":4136},[1881,1882,2076],[1838,4138,2084],{"className":4139},[2083],[1838,4141,4143],{"className":4142},[2056],[1838,4144,4146],{"className":4145,"style":2091},[2060],[1838,4147],{},[1838,4149,4151,4154],{"className":4150},[2159],[1838,4152,1943],{"className":4153},[2159],[1838,4155,4157],{"className":4156},[2047],[1838,4158,4160],{"className":4159},[2051],[1838,4161,4163],{"className":4162},[2056],[1838,4164,4166],{"className":4165,"style":2175},[2060],[1838,4167,4168,4171],{"style":2178},[1838,4169],{"className":4170,"style":2069},[2068],[1838,4172,4174],{"className":4173},[2073,2074,2075,2076],[1838,4175,1947],{"className":4176},[1881,2076],[1838,4178,1997],{"className":4179},[1881],[1798,4181,4182],{},"Then",[1838,4184,4186],{"className":4185},[1888],[1838,4187,4189,4302],{"className":4188},[1841],[1838,4190,4192],{"className":4191},[1845],[1847,4193,4194],{"xmlns":1849,"display":1897},[1851,4195,4196,4299],{},[1854,4197,4198,4200,4202,4204,4206,4208,4210,4252,4254,4256,4258,4260,4262,4264,4266,4268,4297],{},[1857,4199,1956],{"mathvariant":1946},[1907,4201,1909],{},[1857,4203,3731],{},[1907,4205,1913],{"stretchy":1912},[1920,4207,2780],{},[1907,4209,1943],{"stretchy":1912},[1854,4211,4212,4214,4250],{},[1907,4213,2452],{"fence":1868},[4215,4216,4220,4236],"mtable",{"rowspacing":4217,"columnalign":4218,"columnspacing":4219},"0.16em","center center","1em",[4221,4222,4223,4230],"mtr",{},[4224,4225,4226],"mtd",{},[2778,4227,4228],{"scriptlevel":2780,"displaystyle":1912},[1920,4229,1922],{},[4224,4231,4232],{},[2778,4233,4234],{"scriptlevel":2780,"displaystyle":1912},[1857,4235,3792],{},[4221,4237,4238,4244],{},[4224,4239,4240],{},[2778,4241,4242],{"scriptlevel":2780,"displaystyle":1912},[1857,4243,3792],{},[4224,4245,4246],{},[2778,4247,4248],{"scriptlevel":2780,"displaystyle":1912},[1920,4249,1922],{},[1907,4251,2477],{"fence":1868},[1907,4253,1925],{"separator":1868},[1951,4255],{"width":1953},[1857,4257,1977],{"mathvariant":1904},[1907,4259,1909],{},[1857,4261,3731],{},[1907,4263,1913],{"stretchy":1912},[1920,4265,2780],{},[1907,4267,1943],{"stretchy":1912},[1854,4269,4270,4272,4295],{},[1907,4271,2452],{"fence":1868},[4215,4273,4275,4287],{"rowspacing":4217,"columnalign":4274,"columnspacing":4219},"center",[4221,4276,4277],{},[4224,4278,4279],{},[2778,4280,4281],{"scriptlevel":2780,"displaystyle":1912},[1939,4282,4283,4285],{},[1857,4284,3792],{},[1920,4286,2474],{},[4221,4288,4289],{},[4224,4290,4291],{},[2778,4292,4293],{"scriptlevel":2780,"displaystyle":1912},[1857,4294,3792],{},[1907,4296,2477],{"fence":1868},[1857,4298,1997],{"mathvariant":1946},[1861,4300,4301],{"encoding":1863},"\\Gamma=\\gamma(0)\n\\begin{bmatrix}\n1&\\phi\\\\\n\\phi&1\n\\end{bmatrix},\n\\qquad\n\\mathbf g=\\gamma(0)\n\\begin{bmatrix}\n\\phi^2\\\\\n\\phi\n\\end{bmatrix}.",[1838,4303,4305,4323,4496],{"className":4304,"ariaHidden":1868},[1867],[1838,4306,4308,4311,4314,4317,4320],{"className":4307},[1872],[1838,4309],{"className":4310,"style":1877},[1876],[1838,4312,1956],{"className":4313},[1881],[1838,4315],{"className":4316,"style":2018},[1951],[1838,4318,1909],{"className":4319},[2022],[1838,4321],{"className":4322,"style":2018},[1951],[1838,4324,4326,4330,4333,4336,4339,4342,4345,4472,4475,4478,4481,4484,4487,4490,4493],{"className":4325},[1872],[1838,4327],{"className":4328,"style":4329},[1876],"height:2.4em;vertical-align:-0.95em;",[1838,4331,3731],{"className":4332,"style":3753},[1881,1882],[1838,4334,1913],{"className":4335},[2036],[1838,4337,2780],{"className":4338},[1881],[1838,4340,1943],{"className":4341},[2159],[1838,4343],{"className":4344,"style":2101},[1951],[1838,4346,4348,4357,4466],{"className":4347},[2105],[1838,4349,4353],{"className":4350,"style":4352},[2036,4351],"delimcenter","top:0em;",[1838,4354,2452],{"className":4355},[4356,2075],"delimsizing",[1838,4358,4360],{"className":4359},[1881],[1838,4361,4363,4413,4418,4421],{"className":4362},[4215],[1838,4364,4367],{"className":4365},[4366],"col-align-c",[1838,4368,4370,4404],{"className":4369},[2051,2052],[1838,4371,4373,4401],{"className":4372},[2056],[1838,4374,4377,4389],{"className":4375,"style":4376},[2060],"height:1.45em;",[1838,4378,4380,4383],{"style":4379},"top:-3.61em;",[1838,4381],{"className":4382,"style":2356},[2068],[1838,4384,4386],{"className":4385},[1881],[1838,4387,1922],{"className":4388},[1881],[1838,4390,4392,4395],{"style":4391},"top:-2.41em;",[1838,4393],{"className":4394,"style":2356},[2068],[1838,4396,4398],{"className":4397},[1881],[1838,4399,3792],{"className":4400},[1881,1882],[1838,4402,2084],{"className":4403},[2083],[1838,4405,4407],{"className":4406},[2056],[1838,4408,4411],{"className":4409,"style":4410},[2060],"height:0.95em;",[1838,4412],{},[1838,4414],{"className":4415,"style":4417},[4416],"arraycolsep","width:0.5em;",[1838,4419],{"className":4420,"style":4417},[4416],[1838,4422,4424],{"className":4423},[4366],[1838,4425,4427,4458],{"className":4426},[2051,2052],[1838,4428,4430,4455],{"className":4429},[2056],[1838,4431,4433,4444],{"className":4432,"style":4376},[2060],[1838,4434,4435,4438],{"style":4379},[1838,4436],{"className":4437,"style":2356},[2068],[1838,4439,4441],{"className":4440},[1881],[1838,4442,3792],{"className":4443},[1881,1882],[1838,4445,4446,4449],{"style":4391},[1838,4447],{"className":4448,"style":2356},[2068],[1838,4450,4452],{"className":4451},[1881],[1838,4453,1922],{"className":4454},[1881],[1838,4456,2084],{"className":4457},[2083],[1838,4459,4461],{"className":4460},[2056],[1838,4462,4464],{"className":4463,"style":4410},[2060],[1838,4465],{},[1838,4467,4469],{"className":4468,"style":4352},[2159,4351],[1838,4470,2477],{"className":4471},[4356,2075],[1838,4473],{"className":4474,"style":2101},[1951],[1838,4476,1925],{"className":4477},[2097],[1838,4479],{"className":4480,"style":2194},[1951],[1838,4482],{"className":4483,"style":2101},[1951],[1838,4485,1977],{"className":4486,"style":2246},[1881,2014],[1838,4488],{"className":4489,"style":2018},[1951],[1838,4491,1909],{"className":4492},[2022],[1838,4494],{"className":4495,"style":2018},[1951],[1838,4497,4499,4502,4505,4508,4511,4514,4517,4610,4613],{"className":4498},[1872],[1838,4500],{"className":4501,"style":4329},[1876],[1838,4503,3731],{"className":4504,"style":3753},[1881,1882],[1838,4506,1913],{"className":4507},[2036],[1838,4509,2780],{"className":4510},[1881],[1838,4512,1943],{"className":4513},[2159],[1838,4515],{"className":4516,"style":2101},[1951],[1838,4518,4520,4526,4604],{"className":4519},[2105],[1838,4521,4523],{"className":4522,"style":4352},[2036,4351],[1838,4524,2452],{"className":4525},[4356,2075],[1838,4527,4529],{"className":4528},[1881],[1838,4530,4532],{"className":4531},[4215],[1838,4533,4535],{"className":4534},[4366],[1838,4536,4538,4596],{"className":4537},[2051,2052],[1838,4539,4541,4593],{"className":4540},[2056],[1838,4542,4544,4582],{"className":4543,"style":4376},[2060],[1838,4545,4546,4549],{"style":4379},[1838,4547],{"className":4548,"style":2356},[2068],[1838,4550,4552],{"className":4551},[1881],[1838,4553,4555,4558],{"className":4554},[1881],[1838,4556,3792],{"className":4557},[1881,1882],[1838,4559,4561],{"className":4560},[2047],[1838,4562,4564],{"className":4563},[2051],[1838,4565,4567],{"className":4566},[2056],[1838,4568,4571],{"className":4569,"style":4570},[2060],"height:0.8141em;",[1838,4572,4573,4576],{"style":2419},[1838,4574],{"className":4575,"style":2069},[2068],[1838,4577,4579],{"className":4578},[2073,2074,2075,2076],[1838,4580,2474],{"className":4581},[1881,2076],[1838,4583,4584,4587],{"style":4391},[1838,4585],{"className":4586,"style":2356},[2068],[1838,4588,4590],{"className":4589},[1881],[1838,4591,3792],{"className":4592},[1881,1882],[1838,4594,2084],{"className":4595},[2083],[1838,4597,4599],{"className":4598},[2056],[1838,4600,4602],{"className":4601,"style":4410},[2060],[1838,4603],{},[1838,4605,4607],{"className":4606,"style":4352},[2159,4351],[1838,4608,2477],{"className":4609},[4356,2075],[1838,4611],{"className":4612,"style":2101},[1951],[1838,4614,1997],{"className":4615},[1881],[1798,4617,4618,4619,4723,4724,4795,4796,4881],{},"Solving gives ",[1838,4620,4622,4652],{"className":4621},[1841],[1838,4623,4625],{"className":4624},[1845],[1847,4626,4627],{"xmlns":1849},[1851,4628,4629,4649],{},[1854,4630,4631,4633,4635,4637,4639,4641,4643],{},[1857,4632,2319],{"mathvariant":1904},[1907,4634,1909],{},[1907,4636,1913],{"stretchy":1912},[1920,4638,2780],{},[1907,4640,1925],{"separator":1868},[1857,4642,3792],{},[1939,4644,4645,4647],{},[1907,4646,1943],{"stretchy":1912},[1857,4648,1947],{"mathvariant":1946},[1861,4650,4651],{"encoding":1863},"\\mathbf a=(0,\\phi)^\\top",[1838,4653,4655,4673],{"className":4654,"ariaHidden":1868},[1867],[1838,4656,4658,4661,4664,4667,4670],{"className":4657},[1872],[1838,4659],{"className":4660,"style":3004},[1876],[1838,4662,2319],{"className":4663},[1881,2014],[1838,4665],{"className":4666,"style":2018},[1951],[1838,4668,1909],{"className":4669},[2022],[1838,4671],{"className":4672,"style":2018},[1951],[1838,4674,4676,4679,4682,4685,4688,4691,4694],{"className":4675},[1872],[1838,4677],{"className":4678,"style":3453},[1876],[1838,4680,1913],{"className":4681},[2036],[1838,4683,2780],{"className":4684},[1881],[1838,4686,1925],{"className":4687},[2097],[1838,4689],{"className":4690,"style":2101},[1951],[1838,4692,3792],{"className":4693},[1881,1882],[1838,4695,4697,4700],{"className":4696},[2159],[1838,4698,1943],{"className":4699},[2159],[1838,4701,4703],{"className":4702},[2047],[1838,4704,4706],{"className":4705},[2051],[1838,4707,4709],{"className":4708},[2056],[1838,4710,4712],{"className":4711,"style":2398},[2060],[1838,4713,4714,4717],{"style":2419},[1838,4715],{"className":4716,"style":2069},[2068],[1838,4718,4720],{"className":4719},[2073,2074,2075,2076],[1838,4721,1947],{"className":4722},[1881,2076],". Once ",[1838,4725,4727,4745],{"className":4726},[1841],[1838,4728,4730],{"className":4729},[1845],[1847,4731,4732],{"xmlns":1849},[1851,4733,4734,4742],{},[1854,4735,4736],{},[1915,4737,4738,4740],{},[1857,4739,1905],{},[1857,4741,3901],{},[1861,4743,4744],{"encoding":1863},"X_t",[1838,4746,4748],{"className":4747,"ariaHidden":1868},[1867],[1838,4749,4751,4755],{"className":4750},[1872],[1838,4752],{"className":4753,"style":4754},[1876],"height:0.8333em;vertical-align:-0.15em;",[1838,4756,4758,4761],{"className":4757},[1881],[1838,4759,1905],{"className":4760,"style":2043},[1881,1882],[1838,4762,4764],{"className":4763},[2047],[1838,4765,4767,4787],{"className":4766},[2051,2052],[1838,4768,4770,4784],{"className":4769},[2056],[1838,4771,4773],{"className":4772,"style":4125},[2060],[1838,4774,4775,4778],{"style":2064},[1838,4776],{"className":4777,"style":2069},[2068],[1838,4779,4781],{"className":4780},[2073,2074,2075,2076],[1838,4782,3901],{"className":4783},[1881,1882,2076],[1838,4785,2084],{"className":4786},[2083],[1838,4788,4790],{"className":4789},[2056],[1838,4791,4793],{"className":4792,"style":2091},[2060],[1838,4794],{}," is known, ",[1838,4797,4799,4823],{"className":4798},[1841],[1838,4800,4802],{"className":4801},[1845],[1847,4803,4804],{"xmlns":1849},[1851,4805,4806,4820],{},[1854,4807,4808],{},[1915,4809,4810,4812],{},[1857,4811,1905],{},[1854,4813,4814,4816,4818],{},[1857,4815,3901],{},[1907,4817,2459],{},[1920,4819,1922],{},[1861,4821,4822],{"encoding":1863},"X_{t-1}",[1838,4824,4826],{"className":4825,"ariaHidden":1868},[1867],[1838,4827,4829,4832],{"className":4828},[1872],[1838,4830],{"className":4831,"style":3918},[1876],[1838,4833,4835,4838],{"className":4834},[1881],[1838,4836,1905],{"className":4837,"style":2043},[1881,1882],[1838,4839,4841],{"className":4840},[2047],[1838,4842,4844,4873],{"className":4843},[2051,2052],[1838,4845,4847,4870],{"className":4846},[2056],[1838,4848,4850],{"className":4849,"style":2061},[2060],[1838,4851,4852,4855],{"style":2064},[1838,4853],{"className":4854,"style":2069},[2068],[1838,4856,4858],{"className":4857},[2073,2074,2075,2076],[1838,4859,4861,4864,4867],{"className":4860},[1881,2076],[1838,4862,3901],{"className":4863},[1881,1882,2076],[1838,4865,2459],{"className":4866},[2546,2076],[1838,4868,1922],{"className":4869},[1881,2076],[1838,4871,2084],{"className":4872},[2083],[1838,4874,4876],{"className":4875},[2056],[1838,4877,4879],{"className":4878,"style":3966},[2060],[1838,4880],{}," receives zero extra weight. The matrix calculation recovers the Markov property rather than assuming it.",[1798,4883,4884],{},"The error variance is",[1838,4886,4888],{"className":4887},[1888],[1838,4889,4891,4970],{"className":4890},[1841],[1838,4892,4894],{"className":4893},[1845],[1847,4895,4896],{"xmlns":1849,"display":1897},[1851,4897,4898,4967],{},[1854,4899,4900,4902,4904,4906,4908,4910,4916,4926,4928,4930,4932,4934,4936,4938,4940,4942,4944,4950,4952,4954,4965],{},[1857,4901,3731],{},[1907,4903,1913],{"stretchy":1912},[1920,4905,2780],{},[1907,4907,1943],{"stretchy":1912},[1907,4909,2459],{},[1939,4911,4912,4914],{},[1857,4913,1977],{"mathvariant":1904},[1857,4915,1947],{"mathvariant":1946},[1939,4917,4918,4920],{},[1857,4919,1956],{"mathvariant":1946},[1854,4921,4922,4924],{},[1907,4923,2459],{},[1920,4925,1922],{},[1857,4927,1977],{"mathvariant":1904},[1907,4929,1909],{},[1857,4931,3731],{},[1907,4933,1913],{"stretchy":1912},[1920,4935,2780],{},[1907,4937,1943],{"stretchy":1912},[1907,4939,1913],{"stretchy":1912},[1920,4941,1922],{},[1907,4943,2459],{},[1939,4945,4946,4948],{},[1857,4947,3792],{},[1920,4949,2474],{},[1907,4951,1943],{"stretchy":1912},[1907,4953,1909],{},[4955,4956,4957,4960,4963],"msubsup",{},[1857,4958,4959],{},"σ",[1857,4961,4962],{},"ε",[1920,4964,2474],{},[1857,4966,1997],{"mathvariant":1946},[1861,4968,4969],{"encoding":1863},"\\gamma(0)-\\mathbf g^\\top\\Gamma^{-1}\\mathbf g\n=\\gamma(0)(1-\\phi^2)=\\sigma_\\varepsilon^2.",[1838,4971,4973,5000,5082,5115,5163],{"className":4972,"ariaHidden":1868},[1867],[1838,4974,4976,4979,4982,4985,4988,4991,4994,4997],{"className":4975},[1872],[1838,4977],{"className":4978,"style":2216},[1876],[1838,4980,3731],{"className":4981,"style":3753},[1881,1882],[1838,4983,1913],{"className":4984},[2036],[1838,4986,2780],{"className":4987},[1881],[1838,4989,1943],{"className":4990},[2159],[1838,4992],{"className":4993,"style":1883},[1951],[1838,4995,2459],{"className":4996},[2546],[1838,4998],{"className":4999,"style":1883},[1951],[1838,5001,5003,5006,5035,5070,5073,5076,5079],{"className":5002},[1872],[1838,5004],{"className":5005,"style":2666},[1876],[1838,5007,5009,5012],{"className":5008},[1881],[1838,5010,1977],{"className":5011,"style":2246},[1881,2014],[1838,5013,5015],{"className":5014},[2047],[1838,5016,5018],{"className":5017},[2051],[1838,5019,5021],{"className":5020},[2056],[1838,5022,5024],{"className":5023,"style":2175},[2060],[1838,5025,5026,5029],{"style":2178},[1838,5027],{"className":5028,"style":2069},[2068],[1838,5030,5032],{"className":5031},[2073,2074,2075,2076],[1838,5033,1947],{"className":5034},[1881,2076],[1838,5036,5038,5041],{"className":5037},[1881],[1838,5039,1956],{"className":5040},[1881],[1838,5042,5044],{"className":5043},[2047],[1838,5045,5047],{"className":5046},[2051],[1838,5048,5050],{"className":5049},[2056],[1838,5051,5053],{"className":5052,"style":2606},[2060],[1838,5054,5055,5058],{"style":2178},[1838,5056],{"className":5057,"style":2069},[2068],[1838,5059,5061],{"className":5060},[2073,2074,2075,2076],[1838,5062,5064,5067],{"className":5063},[1881,2076],[1838,5065,2459],{"className":5066},[1881,2076],[1838,5068,1922],{"className":5069},[1881,2076],[1838,5071,1977],{"className":5072,"style":2246},[1881,2014],[1838,5074],{"className":5075,"style":2018},[1951],[1838,5077,1909],{"className":5078},[2022],[1838,5080],{"className":5081,"style":2018},[1951],[1838,5083,5085,5088,5091,5094,5097,5100,5103,5106,5109,5112],{"className":5084},[1872],[1838,5086],{"className":5087,"style":2216},[1876],[1838,5089,3731],{"className":5090,"style":3753},[1881,1882],[1838,5092,1913],{"className":5093},[2036],[1838,5095,2780],{"className":5096},[1881],[1838,5098,1943],{"className":5099},[2159],[1838,5101,1913],{"className":5102},[2036],[1838,5104,1922],{"className":5105},[1881],[1838,5107],{"className":5108,"style":1883},[1951],[1838,5110,2459],{"className":5111},[2546],[1838,5113],{"className":5114,"style":1883},[1951],[1838,5116,5118,5122,5151,5154,5157,5160],{"className":5117},[1872],[1838,5119],{"className":5120,"style":5121},[1876],"height:1.1141em;vertical-align:-0.25em;",[1838,5123,5125,5128],{"className":5124},[1881],[1838,5126,3792],{"className":5127},[1881,1882],[1838,5129,5131],{"className":5130},[2047],[1838,5132,5134],{"className":5133},[2051],[1838,5135,5137],{"className":5136},[2056],[1838,5138,5140],{"className":5139,"style":2606},[2060],[1838,5141,5142,5145],{"style":2178},[1838,5143],{"className":5144,"style":2069},[2068],[1838,5146,5148],{"className":5147},[2073,2074,2075,2076],[1838,5149,2474],{"className":5150},[1881,2076],[1838,5152,1943],{"className":5153},[2159],[1838,5155],{"className":5156,"style":2018},[1951],[1838,5158,1909],{"className":5159},[2022],[1838,5161],{"className":5162,"style":2018},[1951],[1838,5164,5166,5170,5224],{"className":5165},[1872],[1838,5167],{"className":5168,"style":5169},[1876],"height:1.1111em;vertical-align:-0.247em;",[1838,5171,5173,5177],{"className":5172},[1881],[1838,5174,4959],{"className":5175,"style":5176},[1881,1882],"margin-right:0.0359em;",[1838,5178,5180],{"className":5179},[2047],[1838,5181,5183,5215],{"className":5182},[2051,2052],[1838,5184,5186,5212],{"className":5185},[2056],[1838,5187,5189,5201],{"className":5188,"style":2606},[2060],[1838,5190,5192,5195],{"style":5191},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[1838,5193],{"className":5194,"style":2069},[2068],[1838,5196,5198],{"className":5197},[2073,2074,2075,2076],[1838,5199,4962],{"className":5200},[1881,1882,2076],[1838,5202,5203,5206],{"style":2178},[1838,5204],{"className":5205,"style":2069},[2068],[1838,5207,5209],{"className":5208},[2073,2074,2075,2076],[1838,5210,2474],{"className":5211},[1881,2076],[1838,5213,2084],{"className":5214},[2083],[1838,5216,5218],{"className":5217},[2056],[1838,5219,5222],{"className":5220,"style":5221},[2060],"height:0.247em;",[1838,5223],{},[1838,5225,1997],{"className":5226},[1881],[1793,5228,5230],{"id":5229},"_4-partitioned-covariance-and-the-schur-complement","4. Partitioned covariance and the Schur complement",[1798,5232,5233],{},"The joint covariance is",[1838,5235,5237],{"className":5236},[1888],[1838,5238,5240,5328],{"className":5239},[1841],[1838,5241,5243],{"className":5242},[1845],[1847,5244,5245],{"xmlns":1849,"display":1897},[1851,5246,5247,5325],{},[1854,5248,5249,5251,5253,5277,5279,5323],{},[1857,5250,1961],{"mathvariant":1946},[1907,5252,1964],{},[1854,5254,5255,5257,5275],{},[1907,5256,1913],{"fence":1868},[4215,5258,5259,5267],{"rowspacing":4217,"columnalign":4274,"columnspacing":4219},[4221,5260,5261],{},[4224,5262,5263],{},[2778,5264,5265],{"scriptlevel":2780,"displaystyle":1912},[1857,5266,1905],{"mathvariant":1904},[4221,5268,5269],{},[4224,5270,5271],{},[2778,5272,5273],{"scriptlevel":2780,"displaystyle":1912},[1857,5274,1859],{},[1907,5276,1943],{"fence":1868},[1907,5278,1909],{},[1854,5280,5281,5283,5321],{},[1907,5282,2452],{"fence":1868},[4215,5284,5285,5299],{"rowspacing":4217,"columnalign":4218,"columnspacing":4219},[4221,5286,5287,5293],{},[4224,5288,5289],{},[2778,5290,5291],{"scriptlevel":2780,"displaystyle":1912},[1857,5292,1956],{"mathvariant":1946},[4224,5294,5295],{},[2778,5296,5297],{"scriptlevel":2780,"displaystyle":1912},[1857,5298,1977],{"mathvariant":1904},[4221,5300,5301,5311],{},[4224,5302,5303],{},[2778,5304,5305],{"scriptlevel":2780,"displaystyle":1912},[1939,5306,5307,5309],{},[1857,5308,1977],{"mathvariant":1904},[1857,5310,1947],{"mathvariant":1946},[4224,5312,5313],{},[2778,5314,5315],{"scriptlevel":2780,"displaystyle":1912},[1915,5316,5317,5319],{},[1857,5318,3731],{},[1857,5320,1859],{},[1907,5322,2477],{"fence":1868},[1857,5324,1997],{"mathvariant":1946},[1861,5326,5327],{"encoding":1863},"\\operatorname{Var}\n\\begin{pmatrix}\n\\mathbf X\\\\Y\n\\end{pmatrix}\n=\n\\begin{bmatrix}\n\\Gamma&\\mathbf g\\\\\n\\mathbf