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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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Its central object is a stochastic process, not a collection of empirical forecasting recipes. Data examples are short simulations or hand-checkable numerical cases used to verify theory.",[1799,1808,1809],{},"For a finite block,",[1811,1812,1815],"span",{"className":1813},[1814],"katex-display",[1811,1816,1819,1951],{"className":1817},[1818],"katex",[1811,1820,1823],{"className":1821},[1822],"katex-mathml",[1824,1825,1828],"math",{"xmlns":1826,"display":1827},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[1829,1830,1831,1946],"semantics",{},[1832,1833,1834,1845,1849,1853,1861,1865,1868,1870,1876,1886,1888,1892,1900,1902,1905,1907,1913,1915,1917,1919,1926,1928,1931,1934,1936,1942,1944],"mrow",{},[1835,1836,1837,1842],"msub",{},[1838,1839,1841],"mi",{"mathvariant":1840},"bold","X",[1838,1843,1844],{},"n",[1846,1847,1848],"mo",{},"=",[1846,1850,1852],{"stretchy":1851},"false","(",[1835,1854,1855,1857],{},[1838,1856,1841],{},[1858,1859,1860],"mn",{},"1",[1846,1862,1864],{"separator":1863},"true",",",[1846,1866,1867],{},"…",[1846,1869,1864],{"separator":1863},[1835,1871,1872,1874],{},[1838,1873,1841],{},[1838,1875,1844],{},[1877,1878,1879,1882],"msup",{},[1846,1880,1881],{"stretchy":1851},")",[1838,1883,1885],{"mathvariant":1884},"normal","⊤",[1846,1887,1864],{"separator":1863},[1889,1890],"mspace",{"width":1891},"2em",[1835,1893,1894,1898],{},[1838,1895,1897],{"mathvariant":1896},"bold-italic","μ",[1838,1899,1844],{},[1846,1901,1848],{},[1838,1903,1904],{},"E",[1846,1906,1852],{"stretchy":1851},[1835,1908,1909,1911],{},[1838,1910,1841],{"mathvariant":1840},[1838,1912,1844],{},[1846,1914,1881],{"stretchy":1851},[1846,1916,1864],{"separator":1863},[1889,1918],{"width":1891},[1835,1920,1921,1924],{},[1838,1922,1923],{"mathvariant":1884},"Γ",[1838,1925,1844],{},[1846,1927,1848],{},[1838,1929,1930],{"mathvariant":1884},"Cov",[1846,1932,1933],{},"⁡",[1846,1935,1852],{"stretchy":1851},[1835,1937,1938,1940],{},[1838,1939,1841],{"mathvariant":1840},[1838,1941,1844],{},[1846,1943,1881],{"stretchy":1851},[1846,1945,1864],{"separator":1863},[1947,1948,1950],"annotation",{"encoding":1949},"application\u002Fx-tex","\\mathbf X_n=(X_1,\\ldots,X_n)^\\top,\\qquad\n\\boldsymbol\\mu_n=E(\\mathbf X_n),\\qquad\n\\Gamma_n=\\operatorname{Cov}(\\mathbf X_n),",[1811,1952,1955,2034,2250,2365],{"className":1953,"ariaHidden":1863},[1954],"katex-html",[1811,1956,1959,1964,2023,2027,2031],{"className":1957},[1958],"base",[1811,1960],{"className":1961,"style":1963},[1962],"strut","height:0.8361em;vertical-align:-0.15em;",[1811,1965,1968,1972],{"className":1966},[1967],"mord",[1811,1969,1841],{"className":1970},[1967,1971],"mathbf",[1811,1973,1976],{"className":1974},[1975],"msupsub",[1811,1977,1981,2014],{"className":1978},[1979,1980],"vlist-t","vlist-t2",[1811,1982,1985,2009],{"className":1983},[1984],"vlist-r",[1811,1986,1990],{"className":1987,"style":1989},[1988],"vlist","height:0.1514em;",[1811,1991,1993,1998],{"style":1992},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1811,1994],{"className":1995,"style":1997},[1996],"pstrut","height:2.7em;",[1811,1999,2005],{"className":2000},[2001,2002,2003,2004],"sizing","reset-size6","size3","mtight",[1811,2006,1844],{"className":2007},[1967,2008,2004],"mathnormal",[1811,2010,2013],{"className":2011},[2012],"vlist-s","​",[1811,2015,2017],{"className":2016},[1984],[1811,2018,2021],{"className":2019,"style":2020},[1988],"height:0.15em;",[1811,2022],{},[1811,2024],{"className":2025,"style":2026},[1889],"margin-right:0.2778em;",[1811,2028,1848],{"className":2029},[2030],"mrel",[1811,2032],{"className":2033,"style":2026},[1889],[1811,2035,2037,2041,2045,2088,2092,2096,2100,2103,2106,2109,2149,2181,2184,2188,2191,2241,2244,2247],{"className":2036},[1958],[1811,2038],{"className":2039,"style":2040},[1962],"height:1.1491em;vertical-align:-0.25em;",[1811,2042,1852],{"className":2043},[2044],"mopen",[1811,2046,2048,2052],{"className":2047},[1967],[1811,2049,1841],{"className":2050,"style":2051},[1967,2008],"margin-right:0.0785em;",[1811,2053,2055],{"className":2054},[1975],[1811,2056,2058,2080],{"className":2057},[1979,1980],[1811,2059,2061,2077],{"className":2060},[1984],[1811,2062,2065],{"className":2063,"style":2064},[1988],"height:0.3011em;",[1811,2066,2068,2071],{"style":2067},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1811,2069],{"className":2070,"style":1997},[1996],[1811,2072,2074],{"className":2073},[2001,2002,2003,2004],[1811,2075,1860],{"className":2076},[1967,2004],[1811,2078,2013],{"className":2079},[2012],[1811,2081,2083],{"className":2082},[1984],[1811,2084,2086],{"className":2085,"style":2020},[1988],[1811,2087],{},[1811,2089,1864],{"className":2090},[2091],"mpunct",[1811,2093],{"className":2094,"style":2095},[1889],"margin-right:0.1667em;",[1811,2097,1867],{"className":2098},[2099],"minner",[1811,2101],{"className":2102,"style":2095},[1889],[1811,2104,1864],{"className":2105},[2091],[1811,2107],{"className":2108,"style":2095},[1889],[1811,2110,2112,2115],{"className":2111},[1967],[1811,2113,1841],{"className":2114,"style":2051},[1967,2008],[1811,2116,2118],{"className":2117},[1975],[1811,2119,2121,2141],{"className":2120},[1979,1980],[1811,2122,2124,2138],{"className":2123},[1984],[1811,2125,2127],{"className":2126,"style":1989},[1988],[1811,2128,2129,2132],{"style":2067},[1811,2130],{"className":2131,"style":1997},[1996],[1811,2133,2135],{"className":2134},[2001,2002,2003,2004],[1811,2136,1844],{"className":2137},[1967,2008,2004],[1811,2139,2013],{"className":2140},[2012],[1811,2142,2144],{"className":2143},[1984],[1811,2145,2147],{"className":2146,"style":2020},[1988],[1811,2148],{},[1811,2150,2153,2156],{"className":2151},[2152],"mclose",[1811,2154,1881],{"className":2155},[2152],[1811,2157,2159],{"className":2158},[1975],[1811,2160,2162],{"className":2161},[1979],[1811,2163,2165],{"className":2164},[1984],[1811,2166,2169],{"className":2167,"style":2168},[1988],"height:0.8991em;",[1811,2170,2172,2175],{"style":2171},"top:-3.113em;margin-right:0.05em;",[1811,2173],{"className":2174,"style":1997},[1996],[1811,2176,2178],{"className":2177},[2001,2002,2003,2004],[1811,2179,1885],{"className":2180},[1967,2004],[1811,2182,1864],{"className":2183},[2091],[1811,2185],{"className