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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 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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. 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1 — Local level with a missing observation",[1798,4515,4516],{},"The local-level model is",[1805,4518,4520],{"className":4519},[1808],[1805,4521,4523,4591],{"className":4522},[1812],[1805,4524,4526],{"className":4525},[1816],[1818,4527,4528],{"xmlns":1820,"display":1821},[1823,4529,4530,4588],{},[1826,4531,4532,4538,4540,4552,4554,4560,4562,4564,4570,4572,4578,4580,4586],{},[1829,4533,4534,4536],{},[1832,4535,1835],{},[1832,4537,1838],{},[1840,4539,1842],{},[1829,4541,4542,4544],{},[1832,4543,1835],{},[1826,4545,4546,4548,4550],{},[1832,4547,1838],{},[1840,4549,1856],{},[1858,4551,1860],{},[1840,4553,1863],{},[1829,4555,4556,4558],{},[1832,4557,1871],{},[1832,4559,1838],{},[1840,4561,1877],{"separator":1876},[1879,4563],{"width":1881},[1829,4565,4566,4568],{},[1832,4567,1886],{},[1832,4569,1838],{},[1840,4571,1842],{},[1829,4573,4574,4576],{},[1832,4575,1835],{},[1832,4577,1838],{},[1840,4579,1863],{},[1829,4581,4582,4584],{},[1832,4583,1906],{},[1832,4585,1838],{},[1840,4587,1877],{"separator":1876},[1912,4589,4590],{"encoding":1914},"\\alpha_t=\\alpha_{t-1}+\\eta_t,\\qquad\nY_t=\\alpha_t+\\varepsilon_t,",[1805,4592,4594,4652,4717,4823,4879],{"className":4593,"ariaHidden":1876},[1919],[1805,4595,4597,4601,4643,4646,4649],{"className":4596},[1923],[1805,4598],{"className":4599,"style":4600},[1927],"height:0.5806em;vertical-align:-0.15em;",[1805,4602,4604,4608],{"className":4603},[1932],[1805,4605,1835],{"className":4606,"style":4607},[1932,1979],"margin-right:0.0037em;",[1805,4609,4611],{"className":4610},[1946],[1805,4612,4614,4635],{"className":4613},[1950,1951],[1805,4615,4617,4632],{"className":4616},[1955],[1805,4618,4620],{"className":4619,"style":1960},[1959],[1805,4621,4623,4626],{"style":4622},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[1805,4624],{"className":4625,"style":1968},[1967],[1805,4627,4629],{"className":4628},[1972,1973,1974,1975],[1805,4630,1838],{"className":4631},[1932,1979,1975],[1805,4633,1984],{"className":4634},[1983],[1805,4636,4638],{"className":4637},[1955],[1805,4639,4641],{"className":4640,"style":1991},[1959],[1805,4642],{},[1805,4644],{"className":4645,"style":1997},[1879],[1805,4647,1842],{"className":4648},[2001],[1805,4650],{"className":4651,"style":1997},[1879],[1805,4653,4655,4659,4708,4711,4714],{"className":4654},[1923],[1805,4656],{"className":4657,"style":4658},[1927],"height:0.7917em;vertical-align:-0.2083em;",[1805,4660,4662,4665],{"className":4661},[1932],[1805,4663,1835],{"className":4664,"style":4607},[1932,1979],[1805,4666,4668],{"className":4667},[1946],[1805,4669,4671,4700],{"className":4670},[1950,1951],[1805,4672,4674,4697],{"className":4673},[1955],[1805,4675,4677],{"className":4676,"style":2040},[1959],[1805,4678,4679,4682],{"style":4622},[1805,4680],{"className":4681,"style":1968},[1967],[1805,4683,4685],{"className":4684},[1972,1973,1974,1975],[1805,4686,4688,4691,4694],{"className":4687},[1932,1975],[1805,4689,1838],{"className":4690},[1932,1979,1975],[1805,4692,1856],{"className":4693},[2058,1975],[1805,4695,1860],{"className":4696},[1932,1975],[1805,4698,1984],{"className":4699},[1983],[1805,4701,4703],{"className":4702},[1955],[1805,4704,4706],{"className":4705,"style":2071},[1959],[1805,4707],{},[1805,4709],{"className":4710,"style":2077},[1879],[1805,4712,1863],{"className":4713},[2058],[1805,4715],{"className":4716,"style":2077},[1879],[1805,4718,4720,4723,4765,4768,4771,4774,4814,4817,4820],{"className":4719},[1923],[1805,4721],{"className":4722,"style":3634},[1927],[1805,4724,4726,4730],{"className":4725},[1932],[1805,4727,1871],{"className":4728,"style":4729},[1932,1979],"margin-right:0.0359em;",[1805,4731,4733],{"className":4732},[1946],[1805,4734,4736,4757],{"className":4735},[1950,1951],[1805,4737,4739,4754],{"className":4738},[1955],[1805,4740,4742],{"className":4741,"style":1960},[1959],[1805,4743,4745,4748],{"style":4744},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1805,4746],{"className