g^\\top&\\gamma_Y\n\\end{bmatrix}.",[1838,5329,5331,5421],{"className":5330,"ariaHidden":1868},[1867],[1838,5332,5334,5337,5343,5346,5412,5415,5418],{"className":5333},[1872],[1838,5335],{"className":5336,"style":4329},[1876],[1838,5338,5340],{"className":5339},[2220],[1838,5341,1961],{"className":5342},[1881,2224],[1838,5344],{"className":5345,"style":2101},[1951],[1838,5347,5349,5355,5406],{"className":5348},[2105],[1838,5350,5352],{"className":5351,"style":4352},[2036,4351],[1838,5353,1913],{"className":5354},[4356,2075],[1838,5356,5358],{"className":5357},[1881],[1838,5359,5361],{"className":5360},[4215],[1838,5362,5364],{"className":5363},[4366],[1838,5365,5367,5398],{"className":5366},[2051,2052],[1838,5368,5370,5395],{"className":5369},[2056],[1838,5371,5373,5384],{"className":5372,"style":4376},[2060],[1838,5374,5375,5378],{"style":4379},[1838,5376],{"className":5377,"style":2356},[2068],[1838,5379,5381],{"className":5380},[1881],[1838,5382,1905],{"className":5383},[1881,2014],[1838,5385,5386,5389],{"style":4391},[1838,5387],{"className":5388,"style":2356},[2068],[1838,5390,5392],{"className":5391},[1881],[1838,5393,1859],{"className":5394,"style":1883},[1881,1882],[1838,5396,2084],{"className":5397},[2083],[1838,5399,5401],{"className":5400},[2056],[1838,5402,5404],{"className":5403,"style":4410},[2060],[1838,5405],{},[1838,5407,5409],{"className":5408,"style":4352},[2159,4351],[1838,5410,1943],{"className":5411},[4356,2075],[1838,5413],{"className":5414,"style":2018},[1951],[1838,5416,1909],{"className":5417},[2022],[1838,5419],{"className":5420,"style":2018},[1951],[1838,5422,5424,5428,5614,5617],{"className":5423},[1872],[1838,5425],{"className":5426,"style":5427},[1876],"height:2.4091em;vertical-align:-0.9546em;",[1838,5429,5431,5437,5608],{"className":5430},[2105],[1838,5432,5434],{"className":5433,"style":4352},[2036,4351],[1838,5435,2452],{"className":5436},[4356,2075],[1838,5438,5440],{"className":5439},[1881],[1838,5441,5443,5518,5521,5524],{"className":5442},[4215],[1838,5444,5446],{"className":5445},[4366],[1838,5447,5449,5509],{"className":5448},[2051,2052],[1838,5450,5452,5506],{"className":5451},[2056],[1838,5453,5456,5468],{"className":5454,"style":5455},[2060],"height:1.4546em;",[1838,5457,5459,5462],{"style":5458},"top:-3.6146em;",[1838,5460],{"className":5461,"style":2356},[2068],[1838,5463,5465],{"className":5464},[1881],[1838,5466,1956],{"className":5467},[1881],[1838,5469,5471,5474],{"style":5470},"top:-2.4054em;",[1838,5472],{"className":5473,"style":2356},[2068],[1838,5475,5477],{"className":5476},[1881],[1838,5478,5480,5483],{"className":5479},[1881],[1838,5481,1977],{"className":5482,"style":2246},[1881,2014],[1838,5484,5486],{"className":5485},[2047],[1838,5487,5489],{"className":5488},[2051],[1838,5490,5492],{"className":5491},[2056],[1838,5493,5495],{"className":5494,"style":2398},[2060],[1838,5496,5497,5500],{"style":2419},[1838,5498],{"className":5499,"style":2069},[2068],[1838,5501,5503],{"className":5502},[2073,2074,2075,2076],[1838,5504,1947],{"className":5505},[1881,2076],[1838,5507,2084],{"className":5508},[2083],[1838,5510,5512],{"className":5511},[2056],[1838,5513,5516],{"className":5514,"style":5515},[2060],"height:0.9546em;",[1838,5517],{},[1838,5519],{"className":5520,"style":4417},[4416],[1838,5522],{"className":5523,"style":4417},[4416],[1838,5525,5527],{"className":5526},[4366],[1838,5528,5530,5600],{"className":5529},[2051,2052],[1838,5531,5533,5597],{"className":5532},[2056],[1838,5534,5536,5547],{"className":5535,"style":5455},[2060],[1838,5537,5538,5541],{"style":5458},[1838,5539],{"className":5540,"style":2356},[2068],[1838,5542,5544],{"className":5543},[1881],[1838,5545,1977],{"className":5546,"style":2246},[1881,2014],[1838,5548,5549,5552],{"style":5470},[1838,5550],{"className":5551,"style":2356},[2068],[1838,5553,5555],{"className":5554},[1881],[1838,5556,5558,5561],{"className":5557},[1881],[1838,5559,3731],{"className":5560,"style":3753},[1881,1882],[1838,5562,5564],{"className":5563},[2047],[1838,5565,5567,5589],{"className":5566},[2051,2052],[1838,5568,5570,5586],{"className":5569},[2056],[1838,5571,5574],{"className":5572,"style":5573},[2060],"height:0.3283em;",[1838,5575,5577,5580],{"style":5576},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[1838,5578],{"className":5579,"style":2069},[2068],[1838,5581,5583],{"className":5582},[2073,2074,2075,2076],[1838,5584,1859],{"className":5585,"style":1883},[1881,1882,2076],[1838,5587,2084],{"className":5588},[2083],[1838,5590,5592],{"className":5591},[2056],[1838,5593,5595],{"className":5594,"style":2091},[2060],[1838,5596],{},[1838,5598,2084],{"className":5599},[2083],[1838,5601,5603],{"className":5602},[2056],[1838,5604,5606],{"className":5605,"style":5515},[2060],[1838,5607],{},[1838,5609,5611],{"className":5610,"style":4352},[2159,4351],[1838,5612,2477],{"className":5613},[4356,2075],[1838,5615],{"className":5616,"style":2101},[1951],[1838,5618,1997],{"className":5619},[1881],[1798,5621,5622,5623,5651,5652,5681],{},"Conditioning removes the part of ",[1838,5624,5626,5639],{"className":5625},[1841],[1838,5627,5629],{"className":5628},[1845],[1847,5630,5631],{"xmlns":1849},[1851,5632,5633,5637],{},[1854,5634,5635],{},[1857,5636,1859],{},[1861,5638,1859],{"encoding":1863},[1838,5640,5642],{"className":5641,"ariaHidden":1868},[1867],[1838,5643,5645,5648],{"className":5644},[1872],[1838,5646],{"className":5647,"style":1877},[1876],[1838,5649,1859],{"className":5650,"style":1883},[1881,1882]," explained linearly by ",[1838,5653,5655,5669],{"className":5654},[1841],[1838,5656,5658],{"className":5657},[1845],[1847,5659,5660],{"xmlns":1849},[1851,5661,5662,5666],{},[1854,5663,5664],{},[1857,5665,1905],{"mathvariant":1904},[1861,5667,5668],{"encoding":1863},"\\mathbf X",[1838,5670,5672],{"className":5671,"ariaHidden":1868},[1867],[1838,5673,5675,5678],{"className":5674},[1872],[1838,5676],{"className":5677,"style":2010},[1876],[1838,5679,1905],{"className":5680},[1881,2014],". The remaining variance,",[1838,5683,5685],{"className":5684},[1888],[1838,5686,5688,5728],{"className":5687},[1841],[1838,5689,5691],{"className":5690},[1845],[1847,5692,5693],{"xmlns":1849,"display":1897},[1851,5694,5695,5725],{},[1854,5696,5697,5703,5705,5711,5721,5723],{},[1915,5698,5699,5701],{},[1857,5700,3731],{},[1857,5702,1859],{},[1907,5704,2459],{},[1939,5706,5707,5709],{},[1857,5708,1977],{"mathvariant":1904},[1857,5710,1947],{"mathvariant":1946},[1939,5712,5713,5715],{},[1857,5714,1956],{"mathvariant":1946},[1854,5716,5717,5719],{},[1907,5718,2459],{},[1920,5720,1922],{},[1857,5722,1977],{"mathvariant":1904},[1907,5724,1925],{"separator":1868},[1861,5726,5727],{"encoding":1863},"\\gamma_Y-\\mathbf g^\\top\\Gamma^{-1}\\mathbf g,",[1838,5729,5731,5786],{"className":5730,"ariaHidden":1868},[1867],[1838,5732,5734,5737,5777,5780,5783],{"className":5733},[1872],[1838,5735],{"className":5736,"style":3358},[1876],[1838,5738,5740,5743],{"className":5739},[1881],[1838,5741,3731],{"className":5742,"style":3753},[1881,1882],[1838,5744,5746],{"className":5745},[2047],[1838,5747,5749,5769],{"className":5748},[2051,2052],[1838,5750,5752,5766],{"className":5751},[2056],[1838,5753,5755],{"className":5754,"style":5573},[2060],[1838,5756,5757,5760],{"style":5576},[1838,5758],{"className":5759,"style":2069},[2068],[1838,5761,5763],{"className":5762},[2073,2074,2075,2076],[1838,5764,1859],{"className":5765,"style":1883},[1881,1882,2076],[1838,5767,2084],{"className":5768},[2083],[1838,5770,5772],{"className":5771},[2056],[1838,5773,5775],{"className":5774,"style":2091},[2060],[1838,5776],{},[1838,5778],{"className":5779,"style":1883},[1951],[1838,5781,2459],{"className":5782},[2546],[1838,5784],{"className":5785,"style":1883},[1951],[1838,5787,5789,5792,5821,5856,5859],{"className":5788},[1872],[1838,5790],{"className":5791,"style":2666},[1876],[1838,5793,5795,5798],{"className":5794},[1881],[1838,5796,1977],{"className":5797,"style":2246},[1881,2014],[1838,5799,5801],{"className":5800},[2047],[1838,5802,5804],{"className":5803},[2051],[1838,5805,5807],{"className":5806},[2056],[1838,5808,5810],{"className":5809,"style":2175},[2060],[1838,5811,5812,5815],{"style":2178},[1838,5813],{"className":5814,"style":2069},[2068],[1838,5816,5818],{"className":5817},[2073,2074,2075,2076],[1838,5819,1947],{"className":5820},[1881,2076],[1838,5822,5824,5827],{"className":5823},[1881],[1838,5825,1956],{"className":5826},[1881],[1838,5828,5830],{"className":5829},[2047],[1838,5831,5833],{"className":5832},[2051],[1838,5834,5836],{"className":5835},[2056],[1838,5837,5839],{"className":5838,"style":2606},[2060],[1838,5840,5841,5844],{"style":2178},[1838,5842],{"className":5843,"style":2069},[2068],[1838,5845,5847],{"className":5846},[2073,2074,2075,2076],[1838,5848,5850,5853],{"className":5849},[1881,2076],[1838,5851,2459],{"className":5852},[1881,2076],[1838,5854,1922],{"className":5855},[1881,2076],[1838,5857,1977],{"className":5858,"style":2246},[1881,2014],[1838,5860,1925],{"className":5861},[2097],[1798,5863,5864,5865,5893],{},"is the Schur complement of ",[1838,5866,5868,5881],{"className":5867},[1841],[1838,5869,5871],{"className":5870},[1845],[1847,5872,5873],{"xmlns":1849},[1851,5874,5875,5879],{},[1854,5876,5877],{},[1857,5878,1956],{"mathvariant":1946},[1861,5880,2904],{"encoding":1863},[1838,5882,5884],{"className":5883,"ariaHidden":1868},[1867],[1838,5885,5887,5890],{"className":5886},[1872],[1838,5888],{"className":5889,"style":1877},[1876],[1838,5891,1956],{"className":5892},[1881],". It is non-negative because the joint covariance matrix is positive semidefinite.",[5895,5896,5898],"h3",{"id":5897},"r-solve-the-projection-and-verify-orthogonality","R: solve the projection and verify orthogonality",[1798,5900,5901,5902,5987,5988,1997],{},"The following cell uses an AR(2). Because the true process uses only two lags, the coefficient on ",[1838,5903,5905,5929],{"className":5904},[1841],[1838,5906,5908],{"className":5907},[1845],[1847,5909,5910],{"xmlns":1849},[1851,5911,5912,5926],{},[1854,5913,5914],{},[1915,5915,5916,5918],{},[1857,5917,1905],{},[1854,5919,5920,5922,5924],{},[1857,5921,3901],{},[1907,5923,2459],{},[1920,5925,2474],{},[1861,5927,5928],{"encoding":1863},"X_{t-2}",[1838,5930,5932],{"className":5931,"ariaHidden":1868},[1867],[1838,5933,5935,5938],{"className":5934},[1872],[1838,5936],{"className":5937,"style":3918},[1876],[1838,5939,5941,5944],{"className":5940},[1881],[1838,5942,1905],{"className":5943,"style":2043},[1881,1882],[1838,5945,5947],{"className":5946},[2047],[1838,5948,5950,5979],{"className":5949},[2051,2052],[1838,5951,5953,5976],{"className":5952},[2056],[1838,5954,5956],{"className":5955,"style":2061},[2060],[1838,5957,5958,5961],{"style":2064},[1838,5959],{"className":5960,"style":2069},[2068],[1838,5962,5964],{"className":5963},[2073,2074,2075,2076],[1838,5965,5967,5970,5973],{"className":5966},[1881,2076],[1838,5968,3901],{"className":5969},[1881,1882,2076],[1838,5971,2459],{"className":5972},[2546,2076],[1838,5974,2474],{"className":5975},[1881,2076],[1838,5977,2084],{"className":5978},[2083],[1838,5980,5982],{"className":5981},[2056],[1838,5983,5985],{"className":5984,"style":3966},[2060],[1838,5986],{}," should be numerically zero when predicting from ",[1838,5989,5991,6045],{"className":5990},[1841],[1838,5992,5994],{"className":5993},[1845],[1847,5995,5996],{"xmlns":1849},[1851,5997,5998,6042],{},[1854,5999,6000,6002,6008,6010,6022,6024,6036],{},[1907,6001,1913],{"stretchy":1912},[1915,6003,6004,6006],{},[1857,6005,1905],{},[1857,6007,3901],{},[1907,6009,1925],{"separator":1868},[1915,6011,6012,6014],{},[1857,6013,1905],{},[1854,6015,6016,6018,6020],{},[1857,6017,3901],{},[1907,6019,2459],{},[1920,6021,1922],{},[1907,6023,1925],{"separator":1868},[1915,6025,6026,6028],{},[1857,6027,1905],{},[1854,6029,6030,6032,6034],{},[1857,6031,3901],{},[1907,6033,2459],{},[1920,6035,2474],{},[1939,6037,6038,6040],{},[1907,6039,1943],{"stretchy":1912},[1857,6041,1947],{"mathvariant":1946},[1861,6043,6044],{"encoding":1863},"(X_t,X_{t-1},X_{t-2})^\\top",[1838,6046,6048],{"className":6047,"ariaHidden":1868},[1867],[1838,6049,6051,6054,6057,6097,6100,6103,6152,6155,6158,6207],{"className":6050},[1872],[1838,6052],{"className":6053,"style":3453},[1876],[1838,6055,1913],{"className":6056},[2036],[1838,6058,6060,6063],{"className":6059},[1881],[1838,6061,1905],{"className":6062,"style":2043},[1881,1882],[1838,6064,6066],{"className":6065},[2047],[1838,6067,6069,6089],{"className":6068},[2051,2052],[1838,6070,6072,6086],{"className":6071},[2056],[1838,6073,6075],{"className":6074,"style":4125},[2060],[1838,6076,6077,6080],{"style":2064},[1838,6078],{"className":6079,"style":2069},[2068],[1838,6081,6083],{"className":6082},[2073,2074,2075,2076],[1838,6084,3901],{"className":6085},[1881,1882,2076],[1838,6087,2084],{"className":6088},[2083],[1838,6090,6092],{"className":6091},[2056],[1838,6093,6095],{"className":6094,"style":2091},[2060],[1838,6096],{},[1838,6098,1925],{"className":6099},[2097],[1838,6101],{"className":6102,"style":2101},[1951],[1838,6104,6106,6109],{"className":6105},[1881],[1838,6107,1905],{"className":6108,"style":2043},[1881,1882],[1838,6110,6112],{"className":6111},[2047],[1838,6113,6115,6144],{"className":6114},[2051,2052],[1838,6116,6118,6141],{"className":6117},[2056],[1838,6119,6121],{"className":6120,"style":2061},[2060],[1838,6122,6123,6126],{"style":2064},[1838,6124],{"className":6125,"style":2069},[2068],[1838,6127,6129],{"className":6128},[2073,2074,2075,2076],[1838,6130,6132,6135,6138],{"className":6131},[1881,2076],[1838,6133,3901],{"className":6134},[1881,1882,2076],[1838,6136,2459],{"className":6137},[2546,2076],[1838,6139,1922],{"className":6140},[1881,2076],[1838,6142,2084],{"className":6143},[2083],[1838,6145,6147],{"className":6146},[2056],[1838,6148,6150],{"className":6149,"style":3966},[2060],[1838,6151],{},[1838,6153,1925],{"className":6154},[2097],[1838,6156],{"className":6157,"style":2101},[1951],[1838,6159,6161,6164],{"className":6160},[1881],[1838,6162,1905],{"className":6163,"style":2043},[1881,1882],[1838,6165,6167],{"className":6166},[2047],[1838,6168,6170,6199],{"className":6169},[2051,2052],[1838,6171,6173,6196],{"className":6172},[2056],[1838,6174,6176],{"className":6175,"style":2061},[2060],[1838,6177,6178,6181],{"style":2064},[1838,6179],{"className":6180,"style":2069},[2068],[1838,6182,6184],{"className":6183},[2073,2074,2075,2076],[1838,6185,6187,6190,6193],{"className":6186},[1881,2076],[1838,6188,3901],{"className":6189},[1881,1882,2076],[1838,6191,2459],{"className":6192},[2546,2076],[1838,6194,2474],{"className":6195},[1881,2076],[1838,6197,2084],{"className":6198},[2083],[1838,6200,6202],{"className":6201},[2056],[1838,6203,6205],{"className":6204,"style":3966},[2060],[1838,6206],{},[1838,6208,6210,6213],{"className":6209},[2159],[1838,6211,1943],{"className":6212},[2159],[1838,6214,6216],{"className":6215},[2047],[1838,6217,6219],{"className":6218},[2051],[1838,6220,6222],{"className":6221},[2056],[1838,6223,6225],{"className":6224,"style":2398},[2060],[1838,6226,6227,6230],{"style":2419},[1838,6228],{"className":6229,"style":2069},[2068],[1838,6231,6233],{"className":6232},[2073,2074,2075,2076],[1838,6234,1947],{"className":6235},[1881,2076],[6237,6238],"web-r",{"code64":6239,"layout":6240,"locale":7,"title":6241},"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","vertical","Best linear prediction from an AR(2) covariance matrix",[1798,6243,6244,6245,6380],{},"The first two weights reproduce ",[1838,6246,6248,6278],{"className":6247},[1841],[1838,6249,6251],{"className":6250},[1845],[1847,6252,6253],{"xmlns":1849},[1851,6254,6255,6275],{},[1854,6256,6257,6259,6265,6267,6273],{},[1907,6258,1913],{"stretchy":1912},[1915,6260,6261,6263],{},[1857,6262,3792],{},[1920,6264,1922],{},[1907,6266,1925],{"separator":1868},[1915,6268,6269,6271],{},[1857,6270,3792],{},[1920,6272,2474],{},[1907,6274,1943],{"stretchy":1912},[1861,6276,6277],{"encoding":1863},"(\\phi_1,\\phi_2)",[1838,6279,6281],{"className":6280,"ariaHidden":1868},[1867],[1838,6282,6284,6287,6290,6331,6334,6337,6377],{"className":6283},[1872],[1838,6285],{"className":6286,"style":2216},[1876],[1838,6288,1913],{"className":6289},[2036],[1838,6291,6293,6296],{"className":6292},[1881],[1838,6294,3792],{"className":6295},[1881,1882],[1838,6297,6299],{"className":6298},[2047],[1838,6300,6302,6323],{"className":6301},[2051,2052],[1838,6303,6305,6320],{"className":6304},[2056],[1838,6306,6308],{"className":6307,"style":2061},[2060],[1838,6309,6311,6314],{"style":6310},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1838,6312],{"className":6313,"style":2069},[2068],[1838,6315,6317],{"className":6316},[2073,2074,2075,2076],[1838,6318,1922],{"className":6319},[1881,2076],[1838,6321,2084],{"className":6322},[2083],[1838,6324,6326],{"className":6325},[2056],[1838,6327,6329],{"className":6328,"style":2091},[2060],[1838,6330],{},[1838,6332,1925],{"className":6333},[2097],[1838,6335],{"className":6336,"style":2101},[1951],[1838,6338,6340,6343],{"className":6339},[1881],[1838,6341,3792],{"className":6342},[1881,1882],[1838,6344,6346],{"className":6345},[2047],[1838,6347,6349,6369],{"className":6348},[2051,2052],[1838,6350,6352,6366],{"className":6351},[2056],[1838,6353,6355],{"className":6354,"style":2061},[2060],[1838,6356,6357,6360],{"style":6310},[1838,6358],{"className":6359,"style":2069},[2068],[1838,6361,6363],{"className":6362},[2073,2074,2075,2076],[1838,6364,2474],{"className":6365},[1881,2076],[1838,6367,2084],{"className":6368},[2083],[1838,6370,6372],{"className":6371},[2056],[1838,6373,6375],{"className":6374,"style":2091},[2060],[1838,6376],{},[1838,6378,1943],{"className":6379},[2159]," and the third is zero up to rounding. The MSPE reproduces the innovation variance.",[1793,6382,6384],{"id":6383},"_5-multi-step-prediction-uses-matrix-powers","5. Multi-step prediction uses matrix powers",[1798,6386,6387,6388,6417],{},"Write an AR(",[1838,6389,6391,6404],{"className":6390},[1841],[1838,6392,6394],{"className":6393},[1845],[1847,6395,6396],{"xmlns":1849},[1851,6397,6398,6402],{},[1854,6399,6400],{},[1857,6401,1798],{},[1861,6403,1798],{"encoding":1863},[1838,6405,6407],{"className":6406,"ariaHidden":1868},[1867],[1838,6408,6410,6414],{"className":6409},[1872],[1838,6411],{"className":6412,"style":6413},[1876],"height:0.625em;vertical-align:-0.1944em;",[1838,6415,1798],{"className":6416},[1881,1882],") in state form:",[1838,6419,6421],{"className":6420},[1888],[1838,6422,6424,6564],{"className":6423},[1841],[1838,6425,6427],{"className":6426},[1845],[1847,6428,6429],{"xmlns":1849,"display":1897},[1851,6430,6431,6561],{},[1854,6432,6433,6440,6442,6521,6523,6525,6531,6533,6536,6548,6550,6553,6559],{},[1915,6434,6435,6438],{},[1857,6436,6437],{"mathvariant":1904},"s",[1857,6439,3901],{},[1907,6441,1909],{},[1854,6443,6444,6446,6519],{},[1907,6445,2452],{"fence":1868},[4215,6447,6448,6460,6478,6497],{"rowspacing":4217,"columnalign":4274,"columnspacing":4219},[4221,6449,6450],{},[4224,6451,6452],{},[2778,6453,6454],{"scriptlevel":2780,"displaystyle":1912},[1915,6455,6456,6458],{},[1857,6457,1905],{},[1857,6459,3901],{},[4221,6461,6462],{},[4224,6463,6464],{},[2778,6465,6466],{"scriptlevel":2780,"displaystyle":1912},[1915,6467,6468,6470],{},[1857,6469,1905],{},[1854,6471,6472,6474,6476],{},[1857,6473,3901],{},[1907,6475,2459],{},[1920,6477,1922],{},[4221,6479,6480],{},[4224,6481,6482],{},[2778,6483,6484],{"scriptlevel":2780,"displaystyle":1912},[1854,6485,6486,6489],{},[1857,6487,6488],{"mathvariant":1946},"⋮",[6490,6491,6493],"mpadded",{"height":6492,"voffset":6492},"0em",[1951,6494],{"mathbackground":6495,"width":6492,"height":6496},"black","1.5em",[4221,6498,6499],{},[4224,6500,6501],{},[2778,6502,6503],{"scriptlevel":2780,"displaystyle":1912},[1915,6504,6505,6507],{},[1857,6506,1905],{},[1854,6508,6509,6511,6513,6515,6517],{},[1857,6510,3901],{},[1907,6512,2459],{},[1857,6514,1798],{},[1907,6516,2504],{},[1920,6518,1922],{},[1907,6520,2477],{"fence":1868},[1907,6522,1925],{"separator":1868},[1951,6524],{"width":1953},[1915,6526,6527,6529],{},[1857,6528,6437],{"mathvariant":1904},[1857,6530,3901],{},[1907,6532,1909],{},[1857,6534,6535],{},"F",[1915,6537,6538,6540],{},[1857,6539,6437],{"mathvariant":1904},[1854,6541,6542,6544,6546],{},[1857,6543,3901],{},[1907,6545,2459],{},[1920,6547,1922],{},[1907,6549,2504],{},[1857,6551,6552],{},"G",[1915,6554,6555,6557],{},[1857,6556,4962],{},[1857,6558,3901],{},[1857,6560,1997],{"mathvariant":1946},[1861,6562,6563],{"encoding":1863},"\\mathbf s_t=\n\\begin{bmatrix}\nX_t\\\\X_{t-1}\\\\ \\vdots\\\\X_{t-p+1}\n\\end{bmatrix},\n\\qquad\n\\mathbf s_t=F\\mathbf s_{t-1}+G\\varepsilon_t.",[1838,6565,6567,6623,7003,7071],{"className":6566,"ariaHidden":1868},[1867],[1838,6568,6570,6574,6614,6617,6620],{"className":6569},[1872],[1838,6571],{"className":6572,"style":6573},[1876],"height:0.5944em;vertical-align:-0.15em;",[1838,6575,6577,6580],{"className":6576},[1881],[1838,6578,6437],{"className":6579},[1881,2014],[1838,6581,6583],{"className":6582},[2047],[1838,6584,6586,6606],{"className":6585},[2051,2052],[1838,6587,6589,6603],{"className":6588},[2056],[1838,6590,6592],{"className":6591,"style":4125},[2060],[1838,6593,6594,6597],{"style":6310},[1838,6595],{"className":6596,"style":2069},[2068],[1838,6598,6600],{"className":6599},[2073,2074,2075,2076],[1838,6601,3901],{"className":6602},[1881,1882,2076],[1838,6604,2084],{"className":6605},[2083],[1838,6607,6609],{"className":6608},[2056],[1838,6610,6612],{"className":6611,"style":2091},[2060],[1838,6613],{},[1838,6615],{"className":6616,"style":2018},[1951],[1838,6618,1909],{"className":6619},[2022],[1838,6621],{"className":6622,"style":2018},[1951],[1838,6624,6626,6630,6942,6945,6948,6951,6954,6994,6997,7000],{"className":6625},[1872],[1838,6627],{"className":6628,"style":6629},[1876],"height:5.46em;vertical-align:-2.48em;",[1838,6631,6633,6680,6904],{"className":6632},[2105],[1838,6634,6636],{"className":6635},[2036],[1838,6637,6640],{"className":6638},[4356,6639],"mult",[1838,6641,6643,6671],{"className":6642},[2051,2052],[1838,6644,6646,6668],{"className":6645},[2056],[1838,6647,6650],{"className":6648,"style":6649},[2060],"height:2.95em;",[1838,6651,6653,6657],{"style":6652},"top:-4.95em;",[1838,6654],{"className":6655,"style":6656},[2068],"height:7.4em;",[1838,6658,6660],{"style":6659},"width:0.667em;height:5.4em;",[2372,6661,6665],{"xmlns":2374,"width":6662,"height":6663,"viewBox":6664},"0.667em","5.4em","0 0 667 5400",[2380,6666],{"d":6667},"M403 1759 V84 H666 V0 H319 V1759 v1800 v1759 v84 h347 v-84\nH403z M403 1759 V0 H319 V1759 v1800 v1759 v84 h84z",[1838,6669,2084],{"className":6670},[2083],[1838,6672,6674],{"className":6673},[2056],[1838,6675,6678],{"className":6676,"style":6677},[2060],"height:2.45em;",[1838,6679],{},[1838,6681,6683],{"className":6682},[1881],[1838,6684,6686],{"className":6685},[4215],[1838,6687,6689],{"className":6688},[4366],[1838,6690,6692,6895],{"className":6691},[2051,2052],[1838,6693,6695,6892],{"className":6694},[2056],[1838,6696,6699,6749,6807,6827],{"className":6697,"style":6698},[2060],"height:2.98em;",[1838,6700,6702,6706],{"style":6701},"top:-5.8275em;",[1838,6703],{"className":6704,"style":6705},[2068],"height:3.6875em;",[1838,6707,6709],{"className":6708},[1881],[1838,6710,6712,6715],{"className":6711},[1881],[1838,6713,1905],{"className":6714,"style":2043},[1881,1882],[1838,6716,6718],{"className":6717},[2047],[1838,6719,6721,6741],{"className":6720},[2051,2052],[1838,6722,6724,6738],{"className":6723},[2056],[1838,6725,6727],{"className":6726,"style":4125},[2060],[1838,6728,6729,6732],{"style":2064},[1838,6730],{"className":6731,"style":2069},[2068],[1838,6733,6735],{"className":6734},[2073,2074,2075,2076],[1838,6736,3901],{"className":6737},[1881,1882,2076],[1838,6739,2084],{"className":6740},[2083],[1838,6742,6744],{"className":6743},[2056],[1838,6745,6747],{"className":6746,"style":2091},[2060],[1838,6748],{},[1838,6750,6752,6755],{"style":6751},"top:-4.6275em;",[1838,6753],{"className":6754,"style":6705},[2068],[1838,6756,6758],{"className":6757},[1881],[1838,6759,6761,6764],{"className":6760},[1881],[1838,6762,1905],{"className":6763,"style":2043},[1881,1882],[1838,6765,6767],{"className":6766},[2047],[1838,6768,6770,6799],{"className":6769},[2051,2052],[1838,6771,6773,6796],{"className":6772},[2056],[1838,6774,6776],{"className":6775,"style":2061},[2060],[1838,6777,6778,6781],{"style":2064},[1838,6779],{"className":6780,"style":2069},[2068],[1838,6782,6784],{"className":6783},[2073,2074,2075,2076],[1838,6785,6787,6790,6793],{"className":6786},[1881,2076],[1838,6788,3901],{"className":6789},[1881,1882,2076],[1838,6791,2459],{"className":6792},[2546,2076],[1838,6794,1922],{"className":6795},[1881,2076],[1838,6797,2084],{"className":6798},[2083],[1838,6800,6802],{"className":6801},[2056],[1838,6803,6805],{"className":6804,"style":3966},[2060],[1838,6806],{},[1838,6808,6810,6813],{"style":6809},"top:-2.7675em;",[1838,6811],{"className":6812,"style":6705},[2068],[1838,6814,6816],{"className":6815},[1881],[1838,6817,6819,6822],{"className":6818},[1881],[1838,6820,6488],{"className":6821},[1881],[1838,6823],{"className":6824,"style":6826},[1881,6825],"rule","border-right-width:0em;border-top-width:1.5em;bottom:0em;",[1838,6828,6830,6833],{"style":6829},"top:-1.5675em;",[1838,6831],{"className":6832,"style":6705},[2068],[1838,6834,6836],{"className":6835},[1881],[1838,6837,6839,6842],{"className":6838},[1881],[1838,6840,1905],{"className":6841,"style":2043},[1881,1882],[1838,6843,6845],{"className":6844},[2047],[1838,6846,6848,6883],{"className":6847},[2051,2052],[1838,6849,6851,6880],{"className":6850},[2056],[1838,6852,6854],{"className":6853,"style":2061},[2060],[1838,6855,6856,6859],{"style":2064},[1838,6857],{"className":6858,"style":2069},[2068],[1838,6860,6862],{"className":6861},[2073,2074,2075,2076],[1838,6863,6865,6868,6871,6874,6877],{"className":6864},[1881,2076],[1838,6866,3901],{"className":6867},[1881,1882,2076],[1838,6869,2459],{"className":6870},[2546,2076],[1838,6872,1798],{"className":6873},[1881,1882,2076],[1838,6875,2504],{"className":6876},[2546,2076],[1838,6878,1922],{"className":6879},[1881,2076],[1838,6881,2084],{"className":6882},[2083],[1838,6884,6886],{"className":6885},[2056],[1838,6887,6890],{"className":6888,"style":6889},[2060],"height:0.2861em;",[1838,6891],{},[1838,6893,2084],{"className":6894},[2083],[1838,6896,6898],{"className":6897},[2056],[1838,6899,6902],{"className":6900,"style":6901},[2060],"height:2.48em;",[1838,6903],{},[1838,6905,6907],{"className":6906},[2159],[1838,6908,6910],{"className":6909},[4356,6639],[1838,6911,6913,6934],{"className":6912},[2051,2052],[1838,6914,6916,6931],{"className":6915},[2056],[1838,6917,6919],{"className":6918,"style":6649},[2060],[1838,6920,6921,6924],{"style":6652},[1838,6922],{"className":6923,"style":6656},[2068],[1838,6925,6926],{"style":6659},[2372,6927,6928],{"xmlns":2374,"width":6662,"height":6663,"viewBox":6664},[2380,6929],{"d":6930},"M347 1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347z\nM347 1759 V0 H263 V1759 v1800 v1759 h84z",[1838,6932,2084],{"className":6933},[2083],[1838,6935,6937],{"className":6936},[2056],[1838,6938,6940],{"className":6939,"style":6677},[2060],[1838,6941],{},[1838,6943],{"className":6944,"style":2101},[1951],[1838,6946,1925],{"className":6947},[2097],[1838,6949],{"className":6950,"style":2194},[1951],[1838,6952],{"className":6953,"style":2101},[1951],[1838,6955,6957,6960],{"className":6956},[1881],[1838,6958,6437],{"className":6959},[1881,2014],[1838,6961,6963],{"className":6962},[2047],[1838,6964,6966,6986],{"className":6965},[2051,2052],[1838,6967,6969,6983],{"className":6968},[2056],[1838,6970,6972],{"className":6971,"style":4125},[2060],[1838,6973,6974,6977],{"style":6310},[1838,6975],{"className":6976,"style":2069},[2068],[1838,6978,6980],{"className":6979},[2073,2074,2075,2076],[1838,6981,3901],{"className":6982},[1881,1882,2076],[1838,6984,2084],{"className":6985},[2083],[1838,6987,6989],{"className":6988},[2056],[1838,6990,6992],{"className":6991,"style":2091},[2060],[1838,6993],{},[1838,6995],{"className":6996,"style":2018},[1951],[1838,6998,1909],{"className":6999},[2022],[1838,7001],{"className":7002,"style":2018},[1951],[1838,7004,7006,7009,7013,7062,7065,7068],{"className":7005},[1872],[1838,7007],{"className":7008,"style":3918},[1876],[1838,7010,6535],{"className":7011,"style":7012},[1881,1882],"margin-right:0.1389em;",[1838,7014,7016,7019],{"className":7015},[1881],[1838,7017,6437],{"className":7018},[1881,2014],[1838,7020,7022],{"className":7021},[2047],[1838,7023,7025,7054],{"className":7024},[2051,2052],[1838,7026,7028,7051],{"className":7027},[2056],[1838,7029,7031],{"className":7030,"style":2061},[2060],[1838,7032,7033,7036],{"style":6310},[1838,7034],{"className":7035,"style":2069},[2068],[1838,7037,7039],{"className":7038},[2073,2074,2075,2076],[1838,7040,7042,7045,7048],{"className":7041},[1881,2076],[1838,7043,3901],{"className":7044},[1881,1882,2076],[1838,7046,2459],{"className":7047},[2546,2076],[1838,7049,1922],{"className":7050},[1881,2076],[1838,7052,2084],{"className":7053},[2083],[1838,7055,7057],{"className":7056},[2056],[1838,7058,7060],{"className":7059,"style":3966},[2060],[1838,7061],{},[1838,7063],{"className":7064,"style":1883},[1951],[1838,7066,2504],{"className":7067},[2546],[1838,7069],{"className":7070,"style":1883},[1951],[1838,7072,7074,7077,7080,7120],{"className":7073},[1872],[1838,7075],{"className":7076,"style":4754},[1876],[1838,7078,6552],{"className":7079},[1881,1882],[1838,7081,7083,7086],{"className":7082},[1881],[1838,7084,4962],{"className":7085},[1881,1882],[1838,7087,7089],{"className":7088},[2047],[1838,7090,7092,7112],{"className":7091},[2051,2052],[1838,7093,7095,7109],{"className":7094},[2056],[1838,7096,7098],{"className":7097,"style":4125},[2060],[1838,7099,7100,7103],{"style":6310},[1838,7101],{"className":7102,"style":2069},[2068],[1838,7104,7106],{"className":7105},[2073,2074,2075,2076],[1838,7107,3901],{"className":7108},[1881,1882,2076],[1838,7110,2084],{"className":7111},[2083],[1838,7113,7115],{"className":7114},[2056],[1838,7116,7118],{"className":7117,"style":2091},[2060],[1838,7119],{},[1838,7121,1997],{"className":7122},[1881],[1798,7124,7125,7126,1925],{},"With ",[1838,7127,7129,7172],{"className":7128},[1841],[1838,7130,7132],{"className":7131},[1845],[1847,7133,7134],{"xmlns":1849},[1851,7135,7136,7169],{},[1854,7137,7138,7145,7147,7149,7151,7153,7155,7157,7159,7161,7163],{},[1915,7139,7140,7143],{},[1857,7141,7142],{"mathvariant":1904},"e",[1920,7144,1922],{},[1907,7146,1909],{},[1907,7148,1913],{"stretchy":1912},[1920,7150,1922],{},[1907,7152,1925],{"separator":1868},[1920,7154,2780],{},[1907,7156,1925],{"separator":1868},[1907,7158,1928],{},[1907,7160,1925],{"separator":1868},[1920,7162,2780],{},[1939,7164,7165,7167],{},[1907,7166,1943],{"stretchy":1912},[1857,7168,1947],{"mathvariant":1946},[1861,7170,7171],{"encoding":1863},"\\mathbf e_1=(1,0,\\ldots,0)^\\top",[1838,7173,7175,7230],{"className":7174,"ariaHidden":1868},[1867],[1838,7176,7178,7181,7221,7224,7227],{"className":7177},[1872],[1838,7179],{"className":7180,"style":6573},[1876],[1838,7182,7184,7187],{"className":7183},[1881],[1838,7185,7142],{"className":7186},[1881,2014],[1838,7188,7190],{"className":7189},[2047],[1838,7191,7193,7213],{"className":7192},[2051,2052],[1838,7194,7196,7210],{"className":7195},[2056],[1838,7197,7199],{"className":7198,"style":2061},[2060],[1838,7200,7201,7204],{"style":6310},[1838,7202],{"className":7203,"style":2069},[2068],[1838,7205,7207],{"className":7206},[2073,2074,2075,2076],[1838,7208,1922],{"className":7209},[1881,2076],[1838,7211,2084],{"className":7212},[2083],[1838,7214,7216],{"className":7215},[2056],[1838,7217,7219],{"className":7218,"style":2091},[2060],[1838,7220],{},[1838,7222],{"className":7223,"style":2018},[1951],[1838,7225,1909],{"className":7226},[2022],[1838,7228],{"className":7229,"style":2018},[1951],[1838,7231,7233,7236,7239,7242,7245,7248,7251,7254,7257,7260,7263,7266,7269,7272],{"className":7232},[1872],[1838,7234],{"className":7235,"style":3453},[1876],[1838,7237,1913],{"className":7238},[2036],[1838,7240,1922],{"className":7241},[1881],[1838,7243,1925],{"className":7244},[2097],[1838,7246],{"className":7247,"style":2101},[1951],[1838,7249,2780],{"className":7250},[1881],[1838,7252,1925],{"className":7253},[2097],[1838,7255],{"className":7256,"style":2101},[1951],[1838,7258,1928],{"className":7259},[2105],[1838,7261],{"className":7262,"style":2101},[1951],[1838,7264,1925],{"className":7265},[2097],[1838,7267],{"className":7268,"style":2101},[1951],[1838,7270,2780],{"className":7271},[1881],[1838,7273,7275,7278],{"className":7274},[2159],[1838,7276,1943],{"className":7277},[2159],[1838,7279,7281],{"className":7280},[2047],[1838,7282,7284],{"className":7283},[2051],[1838,7285,7287],{"className":7286},[2056],[1838,7288,7290],{"className":7289,"style":2398},[2060],[1838,7291,7292,7295],{"style":2419},[1838,7293],{"className":7294,"style":2069},[2068],[1838,7296,7298],{"className":7297},[2073,2074,2075,2076],[1838,7299,1947],{"className":7300},[1881,2076],[1838,7302,7304],{"className":7303},[1888],[1838,7305,7307,7363],{"className":7306},[1841],[1838,7308,7310],{"className":7309},[1845],[1847,7311,7312],{"xmlns":1849,"display":1897},[1851,7313,7314,7360],{},[1854,7315,7316,7336,7338,7346,7352,7358],{},[1915,7317,7318,7324],{},[2306,7319,7320,7322],{"accent":1868},[1857,7321,1905],{},[1907,7323,2312],{"stretchy":1868},[1854,7325,7326,7328,7330,7332,7334],{},[1857,7327,3901],{},[1907,7329,2504],{},[1857,7331,3783],{},[1907,7333,3548],{},[1857,7335,3901],{},[1907,7337,1909],{},[4955,7339,7340,7342,7344],{},[1857,7341,7142],{"mathvariant":1904},[1920,7343,1922],{},[1857,7345,1947],{"mathvariant":1946},[1939,7347,7348,7350],{},[1857,7349,6535],{},[1857,7351,3783],{},[1915,7353,7354,7356],{},[1857,7355,6437],{"mathvariant":1904},[1857,7357,3901],{},[1857,7359,1997],{"mathvariant":1946},[1861,7361,7362],{"encoding":1863},"\\widehat X_{t+h\\mid t}=\\mathbf e_1^\\top F^h\\mathbf s_t.",[1838,7364,7366,7470],{"className":7365,"ariaHidden":1868},[1867],[1838,7367,7369,7373,7461,7464,7467],{"className":7368},[1872],[1838,7370],{"className":7371,"style":7372},[1876],"height:1.2785em;vertical-align:-0.3552em;",[1838,7374,7376,7409],{"className":7375},[1881],[1838,7377,7379],{"className":7378},[1881,2340],[1838,7380,7382],{"className":7381},[2051],[1838,7383,7385],{"className":7384},[2056],[1838,7386,7388,7396],{"className":7387,"style":2336},[2060],[1838,7389,7390,7393],{"style":2352},[1838,7391],{"className":7392,"style":2356},[2068],[1838,7394,1905],{"className":7395,"style":2043},[1881,1882],[1838,7397,7400,7403],{"className":7398,"style":7399},[2363],"width:calc(100% - 0.1667em);margin-left:0.1667em;top:-3.6833em;",[1838,7401],{"className":7402,"style":2356},[2068],[1838,7404,7405],{"style":2370},[2372,7406,7407],{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},[2380,7408],{"d":2382},[1838,7410,7412],{"className":7411},[2047],[1838,7413,7415,7452],{"className":7414},[2051,2052],[1838,7416,7418,7449],{"className":7417},[2056],[1838,7419,7422],{"className":7420,"style":7421},[2060],"height:0.3448em;",[1838,7423,7425,7428],{"style":7424},"top:-2.5198em;margin-left:-0.0785em;margin-right:0.05em;",[1838,7426],{"className":7427,"style":2069},[2068],[1838,7429,7431],{"className":7430},[2073,2074,2075,2076],[1838,7432,7434,7437,7440,7443,7446],{"className":7433},[1881,2076],[1838,7435,3901],{"className":7436},[1881,1882,2076],[1838,7438,2504],{"className":7439},[2546,2076],[1838,7441,3783],{"className":7442},[1881,1882,2076],[1838,7444,3548],{"className":7445},[2022,2076],[1838,7447,3901],{"className":7448},[1881,1882,2076],[1838,7450,2084],{"className":7451},[2083],[1838,7453,7455],{"className":7454},[2056],[1838,7456,7459],{"className":7457,"style":7458},[2060],"height:0.3552em;",[1838,7460],{},[1838,7462],{"className":7463,"style":2018},[1951],[1838,7465,1909],{"className":7466},[2022],[1838,7468],{"className":7469,"style":2018},[1951],[1838,7471,7473,7477,7529,7558,7598],{"className":7472},[1872],[1838,7474],{"className":7475,"style":7476},[1876],"height:1.1461em;vertical-align:-0.247em;",[1838,7478,7480,7483],{"className":7479},[1881],[1838,7481,7142],{"className":7482},[1881,2014],[1838,7484,7486],{"className":7485},[2047],[1838,7487,7489,7521],{"className":7488},[2051,2052],[1838,7490,7492,7518],{"className":7491},[2056],[1838,7493,7495,7507],{"className":7494,"style":2175},[2060],[1838,7496,7498,7501],{"style":7497},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[1838,7499],{"className":7500,"style":2069},[2068],[1838,7502,7504],{"className":7503},[2073,2074,2075,2076],[1838,7505,1922],{"className":7506},[1881,2076],[1838,7508,7509,7512],{"style":2178},[1838,7510],{"className":7511,"style":2069},[2068],[1838,7513,7515],{"className":7514},[2073,2074,2075,2076],[1838,7516,1947],{"className":7517},[1881,2076],[1838,7519,2084],{"className":7520},[2083],[1838,7522,7524],{"className":7523},[2056],[1838,7525,7527],{"className":7526,"style":5221},[2060],[1838,7528],{},[1838,7530,7532,7535],{"className":7531},[1881],[1838,7533,6535],{"className":7534,"style":7012},[1881,1882],[1838,7536,7538],{"className":7537},[2047],[1838,7539,7541],{"className":7540},[2051],[1838,7542,7544],{"className":7543},[2056],[1838,7545,7547],{"className":7546,"style":2175},[2060],[1838,7548,7549,7552],{"style":2178},[1838,7550],{"className":7551,"style":2069},[2068],[1838,7553,7555],{"className":7554},[2073,2074,2075,2076],[1838,7556,3783],{"className":7557},[1881,1882,2076],[1838,7559,7561,7564],{"className":7560},[1881],[1838,7562,6437],{"className":7563},[1881,2014],[1838,7565,7567],{"className":7566},[2047],[1838,7568,7570,7590],{"className":7569},[2051,2052],[1838,7571,7573,7587],{"className":7572},[2056],[1838,7574,7576],{"className":7575,"style":4125},[2060],[1838,7577,7578,7581],{"style":6310},[1838,7579],{"className":7580,"style":2069},[2068],[1838,7582,7584],{"className":7583},[2073,2074,2075,2076],[1838,7585,3901],{"className":7586},[1881,1882,2076],[1838,7588,2084],{"className":7589},[2083],[1838,7591,7593],{"className":7592},[2056],[1838,7594,7596],{"className":7595,"style":2091},[2060],[1838,7597],{},[1838,7599,1997],{"className":7600},[1881],[1798,7602,3410,7603,7745,7746,7869],{},[1838,7604,7606,7640],{"className":7605},[1841],[1838,7607,7609],{"className":7608},[1845],[1847,7610,7611],{"xmlns":1849},[1851,7612,7613,7637],{},[1854,7614,7615,7617,7619,7621,7627,7629,7631],{},[1857,7616,1961],{"mathvariant":1946},[1907,7618,1964],{},[1907,7620,1913],{"stretchy":1912},[1915,7622,7623,7625],{},[1857,7624,4962],{},[1857,7626,3901],{},[1907,7628,1943],{"stretchy":1912},[1907,7630,1909],{},[1939,7632,7633,7635],{},[1857,7634,4959],{},[1920,7636,2474],{},[1861,7638,7639],{"encoding":1863},"\\operatorname{Var}(\\varepsilon_t)=\\sigma^2",[1838,7641,7643,7710],{"className":7642,"ariaHidden":1868},[1867],[1838,7644,7646,7649,7655,7658,7698,7701,7704,7707],{"className":7645},[1872],[1838,7647],{"className":7648,"style":2216},[1876],[1838,7650,7652],{"className":7651},[2220],[1838,7653,1961],{"className":7654},[1881,2224],[1838,7656,1913],{"className":7657},[2036],[1838,7659,7661,7664],{"className":7660},[1881],[1838,7662,4962],{"className":7663},[1881,1882],[1838,7665,7667],{"className":7666},[2047],[1838,7668,7670,7690],{"className":7669},[2051,2052],[1838,7671,7673,7687],{"className":7672},[2056],[1838,7674,7676],{"className":7675,"style":4125},[2060],[1838,7677,7678,7681],{"style":6310},[1838,7679],{"className":7680,"style":2069},[2068],[1838,7682,7684],{"className":7683},[2073,2074,2075,2076],[1838,7685,3901],{"className":7686},[1881,1882,2076],[1838,7688,2084],{"className":7689},[2083],[1838,7691,7693],{"className":7692},[2056],[1838,7694,7696],{"className":7695,"style":2091},[2060],[1838,7697],{},[1838,7699,1943],{"className":7700},[2159],[1838,7702],{"className":7703,"style":2018},[1951],[1838,7705,1909],{"className":7706},[2022],[1838,7708],{"className":7709,"style":2018},[1951],[1838,7711,7713,7716],{"className":7712},[1872],[1838,7714],{"className":7715,"style":4570},[1876],[1838,7717,7719,7722],{"className":7718},[1881],[1838,7720,4959],{"className":7721,"style":5176},[1881,1882],[1838,7723,7725],{"className":7724},[2047],[1838,7726,7728],{"className":7727},[2051],[1838,7729,7731],{"className":7730},[2056],[1838,7732,7734],{"className":7733,"style":4570},[2060],[1838,7735,7736,7739],{"style":2419},[1838,7737],{"className":7738,"style":2069},[2068],[1838,7740,7742],{"className":7741},[2073,2074,2075,2076],[1838,7743,2474],{"className":7744},[1881,2076]," and ",[1838,7747,7749,7780],{"className":7748},[1841],[1838,7750,7752],{"className":7751},[1845],[1847,7753,7754],{"xmlns":1849},[1851,7755,7756,7777],{},[1854,7757,7758,7761,7763,7769,7771],{},[1857,7759,7760],{},"Q",[1907,7762,1909],{},[1939,7764,7765,7767],{},[1857,7766,4959],{},[1920,7768,2474],{},[1857,7770,6552],{},[1939,7772,7773,7775],{},[1857,7774,6552],{},[1857,7776,1947],{"mathvariant":1946},[1861,7778,7779],{"encoding":1863},"Q=\\sigma^2GG^\\top",[1838,7781,7783,7802],{"className":7782,"ariaHidden":1868},[1867],[1838,7784,7786,7790,7793,7796,7799],{"className":7785},[1872],[1838,7787],{"className":7788,"style":7789},[1876],"height:0.8778em;vertical-align:-0.1944em;",[1838,7791,7760],{"className":7792},[1881,1882],[1838,7794],{"className":7795,"style":2018},[1951],[1838,7797,1909],{"className":7798},[2022],[1838,7800],{"className":7801,"style":2018},[1951],[1838,7803,7805,7808,7837,7840],{"className":7804},[1872],[1838,7806],{"className":7807,"style":2398},[1876],[1838,7809,7811,7814],{"className":7810},[1881],[1838,7812,4959],{"className":7813,"style":5176},[1881,1882],[1838,7815,7817],{"className":7816},[2047],[1838,7818,7820],{"className":7819},[2051],[1838,7821,7823],{"className":7822},[2056],[1838,7824,7826],{"className":7825,"style":4570},[2060],[1838,7827,7828,7831],{"style":2419},[1838,7829],{"className":7830,"style":2069},[2068],[1838,7832,7834],{"className":7833},[2073,2074,2075,2076],[1838,7835,2474],{"className":7836},[1881,2076],[1838,7838,6552],{"className":7839},[1881,1882],[1838,7841,7843,7846],{"className":7842},[1881],[1838,7844,6552],{"className":7845},[1881,1882],[1838,7847,7849],{"className":7848},[2047],[1838,7850,7852],{"className":7851},[2051],[1838,7853,7855],{"className":7854},[2056],[1838,7856,7858],{"className":7857,"style":2398},[2060],[1838,7859,7860,7863],{"style":2419},[1838,7861],{"className":7862,"style":2069},[2068],[1838,7864,7866],{"className":7865},[2073,2074,2075,2076],[1838,7867,1947],{"className":7868},[1881,2076],", then",[1838,7871,7873],{"className":7872},[1888],[1838,7874,7876,7979],{"className":7875},[1841],[1838,7877,7879],{"className":7878},[1845],[1847,7880,7881],{"xmlns":1849,"display":1897},[1851,7882,7883,7976],{},[1854,7884,7885,7887,7889,7891,7903,7905,7925,7927,7929,7952,7958,7960,7962,7968,7974],{},[1857,7886,1961],{"mathvariant":1946},[1907,7888,1964],{},[1907,7890,1913],{"stretchy":1912},[1915,7892,7893,7895],{},[1857,7894,6437],{"mathvariant":1904},[1854,7896,7897,7899,7901],{},[1857,7898,3901],{},[1907,7900,2504],{},[1857,7902,3783],{},[1907,7904,2459],{},[1915,7906,7907,7913],{},[2306,7908,7909,7911],{"accent":1868},[1857,7910,6437],{"mathvariant":1904},[1907,7912,2312],{"stretchy":1868},[1854,7914,7915,7917,7919,7921,7923],{},[1857,7916,3901],{},[1907,7918,2504],{},[1857,7920,3783],{},[1907,7922,3548],{},[1857,7924,3901],{},[1907,7926,1943],{"stretchy":1912},[1907,7928,1909],{},[7930,7931,7932,7935,7944],"munderover",{},[1907,7933,7934],{},"∑",[1854,7936,7937,7940,7942],{},[1857,7938,7939],{},"j",[1907,7941,1909],{},[1920,7943,2780],{},[1854,7945,7946,7948,7950],{},[1857,7947,3783],{},[1907,7949,2459],{},[1920,7951,1922],{},[1939,7953,7954,7956],{},[1857,7955,6535],{},[1857,7957,7939],{},[1857,7959,7760],{},[1907,7961,1913],{"stretchy":1912},[1939,7963,7964,7966],{},[1857,7965,6535],{},[1857,7967,1947],{"mathvariant":1946},[1939,7969,7970,7972],{},[1907,7971,1943],{"stretchy":1912},[1857,7973,7939],{},[1857,7975,1997],{"mathvariant":1946},[1861,7977,7978],{"encoding":1863},"\\operatorname{Var}(\\mathbf s_{t+h}-\\widehat{\\mathbf s}_{t+h\\mid t})\n=\\sum_{j=0}^{h-1}F^jQ(F^\\top)^j.",[1838,7980,7982,8056,8162],{"className":7981,"ariaHidden":1868},[1867],[1838,7983,7985,7988,7994,7997,8047,8050,8053],{"className":7984},[1872],[1838,7986],{"className":7987,"style":2216},[1876],[1838,7989,7991],{"className":7990},[2220],[1838,7992,1961],{"className":7993},[1881,2224],[1838,7995,1913],{"className":7996},[2036],[1838,7998,8000,8003],{"className":7999},[1881],[1838,8001,6437],{"className":8002},[1881,2014],[1838,8004,8006],{"className":8005},[2047],[1838,8007,8009,8039],{"className":8008},[2051,2052],[1838,8010,8012,8036],{"className":8011},[2056],[1838,8013,8016],{"className":8014,"style":8015},[2060],"height:0.3361em;",[1838,8017,8018,8021],{"style":6310},[1838,8019],{"className":8020,"style":2069},[2068],[1838,8022,8024],{"className":8023},[2073,2074,2075,2076],[1838,8025,8027,8030,8033],{"className":8026},[1881,2076],[1838,8028,3901],{"className":8029},[1881,1882,2076],[1838,8031,2504],{"className":8032},[2546,2076],[1838,8034,3783],{"className":8035},[1881,1882,2076],[1838,8037,2084],{"className":8038},[2083],[1838,8040,8042],{"className":8041},[2056],[1838,8043,8045],{"className":8044,"style":3966},[2060],[1838,8046],{},[1838,8048],{"className":8049,"style":1883},[1951],[1838,8051,2459],{"className":8052},[2546],[1838,8054],{"className":8055,"style":1883},[1951],[1838,8057,8059,8063,8150,8153,8156,8159],{"className":8058},[1872],[1838,8060],{"className":8061,"style":8062},[1876],"height:1.1052em;vertical-align:-0.3552em;",[1838,8064,8066,8100],{"className":8065},[1881],[1838,8067,8069],{"className":8068},[1881,2340],[1838,8070,8072],{"className":8071},[2051],[1838,8073,8075],{"className":8074},[2056],[1838,8076,8079,8087],{"className":8077,"style":8078},[2060],"height:0.6844em;",[1838,8080,8081,8084],{"style":2352},[1838,8082],{"className":8083,"style":2356},[2068],[1838,8085,6437],{"className":8086},[1881,2014],[1838,8088,8091,8094],{"className":8089,"style":8090},[2363],"top:-3.4444em;",[1838,8092],{"className":8093,"style":2356},[2068],[1838,8095,8096],{"style":2370},[2372,8097,8098],{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},[2380,8099],{"d":2382},[1838,8101,8103],{"className":8102},[2047],[1838,8104,8106,8142],{"className":8105},[2051,2052],[1838,8107,8109,8139],{"className":8108},[2056],[1838,8110,8112],{"className":8111,"style":7421},[2060],[1838,8113,8115,8118],{"style":8114},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[1838,8116],{"className":8117,"style":2069},[2068],[1838,8119,8121],{"className":8120},[2073,2074,2075,2076],[1838,8122,8124,8127,8130,8133,8136],{"className":8123},[1881,2076],[1838,8125,3901],{"className":8126},[1881,1882,2076],[1838,8128,2504],{"className":8129},[2546,2076],[1838,8131,3783],{"className":8132},[1881,1882,2076],[1838,8134,3548],{"className":8135},[2022,2076],[1838,8137,3901],{"className":8138},[1881,1882,2076],[1838,8140,2084],{"className":8141},[2083],[1838,8143,8145],{"className":8144},[2056],[1838,8146,8148],{"className":8147,"style":7458},[2060],[1838,8149],{},[1838,8151,1943],{"className":8152},[2159],[1838,8154],{"className":8155,"style":2018},[1951],[1838,8157,1909],{"className":8158},[2022],[1838,8160],{"className":8161,"style":2018},[1951],[1838,8163,8165,8169,8252,8255,8285,8288,8291,8320,8349],{"className":8164},[1872],[1838,8166],{"className":8167,"style":8168},[1876],"height:3.2499em;vertical-align:-1.4138em;",[1838,8170,8173],{"className":8171},[2220,8172],"op-limits",[1838,8174,8176,8243],{"className":8175},[2051,2052],[1838,8177,8179,8240],{"className":8178},[2056],[1838,8180,8183,8206,8219],{"className":8181,"style":8182},[2060],"height:1.8361em;",[1838,8184,8186,8190],{"style":8185},"top:-1.8723em;margin-left:0em;",[1838,8187],{"className":8188,"style":8189},[2068],"height:3.05em;",[1838,8191,8193],{"className":8192},[2073,2074,2075,2076],[1838,8194,8196,8200,8203],{"className":8195},[1881,2076],[1838,8197,7939],{"className":8198,"style":8199},[1881,1882,2076],"margin-right:0.0572em;",[1838,8201,1909],{"className":8202},[2022,2076],[1838,8204,2780],{"className":8205},[1881,2076],[1838,8207,8209,8212],{"style":8208},"top:-3.05em;",[1838,8210],{"className":8211,"style":8189},[2068],[1838,8213,8214],{},[1838,8215,7934],{"className":8216},[2220,8217,8218],"op-symbol","large-op",[1838,8220,8222,8225],{"style":8221},"top:-4.3em;margin-left:0em;",[1838,8223],{"className":8224,"style":8189},[2068],[1838,8226,8228],{"className":8227},[2073,2074,2075,2076],[1838,8229,8231,8234,8237],{"className":8230},[1881,2076],[1838,8232,3783],{"className":8233},[1881,1882,2076],[1838,8235,2459],{"className":8236},[2546,2076],[1838,8238,1922],{"className":8239},[1881,2076],[1838,8241,2084],{"className":8242},[2083],[1838,8244,8246],{"className":8245},[2056],[1838,8247,8250],{"className":8248,"style":8249},[2060],"height:1.4138em;",[1838,8251],{},[1838,8253],{"className":8254,"style":2101},[1951],[1838,8256,8258,8261],{"className":8257},[1881],[1838,8259,6535],{"className":8260,"style":7012},[1881,1882],[1838,8262,8264],{"className":8263},[2047],[1838,8265,8267],{"className":8266},[2051],[1838,8268,8270],{"className":8269},[2056],[1838,8271,8274],{"className":8272,"style":8273},[2060],"height:0.8747em;",[1838,8275,8276,8279],{"style":2178},[1838,8277],{"className":8278,"style":2069},[2068],[1838,8280,8282],{"className":8281},[2073,2074,2075,2076],[1838,8283,7939],{"className":8284,"style":8199},[1881,1882,2076],[1838,8286,7760],{"className":8287},[1881,1882],[1838,8289,1913],{"className":8290},[2036],[1838,8292,8294,8297],{"className":8293},[1881],[1838,8295,6535],{"className":8296,"style":7012},[1881,1882],[1838,8298,8300],{"className":8299},[2047],[1838,8301,8303],{"className":8302},[2051],[1838,8304,8306],{"className":8305},[2056],[1838,8307,8309],{"className":8308,"style":2175},[2060],[1838,8310,8311,8314],{"style":2178},[1838,8312],{"className":8313,"style":2069},[2068],[1838,8315,8317],{"className":8316},[2073,2074,2075,2076],[1838,8318,1947],{"className":8319},[1881,2076],[1838,8321,8323,8326],{"className":8322},[2159],[1838,8324,1943],{"className":8325},[2159],[1838,8327,8329],{"className":8328},[2047],[1838,8330,8332],{"className":8331},[2051],[1838,8333,8335],{"className":8334},[2056],[1838,8336,8338],{"className":8337,"style":8273},[2060],[1838,8339,8340,8343],{"style":2178},[1838,8341],{"className":8342,"style":2069},[2068],[1838,8344,8346],{"className":8345},[2073,2074,2075,2076],[1838,8347,7939],{"className":8348,"style":8199},[1881,1882,2076],[1838,8350,1997],{"className":8351},[1881],[1798,8353,8354],{},"The point prediction propagates the state; the uncertainty accumulates every future innovation.",[1793,8356,8358],{"id":8357},"_6-innovations-reveal-information-one-observation-at-a-time","6. Innovations reveal information one observation at a time",[1798,8360,8361,8362,8490],{},"Suppose ",[1838,8363,8365,8397],{"className":8364},[1841],[1838,8366,8368],{"className":8367},[1845],[1847,8369,8370],{"xmlns":1849},[1851,8371,8372,8394],{},[1854,8373,8374],{},[1915,8375,8376,8382],{},[2306,8377,8378,8380],{"accent":1868},[1857,8379,1905],{},[1907,8381,2312],{"stretchy":1868},[1854,8383,8384,8386,8388,8390,8392],{},[1857,8385,3901],{},[1907,8387,3548],{},[1857,8389,3901],{},[1907,8391,2459],{},[1920,8393,1922],{},[1861,8395,8396],{"encoding":1863},"\\widehat X_{t\\mid t-1}",[1838,8398,8400],{"className":8399,"ariaHidden":1868},[1867],[1838,8401,8403,8406],{"className":8402},[1872],[1838,8404],{"className":8405,"style":7372},[1876],[1838,8407,8409,8441],{"className":8408},[1881],[1838,8410,8412],{"className":8411},[1881,2340],[1838,8413,8415],{"className":8414},[2051],[1838,8416,8418],{"className":8417},[2056],[1838,8419,8421,8429],{"className":8420,"style":2336},[2060],[1838,8422,8423,8426],{"style":2352},[1838,8424],{"className":8425,"style":2356},[2068],[1838,8427,1905],{"className":8428,"style":2043},[1881,1882],[1838,8430,8432,8435],{"className":8431,"style":7399},[2363],[1838,8433],{"className":8434,"style":2356},[2068],[1838,8436,8437],{"style":2370},[2372,8438,8439],{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},[2380,8440],{"d":2382},[1838,8442,8444],{"className":8443},[2047],[1838,8445,8447,8482],{"className":8446},[2051,2052],[1838,8448,8450,8479],{"className":8449},[2056],[1838,8451,8453],{"className":8452,"style":7421},[2060],[1838,8454,8455,8458],{"style":7424},[1838,8456],{"className":8457,"style":2069},[2068],[1838,8459,8461],{"className":8460},[2073,2074,2075,2076],[1838,8462,8464,8467,8470,8473,8476],{"className":8463},[1881,2076],[1838,8465,3901],{"className":8466},[1881,1882,2076],[1838,8468,3548],{"className":8469},[2022,2076],[1838,8471,3901],{"className":8472},[1881,1882,2076],[1838,8474,2459],{"className":8475},[2546,2076],[1838,8477,1922],{"className":8478},[1881,2076],[1838,8480,2084],{"className":8481},[2083],[1838,8483,8485],{"className":8484},[2056],[1838,8486,8488],{"className":8487,"style":7458},[2060],[1838,8489],{}," is the projection on the past. Define the innovation",[1838,8492,8494],{"className":8493},[1888],[1838,8495,8497,8574],{"className":8496},[1841],[1838,8498,8500],{"className":8499},[1845],[1847,8501,8502],{"xmlns":1849,"display":1897},[1851,8503,8504,8571],{},[1854,8505,8506,8513,8515,8521,8523,8543,8545,8547,8553,8555,8557,8559,8561,8567,8569],{},[1915,8507,8508,8511],{},[1857,8509,8510],{},"v",[1857,8512,3901],{},[1907,8514,1909],{},[1915,8516,8517,8519],{},[1857,8518,1905],{},[1857,8520,3901],{},[1907,8522,2459],{},[1915,8524,8525,8531],{},[2306,8526,8527,8529],{"accent":1868},[1857,8528,1905],{},[1907,8530,2312],{"stretchy":1868},[1854,8532,8533,8535,8537,8539,8541],{},[1857,8534,3901],{},[1907,8536,3548],{},[1857,8538,3901],{},[1907,8540,2459],{},[1920,8542,1922],{},[1907,8544,1925],{"separator":1868},[1951,8546],{"width":1953},[1915,8548,8549,8551],{},[1857,8550,6535],{},[1857,8552,3901],{},[1907,8554,1909],{},[1857,8556,1961],{"mathvariant":1946},[1907,8558,1964],{},[1907,8560,1913],{"stretchy":1912},[1915,8562,8563,8565],{},[1857,8564,8510],{},[1857,8566,3901],{},[1907,8568,1943],{"stretchy":1912},[1857,8570,1997],{"mathvariant":1946},[1861,8572,8573],{"encoding":1863},"v_t=X_t-\\widehat X_{t\\mid t-1},\n\\qquad\nF_t=\\operatorname{Var}(v_t).",[1838,8575,8577,8634,8689,8838],{"className":8576,"ariaHidden":1868},[1867],[1838,8578,8580,8584,8625,8628,8631],{"className":8579},[1872],[1838,8581],{"className":8582,"style":8583},[1876],"height:0.5806em;vertical-align:-0.15em;",[1838,8585,8587,8590],{"className":8586},[1881],[1838,8588,8510],{"className":8589,"style":5176},[1881,1882],[1838,8591,8593],{"className":8592},[2047],[1838,8594,8596,8617],{"className":8595},[2051,2052],[1838,8597,8599,8614],{"className":8598},[2056],[1838,8600,8602],{"className":8601,"style":4125},[2060],[1838,8603,8605,8608],{"style":8604},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1838,8606],{"className":8607,"style":2069},[2068],[1838,8609,8611],{"className":8610},[2073,2074,2075,2076],[1838,8612,3901],{"className":8613},[1881,1882,2076],[1838,8615,2084],{"className":8616},[2083],[1838,8618,8620],{"className":8619},[2056],[1838,8621,8623],{"className":8622,"style":2091},[2060],[1838,8624],{},[1838,8626],{"className":8627,"style":2018},[1951],[1838,8629,1909],{"className":8630},[2022],[1838,8632],{"className":8633,"style":2018},[1951],[1838,8635,8637,8640,8680,8683,8686],{"className":8636},[1872],[1838,8638],{"className":8639,"style":4754},[1876],[1838,8641,8643,8646],{"className":8642},[1881],[1838,8644,1905],{"className":8645,"style":2043},[1881,1882],[1838,8647,8649],{"className":8648},[2047],[1838,8650,8652,8672],{"className":8651},[2051,2052],[1838,8653,8655,8669],{"className":8654},[2056],[1838,8656,8658],{"className":8657,"style":4125},[2060],[1838,8659,8660,8663],{"style":2064},[1838,8661],{"className":8662,"style":2069},[2068],[1838,8664,8666],{"className":8665},[2073,2074,2075,2076],[1838,8667,3901],{"className":8668},[1881,1882,2076],[1838,8670,2084],{"className":8671},[2083],[1838,8673,8675],{"className":8674},[2056],[1838,8676,8678],{"className":8677,"style":2091},[2060],[1838,8679],{},[1838,8681],{"className":8682,"style":1883},[1951],[1838,8684,2459],{"className":8685},[2546],[1838,8687],{"className":8688,"style":1883},[1951],[1838,8690,8692,8695,8779,8782,8785,8788,8829,8832,8835],{"className":8691},[1872],[1838,8693],{"className":8694,"style":7372},[1876],[1838,8696,8698,8730],{"className":8697},[1881],[1838,8699,8701],{"className":8700},[1881,2340],[1838,8702,8704],{"className":8703},[2051],[1838,8705,8707],{"className":8706},[2056],[1838,8708,8710,8718],{"className":8709,"style":2336},[2060],[1838,8711,8712,8715],{"style":2352},[1838,8713],{"className":8714,"style":2356},[2068],[1838,8716,1905],{"className":8717,"style":2043},[1881,1882],[1838,8719,8721,8724],{"className":8720,"style":7399},[2363],[1838,8722],{"className":8723,"style":2356},[2068],[1838,8725,8726],{"style":2370},[2372,8727,8728],{"xmlns":2374,"width":2375,"height":2376,"viewBox":2377,"preserveAspectRatio":2378},[2380,8729],{"d":2382},[1838,8731,8733],{"className":8732},[2047],[1838,8734,8736,8771],{"className":8735},[2051,2052],[1838,8737,8739,8768],{"className":8738},[2056],[1838,8740,8742],{"className":8741,"style":7421},[2060],[1838,8743,8744,8747],{"style":7424},[1838,8745],{"className":8746,"style":2069},[2068],[1838,8748,8750],{"className":8749},[2073,2074,2075,2076],[1838,8751,8753,8756,8759,8762,8765],{"className":8752},[1881,2076],[1838,8754,3901],{"className":8755},[1881,1882,2076],[1838,8757,3548],{"className":8758},[2022,2076],[1838,8760,3901],{"className":8761},[1881,1882,2076],[1838,8763,2459],{"className":8764},[2546,2076],[1838,8766,1922],{"className":8767},[1881,2076],[1838,8769,2084],{"className":8770},[2083],[1838,8772,8774],{"className":8773},[2056],[1838,8775,8777],{"className":8776,"style":7458},[2060],[1838,8778],{},[1838,8780,1925],{"className":8781},[2097],[1838,8783],{"className":8784,"style":2194},[1951],[1838,8786],{"className":8787,"style":2101},[1951],[1838,8789,8791,8794],{"className":8790},[1881],[1838,8792,6535],{"className":8793,"style":7012},[1881,1882],[1838,8795,8797],{"className":8796},[2047],[1838,8798,8800,8821],{"className":8799},[2051,2052],[1838,8801,8803,8818],{"className":8802},[2056],[1838,8804,8806],{"className":8805,"style":4125},[2060],[1838,8807,8809,8812],{"style":8808},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1838,8810],{"className":8811,"style":2069},[2068],[1838,8813,8815],{"className":8814},[2073,2074,2075,2076],[1838,8816,3901],{"className":8817},[1881,1882,2076],[1838,8819,2084],{"className":8820},[2083],[1838,8822,8824],{"className":8823},[2056],[1838,8825,8827],{"className":8826,"style":2091},[2060],[1838,8828],{},[1838,8830],{"className":8831,"style":2018},[1951],[1838,8833,1909],{"className":8834},[2022],[1838,8836],{"className":8837,"style":2018},[1951],[1838,8839,8841,8844,8850,8853,8893,8896],{"className":8840},[1872],[1838,8842],{"className":8843,"style":2216},[1876],[1838,8845,8847],{"className":8846},[2220],[1838,8848,1961],{"className":8849},[1881,2224],[1838,8851,1913],{"className":8852},[2036],[1838,8854,8856,8859],{"className":8855},[1881],[1838,8857,8510],{"className":8858,"style":5176},[1881,1882],[1838,8860,8862],{"className":8861},[2047],[1838,8863,8865,8885],{"className":8864},[2051,2052],[1838,8866,8868,8882],{"className":8867},[2056],[1838,8869,8871],{"className":8870,"style":4125},[2060],[1838,8872,8873,8876],{"style":8604},[1838,8874],{"className":8875,"style":2069},[2068],[1838,8877,8879],{"className":8878},[2073,2074,2075,2076],[1838,8880,3901],{"className":8881},[1881,1882,2076],[1838,8883,2084],{"className":8884},[2083],[1838,8886,8888],{"className":8887},[2056],[1838,8889,8891],{"className":8890,"style":2091},[2060],[1838,8892],{},[1838,8894,1943],{"className":8895},[2159],[1838,8897,1997],{"className":8898},[1881],[1798,8900,8901],{},"Innovations are mutually orthogonal. For Gaussian processes they are independent. A joint Gaussian likelihood can therefore be written as a sum of one-step terms:",[1838,8903,8905],{"className":8904},[1888],[1838,8906,8908,9001],{"className":8907},[1841],[1838,8909,8911],{"className":8910},[1845],[1847,8912,8913],{"xmlns":1849,"display":1897},[1851,8914,8915,8998],{},[1854,8916,8917,8920,8922,8924,8931,8946,8996],{},[1857,8918,8919],{"mathvariant":1946},"ℓ",[1907,8921,1909],{},[1907,8923,2459],{},[8925,8926,8927,8929],"mfrac",{},[1920,8928,1922],{},[1920,8930,2474],{},[7930,8932,8933,8935,8943],{},[1907,8934,7934],{},[1854,8936,8937,8939,8941],{},[1857,8938,3901],{},[1907,8940,1909],{},[1920,8942,1922],{},[1857,8944,8945],{},"n",[1854,8947,8948,8950,8953,8955,8957,8959,8962,8964,8966,8968,8970,8976,8978,8994],{},[1907,8949,2452],{"fence":1868},[1857,8951,8952],{},"log",[1907,8954,1964],{},[1907,8956,1913],{"stretchy":1912},[1920,8958,2474],{},[1857,8960,8961],{},"π",[1907,8963,1943],{"stretchy":1912},[1907,8965,2504],{},[1857,8967,8952],{},[1907,8969,1964],{},[1915,8971,8972,8974],{},[1857,8973,6535],{},[1857,8975,3901],{},[1907,8977,2504],{},[8925,8979,8980,8988],{},[4955,8981,8982,8984,8986],{},[1857,8983,8510],{},[1857,8985,3901],{},[1920,8987,2474],{},[1915,8989,8990,8992],{},[1857,8991,6535],{},[1857,8993,3901],{},[1907,8995,2477],{"fence":1868},[1857,8997,1997],{"mathvariant":1946},[1861,8999,9000],{"encoding":1863},"\\ell=-\\frac12\\sum_{t=1}^n\n\\left[\\log(2\\pi)+\\log