":2186,"style":2187},[1889],"margin-right:2em;",[1811,2189],{"className":2190,"style":2095},[1889],[1811,2192,2194,2204],{"className":2193},[1967],[1811,2195,2197],{"className":2196},[1967],[1811,2198,2200],{"className":2199},[1967],[1811,2201,1897],{"className":2202},[1967,2203],"boldsymbol",[1811,2205,2207],{"className":2206},[1975],[1811,2208,2210,2232],{"className":2209},[1979,1980],[1811,2211,2213,2229],{"className":2212},[1984],[1811,2214,2217],{"className":2215,"style":2216},[1988],"height:0.0573em;",[1811,2218,2220,2223],{"style":2219},"top:-2.4559em;margin-right:0.05em;",[1811,2221],{"className":2222,"style":1997},[1996],[1811,2224,2226],{"className":2225},[2001,2002,2003,2004],[1811,2227,1844],{"className":2228},[1967,2008,2004],[1811,2230,2013],{"className":2231},[2012],[1811,2233,2235],{"className":2234},[1984],[1811,2236,2239],{"className":2237,"style":2238},[1988],"height:0.2441em;",[1811,2240],{},[1811,2242],{"className":2243,"style":2026},[1889],[1811,2245,1848],{"className":2246},[2030],[1811,2248],{"className":2249,"style":2026},[1889],[1811,2251,2253,2257,2261,2264,2304,2307,2310,2313,2316,2356,2359,2362],{"className":2252},[1958],[1811,2254],{"className":2255,"style":2256},[1962],"height:1em;vertical-align:-0.25em;",[1811,2258,1904],{"className":2259,"style":2260},[1967,2008],"margin-right:0.0576em;",[1811,2262,1852],{"className":2263},[2044],[1811,2265,2267,2270],{"className":2266},[1967],[1811,2268,1841],{"className":2269},[1967,1971],[1811,2271,2273],{"className":2272},[1975],[1811,2274,2276,2296],{"className":2275},[1979,1980],[1811,2277,2279,2293],{"className":2278},[1984],[1811,2280,2282],{"className":2281,"style":1989},[1988],[1811,2283,2284,2287],{"style":1992},[1811,2285],{"className":2286,"style":1997},[1996],[1811,2288,2290],{"className":2289},[2001,2002,2003,2004],[1811,2291,1844],{"className":2292},[1967,2008,2004],[1811,2294,2013],{"className":2295},[2012],[1811,2297,2299],{"className":2298},[1984],[1811,2300,2302],{"className":2301,"style":2020},[1988],[1811,2303],{},[1811,2305,1881],{"className":2306},[2152],[1811,2308,1864],{"className":2309},[2091],[1811,2311],{"className":2312,"style":2187},[1889],[1811,2314],{"className":2315,"style":2095},[1889],[1811,2317,2319,2322],{"className":2318},[1967],[1811,2320,1923],{"className":2321},[1967],[1811,2323,2325],{"className":2324},[1975],[1811,2326,2328,2348],{"className":2327},[1979,1980],[1811,2329,2331,2345],{"className":2330},[1984],[1811,2332,2334],{"className":2333,"style":1989},[1988],[1811,2335,2336,2339],{"style":1992},[1811,2337],{"className":2338,"style":1997},[1996],[1811,2340,2342],{"className":2341},[2001,2002,2003,2004],[1811,2343,1844],{"className":2344},[1967,2008,2004],[1811,2346,2013],{"className":2347},[2012],[1811,2349,2351],{"className":2350},[1984],[1811,2352,2354],{"className":2353,"style":2020},[1988],[1811,2355],{},[1811,2357],{"className":2358,"style":2026},[1889],[1811,2360,1848],{"className":2361},[2030],[1811,2363],{"className":2364,"style":2026},[1889],[1811,2366,2368,2371,2380,2383,2423,2426],{"className":2367},[1958],[1811,2369],{"className":2370,"style":2256},[1962],[1811,2372,2375],{"className":2373},[2374],"mop",[1811,2376,1930],{"className":2377,"style":2379},[1967,2378],"mathrm","margin-right:0.0139em;",[1811,2381,1852],{"className":2382},[2044],[1811,2384,2386,2389],{"className":2385},[1967],[1811,2387,1841],{"className":2388},[1967,1971],[1811,2390,2392],{"className":2391},[1975],[1811,2393,2395,2415],{"className":2394},[1979,1980],[1811,2396,2398,2412],{"className":2397},[1984],[1811,2399,2401],{"className":2400,"style":1989},[1988],[1811,2402,2403,2406],{"style":1992},[1811,2404],{"className":2405,"style":1997},[1996],[1811,2407,2409],{"className":2408},[2001,2002,2003,2004],[1811,2410,1844],{"className":2411},[1967,2008,2004],[1811,2413,2013],{"className":2414},[2012],[1811,2416,2418],{"className":2417},[1984],[1811,2419,2421],{"className":2420,"style":2020},[1988],[1811,2422],{},[1811,2424,1881],{"className":2425},[2152],[1811,2427,1864],{"className":2428},[2091],[1799,2430,2431],{},"the course repeatedly asks four questions:",[2433,2434,2435,2511,2585,2588],"ol",{},[2436,2437,2438,2439,2510],"li",{},"What structure must ",[1811,2440,2442,2460],{"className":2441},[1818],[1811,2443,2445],{"className":2444},[1822],[1824,2446,2447],{"xmlns":1826},[1829,2448,2449,2457],{},[1832,2450,2451],{},[1835,2452,2453,2455],{},[1838,2454,1923],{"mathvariant":1884},[1838,2456,1844],{},[1947,2458,2459],{"encoding":1949},"\\Gamma_n",[1811,2461,2463],{"className":2462,"ariaHidden":1863},[1954],[1811,2464,2466,2470],{"className":2465},[1958],[1811,2467],{"className":2468,"style":2469},[1962],"height:0.8333em;vertical-align:-0.15em;",[1811,2471,2473,2476],{"className":2472},[1967],[1811,2474,1923],{"className":2475},[1967],[1811,2477,2479],{"className":2478},[1975],[1811,2480,2482,2502],{"className":2481},[1979,1980],[1811,2483,2485,2499],{"className":2484},[1984],[1811,2486,2488],{"className":2487,"style":1989},[1988],[1811,2489,2490,2493],{"style":1992},[1811,2491],{"className":2492,"style":1997},[1996],[1811,2494,2496],{"className":2495},[2001,2002,2003,2004],[1811,2497,1844],{"className":2498},[1967,2008,2004],[1811,2500,2013],{"className":2501},[2012],[1811,2503,2505],{"className":2504},[1984],[1811,2506,2508],{"className":2507,"style":2020},[1988],[1811,2509],{}," have?",[2436,2512,2513,2514,2584],{},"Which linear operator generates or transforms ",[1811,2515,2517,2535],{"className":2516},[1818],[1811,2518,2520],{"className":2519},[1822],[1824,2521,2522],{"xmlns":1826},[1829,2523,2524,2532],{},[1832,2525,2526],{},[1835,2527,2528,2530],{},[1838,2529,1841],{"mathvariant":1840},[1838,2531,1844],{},[1947,2533,2534],{"encoding":1949},"\\mathbf X_n",[1811,2536,2538],{"className":2537,"ariaHidden":1863},[1954],[1811,2539,2541,2544],{"className":2540},[1958],[1811,2542],{"className":2543,"style":1963},[1962],[1811,2545,2547,2550],{"className":2546},[1967],[1811,2548,1841],{"className":2549},[1967,1971],[1811,2551,2553],{"className":2552},[1975],[1811,2554,2556,2576],{"className":2555},[1979,1980],[1811,2557,2559,2573],{"className":2558},[1984],[1811,2560,2562],{"className":2561,"style":1989},[1988],[1811,2563,2564,2567],{"style":1992},[1811,2565],{"className":2566,"style":1997},[1996],[1811,2568,2570],{"className":2569},[2001,2002,2003,2004],[1811,2571,1844],{"className":2572},[1967,2008,2004],[1811,2574,2013],{"className":2575},[2012],[1811,2577,2579],{"className":2578},[1984],[1811,2580,2582],{"className":2581,"style":2020},[1988],[1811,2583],{},"?",[2436,2586,2587],{},"Which