":4747,"style":1968},[1967],[1805,4749,4751],{"className":4750},[1972,1973,1974,1975],[1805,4752,1838],{"className":4753},[1932,1979,1975],[1805,4755,1984],{"className":4756},[1983],[1805,4758,4760],{"className":4759},[1955],[1805,4761,4763],{"className":4762,"style":1991},[1959],[1805,4764],{},[1805,4766,1877],{"className":4767},[2148],[1805,4769],{"className":4770,"style":2152},[1879],[1805,4772],{"className":4773,"style":2156},[1879],[1805,4775,4777,4780],{"className":4776},[1932],[1805,4778,1886],{"className":4779,"style":2077},[1932,1979],[1805,4781,4783],{"className":4782},[1946],[1805,4784,4786,4806],{"className":4785},[1950,1951],[1805,4787,4789,4803],{"className":4788},[1955],[1805,4790,4792],{"className":4791,"style":1960},[1959],[1805,4793,4794,4797],{"style":2177},[1805,4795],{"className":4796,"style":1968},[1967],[1805,4798,4800],{"className":4799},[1972,1973,1974,1975],[1805,4801,1838],{"className":4802},[1932,1979,1975],[1805,4804,1984],{"className":4805},[1983],[1805,4807,4809],{"className":4808},[1955],[1805,4810,4812],{"className":4811,"style":1991},[1959],[1805,4813],{},[1805,4815],{"className":4816,"style":1997},[1879],[1805,4818,1842],{"className":4819},[2001],[1805,4821],{"className":4822,"style":1997},[1879],[1805,4824,4826,4830,4870,4873,4876],{"className":4825},[1923],[1805,4827],{"className":4828,"style":4829},[1927],"height:0.7333em;vertical-align:-0.15em;",[1805,4831,4833,4836],{"className":4832},[1932],[1805,4834,1835],{"className":4835,"style":4607},[1932,1979],[1805,4837,4839],{"className":4838},[1946],[1805,4840,4842,4862],{"className":4841},[1950,1951],[1805,4843,4845,4859],{"className":4844},[1955],[1805,4846,4848],{"className":4847,"style":1960},[1959],[1805,4849,4850,4853],{"style":4622},[1805,4851],{"className":4852,"style":1968},[1967],[1805,4854,4856],{"className":4855},[1972,1973,1974,1975],[1805,4857,1838],{"className":4858},[1932,1979,1975],[1805,4860,1984],{"className":4861},[1983],[1805,4863,4865],{"className":4864},[1955],[1805,4866,4868],{"className":4867,"style":1991},[1959],[1805,4869],{},[1805,4871],{"className":4872,"style":2077},[1879],[1805,4874,1863],{"className":4875},[2058],[1805,4877],{"className":4878,"style":2077},[1879],[1805,4880,4882,4885,4925],{"className":4881},[1923],[1805,4883],{"className":4884,"style":2279},[1927],[1805,4886,4888,4891],{"className":4887},[1932],[1805,4889,1906],{"className":4890},[1932,1979],[1805,4892,4894],{"className":4893},[1946],[1805,4895,4897,4917],{"className":4896},[1950,1951],[1805,4898,4900,4914],{"className":4899},[1955],[1805,4901,4903],{"className":4902,"style":1960},[1959],[1805,4904,4905,4908],{"style":2300},[1805,4906],{"className":4907,"style":1968},[1967],[1805,4909,4911],{"className":4910},[1972,1973,1974,1975],[1805,4912,1838],{"className":4913},[1932,1979,1975],[1805,4915,1984],{"className":4916},[1983],[1805,4918,4920],{"className":4919},[1955],[1805,4921,4923],{"className":4922,"style":1991},[1959],[1805,4924],{},[1805,4926,1877],{"className":4927},[2148],[1798,4929,4930],{},"so all matrices are scalar:",[1805,4932,4934],{"className":4933},[1808],[1805,4935,4937,4987],{"className":4936},[1812],[1805,4938,4940],{"className":4939},[1816],[1818,4941,4942],{"xmlns":1820,"display":1821},[1823,4943,4944,4984],{},[1826,4945,4946,4948,4950,4952,4954,4956,4958,4960,4962,4964,4966,4968,4971,4973,4975,4977,4979,4982],{},[1832,4947,1845],{},[1840,4949,1842],{},[1832,4951,1866],{},[1840,4953,1842],{},[1832,4955,1893],{},[1840,4957,1842],{},[1858,4959,1860],{},[1840,4961,1877],{"separator":1876},[1879,4963],{"width":1881},[1832,4965,2366],{},[1840,4967,1842],{},[1832,4969,4970],{},"q",[1840,4972,1877],{"separator":1876},[1879,4974],{"width":1881},[1832,4976,2389],{},[1840,4978,1842],{},[1832,4980,4981],{},"h",[1832,4983,2392],{"mathvariant":2344},[1912,4985,4986],{"encoding":1914},"T=R=Z=1,\\qquad Q=q,\\qquad