F_t+\\frac{v_t^2}{F_t}\\right].",[1838,9002,9004,9023],{"className":9003,"ariaHidden":1868},[1867],[1838,9005,9007,9011,9014,9017,9020],{"className":9006},[1872],[1838,9008],{"className":9009,"style":9010},[1876],"height:0.6944em;",[1838,9012,8919],{"className":9013},[1881],[1838,9015],{"className":9016,"style":2018},[1951],[1838,9018,1909],{"className":9019},[2022],[1838,9021],{"className":9022,"style":2018},[1951],[1838,9024,9026,9030,9033,9103,9106,9173,9176,9424,9427],{"className":9025},[1872],[1838,9027],{"className":9028,"style":9029},[1876],"height:2.9185em;vertical-align:-1.2671em;",[1838,9031,2459],{"className":9032},[1881],[1838,9034,9036,9040,9100],{"className":9035},[1881],[1838,9037],{"className":9038},[2036,9039],"nulldelimiter",[1838,9041,9043],{"className":9042},[8925],[1838,9044,9046,9091],{"className":9045},[2051,2052],[1838,9047,9049,9088],{"className":9048},[2056],[1838,9050,9053,9065,9076],{"className":9051,"style":9052},[2060],"height:1.3214em;",[1838,9054,9056,9059],{"style":9055},"top:-2.314em;",[1838,9057],{"className":9058,"style":2356},[2068],[1838,9060,9062],{"className":9061},[1881],[1838,9063,2474],{"className":9064},[1881],[1838,9066,9068,9071],{"style":9067},"top:-3.23em;",[1838,9069],{"className":9070,"style":2356},[2068],[1838,9072],{"className":9073,"style":9075},[9074],"frac-line","border-bottom-width:0.04em;",[1838,9077,9079,9082],{"style":9078},"top:-3.677em;",[1838,9080],{"className":9081,"style":2356},[2068],[1838,9083,9085],{"className":9084},[1881],[1838,9086,1922],{"className":9087},[1881],[1838,9089,2084],{"className":9090},[2083],[1838,9092,9094],{"className":9093},[2056],[1838,9095,9098],{"className":9096,"style":9097},[2060],"height:0.686em;",[1838,9099],{},[1838,9101],{"className":9102},[2159,9039],[1838,9104],{"className":9105,"style":2101},[1951],[1838,9107,9109],{"className":9108},[2220,8172],[1838,9110,9112,9164],{"className":9111},[2051,2052],[1838,9113,9115,9161],{"className":9114},[2056],[1838,9116,9119,9140,9150],{"className":9117,"style":9118},[2060],"height:1.6514em;",[1838,9120,9122,9125],{"style":9121},"top:-1.8829em;margin-left:0em;",[1838,9123],{"className":9124,"style":8189},[2068],[1838,9126,9128],{"className":9127},[2073,2074,2075,2076],[1838,9129,9131,9134,9137],{"className":9130},[1881,2076],[1838,9132,3901],{"className":9133},[1881,1882,2076],[1838,9135,1909],{"className":9136},[2022,2076],[1838,9138,1922],{"className":9139},[1881,2076],[1838,9141,9142,9145],{"style":8208},[1838,9143],{"className":9144,"style":8189},[2068],[1838,9146,9147],{},[1838,9148,7934],{"className":9149},[2220,8217,8218],[1838,9151,9152,9155],{"style":8221},[1838,9153],{"className":9154,"style":8189},[2068],[1838,9156,9158],{"className":9157},[2073,2074,2075,2076],[1838,9159,8945],{"className":9160},[1881,1882,2076],[1838,9162,2084],{"className":9163},[2083],[1838,9165,9167],{"className":9166},[2056],[1838,9168,9171],{"className":9169,"style":9170},[2060],"height:1.2671em;",[1838,9172],{},[1838,9174],{"className":9175,"style":2101},[1951],[1838,9177,9179,9185,9191,9194,9197,9200,9203,9206,9209,9212,9217,9220,9260,9263,9266,9269,9418],{"className":9178},[2105],[1838,9180,9182],{"className":9181,"style":4352},[2036,4351],[1838,9183,2452],{"className":9184},[4356,2075],[1838,9186,9188,9189],{"className":9187},[2220],"lo",[1838,9190,1977],{"style":2268},[1838,9192,1913],{"className":9193},[2036],[1838,9195,2474],{"className":9196},[1881],[1838,9198,8961],{"className":9199,"style":5176},[1881,1882],[1838,9201,1943],{"className":9202},[2159],[1838,9204],{"className":9205,"style":1883},[1951],[1838,9207,2504],{"className":9208},[2546],[1838,9210],{"className":9211,"style":1883},[1951],[1838,9213,9188,9215],{"className":9214},[2220],[1838,9216,1977],{"style":2268},[1838,9218],{"className":9219,"style":2101},[1951],[1838,9221,9223,9226],{"className":9222},[1881],[1838,9224,6535],{"className":9225,"style":7012},[1881,1882],[1838,9227,9229],{"className":9228},[2047],[1838,9230,9232,9252],{"className":9231},[2051,2052],[1838,9233,9235,9249],{"className":9234},[2056],[1838,9236,9238],{"className":9237,"style":4125},[2060],[1838,9239,9240,9243],{"style":8808},[1838,9241],{"className":9242,"style":2069},[2068],[1838,9244,9246],{"className":9245},[2073,2074,2075,2076],[1838,9247,3901],{"className":9248},[1881,1882,2076],[1838,9250,2084],{"className":9251},[2083],[1838,9253,9255],{"className":9254},[2056],[1838,9256,9258],{"className":9257,"style":2091},[2060],[1838,9259],{},[1838,9261],{"className":9262,"style":1883},[1951],[1838,9264,2504],{"className":9265},[2546],[1838,9267],{"className":9268,"style":1883},[1951],[1838,9270,9272,9275,9415],{"className":9271},[1881],[1838,9273],{"className":9274},[2036,9039],[1838,9276,9278],{"className":9277},[8925],[1838,9279,9281,9406],{"className":9280},[2051,2052],[1838,9282,9284,9403],{"className":9283},[2056],[1838,9285,9288,9336,9344],{"className":9286,"style":9287},[2060],"height:1.4911em;",[1838,9289,9290,9293],{"style":9055},[1838,9291],{"className":9292,"style":2356},[2068],[1838,9294,9296],{"className":9295},[1881],[1838,9297,9299,9302],{"className":9298},[1881],[1838,9300,6535],{"className":9301,"style":7012},[1881,1882],[1838,9303,9305],{"className":9304},[2047],[1838,9306,9308,9328],{"className":9307},[2051,2052],[1838,9309,9311,9325],{"className":9310},[2056],[1838,9312,9314],{"className":9313,"style":4125},[2060],[1838,9315,9316,9319],{"style":8808},[1838,9317],{"className":9318,"style":2069},[2068],[1838,9320,9322],{"className":9321},[2073,2074,2075,2076],[1838,9323,3901],{"className":9324},[1881,1882,2076],[1838,9326,2084],{"className":9327},[2083],[1838,9329,9331],{"className":9330},[2056],[1838,9332,9334],{"className":9333,"style":2091},[2060],[1838,9335],{},[1838,9337,9338,9341],{"style":9067},[1838,9339],{"className":9340,"style":2356},[2068],[1838,9342],{"className":9343,"style":9075},[9074],[1838,9345,9346,9349],{"style":9078},[1838,9347],{"className":9348,"style":2356},[2068],[1838,9350,9352],{"className":9351},[1881],[1838,9353,9355,9358],{"className":9354},[1881],[1838,9356,8510],{"className":9357,"style":5176},[1881,1882],[1838,9359,9361],{"className":9360},[2047],[1838,9362,9364,9395],{"className":9363},[2051,2052],[1838,9365,9367,9392],{"className":9366},[2056],[1838,9368,9370,9381],{"className":9369,"style":4570},[2060],[1838,9371,9372,9375],{"style":5191},[1838,9373],{"className":9374,"style":2069},[2068],[1838,9376,9378],{"className":9377},[2073,2074,2075,2076],[1838,9379,3901],{"className":9380},[1881,1882,2076],[1838,9382,9383,9386],{"style":2419},[1838,9384],{"className":9385,"style":2069},[2068],[1838,9387,9389],{"className":9388},[2073,2074,2075,2076],[1838,9390,2474],{"className":9391},[1881,2076],[1838,9393,2084],{"className":9394},[2083],[1838,9396,9398],{"className":9397},[2056],[1838,9399,9401],{"className":9400,"style":5221},[2060],[1838,9402],{},[1838,9404,2084],{"className":9405},[2083],[1838,9407,9409],{"className":9408},[2056],[1838,9410,9413],{"className":9411,"style":9412},[2060],"height:0.836em;",[1838,9414],{},[1838,9416],{"className":9417},[2159,9039],[1838,9419,9421],{"className":9420,"style":4352},[2159,4351],[1838,9422,2477],{"className":9423},[4356,2075],[1838,9425],{"className":9426,"style":2101},[1951],[1838,9428,1997],{"className":9429},[1881],[1798,9431,9432],{},"This is the sequential counterpart of the covariance-matrix likelihood in Module 4.",[1793,9434,9436],{"id":9435},"_7-state-space-avoids-a-growing-covariance-inverse","7. State space avoids a growing covariance inverse",[1798,9438,9439],{},"Use the linear Gaussian model",[1838,9441,9443],{"className":9442},[1888],[1838,9444,9446,9526],{"className":9445},[1841],[1838,9447,9449],{"className":9448},[1845],[1847,9450,9451],{"xmlns":1849,"display":1897},[1851,9452,9453,9523],{},[1854,9454,9455,9463,9465,9468,9480,9482,9485,9492,9494,9496,9502,9504,9507,9513,9515,9521],{},[1915,9456,9457,9461],{},[1857,9458,9460],{"mathvariant":9459},"bold-italic","α",[1857,9462,3901],{},[1907,9464,1909],{},[1857,9466,9467],{},"T",[1915,9469,9470,9472],{},[1857,9471,9460],{"mathvariant":9459},[1854,9473,9474,9476,9478],{},[1857,9475,3901],{},[1907,9477,2459],{},[1920,9479,1922],{},[1907,9481,2504],{},[1857,9483,9484],{},"R",[1915,9486,9487,9490],{},[1857,9488,9489],{"mathvariant":9459},"η",[1857,9491,3901],{},[1907,9493,1925],{"separator":1868},[1951,9495],{"width":1953},[1915,9497,9498,9500],{},[1857,9499,1859],{},[1857,9501,3901],{},[1907,9503,1909],{},[1857,9505,9506],{},"Z",[1915,9508,9509,9511],{},[1857,9510,9460],{"mathvariant":9459},[1857,9512,3901],{},[1907,9514,2504],{},[1915,9516,9517,9519],{},[1857,9518,4962],{},[1857,9520,3901],{},[1907,9522,1925],{"separator":1868},[1861,9524,9525],{"encoding":1863},"\\boldsymbol\\alpha_t=T\\boldsymbol\\alpha_{t-1}+R\\boldsymbol\\eta_t,\n\\qquad\nY_t=Z\\boldsymbol\\alpha_t+\\varepsilon_t,",[1838,9527,9529,9592,9665,9785,9850],{"className":9528,"ariaHidden":1868},[1867],[1838,9530,9532,9535,9583,9586,9589],{"className":9531},[1872],[1838,9533],{"className":9534,"style":6573},[1876],[1838,9536,9538,9548],{"className":9537},[1881],[1838,9539,9541],{"className":9540},[1881],[1838,9542,9544],{"className":9543},[1881],[1838,9545,9460],{"className":9546},[1881,9547],"boldsymbol",[1838,9549,9551],{"className":9550},[2047],[1838,9552,9554,9575],{"className":9553},[2051,2052],[1838,9555,9557,9572],{"className":9556},[2056],[1838,9558,9560],{"className":9559,"style":4125},[2060],[1838,9561,9563,9566],{"style":9562},"top:-2.55em;margin-right:0.05em;",[1838,9564],{"className":9565,"style":2069},[2068],[1838,9567,9569],{"className":9568},[2073,2074,2075,2076],[1838,9570,3901],{"className":9571},[1881,1882,2076],[1838,9573,2084],{"className":9574},[2083],[1838,9576,9578],{"className":9577},[2056],[1838,9579,9581],{"className":9580,"style":2091},[2060],[1838,9582],{},[1838,9584],{"className":9585,"style":2018},[1951],[1838,9587,1909],{"className":9588},[2022],[1838,9590],{"className":9591,"style":2018},[1951],[1838,9593,9595,9598,9601,9656,9659,9662],{"className":9594},[1872],[1838,9596],{"className":9597,"style":3918},[1876],[1838,9599,9467],{"className":9600,"style":7012},[1881,1882],[1838,9602,9604,9613],{"className":9603},[1881],[1838,9605,9607],{"className":9606},[1881],[1838,9608,9610],{"className":9609},[1881],[1838,9611,9460],{"className":9612},[1881,9547],[1838,9614,9616],{"className":9615},[2047],[1838,9617,9619,9648],{"className":9618},[2051,2052],[1838,9620,9622,9645],{"className":9621},[2056],[1838,9623,9625],{"className":9624,"style":2061},[2060],[1838,9626,9627,9630],{"style":9562},[1838,9628],{"className":9629,"style":2069},[2068],[1838,9631,9633],{"className":9632},[2073,2074,2075,2076],[1838,9634,9636,9639,9642],{"className":9635},[1881,2076],[1838,9637,3901],{"className":9638},[1881,1882,2076],[1838,9640,2459],{"className":9641},[2546,2076],[1838,9643,1922],{"className":9644},[1881,2076],[1838,9646,2084],{"className":9647},[2083],[1838,9649,9651],{"className":9650},[2056],[1838,9652,9654],{"className":9653,"style":3966},[2060],[1838,9655],{},[1838,9657],{"className":9658,"style":1883},[1951],[1838,9660,2504],{"className":9661},[2546],[1838,9663],{"className":9664,"style":1883},[1951],[1838,9666,9668,9672,9676,9726,9729,9732,9735,9776,9779,9782],{"className":9667},[1872],[1838,9669],{"className":9670,"style":9671},[1876],"height:0.9275em;vertical-align:-0.2441em;",[1838,9673,9484],{"className":9674,"style":9675},[1881,1882],"margin-right:0.0077em;",[1838,9677,9679,9689],{"className":9678},[1881],[1838,9680,9682],{"className":9681},[1881],[1838,9683,9685],{"className":9684},[1881],[1838,9686,9489],{"className":9687,"style":9688},[1881,9547],"margin-right:0.037em;",[1838,9690,9692],{"className":9691},[2047],[1838,9693,9695,9717],{"className":9694},[2051,2052],[1838,9696,9698,9714],{"className":9697},[2056],[1838,9699,9702],{"className":9700,"style":9701},[2060],"height:0.1864em;",[1838,9703,9705,9708],{"style":9704},"top:-2.4559em;margin-right:0.05em;",[1838,9706],{"className":9707,"style":2069},[2068],[1838,9709,9711],{"className":9710},[2073,2074,2075,2076],[1838,9712,3901],{"className":9713},[1881,1882,2076],[1838,9715,2084],{"className":9716},[2083],[1838,9718,9720],{"className":9719},[2056],[1838,9721,9724],{"className":9722,"style":9723},[2060],"height:0.2441em;",[1838,9725],{},[1838,9727,1925],{"className":9728},[2097],[1838,9730],{"className":9731,"style":2194},[1951],[1838,9733],{"className":9734,"style":2101},[1951],[1838,9736,9738,9741],{"className":9737},[1881],[1838,9739,1859],{"className":9740,"style":1883},[1881,1882],[1838,9742,9744],{"className":9743},[2047],[1838,9745,9747,9768],{"className":9746},[2051,2052],[1838,9748,9750,9765],{"className":9749},[2056],[1838,9751,9753],{"className":9752,"style":4125},[2060],[1838,9754,9756,9759],{"style":9755},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[1838,9757],{"className":9758,"style":2069},[2068],[1838,9760,9762],{"className":9761},[2073,2074,2075,2076],[1838,9763,3901],{"className":9764},[1881,1882,2076],[1838,9766,2084],{"className":9767},[2083],[1838,9769,9771],{"className":9770},[2056],[1838,9772,9774],{"className":9773,"style":2091},[2060],[1838,9775],{},[1838,9777],{"className":9778,"style":2018},[1951],[1838,9780,1909],{"className":9781},[2022],[1838,9783],{"className":9784,"style":2018},[1951],[1838,9786,9788,9791,9795,9841,9844,9847],{"className":9787},[1872],[1838,9789],{"className":9790,"style":4754},[1876],[1838,9792,9506],{"className":9793,"style":9794},[1881,1882],"margin-right:0.0715em;",[1838,9796,9798,9807],{"className":9797},[1881],[1838,9799,9801],{"className":9800},[1881],[1838,9802,9804],{"className":9803},[1881],[1838,9805,9460],{"className":9806},[1881,9547],[1838,9808,9810],{"className":9809},[2047],[1838,9811,9813,9833],{"className":9812},[2051,2052],[1838,9814,9816,9830],{"className":9815},[2056],[1838,9817,9819],{"className":9818,"style":4125},[2060],[1838,9820,9821,9824],{"style":9562},[1838,9822],{"className":9823,"style":2069},[2068],[1838,9825,9827],{"className":9826},[2073,2074,2075,2076],[1838,9828,3901],{"className":9829},[1881,1882,2076],[1838,9831,2084],{"className":9832},[2083],[1838,9834,9836],{"className":9835},[2056],[1838,9837,9839],{"className":9838,"style":2091},[2060],[1838,9840],{},[1838,9842],{"className":9843,"style":1883},[1951],[1838,9845,2504],{"className":9846},[2546],[1838,9848],{"className":9849,"style":1883},[1951],[1838,9851,9853,9856,9896],{"className":9852},[1872],[1838,9854],{"className":9855,"style":6413},[1876],[1838,9857,9859,9862],{"className":9858},[1881],[1838,9860,4962],{"className":9861},[1881,1882],[1838,9863,9865],{"className":9864},[2047],[1838,9866,9868,9888],{"className":9867},[2051,2052],[1838,9869,9871,9885],{"className":9870},[2056],[1838,9872,9874],{"className":9873,"style":4125},[2060],[1838,9875,9876,9879],{"style":6310},[1838,9877],{"className":9878,"style":2069},[2068],[1838,9880,9882],{"className":9881},[2073,2074,2075,2076],[1838,9883,3901],{"className":9884},[1881,1882,2076],[1838,9886,2084],{"className":9887},[2083],[1838,9889,9891],{"className":9890},[2056],[1838,9892,9894],{"className":9893,"style":2091},[2060],[1838,9895],{},[1838,9897,1925],{"className":9898},[2097],[1798,9900,9901],{},"with",[1838,9903,9905],{"className":9904},[1888],[1838,9906,9908,9963],{"className":9907},[1841],[1838,9909,9911],{"className":9910},[1845],[1847,9912,9913],{"xmlns":1849,"display":1897},[1851,9914,9915,9960],{},[1854,9916,9917,9919,9921,9923,9929,9931,9933,9935,9937,9939,9941,9943,9945,9951,9953,9955,9958],{},[1857,9918,1961],{"mathvariant":1946},[1907,9920,1964],{},[1907,9922,1913],{"stretchy":1912},[1915,9924,9925,9927],{},[1857,9926,9489],{"mathvariant":9459},[1857,9928,3901],{},[1907,9930,1943],{"stretchy":1912},[1907,9932,1909],{},[1857,9934,7760],{},[1907,9936,1925],{"separator":1868},[1951,9938],{"width":1953},[1857,9940,1961],{"mathvariant":1946},[1907,9942,1964],{},[1907,9944,1913],{"stretchy":1912},[1915,9946,9947,9949],{},[1857,9948,4962],{},[1857,9950,3901],{},[1907,9952,1943],{"stretchy":1912},[1907,9954,1909],{},[1857,9956,9957],{},"H",[1857,9959,1997],{"mathvariant":1946},[1861,9961,9962],{"encoding":1863},"\\operatorname{Var}(\\boldsymbol\\eta_t)=Q,\\qquad\n\\operatorname{Var}(\\varepsilon_t)=H.",[1838,9964,9966,10039,10118],{"className":9965,"ariaHidden":1868},[1867],[1838,9967,9969,9972,9978,9981,10027,10030,10033,10036],{"className":9968},[1872],[1838,9970],{"className":9971,"style":2216},[1876],[1838,9973,9975],{"className":9974},[2220],[1838,9976,1961],{"className":9977},[1881,2224],[1838,9979,1913],{"className":9980},[2036],[1838,9982,9984,9993],{"className":9983},[1881],[1838,9985,9987],{"className":9986},[1881],[1838,9988,9990],{"className":9989},[1881],[1838,9991,9489],{"className":9992,"style":9688},[1881,9547],[1838,9994,9996],{"className":9995},[2047],[1838,9997,9999,10019],{"className":9998},[2051,2052],[1838,10000,10002,10016],{"className":10001},[2056],[1838,10003,10005],{"className":10004,"style":9701},[2060],[1838,10006,10007,10010],{"style":9704},[1838,10008],{"className":10009,"style":2069},[2068],[1838,10011,10013],{"className":10012},[2073,2074,2075,2076],[1838,10014,3901],{"className":10015},[1881,1882,2076],[1838,10017,2084],{"className":10018},[2083],[1838,10020,10022],{"className":10021},[2056],[1838,10023,10025],{"className":10024,"style":9723},[2060],[1838,10026],{},[1838,10028,1943],{"className":10029},[2159],[1838,10031],{"className":10032,"style":2018},[1951],[1838,10034,1909],{"className":10035},[2022],[1838,10037],{"className":10038,"style":2018},[1951],[1838,10040,10042,10045,10048,10051,10054,10057,10063,10066,10106,10109,10112,10115],{"className":10041},[1872],[1838,10043],{"className":10044,"style":2216},[1876],[1838,10046,7760],{"className":10047},[1881,1882],[1838,10049,1925],{"className":10050},[2097],[1838,10052],{"className":10053,"style":2194},[1951],[1838,10055],{"className":10056,"style":2101},[1951],[1838,10058,10060],{"className":10059},[2220],[1838,10061,1961],{"className":10062},[1881,2224],[1838,10064,1913],{"className":10065},[2036],[1838,10067,10069,10072],{"className":10068},[1881],[1838,10070,4962],{"className":10071},[1881,1882],[1838,10073,10075],{"className":10074},[2047],[1838,10076,10078,10098],{"className":10077},[2051,2052],[1838,10079,10081,10095],{"className":10080},[2056],[1838,10082,10084],{"className":10083,"style":4125},[2060],[1838,10085,10086,10089],{"style":6310},[1838,10087],{"className":10088,"style":2069},[2068],[1838,10090,10092],{"className":10091},[2073,2074,2075,2076],[1838,10093,3901],{"className":10094},[1881,1882,2076],[1838,10096,2084],{"className":10097},[2083],[1838,10099,10101],{"className":10100},[2056],[1838,10102,10104],{"className":10103,"style":2091},[2060],[1838,10105],{},[1838,10107,1943],{"className":10108},[2159],[1838,10110],{"className":10111,"style":2018},[1951],[1838,10113,1909],{"className":10114},[2022],[1838,10116],{"className":10117,"style":2018},[1951],[1838,10119,10121,10124,10128],{"className":10120},[1872],[1838,10122],{"className":10123,"style":1877},[1876],[1838,10125,9957],{"className":10126,"style":10127},[1881,1882],"margin-right:0.0813em;",[1838,10129,1997],{"className":10130},[1881],[1798,10132,3410,10133,10298],{},[1838,10134,10136,10179],{"className":10135},[1841],[1838,10137,10139],{"className":10138},[1845],[1847,10140,10141],{"xmlns":1849},[1851,10142,10143,10176],{},[1854,10144,10145,10147,10159,10161,10174],{},[1907,10146,1913],{"stretchy":1912},[1915,10148,10149,10151],{},[1857,10150,2319],{"mathvariant":1904},[1854,10152,10153,10155,10157],{},[1857,10154,3901],{},[1907,10156,2459],{},[1920,10158,1922],{},[1907,10160,1925],{"separator":1868},[1915,10162,10163,10166],{},[1857,10164,10165],{},"P",[1854,10167,10168,10170,10172],{},[1857,10169,3901],{},[1907,10171,2459],{},[1920,10173,1922],{},[1907,10175,1943],{"stretchy":1912},[1861,10177,10178],{"encoding":1863},"(\\mathbf