normal equations define an estimator or predictor?",[2436,2589,2590],{},"Which matrix recursion propagates uncertainty?",[2592,2593,2595],"tip",{"title":2594},"The matrix-first rule","Whenever a scalar formula appears, identify its vector, matrix, and quadratic-form version. The scalar formula builds intuition; the matrix form reveals what generalises.",[1794,2597,2599],{"id":2598},"preparation","Preparation",[1799,2601,2602],{},"You should already be able to:",[2604,2605,2606,2609,2672,2675,2678],"ul",{},[2436,2607,2608],{},"calculate expectation, variance, covariance, and conditional expectation;",[2436,2610,2611,2612,2671],{},"multiply partitioned matrices and solve ",[1811,2613,2615,2638],{"className":2614},[1818],[1811,2616,2618],{"className":2617},[1822],[1824,2619,2620],{"xmlns":1826},[1829,2621,2622,2635],{},[1832,2623,2624,2627,2630,2632],{},[1838,2625,2626],{"mathvariant":1840},"A",[1838,2628,2629],{"mathvariant":1840},"x",[1846,2631,1848],{},[1838,2633,2634],{"mathvariant":1840},"b",[1947,2636,2637],{"encoding":1949},"\\mathbf A\\mathbf x=\\mathbf b",[1811,2639,2641,2661],{"className":2640,"ariaHidden":1863},[1954],[1811,2642,2644,2648,2652,2655,2658],{"className":2643},[1958],[1811,2645],{"className":2646,"style":2647},[1962],"height:0.6861em;",[1811,2649,2651],{"className":2650},[1967,1971],"Ax",[1811,2653],{"className":2654,"style":2026},[1889],[1811,2656,1848],{"className":2657},[2030],[1811,2659],{"className":2660,"style":2026},[1889],[1811,2662,2664,2668],{"className":2663},[1958],[1811,2665],{"className":2666,"style":2667},[1962],"height:0.6944em;",[1811,2669,2634],{"className":2670},[1967,1971],";",[2436,2673,2674],{},"interpret eigenvalues, positive definiteness, and a Cholesky factor;",[2436,2676,2677],{},"use likelihood, least squares, and asymptotic standard errors;",[2436,2679,2680],{},"read elementary R code.",[1799,2682,2683,2684,2689],{},"If stationarity is unfamiliar, begin with the ",[2685,2686,2688],"a",{"href":2687},".\u002F01-stationary","30-minute preparation note",".",[2691,2692,2694,3126],"legacy-details",{"title":2693},"Ten-minute diagnostic",[2433,2695,2696,2699,2851,2988,3123],{},[2436,2697,2698],{},"Why must every covariance matrix be positive semidefinite?",[2436,2700,2701,2702,2850],{},"If ",[1811,2703,2705,2745],{"className":2704},[1818],[1811,2706,2708],{"className":2707},[1822],[1824,2709,2710],{"xmlns":1826},[1829,2711,2712,2742],{},[1832,2713,2714,2726,2728,2731,2733,2735,2738,2740],{},[1835,2715,2716,2718],{},[1838,2717,1923],{"mathvariant":1884},[1832,2719,2720,2723],{},[1838,2721,2722],{},"i",[1838,2724,2725],{},"j",[1846,2727,1848],{},[1838,2729,2730],{},"γ",[1846,2732,1852],{"stretchy":1851},[1838,2734,2722],{},[1846,2736,2737],{},"−",[1838,2739,2725],{},[1846,2741,1881],{"stretchy":1851},[1947,2743,2744],{"encoding":1949},"\\Gamma_{ij}=\\gamma(i-j)",[1811,2746,2748,2811,2838],{"className":2747,"ariaHidden":1863},[1954],[1811,2749,2751,2755,2802,2805,2808],{"className":2750},[1958],[1811,2752],{"className":2753,"style":2754},[1962],"height:0.9694em;vertical-align:-0.2861em;",[1811,2756,2758,2761],{"className":2757},[1967],[1811,2759,1923],{"className":2760},[1967],[1811,2762,2764],{"className":2763},[1975],[1811,2765,2767,2793],{"className":2766},[1979,1980],[1811,2768,2770,2790],{"className":2769},[1984],[1811,2771,2774],{"className":2772,"style":2773},[1988],"height:0.3117em;",[1811,2775,2776,2779],{"style":1992},[1811,2777],{"className":2778,"style":1997},[1996],[1811,2780,2782],{"className":2781},[2001,2002,2003,2004],[1811,2783,2785],{"className":2784},[1967,2004],[1811,2786,2789],{"className":2787,"style":2788},[1967,2008,2004],"margin-right:0.0572em;","ij",[1811,2791,2013],{"className":2792},[2012],[1811,2794,2796],{"className":2795},[1984],[1811,2797,2800],{"className":2798,"style":2799},[1988],"height:0.2861em;",[1811,2801],{},[1811,2803],{"className":2804,"style":2026},[1889],[1811,2806,1848],{"className":2807},[2030],[1811,2809],{"className":2810,"style":2026},[1889],[1811,2812,2814,2817,2821,2824,2827,2831,2835],{"className":2813},[1958],[1811,2815],{"className":2816,"style":2256},[1962],[1811,2818,2730],{"className":2819,"style":2820},[1967,2008],"margin-right:0.0556em;",[1811,2822,1852],{"className":2823},[2044],[1811,2825,2722],{"className":2826},[1967,2008],[1811,2828],{"className":2829,"style":2830},[1889],"margin-right:0.2222em;",[1811,2832,2737],{"className":2833},[2834],"mbin",[1811,2836],{"className":2837,"style":2830},[1889],[1811,2839,2841,2844,2847],{"className":2840},[1958],[1811,2842],{"className":2843,"style":2256},[1962],[1811,2845,2725],{"className":2846,"style":2788},[1967,2008],[1811,2848,1881],{"className":2849},[2152],", what pattern appears along each diagonal?",[2436,2852,2853,2854,2925,2926,2584],{},"What does ",[1811,2855,2857,2884],{"className":2856},[1818],[1811,2858,2860],{"className":2859},[1822],[1824,2861,2862],{"xmlns":1826},[1829,2863,2864,2881],{},[1832,2865,2866,2869,2871,2874,2876,2879],{},[1838,2867,2868],{},"ρ",[1846,2870,1852],{"stretchy":1851},[1838,2872,2873],{},"F",[1846,2875,1881],{"stretchy":1851},[1846,2877,2878],{},"\u003C",[1858,2880,1860],{},[1947,2882,2883],{"encoding":1949},"\\rho(F)\u003C1",[1811,2885,2887,2915],{"className":2886,"ariaHidden":1863},[1954],[1811,2888,2890,2893,2896,2899,2903,2906,2909,2912],{"className":2889},[1958],[1811,2891],{"className":2892,"style":2256},[1962],[1811,2894,2868],{"className":2895},[1967,2008],[1811,2897,1852],{"className":2898},[2044],[1811,2900,2873],{"className":2901,"style":2902},[1967,2008],"margin-right:0.1389em;",[1811,2904,1881],{"className":2905},[2152],[1811,2907],{"className":2908,"style":2026},[1889],[1811,2910,2878],{"className":2911},[2030],[1811,2913],{"className":2914,"style":2026},[1889],[1811,2916,2918,2922],{"className":2917},[1958],[1811,2919],{"className":2920,"style":2921},[1962],"height:0.6444em;",[1811,2923,1860],{"className":2924},[1967]," mean for powers ",[1811,2927,2929,2948],{"className":2928},[1818],[1811,2930,2932],{"className":2931},[1822],[1824,2933,2934],{"xmlns":1826},[1829,2935,2936,2945],{},[1832,2937,2938],{},[1877,2939,2940,2942],{},[1838,2941,2873],{},[1838,2943,2944],{},"h",[1947,2946,2947],{"encoding":1949},"F^h",[1811,2949,2951],{"className":2950,"ariaHidden":1863},[1954],[1811,2952,2954,2958],{"className":2953},[1958],[1811,2955],{"className":2956,"style":2957},[1962],"height:0.8491em;",[1811,2959,2961,2964],{"className":2960},[1967],[1811,2962,2873],{"className":2963,"style":2902},[1967,2008],[1811,2965,2967],{"className":2966},[1975],[1811,2968,2970],{"className":2969},[1979],[1811,2971,2973],{"className":2972},[1984],[1811,2974,2976],{"className":2975,"style":2957},[1988],[1811,2977,2979,2982],{"style":2978},"top:-3.063em;margin-right:0.05em;",[1811,2980],{"className":2981,"style":1997},[1996],[1811,2983,2985],{"className":2984},[2001,2002,2003,2004],[1811,2986,2944],{"className":2987},[1967,2008,2004],[2436,2989,2990,2991,3051,3052,3122],{},"Why is solving ",[1811,2992,2994,3015],{"className":2993},[1818],[1811,2995,2997],{"className":2996},[1822],[1824,2998,2999],{"xmlns":1826},[1829,3000,3001,3012],{},[1832,3002,3003,3005,3007,3009],{},[1838,3004,1923],{"mathvariant":1884},[1838,3006,2685],{"mathvariant":1840},[1846,3008,1848],{},[1838,3010,3011],{"mathvariant":1840},"g",[1947,3013,3014],{"encoding":1949},"\\Gamma\\mathbf a=\\mathbf g",[1811,3016,3018,3040],{"className":3017,"ariaHidden":1863},[1954],[1811,3019,3021,3025,3028,3031,3034,3037],{"className":3020},[1958],[1811,3022],{"className":3023,"style":3024},[1962],"height:0.6833em;",[1811,3026,1923],{"className":3027},[1967],[1811,3029,2685],{"className":3030},[1967,1971],[1811,3032],{"className":3033,"style":2026},[1889],[1811,3035,1848],{"className":3036},[2030],[1811,3038],{"className":3039,"style":2026},[1889],[1811,3041,3043,3047],{"className":3042},[1958],[1811,3044],{"className":3045,"style":3046},[1962],"height:0.6389em;vertical-align:-0.1944em;",[1811,3048,3011],{"className":3049,"style":3050},[1967,1971],"margin-right:0.016em;"," preferable to computing ",[1811,3053,3055,3077],{"className":3054},[1818],[1811,3056,3058],{"className":3057},[1822],[1824,3059,3060],{"xmlns":1826},[1829,3061,3062,3074],{},[1832,3063,3064],{},[1877,3065,3066,3068],{},[1838,3067,1923],{"mathvariant":1884},[1832,3069,3070,3072],{},[1846,3071,2737],{},[1858,3073,1860],{},[1947,3075,3076],{"encoding":1949},"\\Gamma^{-1}",[1811,3078,3080],{"className":3079,"ariaHidden":1863},[1954],[1811,3081,3083,3087],{"className":3082},[1958],[1811,3084],{"className":3085,"style":3086},[1962],"height:0.8141em;",[1811,3088,3090,3093],{"className":3089},[1967],[1811,3091,1923],{"className":3092},[1967],[1811,3094,3096],{"className":3095},[1975],[1811,3097,3099],{"className":3098},[1979],[1811,3100,3102],{"className":3101},[1984],[1811,3103,3105],{"className":3104,"style":3086},[1988],[1811,3106,3107,3110],{"style":2978},[1811,3108],{"className":3109,"style":1997},[1996],[1811,3111,3113],{"className":3112},[2001,2002,2003,2004],[1811,3114,3116,3119],{"className":3115},[1967,2004],[1811,3117,2737],{"className":3118},[1967,2004],[1811,3120,1860],{"className":3121},[1967,2004]," explicitly?",[2436,3124,3125],{},"What does a Kalman update do when an observation is missing?",[1799,3127,3128],{},"Answers: variances of all linear combinations are non-negative; the matrix is Toeplitz; the powers decay to zero; linear solves are more stable and efficient; it performs prediction but skips measurement correction.",[1794,3130,3132],{"id":3131},"learning-outcomes","Learning outcomes",[1799,3134,3135],{},"By the end, you will be able to:",[2604,3137,3138,3141,3144,3177,3180,3183,3186,3189],{},[2436,3139,3140],{},"construct and test Toeplitz covariance matrices for stationary processes;",[2436,3142,3143],{},"derive ARMA autocovariances from difference equations and lag polynomials;",[2436,3145,3146,3147,3176],{},"move between AR(",[1811,3148,3150,3163],{"className":3149},[1818],[1811,3151,3153],{"className":3152},[1822],[1824,3154,3155],{"xmlns":1826},[1829,3156,3157,3161],{},[1832,3158,3159],{},[1838,3160,1799],{},[1947,3162,1799],{"encoding":1949},[1811,3164,3166],{"className":3165,"ariaHidden":1863},[1954],[1811,3167,3169,3173],{"className":3168},[1958],[1811,3170],{"className":3171,"style":3172},[1962],"height:0.625em;vertical-align:-0.1944em;",[1811,3174,1799],{"className":3175},[1967,2008],") equations, companion matrices, roots, and impulse responses;",[2436,3178,3179],{},"obtain best linear predictors from projection normal equations;",[2436,3181,3182],{},"derive Yule–Walker, conditional least-squares, and Gaussian-likelihood estimators;",[2436,3184,3185],{},"express multivariate, seasonal, spectral, and state-space models with block matrices;",[2436,3187,3188],{},"implement each derivation in base R and check dimensions, symmetry, eigenvalues, and numerical error;",[2436,3190,3191],{},"explain which conclusions require Gaussianity and which use second moments only.",[1794,3193,3195],{"id":3194},"the-matrix-spine","The matrix spine",[3197,3198,3199,3215],"table",{},[3200,3201,3202],"thead",{},[3203,3204,3205,3209,3212],"tr",{},[3206,3207,3208],"th",{},"Statistical idea",[3206,3210,3211],{},"Matrix statement",[3206,3213,3214],{},"R operation",[3216,3217,3218,3371,3518,3709,3982,4149,4252,4263,4515,4629],"tbody",{},[3203,3219,3220,3224,3365],{},[3221,3222,3223],"td",{},"stationary covariance",[3221,3225,3226],{},[1811,3227,3229,3267],{"className":3228},[1818],[1811,3230,3232],{"className":3231},[1822],[1824,3233,3234],{"xmlns":1826},[1829,3235,3236,3264],{},[1832,3237,3238,3244,3246,3249,3251,3253,3255,3257,3259,3261],{},[1835,3239,3240,3242],{},[1838,3241,1923],{"mathvariant":1884},[1838,3243,1844],{},[1846,3245,1848],{},[1846,3247,3248],{"stretchy":1851},"[",[1838,3250,2730],{},[1846,3252,1852],{"stretchy":1851},[1838,3254,2722],{},[1846,3256,2737],{},[1838,3258,2725],{},[1846,3260,1881],{"stretchy":1851},[1846,3262,3263],{"stretchy":1851},"]",[1947,3265,3266],{"encoding":1949},"\\Gamma_n=[\\gamma(i-j)]",[1811,3268,3270,3325,3352],{"className":3269,"ariaHidden":1863},[1954],[1811,3271,3273,3276,3316,3319,3322],{"className":3272},[1958],[1811,3274],{"className":3275,"style":2469},[1962],[1811,3277,3279,3282],{"className":3278},[1967],[1811,3280,1923],{"className":3281},[1967],[1811,3283,3285],{"className":3284},[1975],[1811,3286,3288,3308],{"className":3287},[1979,1980],[1811,3289,3291,3305],{"className":3290},[1984],[1811,3292,3294],{"className":3293,"style":1989},[1988],[1811,3295,3296,3299],{"style":1992},[1811,3297],