H=h.",[1805,4988,4990,5008,5026,5044,5074,5104],{"className":4989,"ariaHidden":1876},[1919],[1805,4991,4993,4996,4999,5002,5005],{"className":4992},[1923],[1805,4994],{"className":4995,"style":2562},[1927],[1805,4997,1845],{"className":4998,"style":2015},[1932,1979],[1805,5000],{"className":5001,"style":1997},[1879],[1805,5003,1842],{"className":5004},[2001],[1805,5006],{"className":5007,"style":1997},[1879],[1805,5009,5011,5014,5017,5020,5023],{"className":5010},[1923],[1805,5012],{"className":5013,"style":2562},[1927],[1805,5015,1866],{"className":5016,"style":2094},[1932,1979],[1805,5018],{"className":5019,"style":1997},[1879],[1805,5021,1842],{"className":5022},[2001],[1805,5024],{"className":5025,"style":1997},[1879],[1805,5027,5029,5032,5035,5038,5041],{"className":5028},[1923],[1805,5030],{"className":5031,"style":2562},[1927],[1805,5033,1893],{"className":5034,"style":2217},[1932,1979],[1805,5036],{"className":5037,"style":1997},[1879],[1805,5039,1842],{"className":5040},[2001],[1805,5042],{"className":5043,"style":1997},[1879],[1805,5045,5047,5050,5053,5056,5059,5062,5065,5068,5071],{"className":5046},[1923],[1805,5048],{"className":5049,"style":3634},[1927],[1805,5051,1860],{"className":5052},[1932],[1805,5054,1877],{"className":5055},[2148],[1805,5057],{"className":5058,"style":2152},[1879],[1805,5060],{"className":5061,"style":2156},[1879],[1805,5063,2366],{"className":5064},[1932,1979],[1805,5066],{"className":5067,"style":1997},[1879],[1805,5069,1842],{"className":5070},[2001],[1805,5072],{"className":5073,"style":1997},[1879],[1805,5075,5077,5080,5083,5086,5089,5092,5095,5098,5101],{"className":5076},[1923],[1805,5078],{"className":5079,"style":3634},[1927],[1805,5081,4970],{"className":5082,"style":4729},[1932,1979],[1805,5084,1877],{"className":5085},[2148],[1805,5087],{"className":5088,"style":2152},[1879],[1805,5090],{"className":5091,"style":2156},[1879],[1805,5093,2389],{"className":5094,"style":2566},[1932,1979],[1805,5096],{"className":5097,"style":1997},[1879],[1805,5099,1842],{"className":5100},[2001],[1805,5102],{"className":5103,"style":1997},[1879],[1805,5105,5107,5111,5114],{"className":5106},[1923],[1805,5108],{"className":5109,"style":5110},[1927],"height:0.6944em;",[1805,5112,4981],{"className":5113},[1932,1979],[1805,5115,2392],{"className":5116},[1932],[5118,5119],"web-r",{"code64":5120,"layout":5121,"locale":7,"title":5122},"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","vertical","Implement a local-level Kalman filter by hand",[1798,5124,5125,5126,5180,5181,5209,5210,5262],{},"At ",[1805,5127,5129,5148],{"className":5128},[1812],[1805,5130,5132],{"className":5131},[1816],[1818,5133,5134],{"xmlns":1820},[1823,5135,5136,5145],{},[1826,5137,5138,5140,5142],{},[1832,5139,1838],{},[1840,5141,1842],{},[1858,5143,5144],{},"4",[1912,5146,5147],{"encoding":1914},"t=4",[1805,5149,5151,5170],{"className":5150,"ariaHidden":1876},[1919],[1805,5152,5154,5158,5161,5164,5167],{"className":5153},[1923],[1805,5155],{"className":5156,"style":5157},[1927],"height:0.6151em;",[1805,5159,1838],{"className":5160},[1932,1979],[1805,5162],{"className":5163,"style":1997},[1879],[1805,5165,1842],{"className":5166},[2001],[1805,5168],{"className":5169,"style":1997},[1879],[1805,5171,5173,5177],{"className":5172},[1923],[1805,5174],{"className":5175,"style":5176},[1927],"height:0.6444em;",[1805,5178,5144],{"className":5179},[1932],", uncertainty grows by ",[1805,5182,5184,5197],{"className":5183},[1812],[1805,5185,5187],{"className":5186},[1816],[1818,5188,5189],{"xmlns":1820},[1823,5190,5191,5195],{},[1826,5192,5193],{},[1832,5194,4970],{},[1912,5196,4970],{"encoding":1914},[1805,5198,5200],{"className":5199,"ariaHidden":1876},[1919],[1805,5201,5203,5206],{"className":5202},[1923],[1805,5204],{"className":5205,"style":2279},[1927],[1805,5207,4970],{"className":5208,"style":4729},[1932,1979]," but no observation reduces it. 