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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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Their difference is evidence about finite-sample conventions, not automatically an error.",[1793,6769,6771],{"id":6770},"_3-exact-gaussian-likelihood-uses-the-whole-covariance-matrix","3. 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log likelihood 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becomes increasingly ill-conditioned. That is both a statistical boundary and a numerical warning.",[1793,8228,8230],{"id":8229},"_4-cholesky-evaluation-is-the-stable-route","4. Cholesky evaluation is the stable route",[1798,8232,8233,8234,8322,8323,8395],{},"If ",[1838,8235,8237,8262],{"className":8236},[1841],[1838,8238,8240],{"className":8239},[1845],[1847,8241,8242],{"xmlns":1849},[1851,8243,8244,8259],{},[1854,8245,8246,8248,8250,8257],{},[1857,8247,6818],{"mathvariant":2539},[1910,8249,1912],{},[3991,8251,8252,8255],{},[1857,8253,8254],{},"U",[1857,8256,3997],{"mathvariant":2539},[1857,8258,8254],{},[1860,8260,8261],{"encoding":1862},"\\Sigma=U^\\top 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then",[1838,8397,8399],{"className":8398},[1886],[1838,8400,8402,8452],{"className":8401},[1841],[1838,8403,8405],{"className":8404},[1845],[1847,8406,8407],{"xmlns":1849,"display":1895},[1851,8408,8409,8449],{},[1854,8410,8411,8413,8415,8417,8419,8421,8423,8425,8433,8435,8437,8447],{},[1857,8412,7046],{},[1910,8414,7049],{},[1857,8416,7067],{"mathvariant":2539},[1857,8418,6818],{"mathvariant":2539},[1857,8420,7067],{"mathvariant":2539},[1910,8422,1912],{},[1919,8424,2528],{},[5010,8426,8427,8430],{},[1910,8428,8429],{},"∑",[1857,8431,8432],{},"i",[1857,8434,7046],{},[1910,8436,7049],{},[1901,8438,8439,8441],{},[1857,8440,8254],{},[1854,8442,8443,8445],{},[1857,8444,8432],{},[1857,8446,8432],{},[1910,8448,1972],{"separator":1867},[1860,8450,8451],{"encoding":1862},"\\log|\\Sigma|=2\\sum_i\\log 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exact AR(1) matrix likelihood",[1798,8913,8914,8915,8918],{},"This cell simulates a centred AR(1), optimises the exact finite-sample likelihood, and compares it with R's ",[4369,8916,8917],{},"arima(..., method = \"ML\")",". Small differences arise from parameterisation and numerical optimisation, but both use the initial-state information that conditional least squares omits.",[8920,8921],"web-r",{"code64":8922,"layout":8923,"locale":7,"title":8924},"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","vertical","Optimise an exact Gaussian covariance-matrix likelihood",[1793,8926,8928],{"id":8927},"_5-joint-and-innovations-likelihoods-are-two-factorizations","5. Joint and innovations likelihoods are two factorizations",[1798,8930,8931],{},"The joint density factors as",[1838,8933,8935],{"className":8934},[1886],[1838,8936,8938,9044],{"className":8937},[1841],[1838,8939,8941],{"className":8940},[1845],[1847,8942,8943],{"xmlns":1849,"display":1895},[1851,8944,8945,9041],{},[1854,8946,8947,8949,8951,8957,8959,8961,8963,8969,8971,8973,8975,8977,8983,8985,9001,9003,9005,9011,9013,9019,9021,9023,9025,9037,9039],{},[1857,8948,1798],{},[1910,8950,3989],{"stretchy":2496},[1901,8952,8953,8955],{},[1857,8954,2501],{},[1919,8956,1921],{},[1910,8958,1972],{"separator":1867},[1910,8960,2371],{},[1910,8962,1972],{"separator":1867},[1901,8964,8965,8967],{},[1857,8966,2501],{},[1857,8968,2376],{},[1910,8970,4004],{"stretchy":2496},[1910,8972,1912],{},[1857,8974,1798],{},[1910,8976,3989],{"stretchy":2496},[1901,8978,8979,8981],{},[1857,8980,2501],{},[1919,8982,1921],{},[1910,8984,4004],{"stretchy":2496},[8986,8987,8988,8991,8999],"munderover",{},[1910,8989,8990],{},"∏",[1854,8992,8993,8995,8997],{},[1857,8994,1908],{},[1910,8996,1912],{},[1919,8998,2528],{},[1857,9000,2376],{},[1857,9002,1798],{},[1910,9004,3989],{"stretchy":2496},[1901,9006,9007,9009],{},[1857,9008,2501],{},[1857,9010,1908],{},[1910,9012,7067],{},[1901,9014,9015,9017],{},[1857,9016,2501],{},[1919,9018,1921],{},[1910,9020,1972],{"separator":1867},[1910,9022,2371],{},[1910,9024,1972],{"separator":1867},[1901,9026,9027,9029],{},[1857,9028,2501],{},[1854,9030,9031,9033,9035],{},[1857,9032,1908],{},[1910,9034,1932],{},[1919,9036,1921],{},[1910,9038,4004],{"stretchy":2496},[1857,9040,2773],{"mathvariant":2539},[1860,9042,9043],{"encoding":1862},"p(x_1,\\ldots,x_n)\n=p(x_1)\\prod_{t=2}^n