{"className":3298,"style":1997},[1996],[1811,3300,3302],{"className":3301},[2001,2002,2003,2004],[1811,3303,1844],{"className":3304},[1967,2008,2004],[1811,3306,2013],{"className":3307},[2012],[1811,3309,3311],{"className":3310},[1984],[1811,3312,3314],{"className":3313,"style":2020},[1988],[1811,3315],{},[1811,3317],{"className":3318,"style":2026},[1889],[1811,3320,1848],{"className":3321},[2030],[1811,3323],{"className":3324,"style":2026},[1889],[1811,3326,3328,3331,3334,3337,3340,3343,3346,3349],{"className":3327},[1958],[1811,3329],{"className":3330,"style":2256},[1962],[1811,3332,3248],{"className":3333},[2044],[1811,3335,2730],{"className":3336,"style":2820},[1967,2008],[1811,3338,1852],{"className":3339},[2044],[1811,3341,2722],{"className":3342},[1967,2008],[1811,3344],{"className":3345,"style":2830},[1889],[1811,3347,2737],{"className":3348},[2834],[1811,3350],{"className":3351,"style":2830},[1889],[1811,3353,3355,3358,3361],{"className":3354},[1958],[1811,3356],{"className":3357,"style":2256},[1962],[1811,3359,2725],{"className":3360,"style":2788},[1967,2008],[1811,3362,3364],{"className":3363},[2152],")]",[3221,3366,3367],{},[3368,3369,3370],"code",{},"toeplitz(gamma)",[3203,3372,3373,3376,3513],{},[3221,3374,3375],{},"valid covariance",[3221,3377,3378],{},[1811,3379,3381,3413],{"className":3380},[1818],[1811,3382,3384],{"className":3383},[1822],[1824,3385,3386],{"xmlns":1826},[1829,3387,3388,3410],{},[1832,3389,3390,3396,3402,3404,3407],{},[1877,3391,3392,3394],{},[1838,3393,2685],{"mathvariant":1840},[1838,3395,1885],{"mathvariant":1884},[1835,3397,3398,3400],{},[1838,3399,1923],{"mathvariant":1884},[1838,3401,1844],{},[1838,3403,2685],{"mathvariant":1840},[1846,3405,3406],{},"≥",[1858,3408,3409],{},"0",[1947,3411,3412],{"encoding":1949},"\\mathbf a^\\top\\Gamma_n\\mathbf a\\ge0",[1811,3414,3416,3504],{"className":3415,"ariaHidden":1863},[1954],[1811,3417,3419,3423,3452,3492,3495,3498,3501],{"className":3418},[1958],[1811,3420],{"className":3421,"style":3422},[1962],"height:0.9991em;vertical-align:-0.15em;",[1811,3424,3426,3429],{"className":3425},[1967],[1811,3427,2685],{"className":3428},[1967,1971],[1811,3430,3432],{"className":3431},[1975],[1811,3433,3435],{"className":3434},[1979],[1811,3436,3438],{"className":3437},[1984],[1811,3439,3441],{"className":3440,"style":2957},[1988],[1811,3442,3443,3446],{"style":2978},[1811,3444],{"className":3445,"style":1997},[1996],[1811,3447,3449],{"className":3448},[2001,2002,2003,2004],[1811,3450,1885],{"className":3451},[1967,2004],[1811,3453,3455,3458],{"className":3454},[1967],[1811,3456,1923],{"className":3457},[1967],[1811,3459,3461],{"className":3460},[1975],[1811,3462,3464,3484],{"className":3463},[1979,1980],[1811,3465,3467,3481],{"className":3466},[1984],[1811,3468,3470],{"className":3469,"style":1989},[1988],[1811,3471,3472,3475],{"style":1992},[1811,3473],{"className":3474,"style":1997},[1996],[1811,3476,3478],{"className":3477},[2001,2002,2003,2004],[1811,3479,1844],{"className":3480},[1967,2008,2004],[1811,3482,2013],{"className":3483},[2012],[1811,3485,3487],{"className":3486},[1984],[1811,3488,3490],{"className":3489,"style":2020},[1988],[1811,3491],{},[1811,3493,2685],{"className":3494},[1967,1971],[1811,3496],{"className":3497,"style":2026},[1889],[1811,3499,3406],{"className":3500},[2030],[1811,3502],{"className":3503,"style":2026},[1889],[1811,3505,3507,3510],{"className":3506},[1958],[1811,3508],{"className":3509,"style":2921},[1962],[1811,3511,3409],{"className":3512},[1967],[3221,3514,3515],{},[3368,3516,3517],{},"eigen(..., symmetric=TRUE)",[3203,3519,3520,3523,3700],{},[3221,3521,3522],{},"Gaussian simulation",[3221,3524,3525,3613,3614],{},[1811,3526,3528,3555],{"className":3527},[1818],[1811,3529,3531],{"className":3530},[1822],[1824,3532,3533],{"xmlns":1826},[1829,3534,3535,3552],{},[1832,3536,3537,3539,3541,3543,3546,3549],{},[1838,3538,1841],{"mathvariant":1840},[1846,3540,1848],{},[1838,3542,1897],{"mathvariant":1896},[1846,3544,3545],{},"+",[1838,3547,3548],{},"L",[1838,3550,3551],{"mathvariant":1840},"z",[1947,3553,3554],{"encoding":1949},"\\mathbf X=\\boldsymbol\\mu+L\\mathbf z",[1811,3556,3558,3576,3601],{"className":3557,"ariaHidden":1863},[1954],[1811,3559,3561,3564,3567,3570,3573],{"className":3560},[1958],[1811,3562],{"className":3563,"style":2647},[1962],[1811,3565,1841],{"className":3566},[1967,1971],[1811,3568],{"className":3569,"style":2026},[1889],[1811,3571,1848],{"className":3572},[2030],[1811,3574],{"className":3575,"style":2026},[1889],[1811,3577,3579,3583,3592,3595,3598],{"className":3578},[1958],[1811,3580],{"className":3581,"style":3582},[1962],"height:0.7778em;vertical-align:-0.1944em;",[1811,3584,3586],{"className":3585},[1967],[1811,3587,3589],{"className":3588},[1967],[1811,3590,1897],{"className":3591},[1967,2203],[1811,3593],{"className":3594,"style":2830},[1889],[1811,3596,3545],{"className":3597},[2834],[1811,3599],{"className":3600,"style":2830},[1889],[1811,3602,3604,3607,3610],{"className":3603},[1958],[1811,3605],{"className":3606,"style":3024},[1962],[1811,3608,3548],{"className":3609},[1967,2008],[1811,3611,3551],{"className":3612},[1967,1971],", ",[1811,3615,3617,3641],{"className":3616},[1818],[1811,3618,3620],{"className":3619},[1822],[1824,3621,3622],{"xmlns":1826},[1829,3623,3624,3638],{},[1832,3625,3626,3628,3634,3636],{},[1838,3627,3548],{},[1877,3629,3630,3632],{},[1838,3631,3548],{},[1838,3633,1885],{"mathvariant":1884},[1846,3635,1848],{},[1838,3637,1923],{"mathvariant":1884},[1947,3639,3640],{"encoding":1949},"LL^\\top=\\Gamma",[1811,3642,3644,3691],{"className":3643,"ariaHidden":1863},[1954],[1811,3645,3647,3650,3653,3682,3685,3688],{"className":3646},[1958],[1811,3648],{"className":3649,"style":2957},[1962],[1811,3651,3548],{"className":3652},[1967,2008],[1811,3654,3656,3659],{"className":3655},[1967],[1811,3657,3548],{"className":3658},[1967,2008],[1811,3660,3662],{"className":3661},[1975],[1811,3663,3665],{"className":3664},[1979],[1811,3666,3668],{"className":3667},[1984],[1811,3669,3671],{"className":3670,"style":2957},[1988],[1811,3672,3673,3676],{"style":2978},[1811,3674],{"className":3675,"style":1997},[1996],[1811,3677,3679],{"className":3678},[2001,2002,2003,2004],[1811,3680,1885],{"className":3681},[1967,2004],[1811,3683],{"className":3684,"style":2026},[1889],[1811,3686,1848],{"className":3687},[2030],[1811,3689],{"className":3690,"style":2026},[1889],[1811,3692,3694,3697],{"className":3693},[1958],[1811,3695],{"className":3696,"style":3024},[1962],[1811,3698,1923],{"className":3699},[1967],[3221,3701,3702,3705,3706],{},[3368,3703,3704],{},"chol"," and ",[3368,3707,3708],{},"%*%",[3203,3710,3711,3743,3974],{},[3221,3712,3713,3714,3742],{},"AR(",[1811,3715,3717,3730],{"className":3716},[1818],[1811,3718,3720],{"className":3719},[1822],[1824,3721,3722],{"xmlns":1826},[1829,3723,3724,3728],{},[1832,3725,3726],{},[1838,3727,1799],{},[1947,3729,1799],{"encoding":1949},[1811,3731,3733],{"className":3732,"ariaHidden":1863},[1954],[1811,3734,3736,3739],{"className":3735},[1958],[1811,3737],{"className":3738,"style":3172},[1962],[1811,3740,1799],{"className":3741},[1967,2008],") recursion",[3221,3744,3745],{},[1811,3746,3748,3796],{"className":3747},[1818],[1811,3749,3751],{"className":3750},[1822],[1824,3752,3753],{"xmlns":1826},[1829,3754,3755,3793],{},[1832,3756,3757,3765,3767,3769,3781,3783,3786],{},[1835,3758,3759,3762],{},[1838,3760,3761],{"mathvariant":1840},"s",[1838,3763,3764],{},"t",[1846,3766,1848],{},[1838,3768,2873],{},[1835,3770,3771,3773],{},[1838,3772,3761],{"mathvariant":1840},[1832,3774,3775,3777,3779],{},[1838,3776,3764],{},[1846,3778,2737],{},[1858,3780,1860],{},[1846,3782,3545],{},[1838,3784,3785],{},"G",[1835,3787,3788,3791],{},[1838,3789,3790],{},"ε",[1838,3792,3764],{},[1947,3794,3795],{"encoding":1949},"\\mathbf s_t=F\\mathbf 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and a Schur complement",[3203,4253,4254,4257,4260],{},[3221,4255,4256],{},"Gaussian likelihood",[3221,4258,4259],{},"$\\log",[3221,4261,4262],{},"\\Sigma_\\theta",[3203,4264,4265,4268,4510],{},[3221,4266,4267],{},"VAR 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a_t,P_t)",[1811,4672,4674],{"className":4673,"ariaHidden":1863},[1954],[1811,4675,4677,4680,4683,4723,4726,4729,4769],{"className":4676},[1958],[1811,4678],{"className":4679,"style":2256},[1962],[1811,4681,1852],{"className":4682},[2044],[1811,4684,4686,4689],{"className":4685},[1967],[1811,4687,2685],{"className":4688},[1967,1971],[1811,4690,4692],{"className":4691},[1975],[1811,4693,4695,4715],{"className":4694},[1979,1980],[1811,4696,4698,4712],{"className":4697},[1984],[1811,4699,4701],{"className":4700,"style":3824},[1988],[1811,4702,4703,4706],{"style":1992},[1811,4704],{"className":4705,"style":1997},[1996],[1811,4707,4709],{"className":4708},[2001,2002,2003,2004],[1811,4710,3764],{"className":4711},[1967,2008,2004],[1811,4713,2013],{"className":4714},[2012],[1811,4716,4718],{"className":4717},[1984],[1811,4719,4721],{"className":4720,"style":2020},[1988],[1811,4722],{},[1811,4724,1864],{"className":4725},[2091],[1811,4727],{"className":4728,"style":2095},[1889],[1811,4730,4732,4735],{"className":4731},[1967],[1811,4733,4663],{"className":4734,"style":2902},[1967,2008],[1811,4736,4738],{"className":4737},[1975],[1811,4739,4741,4761],{"className":4740},[1979,1980],[1811,4742,4744,4758],{"className":4743},[1984],[1811,4745,4747],{"className":4746,"style":3824},[1988],[1811,4748,4749,4752],{"style":4597},[1811,4750],{"className":4751,"style":1997},[1996],[1811,4753,4755],{"className":4754},[2001,2002,2003,2004],[1811,4756,3764],{"className":4757},[1967,2008,2004],[1811,4759,2013],{"className":4760},[2012],[1811,4762,4764],{"className":4763},[1984],[1811,4765,4767],{"className":4766,"style":2020},[1988],[1811,4768],{},[1811,4770,1881],{"className":4771},[2152],[3221,4773,4774],{},"matrix 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Processes and covariance",[3221,4818,4819,4820,4889],{},"Toeplitz ",[1811,4821,4823,4840],{"className":4822},[1818],[1811,4824,4826],{"className":4825},[1822],[1824,4827,4828],{"xmlns":1826},[1829,4829,4830,4838],{},[1832,4831,4832],{},[1835,4833,4834,4836],{},[1838,4835,1923],{"mathvariant":1884},[1838,4837,1844],{},[1947,4839,2459],{"encoding":1949},[1811,4841,4843],{"className":4842,"ariaHidden":1863},[1954],[1811,4844,4846,4849],{"className":4845},[1958],[1811,4847],{"className":4848,"style":2469},[1962],[1811,4850,4852,4855],{"className":4851},[1967],[1811,4853,1923],{"className":4854},[1967],[1811,4856,4858],{"className":4857},[1975],[1811,4859,4861,4881],{"className":4860},[1979,1980],[1811,4862,4864,4878],{"className":4863},[1984],[1811,4865,4867],{"className":4866,"style":1989},[1988],[1811,4868,4869,4872],{"style":1992},[1811,4870],{"className":4871,"style":1997},[1996],[1811,4873,4875],{"className":4874},[2001,2002,2003,2004],[1811,4876,1844],{"className":4877},[1967,2008,2004],[1811,4879,2013],{"className":4880},[2012],[1811,4882,4884],{"className":4883},[1984],[1811,4885,4887],{"className":4886,"style":2020},[1988],[1811,4888],{}," and Wold representation",[3221,4891,4892],{},"construct, factor, and simulate a covariance matrix",[3203,4894,4895,4901,4904],{},[3221,4896,4897],{},[2685,4898,4900],{"href":4899},"..\u002F02-arma\u002F","2. ARMA difference equations",[3221,4902,4903],{},"lag polynomials, companion form, Yule–Walker equations",[3221,4905,4906],{},"compare roots, eigenvalues, and theoretical ACF",[3203,4908,4909,4915,4918],{},[3221,4910,4911],{},[2685,4912,4914],{"href":4913},"..\u002F03-prediction\u002F","3. Linear prediction",[3221,4916,4917],{},"projection equations and state-space recursion",[3221,4919,4920],{},"compute predictor weights and error variance",[3203,4922,4923,4929,4932],{},[3221,4924,4925],{},[2685,4926,4928],{"href":4927},"..\u002F04-estimation\u002F","4. Estimation and inference",[3221,4930,4931],{},"OLS, moments, exact Gaussian likelihood",[3221,4933,4934],{},"optimise a covariance-matrix likelihood",[3203,4936,4937,4943,4946],{},[3221,4938,4939],{},[2685,4940,4942],{"href":4941},"..\u002F05-multi-ar\u002F","5. Multivariate systems",[3221,4944,4945],{},"block companion and Kronecker equations",[3221,4947,4948],{},"solve a stationary VAR covariance",[3203,4950,4951,4957,4960],{},[3221,4952,4953],{},[2685,4954,4956],{"href":4955},"..\u002F06-spectral-analysis\u002F","6. Spectral analysis",[3221,4958,4959],{},"Fourier matrix and spectral density",[3221,4961,4962],{},"recover frequencies with a discrete transform",[3203,4964,4965,4971,4974],{},[3221,4966,4967],{},[2685,4968,4970],{"href":4969},"..\u002F07-r-implementation\u002F","7. R matrix laboratory",[3221,4972,4973],{},"four integrated derivation labs",[3221,4975,4976],{},"browser-based base R notebook",[3203,4978,4979,4985,4988],{},[3221,4980,4981],{},[2685,4982,4984],{"href":4983},"..