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turns it into one linear solve:",[1805,6340,6342],{"className":6341},[1808],[1805,6343,6345,6422],{"className":6344},[1812],[1805,6346,6348],{"className":6347},[1816],[1818,6349,6350],{"xmlns":1820,"display":1821},[1823,6351,6352,6419],{},[1826,6353,6354,6357,6359,6361,6363,6365,6367,6393,6395,6397,6399,6407,6409,6415,6417],{},[1832,6355,6356],{"mathvariant":2344},"vec",[1840,6358,2348],{},[1840,6360,2352],{"stretchy":2351},[1832,6362,2663],{},[1840,6364,2361],{"stretchy":2351},[1840,6366,1842],{},[2811,6368,6369,6387],{},[1826,6370,6371,6373,6376,6378,6380,6383,6385],{},[1840,6372,2352],{"fence":1876},[1832,6374,6375],{},"I",[1840,6377,1856],{},[1832,6379,1845],{},[1840,6381,6382],{},"⊗",[1832,6384,1845],{},[1840,6386,2361],{"fence":1876},[1826,6388,6389,6391],{},[1840,6390,1856],{},[1858,6392,1860],{},[1832,6394,6356],{"mathvariant":2344},[1840,6396,2348],{},[1840,6398,2352],{"stretchy":2351},[3688,6400,6401,6403,6405],{},[1832,6402,6153],{},[1832,6404,1906],{},[1858,6406,5326],{},[1832,6408,1866],{},[2811,6410,6411,6413],{},[1832,6412,1866],{},[1832,6414,2817],{"mathvariant":2344},[1840,6416,2361],{"stretchy":2351},[1832,6418,2392],{"mathvariant":2344},[1912,6420,6421],{"encoding":1914},"\\operatorname{vec}(P)\n=\n\\left(I-T\\otimes T\\right)^{-1}\n\\operatorname{vec}(\\sigma_\\varepsilon^2RR^\\top).",[1805,6423,6425,6455],{"className":6424,"ariaHidden":1876},[1919],[1805,6426,6428,6431,6437,6440,6443,6446,6449,6452],{"className":6427},[1923],[1805,6429],{"className":6430,"style":2405},[1927],[1805,6432,6434],{"className":6433},[2409],[1805,6435,6356],{"className":6436},[1932,2413],[1805,6438,2352],{"className":6439},[2417],[1805,6441,2663],{"className":6442,"style":2015},[1932,1979],[1805,6444,2361],{"className":6445},[2467],[1805,6447],{"className":6448,"style":1997},[1879],[1805,6450,1842],{"className":6451},[2001],[1805,6453],{"className":6454,"style":1997},[1879],[1805,6456,6458,6462,6532,6535,6541,6544,6595,6598,6627,6630],{"className":6457},[1923],[1805,6459],{"className":6460,"style":6461},[1927],"height:1.204em;vertical-align:-0.25em;",[1805,6463,6465,6501],{"className":6464},[5754],[1805,6466,6468,6471,6474,6477,6480,6483,6486,6489,6492,6495,6498],{"className":6467},[5754],[1805,6469,2352],{"className":6470,"style":5759},[2417,5758],[1805,6472,6375],{"className":6473,"style":5355},[1932,1979],[1805,6475],{"className":6476,"style":2077},[1879],[1805,6478,1856],{"className":6479},[2058],[1805,6481],{"className":6482,"style":2077},[1879],[1805,6484,1845],{"className":6485,"style":2015},[1932,1979],[1805,6487],{"className":6488,"style":2077},[1879],[1805,6490,6382],{"className":6491},[2058],[1805,6493],{"className":6494,"style":2077},[1879],[1805,6496,1845],{"className":6497,"style":2015},[1932,1979],[1805,6499,2361],{"className":6500,"style":5759},[2467,5758],[1805,6502,6504],{"className":6503},[1946],[1805,6505,6507],{"className":6506},[1950],[1805,6508,6510],{"className":6509},[1955],[1805,6511,6514],{"className":6512,"style":6513},[1959],"height:0.954em;",[1805,6515,6517,6520],{"style":6516},"top:-3.2029em;margin-right:0.05em;",[1805,6518],{"className":6519,"style":1968},[1967],[1805,6521,6523],{"className":6522},[1972,1973,1974,1975],[1805,6524,6526,6529],{"className":6525},[1932,1975],[1805,6527,1856],{"className":6528},[1932,1975],[1805,6530,1860],{"className":6531},[1932,1975],[1805,6533],{"className":6534,"style":2156},[1879],[1805,6536,6538],{"className":6537},[2409],[1805,6539,6356],{"className":6540},[1932,2413],[1805,6542,2352],{"className":6543},[2417],[1805,6545,6547,6550],{"className":6546},[1932],[1805,6548,6153],{"className":6549,"style":4729},[1932,1979],[1805,6551,6553],{"className":6552},[1946],[1805,6554,6556,6587],{"className":6555},[1950,1951],[1805,6557,6559,6584],{"className":6558},[1955],[1805,6560,6562,6573],{"className":6561,"style":3872},[1959],[1805,6563,6564,6567],{"style":6269},[1805,6565],{"className":6566,"style":1968},[1967],[1805,6568,6570],{"className":6569},[1972,1973,1974,1975],[1805,6571,1906],{"className":6572},[1932,1979,1975],[1805,6574,6575,6578],{"style":3125},[1805,6576],{"className":6577,"style":1968},[1967],[1805,6579,6581],{"className":6580},[1972,1973,1974,1975],[1805,6582,5326],{"className":6583},[1932,1975],[1805,6585,1984],{"className":6586},[1983],[1805,6588,6590],{"className":6589},[1955],[1805,6591,6593],{"className":6592,"style":3911},[1959],[1805,6594],{},[1805,6596,1866],{"className":6597,"style":2094},[1932,1979],[1805,6599,6601,6604],{"className":6600},[1932],[1805,6602,1866],{"className":6603,"style":2094},[1932,1979],[1805,6605,6607],{"className":6606},[1946],[1805,6608,6610],{"className":6609},[1950],[1805,6611,6613],{"className":6612},[1955],[1805,6614,6616],{"className":6615,"style":3122},[1959],[1805,6617,6618,6621],{"style":3125},[1805,6619],{"className":6620,"style":1968},[1967],[1805,6622,6624],{"className":6623},[1972,1973,1974,1975],[1805,6625,2817],{"className":6626},[1932,1975],[1805,6628,2361],{"className":6629},[2467],[1805,6631,2392],{"className":6632},[1932],[5118,6634],{"code64":6635,"layout":5121,"locale":7,"title":6636},"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an