a_{t-1},P_{t-1})",[1838,10180,10182],{"className":10181,"ariaHidden":1868},[1867],[1838,10183,10185,10188,10191,10240,10243,10246,10295],{"className":10184},[1872],[1838,10186],{"className":10187,"style":2216},[1876],[1838,10189,1913],{"className":10190},[2036],[1838,10192,10194,10197],{"className":10193},[1881],[1838,10195,2319],{"className":10196},[1881,2014],[1838,10198,10200],{"className":10199},[2047],[1838,10201,10203,10232],{"className":10202},[2051,2052],[1838,10204,10206,10229],{"className":10205},[2056],[1838,10207,10209],{"className":10208,"style":2061},[2060],[1838,10210,10211,10214],{"style":6310},[1838,10212],{"className":10213,"style":2069},[2068],[1838,10215,10217],{"className":10216},[2073,2074,2075,2076],[1838,10218,10220,10223,10226],{"className":10219},[1881,2076],[1838,10221,3901],{"className":10222},[1881,1882,2076],[1838,10224,2459],{"className":10225},[2546,2076],[1838,10227,1922],{"className":10228},[1881,2076],[1838,10230,2084],{"className":10231},[2083],[1838,10233,10235],{"className":10234},[2056],[1838,10236,10238],{"className":10237,"style":3966},[2060],[1838,10239],{},[1838,10241,1925],{"className":10242},[2097],[1838,10244],{"className":10245,"style":2101},[1951],[1838,10247,10249,10252],{"className":10248},[1881],[1838,10250,10165],{"className":10251,"style":7012},[1881,1882],[1838,10253,10255],{"className":10254},[2047],[1838,10256,10258,10287],{"className":10257},[2051,2052],[1838,10259,10261,10284],{"className":10260},[2056],[1838,10262,10264],{"className":10263,"style":2061},[2060],[1838,10265,10266,10269],{"style":8808},[1838,10267],{"className":10268,"style":2069},[2068],[1838,10270,10272],{"className":10271},[2073,2074,2075,2076],[1838,10273,10275,10278,10281],{"className":10274},[1881,2076],[1838,10276,3901],{"className":10277},[1881,1882,2076],[1838,10279,2459],{"className":10280},[2546,2076],[1838,10282,1922],{"className":10283},[1881,2076],[1838,10285,2084],{"className":10286},[2083],[1838,10288,10290],{"className":10289},[2056],[1838,10291,10293],{"className":10292,"style":3966},[2060],[1838,10294],{},[1838,10296,1943],{"className":10297},[2159]," summarises the filtered state, prediction is",[1838,10300,10302],{"className":10301},[1888],[1838,10303,10305,10405],{"className":10304},[1841],[1838,10306,10308],{"className":10307},[1845],[1847,10309,10310],{"xmlns":1849,"display":1897},[1851,10311,10312,10402],{},[1854,10313,10314,10330,10332,10334,10346,10348,10350,10366,10368,10370,10382,10388,10390,10392,10394,10400],{},[1915,10315,10316,10318],{},[1857,10317,2319],{"mathvariant":1904},[1854,10319,10320,10322,10324,10326,10328],{},[1857,10321,3901],{},[1907,10323,3548],{},[1857,10325,3901],{},[1907,10327,2459],{},[1920,10329,1922],{},[1907,10331,1909],{},[1857,10333,9467],{},[1915,10335,10336,10338],{},[1857,10337,2319],{"mathvariant":1904},[1854,10339,10340,10342,10344],{},[1857,10341,3901],{},[1907,10343,2459],{},[1920,10345,1922],{},[1907,10347,1925],{"separator":1868},[1951,10349],{"width":1953},[1915,10351,10352,10354],{},[1857,10353,10165],{},[1854,10355,10356,10358,10360,10362,10364],{},[1857,10357,3901],{},[1907,10359,3548],{},[1857,10361,3901],{},[1907,10363,2459],{},[1920,10365,1922],{},[1907,10367,1909],{},[1857,10369,9467],{},[1915,10371,10372,10374],{},[1857,10373,10165],{},[1854,10375,10376,10378,10380],{},[1857,10377,3901],{},[1907,10379,2459],{},[1920,10381,1922],{},[1939,10383,10384,10386],{},[1857,10385,9467],{},[1857,10387,1947],{"mathvariant":1946},[1907,10389,2504],{},[1857,10391,9484],{},[1857,10393,7760],{},[1939,10395,10396,10398],{},[1857,10397,9484],{},[1857,10399,1947],{"mathvariant":1946},[1857,10401,1997],{"mathvariant":1946},[1861,10403,10404],{"encoding":1863},"\\mathbf a_{t\\mid t-1}=T\\mathbf a_{t-1},\\qquad\nP_{t\\mid t-1}=TP_{t-1}T^\\top+RQR^\\top.",[1838,10406,10408,10479,10612,10709],{"className":10407,"ariaHidden":1868},[1867],[1838,10409,10411,10415,10470,10473,10476],{"className":10410},[1872],[1838,10412],{"className":10413,"style":10414},[1876],"height:0.7996em;vertical-align:-0.3552em;",[1838,10416,10418,10421],{"className":10417},[1881],[1838,10419,2319],{"className":10420},[1881,2014],[1838,10422,10424],{"className":10423},[2047],[1838,10425,10427,10462],{"className":10426},[2051,2052],[1838,10428,10430,10459],{"className":10429},[2056],[1838,10431,10433],{"className":10432,"style":7421},[2060],[1838,10434,10435,10438],{"style":8114},[1838,10436],{"className":10437,"style":2069},[2068],[1838,10439,10441],{"className":10440},[2073,2074,2075,2076],[1838,10442,10444,10447,10450,10453,10456],{"className":10443},[1881,2076],[1838,10445,3901],{"className":10446},[1881,1882,2076],[1838,10448,3548],{"className":10449},[2022,2076],[1838,10451,3901],{"className":10452},[1881,1882,2076],[1838,10454,2459],{"className":10455},[2546,2076],[1838,10457,1922],{"className":10458},[1881,2076],[1838,10460,2084],{"className":10461},[2083],[1838,10463,10465],{"className":10464},[2056],[1838,10466,10468],{"className":10467,"style":7458},[2060],[1838,10469],{},[1838,10471],{"className":10472,"style":2018},[1951],[1838,10474,1909],{"className":10475},[2022],[1838,10477],{"className":10478,"style":2018},[1951],[1838,10480,10482,10486,10489,10538,10541,10544,10547,10603,10606,10609],{"className":10481},[1872],[1838,10483],{"className":10484,"style":10485},[1876],"height:1.0385em;vertical-align:-0.3552em;",[1838,10487,9467],{"className":10488,"style":7012},[1881,1882],[1838,10490,10492,10495],{"className":10491},[1881],[1838,10493,2319],{"className":10494},[1881,2014],[1838,10496,10498],{"className":10497},[2047],[1838,10499,10501,10530],{"className":10500},[2051,2052],[1838,10502,10504,10527],{"className":10503},[2056],[1838,10505,10507],{"className":10506,"style":2061},[2060],[1838,10508,10509,10512],{"style":6310},[1838,10510],{"className":10511,"style":2069},[2068],[1838,10513,10515],{"className":10514},[2073,2074,2075,2076],[1838,10516,10518,10521,10524],{"className":10517},[1881,2076],[1838,10519,3901],{"className":10520},[1881,1882,2076],[1838,10522,2459],{"className":10523},[2546,2076],[1838,10525,1922],{"className":10526},[1881,2076],[1838,10528,2084],{"className":10529},[2083],[1838,10531,10533],{"className":10532},[2056],[1838,10534,10536],{"className":10535,"style":3966},[2060],[1838,10537],{},[1838,10539,1925],{"className":10540},[2097],[1838,10542],{"className":10543,"style":2194},[1951],[1838,10545],{"className":10546,"style":2101},[1951],[1838,10548,10550,10553],{"className":10549},[1881],[1838,10551,10165],{"className":10552,"style":7012},[1881,1882],[1838,10554,10556],{"className":10555},[2047],[1838,10557,10559,10595],{"className":10558},[2051,2052],[1838,10560,10562,10592],{"className":10561},[2056],[1838,10563,10565],{"className":10564,"style":7421},[2060],[1838,10566,10568,10571],{"style":10567},"top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;",[1838,10569],{"className":10570,"style":2069},[2068],[1838,10572,10574],{"className":10573},[2073,2074,2075,2076],[1838,10575,10577,10580,10583,10586,10589],{"className":10576},[1881,2076],[1838,10578,3901],{"className":10579},[1881,1882,2076],[1838,10581,3548],{"className":10582},[2022,2076],[1838,10584,3901],{"className":10585},[1881,1882,2076],[1838,10587,2459],{"className":10588},[2546,2076],[1838,10590,1922],{"className":10591},[1881,2076],[1838,10593,2084],{"className":10594},[2083],[1838,10596,10598],{"className":10597},[2056],[1838,10599,10601],{"className":10600,"style":7458},[2060],[1838,10602],{},[1838,10604],{"className":10605,"style":2018},[1951],[1838,10607,1909],{"className":10608},[2022],[1838,10610],{"className":10611,"style":2018},[1951],[1838,10613,10615,10619,10622,10671,10700,10703,10706],{"className":10614},[1872],[1838,10616],{"className":10617,"style":10618},[1876],"height:1.1074em;vertical-align:-0.2083em;",[1838,10620,9467],{"className":10621,"style":7012},[1881,1882],[1838,10623,10625,10628],{"className":10624},[1881],[1838,10626,10165],{"className":10627,"style":7012},[1881,1882],[1838,10629,10631],{"className":10630},[2047],[1838,10632,10634,10663],{"className":10633},[2051,2052],[1838,10635,10637,10660],{"className":10636},[2056],[1838,10638,10640],{"className":10639,"style":2061},[2060],[1838,10641,10642,10645],{"style":8808},[1838,10643],{"className":10644,"style":2069},[2068],[1838,10646,10648],{"className":10647},[2073,2074,2075,2076],[1838,10649,10651,10654,10657],{"className":10650},[1881,2076],[1838,10652,3901],{"className":10653},[1881,1882,2076],[1838,10655,2459],{"className":10656},[2546,2076],[1838,10658,1922],{"className":10659},[1881,2076],[1838,10661,2084],{"className":10662},[2083],[1838,10664,10666],{"className":10665},[2056],[1838,10667,10669],{"className":10668,"style":3966},[2060],[1838,10670],{},[1838,10672,10674,10677],{"className":10673},[1881],[1838,10675,9467],{"className":10676,"style":7012},[1881,1882],[1838,10678,10680],{"className":10679},[2047],[1838,10681,10683],{"className":10682},[2051],[1838,10684,10686],{"className":10685},[2056],[1838,10687,10689],{"className":10688,"style":2175},[2060],[1838,10690,10691,10694],{"style":2178},[1838,10692],{"className":10693,"style":2069},[2068],[1838,10695,10697],{"className":10696},[2073,2074,2075,2076],[1838,10698,1947],{"className":10699},[1881,2076],[1838,10701],{"className":10702,"style":1883},[1951],[1838,10704,2504],{"className":10705},[2546],[1838,10707],{"className":10708,"style":1883},[1951],[1838,10710,10712,10715,10718,10721,10750],{"className":10711},[1872],[1838,10713],{"className":10714,"style":2666},[1876],[1838,10716,9484],{"className":10717,"style":9675},[1881,1882],[1838,10719,7760],{"className":10720},[1881,1882],[1838,10722,10724,10727],{"className":10723},[1881],[1838,10725,9484],{"className":10726,"style":9675},[1881,1882],[1838,10728,10730],{"className":10729},[2047],[1838,10731,10733],{"className":10732},[2051],[1838,10734,10736],{"className":10735},[2056],[1838,10737,10739],{"className":10738,"style":2175},[2060],[1838,10740,10741,10744],{"style":2178},[1838,10742],{"className":10743,"style":2069},[2068],[1838,10745,10747],{"className":10746},[2073,2074,2075,2076],[1838,10748,1947],{"className":10749},[1881,2076],[1838,10751,1997],{"className":10752},[1881],[1798,10754,10755],{},"The observation innovation and its covariance are",[1838,10757,10759],{"className":10758},[1888],[1838,10760,10762,10850],{"className":10761},[1841],[1838,10763,10765],{"className":10764},[1845],[1847,10766,10767],{"xmlns":1849,"display":1897},[1851,10768,10769,10847],{},[1854,10770,10771,10777,10779,10785,10787,10789,10805,10807,10809,10815,10817,10819,10835,10841,10843,10845],{},[1915,10772,10773,10775],{},[1857,10774,8510],{"mathvariant":1904},[1857,10776,3901],{},[1907,10778,1909],{},[1915,10780,10781,10783],{},[1857,10782,1859],{},[1857,10784,3901],{},[1907,10786,2459],{},[1857,10788,9506],{},[1915,10790,10791,10793],{},[1857,10792,2319],{"mathvariant":1904},[1854,10794,10795,10797,10799,10801,10803],{},[1857,10796,3901],{},[1907,10798,3548],{},[1857,10800,3901],{},[1907,10802,2459],{},[1920,10804,1922],{},[1907,10806,1925],{"separator":1868},[1951,10808],{"width":1953},[1915,10810,10811,10813],{},[1857,10812,6535],{},[1857,10814,3901],{},[1907,10816,1909],{},[1857,10818,9506],{},[1915,10820,10821,10823],{},[1857,10822,10165],{},[1854,10824,10825,10827,10829,10831,10833],{},[1857,10826,3901],{},[1907,10828,3548],{},[1857,10830,3901],{},[1907,10832,2459],{},[1920,10834,1922],{},[1939,10836,10837,10839],{},[1857,10838,9506],{},[1857,10840,1947],{"mathvariant":1946},[1907,10842,2504],{},[1857,10844,9957],{},[1857,10846,1997],{"mathvariant":1946},[1861,10848,10849],{"encoding":1863},"\\mathbf v_t=Y_t-Z\\mathbf a_{t\\mid t-1},\\qquad\nF_t=ZP_{t\\mid t-1}Z^\\top+H.",[1838,10851,10853,10909,10964,11086,11189],{"className":10852,"ariaHidden":1868},[1867],[1838,10854,10856,10859,10900,10903,10906],{"className":10855},[1872],[1838,10857],{"className":10858,"style":6573},[1876],[1838,10860,10862,10865],{"className":10861},[1881],[1838,10863,8510],{"className":10864,"style":2246},[1881,2014],[1838,10866,10868],{"className":10867},[2047],[1838,10869,10871,10892],{"className":10870},[2051,2052],[1838,10872,10874,10889],{"className":10873},[2056],[1838,10875,10877],{"className":10876,"style":4125},[2060],[1838,10878,10880,10883],{"style":10879},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[1838,10881],{"className":10882,"style":2069},[2068],[1838,10884,10886],{"className":10885},[2073,2074,2075,2076],[1838,10887,3901],{"className":10888},[1881,1882,2076],[1838,10890,2084],{"className":10891},[2083],[1838,10893,10895],{"className":10894},[2056],[1838,10896,10898],{"className":10897,"style":2091},[2060],[1838,10899],{},[1838,10901],{"className":10902,"style":2018},[1951],[1838,10904,1909],{"className":10905},[2022],[1838,10907],{"className":10908,"style":2018},[1951],[1838,10910,10912,10915,10955,10958,10961],{"className":10911},[1872],[1838,10913],{"className":10914,"style":4754},[1876],[1838,10916,10918,10921],{"className":10917},[1881],[1838,10919,1859],{"className":10920,"style":1883},[1881,1882],[1838,10922,10924],{"className":10923},[2047],[1838,10925,10927,10947],{"className":10926},[2051,2052],[1838,10928,10930,10944],{"className":10929},[2056],[1838,10931,10933],{"className":10932,"style":4125},[2060],[1838,10934,10935,10938],{"style":9755},[1838,10936],{"className":10937,"style":2069},[2068],[1838,10939,10941],{"className":10940},[2073,2074,2075,2076],[1838,10942,3901],{"className":10943},[1881,1882,2076],[1838,10945,2084],{"className":10946},[2083],[1838,10948,10950],{"className":10949},[2056],[1838,10951,10953],{"className":10952,"style":2091},[2060],[1838,10954],{},[1838,10956],{"className":10957,"style":1883},[1951],[1838,10959,2459],{"className":10960},[2546],[1838,10962],{"className":10963,"style":1883},[1951],[1838,10965,10967,10970,10973,11028,11031,11034,11037,11077,11080,11083],{"className":10966},[1872],[1838,10968],{"className":10969,"style":10485},[1876],[1838,10971,9506],{"className":10972,"style":9794},[1881,1882],[1838,10974,10976,10979],{"className":10975},[1881],[1838,10977,2319],{"className":10978},[1881,2014],[1838,10980,10982],{"className":10981},[2047],[1838,10983,10985,11020],{"className":10984},[2051,2052],[1838,10986,10988,11017],{"className":10987},[2056],[1838,10989,10991],{"className":10990,"style":7421},[2060],[1838,10992,10993,10996],{"style":8114},[1838,10994],{"className":10995,"style":2069},[2068],[1838,10997,10999],{"className":10998},[2073,2074,2075,2076],[1838,11000,11002,11005,11008,11011,11014],{"className":11001},[1881,2076],[1838,11003,3901],{"className":11004},[1881,1882,2076],[1838,11006,3548],{"className":11007},[2022,2076],[1838,11009,3901],{"className":11010},[1881,1882,2076],[1838,11012,2459],{"className":11013},[2546,2076],[1838,11015,1922],{"className":11016},[1881,2076],[1838,11018,2084],{"className":11019},[2083],[1838,11021,11023],{"className":11022},[2056],[1838,11024,11026],{"className":11025,"style":7458},[2060],[1838,11027],{},[1838,11029,1925],{"className":11030},[2097],[1838,11032],{"className":11033,"style":2194},[1951],[1838,11035],{"className":11036,"style":2101},[1951],[1838,11038,11040,11043],{"className":11039},[1881],[1838,11041,6535],{"className":11042,"style":7012},[1881,1882],[1838,11044,11046],{"className":11045},[2047],[1838,11047,11049,11069],{"className":11048},[2051,2052],[1838,11050,11052,11066],{"className":11051},[2056],[1838,11053,11055],{"className":11054,"style":4125},[2060],[1838,11056,11057,11060],{"style":8808},[1838,11058],{"className":11059,"style":2069},[2068],[1838,11061,11063],{"className":11062},[2073,2074,2075,2076],[1838,11064,3901],{"className":11065},[1881,1882,2076],[1838,11067,2084],{"className":11068},[2083],[1838,11070,11072],{"className":11071},[2056],[1838,11073,11075],{"className":11074,"style":2091},[2060],[1838,11076],{},[1838,11078],{"className":11079,"style":2018},[1951],[1838,11081,1909],{"className":11082},[2022],[1838,11084],{"className":11085,"style":2018},[1951],[1838,11087,11089,11093,11096,11151,11180,11183,11186],{"className":11088},[1872],[1838,11090],{"className":11091,"style":11092},[1876],"height:1.2543em;vertical-align:-0.3552em;",[1838,11094,9506],{"className":11095,"style":9794},[1881,1882],[1838,11097,11099,11102],{"className":11098},[1881],[1838,11100,10165],{"className":11101,"style":7012},[1881,1882],[1838,11103,11105],{"className":11104},[2047],[1838,11106,11108,11143],{"className":11107},[2051,2052],[1838,11109,11111,11140],{"className":11110},[2056],[1838,11112,11114],{"className":11113,"style":7421},[2060],[1838,11115,11116,11119],{"style":10567},[1838,11117],{"className":11118,"style":2069},[2068],[1838,11120,11122],{"className":11121},[2073,2074,2075,2076],[1838,11123,11125,11128,11131,11134,11137],{"className":11124},[1881,2076],[1838,11126,3901],{"className":11127},[1881,1882,2076],[1838,11129,3548],{"className":11130},[2022,2076],[1838,11132,3901],{"className":11133},[1881,1882,2076],[1838,11135,2459],{"className":11136},[2546,2076],[1838,11138,1922],{"className":11139},[1881,2076],[1838,11141,2084],{"className":11142},[2083],[1838,11144,11146],{"className":11145},[2056],[1838,11147,11149],{"className":11148,"style":7458},[2060],[1838,11150],{},[1838,11152,11154,11157],{"className":11153},[1881],[1838,11155,9506],{"className":11156,"style":9794},[1881,1882],[1838,11158,11160],{"className":11159},[2047],[1838,11161,11163],{"className":11162},[2051],[1838,11164,11166],{"className":11165},[2056],[1838,11167,11169],{"className":11168,"style":2175},[2060],[1838,11170,11171,11174],{"style":2178},[1838,11172],{"className":11173,"style":2069},[2068],[1838,11175,11177],{"className":11176},[2073,2074,2075,2076],[1838,11178,1947],{"className":11179},[1881,2076],[1838,11181],{"className":11182,"style":1883},[1951],[1838,11184,2504],{"className":11185},[2546],[1838,11187],{"className":11188,"style":1883},[1951],[1838,11190,11192,11195,11198],{"className":11191},[1872],[1838,11193],{"className":11194,"style":1877},[1876],[1838,11196,9957],{"className":11197,"style":10127},[1881,1882],[1838,11199,1997],{"className":11200},[1881],[1798,11202,11203],{},"Correction is",[1838,11205,11207],{"className":11206},[1888],[1838,11208,11210,11267],{"className":11209},[1841],[1838,11211,11213],{"className":11212},[1845],[1847,11214,11215],{"xmlns":1849,"display":1897},[1851,11216,11217,11264],{},[1854,11218,11219,11226,11228,11244,11250,11262],{},[1915,11220,11221,11224],{},[1857,11222,11223],{},"K",[1857,11225,3901],{},[1907,11227,1909],{},[1915,11229,11230,11232],{},[1857,11231,10165],{},[1854,11233,11234,11236,11238,11240,11242],{},[1857,11235,3901],{},[1907,11237,3548],{},[1857,11239,3901],{},[1907,11241,2459],{},[1920,11243,1922],{},[1939,11245,11246,11248],{},[1857,11247,9506],{},[1857,11249,1947],{"mathvariant":1946},[4955,11251,11252,11254,11256],{},[1857,11253,6535],{},[1857,11255,3901],{},[1854,11257,11258,11260],{},[1907,11259,2459],{},[1920,11261,1922],{},[1907,11263,1925],{"separator":1868},[1861,11265,11266],{"encoding":1863},"K_t=P_{t\\mid t-1}Z^\\top