\u002F08-python-implementation\u002F","8. Optional Python appendix",[3221,4986,4987],{},"empirical implementation contrast",[3221,4989,4990],{},"forecasting workflow",[1799,4992,4993],{},"The Python appendix is supplementary. It is useful for comparing software conventions, but it is not the conceptual spine of this course.",[1794,4995,4997],{"id":4996},"a-finite-block-view-of-one-process","A finite-block view of one process",[1799,4999,5000],{},"Weak stationarity means that every finite block 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1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347z\nM347 1759 V0 H263 V1759 v1800 v1759 h84z",[1811,6008,2013],{"className":6009},[2012],[1811,6011,6013],{"className":6012},[1984],[1811,6014,6016],{"className":6015,"style":5521},[1988],[1811,6017],{},[1811,6019],{"className":6020,"style":2095},[1889],[1811,6022,2689],{"className":6023},[1967],[1799,6025,6026],{},"This one matrix connects the course:",[2604,6028,6029,6032,6035,6038,6041],{},[2436,6030,6031],{},"ARMA equations restrict its entries;",[2436,6033,6034],{},"prediction partitions it;",[2436,6036,6037],{},"Gaussian likelihood evaluates its determinant and quadratic form;",[2436,6039,6040],{},"the spectrum diagonalises its large-sample analogue;",[2436,6042,6043],{},"state-space recursions avoid constructing it all at once.",[1794,6045,6047],{"id":6046},"how-to-read-every-r-example","How to read every R example",[1799,6049,6050],{},"Each webR cell is self-contained. Read it in this order:",[2433,6052,6053,6059,6065,6071,6077],{},[2436,6054,6055,6058],{},[1803,6056,6057],{},"Dimensions:"," write the size of every vector and matrix.",[2436,6060,6061,6064],{},[1803,6062,6063],{},"Identity:"," name the equation the code implements.",[2436,6066,6067,6070],{},[1803,6068,6069],{},"Numerical check:"," inspect symmetry, eigenvalues, or reconstruction error.",[2436,6072,6073,6076],{},[1803,6074,6075],{},"Perturbation:"," change one parameter and predict the direction of change first.",[2436,6078,6079,6082],{},[1803,6080,6081],{},"Conclusion:"," state a mathematical claim, not merely “the code ran.”",[1799,6084,6085],{},"Example conclusion:",[6087,6088,6089],"blockquote",{},[1799,6090,6091,6092,6163,6164,6194,6195,6224,6225,6254],{},"The smallest eigenvalue of ",[1811,6093,6095,6114],{"className":6094},[1818],[1811,6096,6098],{"className":6097},[1822],[1824,6099,6100],{"xmlns":1826},[1829,6101,6102,6111],{},[1832,6103,6104],{},[1835,6105,6106,6108],{},[1838,6107,1923],{"mathvariant":1884},[1858,6109,6110],{},"6",[1947,6112,6113],{"encoding":1949},"\\Gamma_6",[1811,6115,6117],{"className":6116,"ariaHidden":1863},[1954],[1811,6118,6120,6123],{"className":6119},[1958],[1811,6121],{"className":6122,"style":2469},[1962],[1811,6124,6126,6129],{"className":6125},[1967],[1811,6127,1923],{"className":6128},[1967],[1811,6130,6132],{"className":6131},[1975],[1811,6133,6135,6155],{"className":6134},[1979,1980],[1811,6136,6138,6152],{"className":6137},[1984],[1811,6139,6141],{"className":6140,"style":2064},[1988],[1811,6142,6143,6146],{"style":1992},[1811,6144],{"className":6145,"style":1997},[1996],[1811,6147,6149],{"className":6148},[2001,2002,2003,2004],[1811,6150,6110],{"className":6151},[1967,2004],[1811,6153,2013],{"className":6154},[2012],[1811,6156,6158],{"className":6157},[1984],[1811,6159,6161],{"className":6160,"style":2020},[1988],[1811,6162],{}," is positive, so this six-dimensional covariance block is positive definite. As ",[1811,6165,6167,6181],{"className":6166},[1818],[1811,6168,6170],{"className":6169},[1822],[1824,6171,6172],{"xmlns":1826},[1829,6173,6174,6178],{},[1832,6175,6176],{},[1838,6177,4009],{},[1947,6179,6180],{"encoding":1949},"\\phi",[1811,6182,6184],{"className":6183,"ariaHidden":1863},[1954],[1811,6185,6187,6191],{"className":6186},[1958],[1811,6188],{"className":6189,"style":6190},[1962],"height:0.8889em;vertical-align:-0.1944em;",[1811,6192,4009],{"className":6193},[1967,2008]," moves from ",[1811,6196,6198,6212],{"className":6197},[1818],[1811,6199,6201],{"className":6200},[1822],[1824,6202,6203],{"xmlns":1826},[1829,6204,6205,6210],{},[1832,6206,6207],{},[1858,6208,6209],{},"0.6",[1947,6211,6209],{"encoding":1949},[1811,6213,6215],{"className":6214,"ariaHidden":1863},[1954],[1811,6216,6218,6221],{"className":6217},[1958],[1811,6219],{"className":6220,"style":2921},[1962],[1811,6222,6209],{"className":6223},[1967]," to ",[1811,6226,6228,6242],{"className":6227},[1818],[1811,6229,6231],{"className":6230},[1822],[1824,6232,6233],{"xmlns":1826},[1829,6234,6235,6240],{},[1832,6236,6237],{},[1858,6238,6239],{},"0.95",[1947,6241,6239],{"encoding":1949},[1811,6243,6245],{"className":6244,"ariaHidden":1863},[1954],[1811,6246,6248,6251],{"className":6247},[1958],[1811,6249],{"className":6250,"style":2921},[1962],[1811,6252,6239],{"className":6253},[1967],", the condition number increases; near-unit-root covariance matrices are harder to distinguish numerically.",[1794,6256,6258],{"id":6257},"ten-week-route","Ten-week route",[3197,6260,6261,6274],{},[3200,6262,6263],{},[3203,6264,6265,6269,6271],{},[3206,6266,6268],{"align":6267},"right","Week",[3206,6270,2599],{},[3206,6272,6273],{},"Seminar or laboratory output",[3216,6275,6276,6286,6296,6336,6347,6358,6368,6379,6390,6401],{},[3203,6277,6278,6280,6283],{},[3221,6279,1860],{"align":6267},[3221,6281,6282],{},"random vectors and covariance",[3221,6284,6285],{},"build a Toeplitz covariance by hand",[3203,6287,6288,6290,6293],{},[3221,6289,5171],{"align":6267},[3221,6291,6292],{},"stationarity and Wold representation",[3221,6294,6295],{},"prove a covariance sequence is admissible or find a counterexample",[3203,6297,6298,6301,6304],{},[3221,6299,6300],{"align":6267},"3",[3221,6302,6303],{},"AR, MA, and lag polynomials",[3221,6305,6306,6307,6335],{},"convert an AR(",[1811,6308,6310,6323],{"className":6309},[1818],[1811,6311,6313],{"className":6312},[1822],[1824,6314,6315],{"xmlns":1826},[1829,6316,6317,6321],{},[1832,6318,6319],{},[1838,6320,1799],{},[1947,6322,1799],{"encoding":1949},[1811,6324,6326],{"className":6325,"ariaHidden":1863},[1954],[1811,6327,6329,6332],{"className":6328},[1958],[1811,6330],{"className":6331,"style":3172},[1962],[1811,6333,1799],{"className":6334},[1967,2008],") to companion