AR(2) state mean and forecast covariance",[1798,6638,6639,6640,6753],{},"The mean decays toward zero because the companion eigenvalues lie inside the unit circle. Forecast variance rises toward ",[1805,6641,6643,6673],{"className":6642},[1812],[1805,6644,6646],{"className":6645},[1816],[1818,6647,6648],{"xmlns":1820},[1823,6649,6650,6670],{},[1826,6651,6652,6659,6661,6664,6666,6668],{},[1829,6653,6654,6656],{},[1832,6655,2663],{},[1858,6657,6658],{},"11",[1840,6660,1842],{},[1832,6662,6663],{},"γ",[1840,6665,2352],{"stretchy":2351},[1858,6667,5625],{},[1840,6669,2361],{"stretchy":2351},[1912,6671,6672],{"encoding":1914},"P_{11}=\\gamma(0)",[1805,6674,6676,6734],{"className":6675,"ariaHidden":1876},[1919],[1805,6677,6679,6682,6725,6728,6731],{"className":6678},[1923],[1805,6680],{"className":6681,"style":2213},[1927],[1805,6683,6685,6688],{"className":6684},[1932],[1805,6686,2663],{"className":6687,"style":2015},[1932,1979],[1805,6689,6691],{"className":6690},[1946],[1805,6692,6694,6717],{"className":6693},[1950,1951],[1805,6695,6697,6714],{"className":6696},[1955],[1805,6698,6700],{"className":6699,"style":2040},[1959],[1805,6701,6702,6705],{"style":2698},[1805,6703],{"className":6704,"style":1968},[1967],[1805,6706,6708],{"className":6707},[1972,1973,1974,1975],[1805,6709,6711],{"className":6710},[1932,1975],[1805,6712,6658],{"className":6713},[1932,1975],[1805,6715,1984],{"className":6716},[1983],[1805,6718,6720],{"className":6719},[1955],[1805,6721,6723],{"className":6722,"style":1991},[1959],[1805,6724],{},[1805,6726],{"className":6727,"style":1997},[1879],[1805,6729,1842],{"className":6730},[2001],[1805,6732],{"className":6733,"style":1997},[1879],[1805,6735,6737,6740,6744,6747,6750],{"className":6736},[1923],[1805,6738],{"className":6739,"style":2405},[1927],[1805,6741,6663],{"className":6742,"style":6743},[1932,1979],"margin-right:0.0556em;",[1805,6745,2352],{"className":6746},[2417],[1805,6748,5625],{"className":6749},[1932],[1805,6751,2361],{"className":6752},[2467]," because future innovations accumulate until the initial state is no longer informative.",[1793,6755,6757],{"id":6756},"from-hand-recursion-to-rs-implementation","From hand recursion to R's implementation",[1798,6759,6760,6761,6765,6766,6765,6769,6772,6773,6776,6777,6780],{},"Base R exposes lower-level functions ",[6762,6763,6764],"code",{},"KalmanRun",", ",[6762,6767,6768],{},"KalmanForecast",[6762,6770,6771],{},"KalmanSmooth",", and ",[6762,6774,6775],{},"KalmanLike",". R's exact ",[6762,6778,6779],{},"arima"," likelihood builds a state-space representation and evaluates innovations with a Kalman filter.",[1798,6782,6783,6784,6765,6812,6765,6840,6765,6912,6940],{},"The hand implementation remains important: software output is interpretable only when you can identify ",[1805,6785,6787,6800],{"className":6786},[1812],[1805,6788,6790],{"className":6789},[1816],[1818,6791,6792],{"xmlns":1820},[1823,6793,6794,6798],{},[1826,6795,6796],{},[1832,6797,1845],{},[1912,6799,1845],{"encoding":1914},[1805,6801,6803],{"className":6802,"ariaHidden":1876},[1919],[1805,6804,6806,6809],{"className":6805},[1923],[1805,6807],{"className":6808,"style":2562},[1927],[1805,6810,1845],{"className":6811,"style":2015},[1932,1979],[1805,6813,6815,6828],{"className":6814},[1812],[1805,6816,6818],{"className":6817},[1816],[1818,6819,6820],{"xmlns":1820},[1823,6821,6822,6826],{},[1826,6823,6824],{},[1832,6825,1893],{},[1912,6827,1893],{"encoding":1914},[1805,6829,6831],{"className":6830,"ariaHidden":1876},[1919],[1805,6832,6834,6837],{"className":6833},[1923],[1805,6835],{"className":6836,"style":2562},[1927],[1805,6838,1893],{"className":6839,"style":2217},[1932,1979],[1805,6841,6843,6865],{"className":6842},[1812],[1805,6844,6846],{"className":6845},[1816],[1818,6847,6848],{"xmlns":1820},[1823,6849,6850,6862],{},[1826,6851,