F_t^{-1},",[1838,11268,11270,11326],{"className":11269,"ariaHidden":1868},[1867],[1838,11271,11273,11276,11317,11320,11323],{"className":11272},[1872],[1838,11274],{"className":11275,"style":4754},[1876],[1838,11277,11279,11282],{"className":11278},[1881],[1838,11280,11223],{"className":11281,"style":9794},[1881,1882],[1838,11283,11285],{"className":11284},[2047],[1838,11286,11288,11309],{"className":11287},[2051,2052],[1838,11289,11291,11306],{"className":11290},[2056],[1838,11292,11294],{"className":11293,"style":4125},[2060],[1838,11295,11297,11300],{"style":11296},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[1838,11298],{"className":11299,"style":2069},[2068],[1838,11301,11303],{"className":11302},[2073,2074,2075,2076],[1838,11304,3901],{"className":11305},[1881,1882,2076],[1838,11307,2084],{"className":11308},[2083],[1838,11310,11312],{"className":11311},[2056],[1838,11313,11315],{"className":11314,"style":2091},[2060],[1838,11316],{},[1838,11318],{"className":11319,"style":2018},[1951],[1838,11321,1909],{"className":11322},[2022],[1838,11324],{"className":11325,"style":2018},[1951],[1838,11327,11329,11332,11387,11416,11474],{"className":11328},[1872],[1838,11330],{"className":11331,"style":11092},[1876],[1838,11333,11335,11338],{"className":11334},[1881],[1838,11336,10165],{"className":11337,"style":7012},[1881,1882],[1838,11339,11341],{"className":11340},[2047],[1838,11342,11344,11379],{"className":11343},[2051,2052],[1838,11345,11347,11376],{"className":11346},[2056],[1838,11348,11350],{"className":11349,"style":7421},[2060],[1838,11351,11352,11355],{"style":10567},[1838,11353],{"className":11354,"style":2069},[2068],[1838,11356,11358],{"className":11357},[2073,2074,2075,2076],[1838,11359,11361,11364,11367,11370,11373],{"className":11360},[1881,2076],[1838,11362,3901],{"className":11363},[1881,1882,2076],[1838,11365,3548],{"className":11366},[2022,2076],[1838,11368,3901],{"className":11369},[1881,1882,2076],[1838,11371,2459],{"className":11372},[2546,2076],[1838,11374,1922],{"className":11375},[1881,2076],[1838,11377,2084],{"className":11378},[2083],[1838,11380,11382],{"className":11381},[2056],[1838,11383,11385],{"className":11384,"style":7458},[2060],[1838,11386],{},[1838,11388,11390,11393],{"className":11389},[1881],[1838,11391,9506],{"className":11392,"style":9794},[1881,1882],[1838,11394,11396],{"className":11395},[2047],[1838,11397,11399],{"className":11398},[2051],[1838,11400,11402],{"className":11401},[2056],[1838,11403,11405],{"className":11404,"style":2175},[2060],[1838,11406,11407,11410],{"style":2178},[1838,11408],{"className":11409,"style":2069},[2068],[1838,11411,11413],{"className":11412},[2073,2074,2075,2076],[1838,11414,1947],{"className":11415},[1881,2076],[1838,11417,11419,11422],{"className":11418},[1881],[1838,11420,6535],{"className":11421,"style":7012},[1881,1882],[1838,11423,11425],{"className":11424},[2047],[1838,11426,11428,11466],{"className":11427},[2051,2052],[1838,11429,11431,11463],{"className":11430},[2056],[1838,11432,11434,11446],{"className":11433,"style":2606},[2060],[1838,11435,11437,11440],{"style":11436},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[1838,11438],{"className":11439,"style":2069},[2068],[1838,11441,11443],{"className":11442},[2073,2074,2075,2076],[1838,11444,3901],{"className":11445},[1881,1882,2076],[1838,11447,11448,11451],{"style":2178},[1838,11449],{"className":11450,"style":2069},[2068],[1838,11452,11454],{"className":11453},[2073,2074,2075,2076],[1838,11455,11457,11460],{"className":11456},[1881,2076],[1838,11458,2459],{"className":11459},[1881,2076],[1838,11461,1922],{"className":11462},[1881,2076],[1838,11464,2084],{"className":11465},[2083],[1838,11467,11469],{"className":11468},[2056],[1838,11470,11472],{"className":11471,"style":5221},[2060],[1838,11473],{},[1838,11475,1925],{"className":11476},[2097],[1838,11478,11480],{"className":11479},[1888],[1838,11481,11483,11585],{"className":11482},[1841],[1838,11484,11486],{"className":11485},[1845],[1847,11487,11488],{"xmlns":1849,"display":1897},[1851,11489,11490,11582],{},[1854,11491,11492,11498,11500,11516,11518,11524,11530,11532,11534,11540,11542,11558,11560,11566,11572,11580],{},[1915,11493,11494,11496],{},[1857,11495,2319],{"mathvariant":1904},[1857,11497,3901],{},[1907,11499,1909],{},[1915,11501,11502,11504],{},[1857,11503,2319],{"mathvariant":1904},[1854,11505,11506,11508,11510,11512,11514],{},[1857,11507,3901],{},[1907,11509,3548],{},[1857,11511,3901],{},[1907,11513,2459],{},[1920,11515,1922],{},[1907,11517,2504],{},[1915,11519,11520,11522],{},[1857,11521,11223],{},[1857,11523,3901],{},[1915,11525,11526,11528],{},[1857,11527,8510],{"mathvariant":1904},[1857,11529,3901],{},[1907,11531,1925],{"separator":1868},[1951,11533],{"width":1953},[1915,11535,11536,11538],{},[1857,11537,10165],{},[1857,11539,3901],{},[1907,11541,1909],{},[1915,11543,11544,11546],{},[1857,11545,10165],{},[1854,11547,11548,11550,11552,11554,11556],{},[1857,11549,3901],{},[1907,11551,3548],{},[1857,11553,3901],{},[1907,11555,2459],{},[1920,11557,1922],{},[1907,11559,2459],{},[1915,11561,11562,11564],{},[1857,11563,11223],{},[1857,11565,3901],{},[1915,11567,11568,11570],{},[1857,11569,6535],{},[1857,11571,3901],{},[4955,11573,11574,11576,11578],{},[1857,11575,11223],{},[1857,11577,3901],{},[1857,11579,1947],{"mathvariant":1946},[1857,11581,1997],{"mathvariant":1946},[1861,11583,11584],{"encoding":1863},"\\mathbf a_t=\\mathbf a_{t\\mid t-1}+K_t\\mathbf v_t,\\qquad\nP_t=P_{t\\mid t-1}-K_tF_tK_t^\\top.",[1838,11586,11588,11643,11714,11858,11928],{"className":11587,"ariaHidden":1868},[1867],[1838,11589,11591,11594,11634,11637,11640],{"className":11590},[1872],[1838,11592],{"className":11593,"style":6573},[1876],[1838,11595,11597,11600],{"className":11596},[1881],[1838,11598,2319],{"className":11599},[1881,2014],[1838,11601,11603],{"className":11602},[2047],[1838,11604,11606,11626],{"className":11605},[2051,2052],[1838,11607,11609,11623],{"className":11608},[2056],[1838,11610,11612],{"className":11611,"style":4125},[2060],[1838,11613,11614,11617],{"style":6310},[1838,11615],{"className":11616,"style":2069},[2068],[1838,11618,11620],{"className":11619},[2073,2074,2075,2076],[1838,11621,3901],{"className":11622},[1881,1882,2076],[1838,11624,2084],{"className":11625},[2083],[1838,11627,11629],{"className":11628},[2056],[1838,11630,11632],{"className":11631,"style":2091},[2060],[1838,11633],{},[1838,11635],{"className":11636,"style":2018},[1951],[1838,11638,1909],{"className":11639},[2022],[1838,11641],{"className":11642,"style":2018},[1951],[1838,11644,11646,11650,11705,11708,11711],{"className":11645},[1872],[1838,11647],{"className":11648,"style":11649},[1876],"height:0.9385em;vertical-align:-0.3552em;",[1838,11651,11653,11656],{"className":11652},[1881],[1838,11654,2319],{"className":11655},[1881,2014],[1838,11657,11659],{"className":11658},[2047],[1838,11660,11662,11697],{"className":11661},[2051,2052],[1838,11663,11665,11694],{"className":11664},[2056],[1838,11666,11668],{"className":11667,"style":7421},[2060],[1838,11669,11670,11673],{"style":8114},[1838,11671],{"className":11672,"style":2069},[2068],[1838,11674,11676],{"className":11675},[2073,2074,2075,2076],[1838,11677,11679,11682,11685,11688,11691],{"className":11678},[1881,2076],[1838,11680,3901],{"className":11681},[1881,1882,2076],[1838,11683,3548],{"className":11684},[2022,2076],[1838,11686,3901],{"className":11687},[1881,1882,2076],[1838,11689,2459],{"className":11690},[2546,2076],[1838,11692,1922],{"className":11693},[1881,2076],[1838,11695,2084],{"className":11696},[2083],[1838,11698,11700],{"className":11699},[2056],[1838,11701,11703],{"className":11702,"style":7458},[2060],[1838,11704],{},[1838,11706],{"className":11707,"style":1883},[1951],[1838,11709,2504],{"className":11710},[2546],[1838,11712],{"className":11713,"style":1883},[1951],[1838,11715,11717,11720,11760,11800,11803,11806,11809,11849,11852,11855],{"className":11716},[1872],[1838,11718],{"className":11719,"style":7789},[1876],[1838,11721,11723,11726],{"className":11722},[1881],[1838,11724,11223],{"className":11725,"style":9794},[1881,1882],[1838,11727,11729],{"className":11728},[2047],[1838,11730,11732,11752],{"className":11731},[2051,2052],[1838,11733,11735,11749],{"className":11734},[2056],[1838,11736,11738],{"className":11737,"style":4125},[2060],[1838,11739,11740,11743],{"style":11296},[1838,11741],{"className":11742,"style":2069},[2068],[1838,11744,11746],{"className":11745},[2073,2074,2075,2076],[1838,11747,3901],{"className":11748},[1881,1882,2076],[1838,11750,2084],{"className":11751},[2083],[1838,11753,11755],{"className":11754},[2056],[1838,11756,11758],{"className":11757,"style":2091},[2060],[1838,11759],{},[1838,11761,11763,11766],{"className":11762},[1881],[1838,11764,8510],{"className":11765,"style":2246},[1881,2014],[1838,11767,11769],{"className":11768},[2047],[1838,11770,11772,11792],{"className":11771},[2051,2052],[1838,11773,11775,11789],{"className":11774},[2056],[1838,11776,11778],{"className":11777,"style":4125},[2060],[1838,11779,11780,11783],{"style":10879},[1838,11781],{"className":11782,"style":2069},[2068],[1838,11784,11786],{"className":11785},[2073,2074,2075,2076],[1838,11787,3901],{"className":11788},[1881,1882,2076],[1838,11790,2084],{"className":11791},[2083],[1838,11793,11795],{"className":11794},[2056],[1838,11796,11798],{"className":11797,"style":2091},[2060],[1838,11799],{},[1838,11801,1925],{"className":11802},[2097],[1838,11804],{"className":11805,"style":2194},[1951],[1838,11807],{"className":11808,"style":2101},[1951],[1838,11810,11812,11815],{"className":11811},[1881],[1838,11813,10165],{"className":11814,"style":7012},[1881,1882],[1838,11816,11818],{"className":11817},[2047],[1838,11819,11821,11841],{"className":11820},[2051,2052],[1838,11822,11824,11838],{"className":11823},[2056],[1838,11825,11827],{"className":11826,"style":4125},[2060],[1838,11828,11829,11832],{"style":8808},[1838,11830],{"className":11831,"style":2069},[2068],[1838,11833,11835],{"className":11834},[2073,2074,2075,2076],[1838,11836,3901],{"className":11837},[1881,1882,2076],[1838,11839,2084],{"className":11840},[2083],[1838,11842,11844],{"className":11843},[2056],[1838,11845,11847],{"className":11846,"style":2091},[2060],[1838,11848],{},[1838,11850],{"className":11851,"style":2018},[1951],[1838,11853,1909],{"className":11854},[2022],[1838,11856],{"className":11857,"style":2018},[1951],[1838,11859,11861,11864,11919,11922,11925],{"className":11860},[1872],[1838,11862],{"className":11863,"style":10485},[1876],[1838,11865,11867,11870],{"className":11866},[1881],[1838,11868,10165],{"className":11869,"style":7012},[1881,1882],[1838,11871,11873],{"className":11872},[2047],[1838,11874,11876,11911],{"className":11875},[2051,2052],[1838,11877,11879,11908],{"className":11878},[2056],[1838,11880,11882],{"className":11881,"style":7421},[2060],[1838,11883,11884,11887],{"style":10567},[1838,11885],{"className":11886,"style":2069},[2068],[1838,11888,11890],{"className":11889},[2073,2074,2075,2076],[1838,11891,11893,11896,11899,11902,11905],{"className":11892},[1881,2076],[1838,11894,3901],{"className":11895},[1881,1882,2076],[1838,11897,3548],{"className":11898},[2022,2076],[1838,11900,3901],{"className":11901},[1881,1882,2076],[1838,11903,2459],{"className":11904},[2546,2076],[1838,11906,1922],{"className":11907},[1881,2076],[1838,11909,2084],{"className":11910},[2083],[1838,11912,11914],{"className":11913},[2056],[1838,11915,11917],{"className":11916,"style":7458},[2060],[1838,11918],{},[1838,11920],{"className":11921,"style":1883},[1951],[1838,11923,2459],{"className":11924},[2546],[1838,11926],{"className":11927,"style":1883},[1951],[1838,11929,11931,11934,11974,12014,12066],{"className":11930},[1872],[1838,11932],{"className":11933,"style":7476},[1876],[1838,11935,11937,11940],{"className":11936},[1881],[1838,11938,11223],{"className":11939,"style":9794},[1881,1882],[1838,11941,11943],{"className":11942},[2047],[1838,11944,11946,11966],{"className":11945},[2051,2052],[1838,11947,11949,11963],{"className":11948},[2056],[1838,11950,11952],{"className":11951,"style":4125},[2060],[1838,11953,11954,11957],{"style":11296},[1838,11955],{"className":11956,"style":2069},[2068],[1838,11958,11960],{"className":11959},[2073,2074,2075,2076],[1838,11961,3901],{"className":11962},[1881,1882,2076],[1838,11964,2084],{"className":11965},[2083],[1838,11967,11969],{"className":11968},[2056],[1838,11970,11972],{"className":11971,"style":2091},[2060],[1838,11973],{},[1838,11975,11977,11980],{"className":11976},[1881],[1838,11978,6535],{"className":11979,"style":7012},[1881,1882],[1838,11981,11983],{"className":11982},[2047],[1838,11984,11986,12006],{"className":11985},[2051,2052],[1838,11987,11989,12003],{"className":11988},[2056],[1838,11990,11992],{"className":11991,"style":4125},[2060],[1838,11993,11994,11997],{"style":8808},[1838,11995],{"className":11996,"style":2069},[2068],[1838,11998,12000],{"className":11999},[2073,2074,2075,2076],[1838,12001,3901],{"className":12002},[1881,1882,2076],[1838,12004,2084],{"className":12005},[2083],[1838,12007,12009],{"className":12008},[2056],[1838,12010,12012],{"className":12011,"style":2091},[2060],[1838,12013],{},[1838,12015,12017,12020],{"className":12016},[1881],[1838,12018,11223],{"className":12019,"style":9794},[1881,1882],[1838,12021,12023],{"className":12022},[2047],[1838,12024,12026,12058],{"className":12025},[2051,2052],[1838,12027,12029,12055],{"className":12028},[2056],[1838,12030,12032,12044],{"className":12031,"style":2175},[2060],[1838,12033,12035,12038],{"style":12034},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[1838,12036],{"className":12037,"style":2069},[2068],[1838,12039,12041],{"className":12040},[2073,2074,2075,2076],[1838,12042,3901],{"className":12043},[1881,1882,2076],[1838,12045,12046,12049],{"style":2178},[1838,12047],{"className":12048,"style":2069},[2068],[1838,12050,12052],{"className":12051},[2073,2074,2075,2076],[1838,12053,1947],{"className":12054},[1881,2076],[1838,12056,2084],{"className":12057},[2083],[1838,12059,12061],{"className":12060},[2056],[1838,12062,12064],{"className":12063,"style":5221},[2060],[1838,12065],{},[1838,12067,1997],{"className":12068},[1881],[12070,12071,12072,12085],"table",{},[12073,12074,12075],"thead",{},[12076,12077,12078,12082],"tr",{},[12079,12080,12081],"th",{},"Task",[12079,12083,12084],{},"Information used",[12086,12087,12088,12149,12185],"tbody",{},[12076,12089,12090,12094],{},[12091,12092,12093],"td",{},"prediction",[12091,12095,12096,12097],{},"observations through ",[1838,12098,12100,12118],{"className":12099},[1841],[1838,12101,12103],{"className":12102},[1845],[1847,12104,12105],{"xmlns":1849},[1851,12106,12107,12115],{},[1854,12108,12109,12111,12113],{},[1857,12110,3901],{},[1907,12112,2459],{},[1920,12114,1922],{},[1861,12116,12117],{"encoding":1863},"t-1",[1838,12119,12121,12140],{"className":12120,"ariaHidden":1868},[1867],[1838,12122,12124,12128,12131,12134,12137],{"className":12123},[1872],[1838,12125],{"className":12126,"style":12127},[1876],"height:0.6984em;vertical-align:-0.0833em;",[1838,12129,3901],{"className":12130},[1881,1882],[1838,12132],{"className":12133,"style":1883},[1951],[1838,12135,2459],{"className":12136},[2546],[1838,12138],{"className":12139,"style":1883},[1951],[1838,12141,12143,12146],{"className":12142},[1872],[1838,12144],{"className":12145,"style":3398},[1876],[1838,12147,1922],{"className":12148},[1881],[12076,12150,12151,12154],{},[12091,12152,12153],{},"filtering",[12091,12155,12096,12156],{},[1838,12157,12159,12172],{"className":12158},[1841],[1838,12160,12162],{"className":12161},[1845],[1847,12163,12164],{"xmlns":1849},[1851,12165,12166,12170],{},[1854,12167,12168],{},[1857,12169,3901],{},[1861,12171,3901],{"encoding":1863},[1838,12173,12175],{"className":12174,"ariaHidden":1868},[1867],[1838,12176,12178,12182],{"className":12177},[1872],[1838,12179],{"className":12180,"style":12181},[1876],"height:0.6151em;",[1838,12183,3901],{"className":12184},[1881,1882],[12076,12186,12187,12190],{},[12091,12188,12189],{},"smoothing",[12091,12191,12192,12193],{},"the complete sample, including observations after ",[1838,12194,12196,12209],{"className":12195},[1841],[1838,12197,12199],{"className":12198},[1845],[1847,12200,12201],{"xmlns":1849},[1851,12202,12203,12207],{},[1854,12204,12205],{},[1857,12206,3901],{},[1861,12208,3901],{"encoding":1863},[1838,12210,12212],{"className":12211,"ariaHidden":1868},[1867],[1838,12213,12215,12218],{"className":12214},[1872],[1838,12216],{"className":12217,"style":12181},[1876],[1838,12219,3901],{"className":12220},[1881,1882],[1798,12222,2887,12223,12293],{},[1838,12224,12226,12244],{"className":12225},[1841],[1838,12227,12229],{"className":12228},[1845],[1847,12230,12231],{"xmlns":1849},[1851,12232,12233,12241],{},[1854,12234,12235],{},[1915,12236,12237,12239],{},[1857,12238,1859],{},[1857,12240,3901],{},[1861,12242,12243],{"encoding":1863},"Y_t",[1838,12245,12247],{"className":12246,"ariaHidden":1868},[1867],[1838,12248,12250,12253],{"className":12249},[1872],[1838,12251],{"className":12252,"style":4754},[1876],[1838,12254,12256,12259],{"className":12255},[1881],[1838,12257,1859],{"className":12258,"style":1883},[1881,1882],[1838,12260,12262],{"className":12261},[2047],[1838,12263,12265,12285],{"className":12264},[2051,2052],[1838,12266,12268,12282],{"className":12267},[2056],[1838,12269,12271],{"className":12270,"style":4125},[2060],[1838,12272,12273,12276],{"style":9755},[1838,12274],{"className":12275,"style":2069},[2068],[1838,12277,12279],{"className":12278},[2073,2074,2075,2076],[1838,12280,3901],{"className":12281},[1881,1882,2076],[1838,12283,2084],{"className":12284},[2083],[1838,12286,12288],{"className":12287},[2056],[1838,12289,12291],{"className":12290,"style":2091},[2060],[1838,12292],{}," is missing, keep the prediction and skip correction. No ad hoc imputation is required.",[1793,12295,12297],{"id":12296},"practice","Practice",[12299,12300,12301,12513,12516,12606,12609],"ol",{},[1812,12302,12303,12304,7745,12357,12424,12425,7745,12453,12483,12484,1997],{},"For AR(1) with ",[1838,12305,12307,12326],{"className":12306},[1841],[1838,12308,12310],{"className":12309},[1845],[1847,12311,12312],{"xmlns":1849},[1851,12313,12314,12323],{},[1854,12315,12316,12318,12320],{},[1857,12317,3792],{},[1907,12319,1909],{},[1920,12321,12322],{},"0.6",[1861,12324,12325],{"encoding":1863},"\\phi=0.6",[1838,12327,12329,12348],{"className":12328,"ariaHidden":1868},[1867],[1838,12330,12332,12336,12339,12342,12345],{"className":12331},[1872],[1838,12333],{"className":12334,"style":12335},[1876],"height:0.8889em;vertical-align:-0.1944em;",[1838,12337,3792],{"className":12338},[1881,1882],[1838,12340],{"className":12341,"style":2018},[1951],[1838,12343,1909],{"className":12344},[2022],[1838,12346],{"className":12347,"style":2018},[1951],[1838,12349,12351,12354],{"className":12350},[1872],[1838,12352],{"className":12353,"style":3398},[1876],[1838,12355,12322],{"className":12356},[1881],[1838,12358,12360,12385],{"className":12359},[1841],[1838,12361,12363],{"className":12362},[1845],[1847,12364,12365],{"xmlns":1849},[1851,12366,12367,12382],{},[1854,12368,12369,12371,12373,12375,12377,12379],{},[1857,12370,3731],{},[1907,12372,1913],{"stretchy":1912},[1920,12374,2780],{},[1907,12376,1943],{"stretchy":1912},[1907,12378,1909],{},[1920,12380,12381],{},"4",[1861,12383,12384],{"encoding":1863},"\\gamma(0)=4",[1838,12386,12388,12415],{"className":12387,"ariaHidden":1868},[1867],[1838,12389,12391,12394,12397,12400,12403,12406,12409,12412],{"className":12390},[1872],[1838,12392],{"className":12393,"style":2216},[1876],[1838,12395,3731],{"className":12396,"style":3753},[1881,1882],[1838,12398,1913],{"className":12399},[2036],[1838,12401,2780],{"className":12402},[1881],[1838,12404,1943],{"className":12405},[2159],[1838,12407],{"className":12408,"style":2018},[1951],[1838,12410,1909],{"className":12411},[2022],[1838,12413],{"className":12414,"style":2018},[1951],[1838,12416,12418,12421],{"className":12417},[1872],[1838,12419],{"className":12420,"style":3398},[1876],[1838,12422,12381],{"className":12423},[1881],", construct ",[1838,12426,12428,12441],{"className":12427},[1841],[1838,12429,12431],{"className":12430},[1845],[1847,12432,12433],{"xmlns":1849},[1851,12434,12435,12439],{},[1854,12436,12437],{},[1857,12438,1956],{"mathvariant":1946},[1861,12440,2904],{"encoding":1863},[1838,12442,12444],{"className":12443,"ariaHidden":1868},[1867],[1838,12445,12447,12450],{"className":12446},[1872],[1838,12448],{"className":12449,"style":1877},[1876],[1838,12451,1956],{"className":12452},[1881],[1838,12454,12456,12470],{"className":12455},[1841],[1838,12457,12459],{"className":12458},[1845],[1847,12460,12461],{"xmlns":1849},[1851,12462,12463,12467],{},[1854,12464,12465],{},[1857,12466,1977],{"mathvariant":1904},[1861,12468,12469],{"encoding":1863},"\\mathbf g",[1838,12471,12473],{"className":12472,"ariaHidden":1868},[1867],[1838,12474,12476,12480],{"className":12475},[1872],[1838,12477],{"className":12478,"style":12479},[1876],"height:0.6389em;vertical-align:-0.1944em;",[1838,12481,1977],{"className":12482,"style":2246},[1881,2014]," above and solve for ",[1838,12485,12487,12501],{"className":12486},[1841],[1838,12488,12490],{"className":12489},[1845],[1847,12491,12492],{"xmlns":1849},[1851,12493,12494,12498],{},[1854,12495,12496],{},[1857,12497,2319],{"mathvariant":1904},[1861,12499,12500],{"encoding":1863},"\\mathbf a",[1838,12502,12504],{"className":12503,"ariaHidden":1868},[1867],[1838,12505,12507,12510],{"className":12506},[1872],[1838,12508],{"className":12509,"style":3004},[1876],[1838,12511,2319],{"className":12512},[1881,2014],[1812,12514,12515],{},"Show that adding a predictor cannot increase the minimum MSPE.",[1812,12517,12518,12519,12547,12548,12576,12577,12605],{},"For an AR(2), write ",[1838,12520,12522,12535],{"className":12521},[1841],[1838,12523,12525],{"className":12524},[1845],[1847,12526,12527],{"xmlns":1849},[1851,12528,12529,12533],{},[1854,12530,12531],{},[1857,12532,6535],{},[1861,12534,6535],{"encoding":1863},[1838,12536,12538],{"className":12537,"ariaHidden":1868},[1867],[1838,12539,12541,12544],{"className":12540},[1872],[1838,12542],{"className":12543,"style":1877},[1876],[1838,12545,6535],{"className":12546,"style":7012},[1881,1882],", ",[1838,12549,12551,12564],{"className":12550},[1841],[1838,12552,12554],{"className":12553},[1845],[1847,12555,12556],{"xmlns":1849},[1851,12557,12558,12562],{},[1854,12559,12560],{},[1857,12561,6552],{},[1861,12563,6552],{"encoding":1863},[1838,12565,12567],{"className":12566,"ariaHidden":1868},[1867],[1838,12568,12570,12573],{"className":12569},[1872],[1838,12571],{"className":12572,"style":1877},[1876],[1838,12574,6552],{"className":12575},[1881,1882],", and ",[1838,12578,12580,12593],{"className":12579},[1841],[1838,12581,12583],{"className":12582},[1845],[1847,12584,12585],{"xmlns":1849},[1851,12586,12587,12591],{},[1854,12588,12589],{},[1857,12590,7760],{},[1861,12592,7760],{"encoding":1863},[1838,12594,12596],{"className":12595,"ariaHidden":1868},[1867],[1838,12597,12599,12602],{"className":12598},[1872],[1838,12600],{"className":12601,"style":7789},[1876],[1838,12603,7760],{"className":12604},[1881,1882]," explicitly.",[1812,12607,12608],{},"Explain why the innovations likelihood and joint covariance likelihood agree for a Gaussian model.",[1812,12610,12611,12612,12707],{},"In a local-level model, what happens to ",[1838,12613,12615,12643],{"className":12614},[1841],[1838,12616,12618],{"className":12617},[1845],[1847,12619,12620],{"xmlns":1849},[1851,12621,12622,12640],{},[1854,12623,12624],{},[1915,12625,12626,12628],{},[1857,12627,10165],{},[1854,12629,12630,12632,12634,12636,12638],{},[1857,12631,3901],{},[1907,12633,3548],{},[1857,12635,3901],{},[1907,12637,2459],{},[1920,12639,1922],{},[1861,12641,12642],{"encoding":1863},"P_{t\\mid t-1}",[1838,12644,12646],{"className":12645,"ariaHidden":1868},[1867],[1838,12647,12649,12652],{"className":12648},[1872],[1838,12650],{"className":12651,"style":10485},[1876],[1838,12653,12655,12658],{"className":12654},[1881],[1838,12656,10165],{"className":12657,"style":7012},[1881,1882],[1838,12659,12661],{"className":12660},[2047],[1838,12662,12664,12699],{"className":12663},[2051,2052],[1838,12665,12667,12696],{"className":12666},[2056],[1838,12668,12670],{"className":12669,"style":7421},[2060],[1838,12671,12672,12675],{"style":10567},[1838,12673],{"className":12674,"style":2069},[2068],[1838,12676,12678],{"className":12677},[2073,2074,2075,2076],[1838,12679,12681,12684,12687,12690,12693],{"className":12680},[1881,2076],[1838,12682,3901],{"className":12683},[1881,1882,2076],[1838,12685,3548],{"className":12686},[2022,2076],[1838,12688,3901],{"className":12689},[1881,1882,2076],[1838,12691,2459],{"className":12692},[2546,2076],[1838,12694,1922],{"className":12695},[1881,2076],[1838,12697,2084],{"className":12698},[2083],[1838,12700,12702],{"className":12701},[2056],[1838,12703,12705],{"className":12704,"style":7458},[2060],[1838,12706],{}," during several consecutive missing observations?",[12709,12710,12712],"legacy-details",{"title":12711},"Answers",[12299,12713,12714,13138,13141,13583,13586],{},[1812,12715,12716,12923,12924,12576,13034,1997],{},[1838,12717,12719,12773],{"className":12718},[1841],[1838,12720,12722],{"className":12721},[1845],[1847,12723,12724],{"xmlns":1849},[1851,12725,12726,12770],{},[1854,12727,12728,12730,12732,12734],{},[1857,12729,1956],{"mathvariant":1946},[1907,12731,1909],{},[1920,12733,12381],{},[1854,12735,12736,12738,12768],{},[1907,12737,2452],{"fence":1868},[4215,12739,12740,12754],{"rowspacing":4217,"columnalign":4218,"columnspacing":4219},[4221,12741,12742,12748],{},[4224,12743,12744],{},[2778,12745,12746],{"scriptlevel":2780,"displaystyle":1912},[1920,12747,1922],{},[4224,12749,12750],{},[2778,12751,12752],{"scriptlevel":2780,"displaystyle":1912},[1920,12753,12322],{},[4221,12755,12756,12762],{},[4224,12757,12758],{},[2778,12759,12760],{"scriptlevel":2780,"displaystyle":1912},[1920,12761,12322],{},[4224,12763,12764],{},[2778,12765,12766],{"scriptlevel":2780,"displaystyle":1912},[1920,12767,1922],{},[1907,12769,2477],{"fence":1868},[1861,12771,12772],{"encoding":1863},"\\Gamma=4\\begin{bmatrix}1&0.6\\\\0.6&1\\end{bmatrix}",[1838,12774,12776,12794],{"className":12775,"ariaHidden":1868},[1867],[1838,12777,12779,12782,12785,12788,12791],{"className":12778},[1872],[1838,12780],{"className":12781,"style":1877},[1876],[1838,12783,1956],{"className":12784},[1881],[1838,12786],{"className":12787,"style":2018},[1951],[1838,12789,1909],{"className":12790},[2022],[1838,12792],{"className":12793,"style":2018},[1951],[1838,12795,12797,12800,12803,12806],{"className":12796},[1872],[1838,12798],{"className":12799,"style":4329},[1876],[1838,12801,12381],{"className":12802},[1881],[1838,12804],{"className":12805,"style":2101},[1951],[1838,12807,12809,12815,12917],{"className":12808},[2105],[1838,12810,12812],{"className":12811,"style":4352},[2036,4351],[1838,12813,2452],{"className":12814},[4356,2075],[1838,12816,12818],{"className":12817},[1881],[1838,12819,12821,12866,12869,12872],{"className":12820},[4215],[1838,12822,12824],{"className":12823},[4366],[1838,12825,12827,12858],{"className":12826},[2051,2052],[1838,12828,12830,12855],{"className":12829},[2056],[1838,12831,12833,12844],{"className":12832,"style":4376},[2060],[1838,12834,12835,12838],{"style":4379},[1838,12836],{"className":12837,"style":2356},[2068],[1838,12839,12841],{"className":12840},[1881],[1838,12842,1922],{"className":12843},[1881],[1838,12845,12846,12849],{"style":4391},[1838,12847],{"className":12848,"style":2356},[2068],[1838,12850,12852],{"className":12851},[1881],[1838,12853,12322],{"className":12854},[1881],[1838,12856,2084],{"className":12857},[2083],[1838,12859,12861],{"className":12860},[2056],[1838,12862,12864],{"className":12863,"style":4410},[2060],[1838,12865],{},[1838,12867],{"className":12868,"style":4417},[4416],[1838,12870],{"className":12871,"style":4417},[4416],[1838,12873,12875],{"className":12874},[4366],[1838,12876,12878,12909],{"className":12877},[2051,2052],[1838,12879,12881,12906],{"className":12880},[2056],[1838,12882,12884,12895],{"className":12883,"style":4376},[2060],[1838,12885,12886,12889],{"style":4379},[1838,12887],{"className":12888,"style":2356},[2068],[1838,12890,12892],{"className":12891},[1881],[1838,12893,12322],{"className":12894},[1881],[1838,12896,12897,12900],{"style":4391},[1838,12898],{"className":12899,"style":2356},[2068],[1838,12901,12903],{"className":12902},[1881],[1838,12904,1922],{"className":12905},[1881],[1838,12907,2084],{"className":12908},[2083],[1838,12910,12912],{"className":12911},[2056],[1838,12913,12915],{"className":12914,"style":4410},[2060],[1838,12916],{},[1838,12918,12920],{"className":12919,"style":4352},[2159,4351],[1838,12921,2477],{"className":12922},[4356,2075],",\n",[1838,12925,12927,12960],{"className":12926},[1841],[1838,12928,12930],{"className":12929},[1845],[1847,12931,12932],{"xmlns":1849},[1851,12933,12934,12957],{},[1854,12935,12936,12938,12940,12942,12944,12947,12949,12951],{},[1857,12937,1977],{"mathvariant":1904},[1907,12939,1909],{},[1920,12941,12381],{},[1907,12943,1913],{"stretchy":1912},[1920,12945,12946],{},"0.36",[1907,12948,1925],{"separator":1868},[1920,12950,12322],{},[1939,12952,12953,12955],{},[1907,12954,1943],{"stretchy":1912},[1857,12956,1947],{"mathvariant":1946},[1861,12958,12959],{"encoding":1863},"\\mathbf g=4(0.36,0.6)^\\top",[1838,12961,12963,12981],{"className":12962,"ariaHidden":1868},[1867],[1838,12964,12966,12969,12972,12975,12978],{"className":12965},[1872],[1838,12967],{"className":12968,"style":12479},[1876],[1838,12970,1977],{"className":12971,"style":2246},[1881,2014],[1838,12973],{"className":12974,"style":2018},[1951],[1838,12976,1909],{"className":12977},[2022],[1838,12979],{"className":12980,"style":2018},[1951],[1838,12982,12984,12987,12990,12993,12996,12999,13002,13005],{"className":12983},[1872],[1838,12985],{"className":12986,"style":3453},[1876],[1838,12988,12381],{"className":12989},[1881],[1838,12991,1913],{"className":12992},[2036],[1838,12994,12946],{"className":12995},[1881],[1838,12997,1925],{"className":12998},[2097],[1838,13000],{"className":13001,"style":2101},[1951],[1838,13003,12322],{"className":13004},[1881],[1838,13006,13008,13011],{"className":13007},[2159],[1838,13009,1943],{"className":13010},[2159],[1838,13012,13014],{"className":13013},[2047],[1838,13015,13017],{"className":13016},[2051],[1838,13018,13020],{"className":13019},[2056],[1838,13021,13023],{"className":13022,"style":2398},[2060],[1838,13024,13025,13028],{"style":2419},[1838,13026],{"className":13027,"style":2069},[2068],[1838,13029,13031],{"className":13030},[2073,2074,2075,2076],[1838,13032,1947],{"className":13033},[1881,2076],[1838,13035,13037,13067],{"className":13036},[1841],[1838,13038,13040],{"className":13039},[1845],[1847,13041,13042],{"xmlns":1849},[1851,13043,13044,13064],{},[1854,13045,13046,13048,13050,13052,13054,13056,13058],{},[1857,13047,2319],{"mathvariant":1904},[1907,13049,1909],{},[1907,13051,1913],{"stretchy":1912},[1920,13053,2780],{},[1907,13055,1925],{"separator":1868},[1920,13057,12322],{},[1939,13059,13060,13062],{},[1907,13061,1943],{"stretchy":1912},[1857,13063,1947],{"mathvariant":1946},[1861,13065,13066],{"encoding":1863},"\\mathbf a=(0,0.6)^\\top",[1838,13068,13070,13088],{"className":13069,"ariaHidden":1868},[1867],[1838,13071,13073,13076,13079,13082,13085],{"className":13072},[1872],[1838,13074],{"className":13075,"style":3004},[1876],[1838,13077,2319],{"className":13078},[1881,2014],[1838,13080],{"className":13081,"style":2018},[1951],[1838,13083,1909],{"className":13084},[2022],[1838,13086],{"className":13087,"style":2018},[1951],[1838,13089,13091,13094,13097,13100,13103,13106,13109],{"className":13090},[1872],[1838,13092],{"className":13093,"style":3453},[1876],[1838,13095,1913],{"className":13096},[2036],[1838,13098,2780],{"className":13099},[1881],[1838,13101,1925],{"className":13102},[2097],[1838,13104],{"className":13105,"style":2101},[1951],[1838,13107,12322],{"className":13108},[1881],[1838,13110,13112,13115],{"className":13111},[2159],[1838,13113,1943],{"className":13114},[2159],[1838,13116,13118],{"className":13117},[2047],[1838,13119,13121],{"className":13120},[2051],[1838,13122,13124],{"className":13123},[2056],[1838,13125,13127],{"className":13126,"style":2398},[2060],[1838,13128,13129,13132],{"style":2419},[1838,13130],{"className":13131,"style":2069},[2068],[1838,13133,13135],{"className":13134},[2073,2074,2075,2076],[1838,13136,1947],{"className":13137},[1881,2076],[1812,13139,13140],{},"The old predictor remains available in the larger span, so minimisation over the larger set cannot be worse.",[1812,13142,13143,13144,13172,13173,13306,13307,12547,13359,12576,13463,1997],{},"The first row of ",[1838,13145,13147,13160],{"className":13146},[1841],[1838,13148,13150],{"className":13149},[1845],[1847,13151,13152],{"xmlns":1849},[1851,13153,13154,13158],{},[1854,13155,13156],{},[1857,13157,6535],{},[1861,13159,6535],{"encoding":1863},[1838,13161,13163],{"className":13162,"ariaHidden":1868},[1867],[1838,13164,13166,13169],{"className":13165},[1872],[1838,13167],{"className":13168,"style":1877},[1876],[1838,13170,6535],{"className":13171,"style":7012},[1881,1882]," is ",[1838,13174,13176,13205],{"className":13175},[1841],[1838,13177,13179],{"className":13178},[1845],[1847,13180,13181],{"xmlns":1849},[1851,13182,13183,13203],{},[1854,13184,13185,13187,13193,13195,13201],{},[1907,13186,1913],{"stretchy":1912},[1915,13188,13189,13191],{},[1857,13190,3792],{},[1920,13192,1922],{},[1907,13194,1925],{"separator":1868},[1915,13196,13197,13199],{},[1857,13198,3792],{},[1920,13200,2474],{},[1907,13202,1943],{"stretchy":1912},[1861,13204,6277],{"encoding":1863},[1838,13206,13208],{"className":13207,"ariaHidden":1868},[1867],[1838,13209,13211,13214,13217,13257,13260,13263,13303],{"className":13210},[1872],[1838,13212],{"className":13213,"style":2216},[1876],[1838,13215,1913],{"className":13216},[2036],[1838,13218,13220,13223],{"className":13219},[1881],[1838,13221,3792],{"className":13222},[1881,1882],[1838,13224,13226],{"className":13225},[2047],[1838,13227,13229,13249],{"className":13228},[2051,2052],[1838,13230,13232,13246],{"className":13231},[2056],[1838,13233,13235],{"className":13234,"style":2061},[2060],[1838,13236,13237,13240],{"style":6310},[1838,13238],{"className":13239,"style":2069},[2068],[1838,13241,13243],{"className":13242},[2073,2074,2075,2076],[1838,13244,1922],{"className":13245},[1881,2076],[1838,13247,2084],{"className":13248},[2083],[1838,13250,13252],{"className":13251},[2056],[1838,13253,13255],{"className":13254,"style":2091},[2060],[1838,13256],{},[1838,13258,1925],{"className":13259},[2097],[1838,13261],{"className":13262,"style":2101},[1951],[1838,13264,13266,13269],{"className":13265},[1881],[1838,13267,3792],{"className":13268},[1881,1882],[1838,13270,13272],{"className":13271},[2047],[1838,13273,13275,13295],{"className":13274},[2051,2052],[1838,13276,13278,13292],{"className":13277},[2056],[1838,13279,13281],{"className":13280,"style":2061},[2060],[1838,13282,13283,13286],{"style":6310},[1838,13284],{"className":13285,"style":2069},[2068],[1838,13287,13289],{"className":13288},[2073,2074,2075,2076],[1838,13290,2474],{"className":13291},[1881,2076],[1838,13293,2084],{"className":13294},[2083],[1838,13296,13298],{"className":13297},[2056],[1838,13299,13301],{"className":13300,"style":2091},[2060],[1838,13302],{},[1838,13304,1943],{"className":13305},[2159],", the second is ",[1838,13308,13310,13332],{"className":13309},[1841],[1838,13311,13313],{"className":13312},[1845],[1847,13314,13315],{"xmlns":1849},[1851,13316,13317,13329],{},[1854,13318,13319,13321,13323,13325,13327],{},[1907,13320,1913],{"stretchy":1912},[1920,13322,1922],{},[1907,13324,1925],{"separator":1868},[1920,13326,2780],{},[1907,13328,1943],{"stretchy":1912},[1861,13330,13331],{"encoding":1863},"(1,0)",[1838,13333,13335],{"className":13334,"ariaHidden":1868},[1867],[1838,13336,13338,13341,13344,13347,13350,13353,13356],{"className":13337},[1872],[1838,13339],{"className":13340,"style":2216},[1876],[1838,13342,1913],{"className":13343},[2036],[1838,13345,1922],{"className":13346},[1881],[1838,13348,1925],{"className":13349},[2097],[1838,13351],{"className":13352,"style":2101},[1951],[1838,13354,2780],{"className":13355},[1881],[1838,13357,1943],{"className":13358},[2159],[1838,13360,13362,13392],{"className":13361},[1841],[1838,13363,13365],{"className":13364},[1845],[1847,13366,13367],{"xmlns":1849},[1851,13368,13369,13389],{},[1854,13370,13371,13373,13375,13377,13379,13381,13383],{},[1857,13372,6552],{},[1907,13374,1909],{},[1907,13376,1913],{"stretchy":1912},[1920,13378,1922],{},[1907,13380,1925],{"separator":1868},[1920,13382,2780],{},[1939,13384,13385,13387],{},[1907,13386,1943],{"stretchy":1912},[1857,13388,1947],{"mathvariant":1946},[1861,13390,13391],{"encoding":1863},"G=(1,0)^\\top",[1838,13393,13395,13413],{"className":13394,"ariaHidden":1868},[1867],[1838,13396,13398,13401,13404,13407,13410],{"className":13397},[1872],[1838,13399],{"className":13400,"style":1877},[1876],[1838,13402,6552],{"className":13403},[1881,1882],[1838,13405],{"className":13406,"style":2018},[1951],[1838,13408,1909],{"className":13409},[2022],[1838,13411],{"className":13412,"style":2018},[1951],[1838,13414,13416,13419,13422,13425,13428,13431,13434],{"className":13415},[1872],[1838,13417],{"className":13418,"style":3453},[1876],[1838,13420,1913],{"className":13421},[2036],[1838,13423,1922],{"className":13424},[1881],[1838,13426,1925],{"className":13427},[2097],[1838,13429],{"className":13430,"style":2101},[1951],[1838,13432,2780],{"className":13433},[1881],[1838,13435,13437,13440],{"className":13436},[2159],[1838,13438,1943],{"className":13439},[2159],[1838,13441,13443],{"className":13442},[2047],[1838,13444,13446],{"className":13445},[2051],[1838,13447,13449],{"className":13448},[2056],[1838,13450,13452],{"className":13451,"style":2398},[2060],[1838,13453,13454,13457],{"style":2419},[1838,13455],{"className":13456,"style":2069},[2068],[1838,13458,13460],{"className":13459},[2073,2074,2075,2076],[1838,13461,1947],{"className":13462},[1881,2076],[1838,13464,13466,13495],{"className":13465},[1841],[1838,13467,13469],{"className":13468},[1845],[1847,13470,13471],{"xmlns":1849},[1851,13472,13473,13493],{},[1854,13474,13475,13477,13479,13485,13487],{},[1857,13476,7760],{},[1907,13478,1909],{},[1939,13480,13481,13483],{},[1857,13482,4959],{},[1920,13484,2474],{},[1857,13486,6552],{},[1939,13488,13489,13491],{},[1857,13490,6552],{},[1857,13492,1947],{"mathvariant":1946},[1861,13494,7779],{"encoding":1863},[1838,13496,13498,13516],{"className":13497,"ariaHidden":1868},[1867],[1838,13499,13501,13504,13507,13510,13513],{"className":13500},[1872],[1838,13502],{"className":13503,"style":7789},[1876],[1838,13505,7760],{"className":13506},[1881,1882],[1838,13508],{"className":13509,"style":2018},[1951],[1838,13511,1909],{"className":13512},[2022],[1838,13514],{"className":13515,"style":2018},[1951],[1838,13517,13519,13522,13551,13554],{"className":13518},[1872],[1838,13520],{"className":13521,"style":2398},[1876],[1838,13523,13525,13528],{"className":13524},[1881],[1838,13526,4959],{"className":13527,"style":5176},[1881,1882],[1838,13529,13531],{"className":13530},[2047],[1838,13532,13534],{"className":13533},[2051],[1838,13535,13537],{"className":13536},[2056],[1838,13538,13540],{"className":13539,"style":4570},[2060],[1838,13541,13542,13545],{"style":2419},[1838,13543],{"className":13544,"style":2069},[2068],[1838,13546,13548],{"className":13547},[2073,2074,2075,2076],[1838,13549,2474],{"className":13550},[1881,2076],[1838,13552,6552],{"className":13553},[1881,1882],[1838,13555,13557,13560],{"className":13556},[1881],[1838,13558,6552],{"className":13559},[1881,1882],[1838,13561,13563],{"className":13562},[2047],[1838,13564,13566],{"className":13565},[2051],[1838,13567,13569],{"className":13568},[2056],[1838,13570,13572],{"className":13571,"style":2398},[2060],[1838,13573,13574,13577],{"style":2419},[1838,13575],{"className":13576,"style":2069},[2068],[1838,13578,13580],{"className":13579},[2073,2074,2075,2076],[1838,13581,1947],{"className":13582},[1881,2076],[1812,13584,13585],{},"Orthogonal sequential innovations factor the same multivariate normal density into conditional normal densities.",[1812,13587,13588],{},"Prediction uncertainty accumulates process noise because no measurement correction reduces it.",[1793,13590,13592],{"id":13591},"takeaway","Takeaway",[1798,13594,13595,13596,1997],{},"Linear prediction is a covariance problem: solve normal equations, verify orthogonality, and read uncertainty from a Schur complement. State-space recursions perform the same conditioning sequentially. Continue to ",[2319,13597,13599],{"href":13598},"..\u002F04-estimation\u002F","Estimation and Inference",{"title":10,"searchDepth":13601,"depth":13601,"links":13602},2,[13603,13604,13605,13606,13607,13608,13612,13613,13614,13615,13616],{"id":1795,"depth":13601,"text":1796},{"id":1803,"depth":13601,"text":1804},{"id":1832,"depth":13601,"text":1833},{"id":3202,"depth":13601,"text":3203},{"id":3712,"depth":13601,"text":3713},{"id":5229,"depth":13601,"text":5230,"children":13609},[13610],{"id":5897,"depth":13611,"text":5898},3,{"id":6383,"depth":13601,"text":6384},{"id":8357,"depth":13601,"text":8358},{"id":9435,"depth":13601,"text":9436},{"id":12296,"depth":13601,"text":12297},{"id":13591,"depth":13601,"text":13592},"Derive best linear predictors from partitioned covariance matrices, quantify error with Schur complements, and introduce Kalman recursions.","md",{"sidebar":13620},{"order":13611},true,{"title":971,"description":13617},"HjJw6bYot3njiTxywLrrGvdiKNx1RO_ACQ7KjZn-GSQ",[13625,13627],{"title":965,"path":966,"stem":967,"description":13626,"children":-1},"Build AR, MA, and ARIMA models from lag polynomials, roots, shock propagation, and finite-sample diagnostics.",{"title":977,"path":978,"stem":979,"description":13628,"children":-1},"Derive AR estimators from regression and moment equations, evaluate exact Gaussian likelihoods with covariance matrices, and interpret uncertainty.",1785754738037]