p(x_t\\mid x_1,\\ldots,x_{t-1}).",[1838,9045,9047,9169,9354],{"className":9046,"ariaHidden":1867},[1866],[1838,9048,9050,9053,9056,9059,9099,9102,9105,9108,9111,9114,9117,9157,9160,9163,9166],{"className":9049},[1871],[1838,9051],{"className":9052,"style":4756},[1875],[1838,9054,1798],{"className":9055},[1880,1881],[1838,9057,3989],{"className":9058},[2814],[1838,9060,9062,9065],{"className":9061},[1880],[1838,9063,2501],{"className":9064},[1880,1881],[1838,9066,9068],{"className":9067},[1996],[1838,9069,9071,9091],{"className":9070},[2000,2001],[1838,9072,9074,9088],{"className":9073},[2005],[1838,9075,9077],{"className":9076,"style":2080},[2009],[1838,9078,9079,9082],{"style":2083},[1838,9080],{"className":9081,"style":2018},[2017],[1838,9083,9085],{"className":9084},[2022,2023,2024,2025],[1838,9086,1921],{"className":9087},[1880,2025],[1838,9089,2033],{"className":9090},[2032],[1838,9092,9094],{"className":9093},[2005],[1838,9095,9097],{"className":9096,"style":2040},[2009],[1838,9098],{},[1838,9100,1972],{"className":9101},[2341],[1838,9103],{"className":9104,"style":2437},[2046],[1838,9106,2371],{"className":9107},[2175],[1838,9109],{"className":9110,"style":2437},[2046],[1838,9112,1972],{"className":9113},[2341],[1838,9115],{"className":9116,"style":2437},[2046],[1838,9118,9120,9123],{"className":9119},[1880],[1838,9121,2501],{"className":9122},[1880,1881],[1838,9124,9126],{"className":9125},[1996],[1838,9127,9129,9149],{"className":9128},[2000,2001],[1838,9130,9132,9146],{"className":9131},[2005],[1838,9133,9135],{"className":9134,"style":2210},[2009],[1838,9136,9137,9140],{"style":2083},[1838,9138],{"className":9139,"style":2018},[2017],[1838,9141,9143],{"className":9142},[2022,2023,2024,2025],[1838,9144,2376],{"className":9145},[1880,1881,2025],[1838,9147,2033],{"className":9148},[2032],[1838,9150,9152],{"className":9151},[2005],[1838,9153,9155],{"className":9154,"style":2040},[2009],[1838,9156],{},[1838,9158,4004],{"className":9159},[3084],[1838,9161],{"className":9162,"style":2047},[2046],[1838,9164,1912],{"className":9165},[2051],[1838,9167],{"className":9168,"style":2047},[2046],[1838,9170,9172,9176,9179,9182,9222,9225,9228,9296,9299,9302,9305,9345,9348,9351],{"className":9171},[1871],[1838,9173],{"className":9174,"style":9175},[1875],"height:2.9185em;vertical-align:-1.2671em;",[1838,9177,1798],{"className":9178},[1880,1881],[1838,9180,3989],{"className":9181},[2814],[1838,9183,9185,9188],{"className":9184},[1880],[1838,9186,2501],{"className":9187},[1880,1881],[1838,9189,9191],{"className":9190},[1996],[1838,9192,9194,9214],{"className":9193},[2000,2001],[1838,9195,9197,9211],{"className":9196},[2005],[1838,9198,9200],{"className":9199,"style":2080},[2009],[1838,9201,9202,9205],{"style":2083},[1838,9203],{"className":9204,"style":2018},[2017],[1838,9206,9208],{"className":9207},[2022,2023,2024,2025],[1838,9209,1921],{"className":9210},[1880,2025],[1838,9212,2033],{"className":9213},[2032],[1838,9215,9217],{"className":9216},[2005],[1838,9218,9220],{"className":9219,"style":2040},[2009],[1838,9221],{},[1838,9223,4004],{"className":9224},[3084],[1838,9226],{"className":9227,"style":2437},[2046],[1838,9229,9231],{"className":9230},[7262,8498],[1838,9232,9234,9287],{"className":9233},[2000,2001],[1838,9235,9237,9284],{"className":9236},[2005],[1838,9238,9241,9262,9272],{"className":9239,"style":9240},[2009],"height:1.6514em;",[1838,9242,9244,9247],{"style":9243},"top:-1.8829em;margin-left:0em;",[1838,9245],{"className":9246,"style":8515},[2017],[1838,9248,9250],{"className":9249},[2022,2023,2024,2025],[1838,9251,9253,9256,9259],{"className":9252},[1880,2025],[1838,9254,1908],{"className":9255},[1880,1881,2025],[1838,9257,1912],{"className":9258},[2051,2025],[1838,9260,2528],{"className":9261},[1880,2025],[1838,9263,9264,9267],{"style":8524},[1838,9265],{"className":9266,"style":8515},[2017],[1838,9268,9269