6852,6854,6856],{},[1832,6853,1866],{},[1832,6855,2366],{},[2811,6857,6858,6860],{},[1832,6859,1866],{},[1832,6861,2817],{"mathvariant":2344},[1912,6863,6864],{"encoding":1914},"RQR^\\top",[1805,6866,6868],{"className":6867,"ariaHidden":1876},[1919],[1805,6869,6871,6875,6878,6881],{"className":6870},[1923],[1805,6872],{"className":6873,"style":6874},[1927],"height:1.0435em;vertical-align:-0.1944em;",[1805,6876,1866],{"className":6877,"style":2094},[1932,1979],[1805,6879,2366],{"className":6880},[1932,1979],[1805,6882,6884,6887],{"className":6883},[1932],[1805,6885,1866],{"className":6886,"style":2094},[1932,1979],[1805,6888,6890],{"className":6889},[1946],[1805,6891,6893],{"className":6892},[1950],[1805,6894,6896],{"className":6895},[1955],[1805,6897,6900],{"className":6898,"style":6899},[1959],"height:0.8491em;",[1805,6901,6903,6906],{"style":6902},"top:-3.063em;margin-right:0.05em;",[1805,6904],{"className":6905,"style":1968},[1967],[1805,6907,6909],{"className":6908},[1972,1973,1974,1975],[1805,6910,2817],{"className":6911},[1932,1975],[1805,6913,6915,6928],{"className":6914},[1812],[1805,6916,6918],{"className":6917},[1816],[1818,6919,6920],{"xmlns":1820},[1823,6921,6922,6926],{},[1826,6923,6924],{},[1832,6925,2389],{},[1912,6927,2389],{"encoding":1914},[1805,6929,6931],{"className":6930,"ariaHidden":1876},[1919],[1805,6932,6934,6937],{"className":6933},[1923],[1805,6935],{"className":6936,"style":2562},[1927],[1805,6938,2389],{"className":6939,"style":2566},[1932,1979],", and the initial covariance.",[1793,6942,6944],{"id":6943},"exercises","Exercises",[6946,6947,6948,7004,7036,7185,7188,7191],"ol",{},[6949,6950,6951,6952,7003],"li",{},"In Example 1, set ",[1805,6953,6955,6973],{"className":6954},[1812],[1805,6956,6958],{"className":6957},[1816],[1818,6959,6960],{"xmlns":1820},[1823,6961,6962,6970],{},[1826,6963,6964,6966,6968],{},[1832,6965,4970],{},[1840,6967,1842],{},[1858,6969,5625],{},[1912,6971,6972],{"encoding":1914},"q=0",[1805,6974,6976,6994],{"className":6975,"ariaHidden":1876},[1919],[1805,6977,6979,6982,6985,6988,6991],{"className":6978},[1923],[1805,6980],{"className":6981,"style":2279},[1927],[1805,6983,4970],{"className":6984,"style":4729},[1932,1979],[1805,6986],{"className":6987,"style":1997},[1879],[1805,6989,1842],{"className":6990},[2001],[1805,6992],{"className":6993,"style":1997},[1879],[1805,6995,6997,7000],{"className":6996},[1923],[1805,6998],{"className":6999,"style":5176},[1927],[1805,7001,5625],{"className":7002},[1932],". What happens to the gain over time, and why?",[6949,7005,7006,7007,7035],{},"Increase ",[1805,7008,7010,7023],{"className":7009},[1812],[1805,7011,7013],{"className":7012},[1816],[1818,7014,7015],{"xmlns":1820},[1823,7016,7017,7021],{},[1826,7018,7019],{},[1832,7020,4981],{},[1912,7022,4981],{"encoding":1914},[1805,7024,7026],{"className":7025,"ariaHidden":1876},[1919],[1805,7027,7029,7032],{"className":7028},[1923],[1805,7030],{"className":7031,"style":5110},[1927],[1805,7033,4981],{"className":7034},[1932,1979]," from 0.5 to 5. Predict the direction of change in the gain.",[6949,7037,7038,7039,7089,7090,2392],{},"Replace the missing ",[1805,7040,7042,7059],{"className":7041},[1812],[1805,7043,7045],{"className":7044},[1816],[1818,7046,7047],{"xmlns":1820},[1823,7048,7049,7057],{},[1826,7050,7051,7053,7055],{},[1832,7052,1838],{},[1840,7054,1842],{},[1858,7056,5144],{},[1912,7058,5147],{"encoding":1914},[1805,7060,7062,7080],{"className":7061,"ariaHidden":1876},[1919],[1805,7063,7065,7068,7071,7074,7077],{"className":7064},[1923],[1805,7066],{"className":7067,"style":5157},[1927],[1805,7069,1838],{"className":7070},[1932,1979],[1805,7072],{"className":7073,"style":1997},[1879],[1805,7075,1842],{"className":7076},[2001],[1805,7078],{"className":7079,"style":1997},[1879],[1805,7081,7083,7086],{"className":7082},[1923],[1805,7084],{"className":7085,"style":5176},[1927],[1805,7087,5144],{"className":7088},[1932]," observation with three consecutive missing values. Track ",[1805,7091,7093,7121],{"className":7092},[1812],[1805,7094,7096],{"className":7095},[1816],[1818,7097,7098],{"xmlns":1820},[1823,7099,7100,7118],{},[1826,7101,7102],{},[1829,7103,7104,7106],{},[1832,7105,2663],{},[1826,7107,7108,7110,7112,7114,7116],{},[1832,7109,1838],{},[1840,7111,2751],{},[1832,7113,1838],{},[1840,7115,1856],{},[1858,7117,1860],{},[1912,7119,7120],{"encoding":1914},"P_{t\\mid