form",[3203,6337,6338,6341,6344],{},[3221,6339,6340],{"align":6267},"4",[3221,6342,6343],{},"Yule–Walker equations",[3221,6345,6346],{},"recover AR coefficients from autocovariances",[3203,6348,6349,6352,6355],{},[3221,6350,6351],{"align":6267},"5",[3221,6353,6354],{},"Hilbert-space projection",[3221,6356,6357],{},"derive a best linear predictor and its error variance",[3203,6359,6360,6362,6365],{},[3221,6361,6110],{"align":6267},[3221,6363,6364],{},"estimation and Gaussian likelihood",[3221,6366,6367],{},"compare conditional and exact estimators",[3203,6369,6370,6373,6376],{},[3221,6371,6372],{"align":6267},"7",[3221,6374,6375],{},"VAR and Kronecker products",[3221,6377,6378],{},"solve a discrete Lyapunov equation",[3203,6380,6381,6384,6387],{},[3221,6382,6383],{"align":6267},"8",[3221,6385,6386],{},"Fourier representation",[3221,6388,6389],{},"connect a periodogram to a Fourier matrix",[3203,6391,6392,6395,6398],{},[3221,6393,6394],{"align":6267},"9",[3221,6396,6397],{},"state-space models",[3221,6399,6400],{},"implement Kalman prediction and correction",[3203,6402,6403,6406,6409],{},[3221,6404,6405],{"align":6267},"10",[3221,6407,6408],{},"synthesis",[3221,6410,6411],{},"defend every line of one R matrix computation",[1794,6413,6415],{"id":6414},"assessment-alignment","Assessment alignment",[3197,6417,6418,6431],{},[3200,6419,6420],{},[3203,6421,6422,6425,6428],{},[3206,6423,6424],{},"Task",[3206,6426,6427],{"align":6267},"Weight",[3206,6429,6430],{},"Evidence expected",[3216,6432,6433,6444,6455,6466],{},[3203,6434,6435,6438,6441],{},[3221,6436,6437],{},"derivations",[3221,6439,6440],{"align":6267},"35%",[3221,6442,6443],{},"correct assumptions, dimensions, and algebra",[3203,6445,6446,6449,6452],{},[3221,6447,6448],{},"R matrix labs",[3221,6450,6451],{"align":6267},"30%",[3221,6453,6454],{},"reproducible code plus numerical verification",[3203,6456,6457,6460,6463],{},[3221,6458,6459],{},"theory critique",[3221,6461,6462],{"align":6267},"15%",[3221,6464,6465],{},"identify what fails without stationarity or Gaussianity",[3203,6467,6468,6471,6474],{},[3221,6469,6470],{},"final synthesis",[3221,6472,6473],{"align":6267},"20%",[3221,6475,6476],{},"connect covariance, prediction, likelihood, and state space",[1799,6478,6479],{},"Graduate extensions ask for proofs, asymptotic arguments, multivariate identification, or computational complexity. Undergraduate solutions may use stated theorems but must still explain dimensions and assumptions.",[1794,6481,6483],{"id":6482},"reading-ladder","Reading ladder",[3197,6485,6486,6496],{},[3200,6487,6488],{},[3203,6489,6490,6493],{},[3206,6491,6492],{},"Reading",[3206,6494,6495],{},"Use in this course",[3216,6497,6498,6515,6530,6546,6560,6572],{},[3203,6499,6500,6512],{},[3221,6501,6502],{},[2685,6503,6507,6508],{"href":6504,"rel":6505},"https:\u002F\u002Fdoi.org\u002F10.1007\u002F978-1-4419-0320-4",[6506],"nofollow","Brockwell & Davis, ",[6509,6510,6511],"em",{},"Time Series: Theory and Methods",[3221,6513,6514],{},"stationary processes, Hilbert-space prediction, ARMA, spectra, and asymptotics",[3203,6516,6517,6527],{},[3221,6518,6519],{},[2685,6520,6523,6524],{"href":6521,"rel":6522},"https:\u002F\u002Fdoi.org\u002F10.1093\u002Facprof:oso\u002F9780199641178.001.0001",[6506],"Durbin & Koopman, ",[6509,6525,6526],{},"Time Series Analysis by State Space Methods",[3221,6528,6529],{},"matrix filtering, smoothing, likelihood, and initialisation",[3203,6531,6532,6543],{},[3221,6533,6534],{},[2685,6535,6538,6539,6542],{"href":6536,"rel":6537},"https:\u002F\u002Fstat.ethz.ch\u002FR-manual\u002FR-devel\u002Flibrary\u002Fstats\u002Fhtml\u002FARMAacf.html",[6506],"R ",[3368,6540,6541],{},"stats::ARMAacf"," documentation",[3221,6544,6545],{},"implementation of theoretical ARMA autocorrelations from difference equations",[3203,6547,6548,6557],{},[3221,6549,6550],{},[2685,6551,6538,6554,6542],{"href":6552,"rel":6553},"https:\u002F\u002Fstat.ethz.ch\u002FR-manual\u002FR-devel\u002FRHOME\u002Flibrary\u002Fstats\u002Fhtml\u002FKalmanLike.html",[6506],[3368,6555,6556],{},"stats::KalmanLike",[3221,6558,6559],{},"the state-space machinery used by R's exact ARIMA likelihood",[3203,6561,6562,6569],{},[3221,6563,6564],{},[2685,6565,6568],{"href":6566,"rel":6567},"https:\u002F\u002Fpmc.ncbi.nlm.nih.gov\u002Farticles\u002FPMC11729849\u002F",[6506],"Düker et al. (2024\u002F25), VARMA review",[3221,6570,6571],{},"graduate bridge from classical matrix equations to modern identification and estimation",[3203,6573,6574,6581],{},[3221,6575,6576],{},[2685,6577,6580],{"href":6578,"rel":6579},"https:\u002F\u002Fdoi.org\u002F10.1080\u002F01621459.2024.2311365",[6506],"Zheng (2024\u002F25), infinite-order high-dimensional VAR",[3221,6582,6583],{},"recent example of preserving interpretable linear dynamics under high dimensionality",[1799,6585,6586],{},"The first four readings establish the course. The recent papers show where the same matrix ideas lead; they do not replace the foundations.",[1794,6588,6590],{"id":6589},"start","Start",[1799,6592,6593,6594,6597,6598,6601],{},"Complete the ",[2685,6595,6596],{"href":2687},"stationarity preparation",", then open ",[2685,6599,6600],{"href":4815},"Module 1",". Keep one page of matrix dimensions beside every derivation.",{"title":10,"searchDepth":6603,"depth":6603,"links":6604},2,[6605,6606,6607,6608,6609,6610,6611,6612,6613,6614,6615],{"id":1796,"depth":6603,"text":1797},{"id":2598,"depth":6603,"text":2599},{"id":3131,"depth":6603,"text":3132},{"id":3194,"depth":6603,"text":3195},{"id":4777,"depth":6603,"text":4778},{"id":4996,"depth":6603,"text":4997},{"id":6046,"depth":6603,"text":6047},{"id":6257,"depth":6603,"text":6258},{"id":6414,"depth":6603,"text":6415},{"id":6482,"depth":6603,"text":6483},{"id":6589,"depth":6603,"text":6590},"A matrix-first course in covariance-stationary processes, ARMA equations, linear prediction, likelihood, state space, and spectra.","md",{"sidebar":6619},{"order":6620},0,true,{"title":949,"description":6616},"wQQ8W7ZZqzh8ejaV9_RbZZeZPApKLdEldxTcALZXZgE",[6625,6627],{"title":938,"path":939,"stem":940,"description":6626,"children":-1},"Integrate reserving, frequency, severity, reinsurance, aggregate capital, and ruin in one auditable recommendation.",{"title":955,"path":956,"stem":957,"description":6628,"children":-1},"A compact diagnostic primer on stable dependence, trends, unit roots, seasonality, and breaks.",1785754737613]