t-1}",[1805,7122,7124],{"className":7123,"ariaHidden":1876},[1919],[1805,7125,7127,7130],{"className":7126},[1923],[1805,7128],{"className":7129,"style":2918},[1927],[1805,7131,7133,7136],{"className":7132},[1932],[1805,7134,2663],{"className":7135,"style":2015},[1932,1979],[1805,7137,7139],{"className":7138},[1946],[1805,7140,7142,7177],{"className":7141},[1950,1951],[1805,7143,7145,7174],{"className":7144},[1955],[1805,7146,7148],{"className":7147,"style":2863},[1959],[1805,7149,7150,7153],{"style":3000},[1805,7151],{"className":7152,"style":1968},[1967],[1805,7154,7156],{"className":7155},[1972,1973,1974,1975],[1805,7157,7159,7162,7165,7168,7171],{"className":7158},[1932,1975],[1805,7160,1838],{"className":7161},[1932,1979,1975],[1805,7163,2751],{"className":7164},[2001,1975],[1805,7166,1838],{"className":7167},[1932,1979,1975],[1805,7169,1856],{"className":7170},[2058,1975],[1805,7172,1860],{"className":7173},[1932,1975],[1805,7175,1984],{"className":7176},[1983],[1805,7178,7180],{"className":7179},[1955],[1805,7181,7183],{"className":7182,"style":2900},[1959],[1805,7184],{},[6949,7186,7187],{},"Verify the first two AR(2) point forecasts by scalar recursion.",[6949,7189,7190],{},"Change the AR coefficients so the companion spectral radius exceeds one. Which step of the stationary covariance solve fails conceptually, even if a numerical answer is returned?",[6949,7192,7193,7194,7247],{},"Graduate extension: add measurement error ",[1805,7195,7197,7216],{"className":7196},[1812],[1805,7198,7200],{"className":7199},[1816],[1818,7201,7202],{"xmlns":1820},[1823,7203,7204,7213],{},[1826,7205,7206,7208,7211],{},[1832,7207,2389],{},[1840,7209,7210],{},">",[1858,7212,5625],{},[1912,7214,7215],{"encoding":1914},"H>0",[1805,7217,7219,7238],{"className":7218,"ariaHidden":1876},[1919],[1805,7220,7222,7226,7229,7232,7235],{"className":7221},[1923],[1805,7223],{"className":7224,"style":7225},[1927],"height:0.7224em;vertical-align:-0.0391em;",[1805,7227,2389],{"className":7228,"style":2566},[1932,1979],[1805,7230],{"className":7231,"style":1997},[1879],[1805,7233,7210],{"className":7234},[2001],[1805,7236],{"className":7237,"style":1997},[1879],[1805,7239,7241,7244],{"className":7240},[1923],[1805,7242],{"className":7243,"style":5176},[1927],[1805,7245,5625],{"className":7246},[1932]," to the AR(2) observation equation and implement filtering rather than forecasting from an exactly observed state.",[7249,7250,7252],"legacy-details",{"title":7251},"Checkpoints",[6946,7253,7254,7257,7260,7263,7373],{},[6949,7255,7256],{},"With no state noise, repeated observations make state uncertainty and gain decline toward zero.",[6949,7258,7259],{},"Noisier measurements receive less weight, so the gain decreases.",[6949,7261,7262],{},"Each missing time adds process uncertainty without correction.",[6949,7264,7265,7266,7372],{},"The first forecast is ",[1805,7267,7269,7306],{"className":7268},[1812],[1805,7270,7272],{"className":7271},[1816],[1818,7273,7274],{"xmlns":1820},[1823,7275,7276,7303],{},[1826,7277,7278,7280,7282,7285,7287,7289,7291,7293,7296,7298,7300],{},[1858,7279,5296],{},[1840,7281,2352],{"stretchy":2351},[1858,7283,7284],{},"1.2",[1840,7286,2361],{"stretchy":2351},[1840,7288,1856],{},[1858,7290,5313],{},[1840,7292,2352],{"stretchy":2351},[1858,7294,7295],{},"0.4",[1840,7297,2361],{"stretchy":2351},[1840,7299,1842],{},[1858,7301,7302],{},"0.70",[1912,7304,7305],{"encoding":1914},"0.65(1.2)-0.20(0.4)=0.70",[1805,7307,7309,7336,7363],{"className":7308,"ariaHidden":1876},[1919],[1805,7310,7312,7315,7318,7321,7324,7327,7330,7333],{"className":7311},[1923],[1805,7313],{"className":7314,"style":2405},[1927],[1805,7316,5296],{"className":7317},[1932],[1805,7319,2352],{"className":7320},[2417],[1805,7322,7284],{"className":7323},[1932],[1805,7325,2361],{"className":7326},[2467],[1805,7328],{"className":7329,"style":2077},[1879],[1805,7331,1856],{"className":7332},[2058],[1805,7334],{"className":7335,"style":2077},[1879],[1805,7337,7339,7342,7345,7348,7351,7354,7357,7360],{"className":7338},[1923],[1805,7340],{"className":7341,"style":2405},[1927],[1805,7343,5313],{"className":7344},[1932],[1805,7346,2352],{"className":7347},[2417],[1805,7349,7295],{"className":7350},[1932],[1805,7352,2361],{"className":7353},[2467],[1805,7355],{"className":7356,"style":1997},[1879],[1805,7358,1842],{"className":7359},[2001],[1805,7361],{"className":7362,"style":1997},[1879],[1805,7364,7366,7369],{"className":7365},[1923],[1805,7367],{"className":7368,"style":5176},[1927],[1805,7370,7302],{"className":7371},[1932],"; insert that value and 1.2 for the second.",[6949,7374,7375,7376,7490],{},"A finite stationary solution to ",[1805,7377,7379,7409],{"className":7378},[1812],[1805,7380,7382],{"className":7381},[1816],[1818,7383,7384],{"xmlns":1820},[1823,7385,7386,7406],{},[1826,7387,7388,7390,7392,7394,7396,7402,7404],{},[1832,7389,2663],{},[1840,7391,1842],{},[1832,7393,1845],{},[1832,7395,2663],{},[2811,7397,7398,7400],{},[1832,7399,1845],{},[1832,7401,2817],{"mathvariant":2344},[1840,7403,1863],{},[1832,7405,2366],{},[1912,7407,7408],{"encoding":1914},"P=TPT^\\top+Q",[1805,7410,7412,7430,7481],{"className":7411,"ariaHidden":1876},[1919],[1805,7413,7415,7418,7421,7424,7427],{"className":7414},[1923],[1805,7416],{"className":7417,"style":2562},[1927],[1805,7419,2663],{"className":7420,"style":2015},[1932,1979],[1805,7422],{"className":7423,"style":1997},[1879],[1805,7425,1842],{"className":7426},[2001],[1805,7428],{"className":7429,"style":1997},[1879],[1805,7431,7433,7437,7440,7443,7472,7475,7478],{"className":7432},[1923],[1805,7434],{"className":7435,"style":7436},[1927],"height:0.9324em;vertical-align:-0.0833em;",[1805,7438,1845],{"className":7439,"style":2015},[1932,1979],[1805,7441,2663],{"className":7442,"style":2015},[1932,1979],[1805,7444,7446,7449],{"className":7445},[1932],[1805,7447,1845],{"className":7448,"style":2015},[1932,1979],[1805,7450,7452],{"className":7451},[1946],[1805,7453,7455],{"className":7454},[1950],[1805,7456,7458],{"className":7457},[1955],[1805,7459,7461],{"className":7460,"style":6899},[1959],[1805,7462,7463,7466],{"style":6902},[1805,7464],{"className":7465,"style":1968},[1967],[1805,7467,7469],{"className":7468},[1972,1973,1974,1975],[1805,7470,2817],{"className":7471},[1932,1975],[1805,7473],{"className":7474,"style":2077},[1879],[1805,7476,1863],{"className":7477},[2058],[1805,7479],{"className":7480,"style":2077},[1879],[1805,7482,7484,7487],{"className":7483},[1923],[1805,7485],{"className":7486,"style":3634},[1927],[1805,7488,2366],{"className":7489},[1932,1979]," requires a stable transition matrix.",[1793,7492,7494],{"id":7493},"completion-standard","Completion standard",[1798,7496,7497],{},"You should be able to derive every line of a Kalman recursion, explain missing-data handling, and connect AR companion propagation with state-space prediction.",[1798,7499,7500,7501,7505,7506,7510],{},"Return to the ",[2590,7502,7504],{"href":7503},"..\u002F00-intro\u002F","course guide"," or use the ",[2590,7507,7509],{"href":7508},"..\u002F08-python-implementation\u002F","optional Python appendix"," only after the matrix derivations are secure.",{"title":10,"searchDepth":7512,"depth":7512,"links":7513},2,[7514,7515,7516,7517,7518,7519,7520],{"id":1795,"depth":7512,"text":1796},{"id":2721,"depth":7512,"text":2722},{"id":4512,"depth":7512,"text":4513},{"id":5265,"depth":7512,"text":5266},{"id":6756,"depth":7512,"text":6757},{"id":6943,"depth":7512,"text":6944},{"id":7493,"depth":7512,"text":7494},"Implement Kalman prediction and correction from matrix equations, handle missing observations, and express AR forecasts as state propagation.","md",{"sidebar":7524},{"order":7525},4,true,{"title":1013,"description":7521},"pfisY9kM15qgeCYcAzsRxXfwL32cPOJd7wigHAWDYyI",[7530,7532],{"title":1009,"path":1010,"stem":1011,"description":7531,"children":-1},"Solve best-linear-prediction equations, verify Schur-complement uncertainty, and compare conditional, moment, and exact Gaussian AR estimates.",{"title":1017,"path":1018,"stem":1019,"description":7533,"children":-1},"A supplementary Pyodide forecasting workflow for students who want to contrast matrix theory with an applied software pipeline.",1785754738929]