],{},[1838,9270,8990],{"className":9271},[7262,8533,8534],[1838,9273,9275,9278],{"style":9274},"top:-4.3em;margin-left:0em;",[1838,9276],{"className":9277,"style":8515},[2017],[1838,9279,9281],{"className":9280},[2022,2023,2024,2025],[1838,9282,2376],{"className":9283},[1880,1881,2025],[1838,9285,2033],{"className":9286},[2032],[1838,9288,9290],{"className":9289},[2005],[1838,9291,9294],{"className":9292,"style":9293},[2009],"height:1.2671em;",[1838,9295],{},[1838,9297],{"className":9298,"style":2437},[2046],[1838,9300,1798],{"className":9301},[1880,1881],[1838,9303,3989],{"className":9304},[2814],[1838,9306,9308,9311],{"className":9307},[1880],[1838,9309,2501],{"className":9310},[1880,1881],[1838,9312,9314],{"className":9313},[1996],[1838,9315,9317,9337],{"className":9316},[2000,2001],[1838,9318,9320,9334],{"className":9319},[2005],[1838,9321,9323],{"className":9322,"style":2010},[2009],[1838,9324,9325,9328],{"style":2083},[1838,9326],{"className":9327,"style":2018},[2017],[1838,9329,9331],{"className":9330},[2022,2023,2024,2025],[1838,9332,1908],{"className":9333},[1880,1881,2025],[1838,9335,2033],{"class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a Gaussian process, each conditional density is determined by an innovation 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Residuals estimate innovations, not truth",[1798,11031,11032,11033,11101],{},"For residual vector 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inspect",[1838,11103,11105],{"className":11104},[1886],[1838,11106,11108,11162],{"className":11107},[1841],[1838,11109,11111],{"className":11110},[1845],[1847,11112,11113],{"xmlns":1849,"display":1895},[1851,11114,11115,11159],{},[1854,11116,11117,11128,11130,11157],{},[1901,11118,11119,11125],{},[3967,11120,11121,11123],{"accent":1867},[1857,11122,5240],{"mathvariant":2539},[1910,11124,3973],{"stretchy":1867},[1857,11126,11127],{},"e",[1910,11129,1912],{},[1854,11131,11132,11134,11144,11146,11148,11150,11153,11155],{},[1910,11133,2479],{"fence":1867},[1901,11135,11136,11142],{},[3967,11137,11138,11140],{"accent":1867},[1857,11139,4691],{},[1910,11141,3973],{"stretchy":1867},[1857,11143,11127],{},[1910,11145,3989],{"stretchy":2496},[1857,11147,8432],{},[1910,11149,1932],{},[1857,11151,11152],{},"j",[1910,11154,4004],{"stretchy":2496},[1910,11156,2563],{"fence":1867},[1857,11158,2773],{"mathvariant":2539},[1860,11160,11161],{"encoding":1862},"\\widehat\\Gamma_e\n=\\left[\\widehat\\gamma_e(i-j)\\right].",[1838,11163,11165,11250],{"className":11164,"ariaHidden":1867},[1866],[1838,11166,11168,11172,11241,11244,11247],{"className":11167},[1871],[1838,11169],{"className":11170,"style":11171},[1875],"height:1.0733em;vertical-align:-0.15em;",[1838,11173,11175,11207],{"className":11174},[1880],[1838,11176,11178],{"className":11177},[1880,4105],[1838,11179,11181],{"className":11180},[2000],[1838,11182,11184],{"className":11183},[2005],[1838,11185,11187,11195],{"className":11186,"style":6597},[2009],[1838,11188,11189,11192],{"style":4118},[1838,11190],{"className":11191,"style":4122},[2017],[1838,11193,5240],{"className":11194},[1880],[1838,11196,11198,11201],{"className":11197,"style":6609},[4135],[1838,11199],{"className":11200,"style":4122},[2017],[1838,11202,11203],{"style":4142},[2841,11204,11205],{"xmlns":2843,"width":4145,"height":4146,"viewBox":4147,"preserveAspectRatio":4148},[2848,11206],{"d":4151},[1838,11208,11210],{"className":11209},[1996],[1838,11211,11213,11233],{"className":11212},[2000,2001],[1838,11214,11216,11230],{"className":11215},[2005],[1838,11217,11219],{"className":11218,"style":2210},[2009],[1838,11220,11221,11224],{"style":2083},[1838,11222],{"className":11223,"style":2018},[2017],[1838,11225,11227],{"className":11226},[2022,2023,2024,2025],[1838,11228,11127],{"className":11229},[1880,1881,2025],[1838,11231,2033],{"className":11232},[2032],[1838,11234,11236],{"className":11235},[2005],[1838,11237,11239],{"className":11238,"style":2040},[2009],[1838,11240],{},[1838,11242],{"className":11243,"style":2047},[2046],[1838,11245,1912],{"className":11246},[2051],[1838,11248],{"className":11249,"style":2047},[2046],[1838,11251,11253,11256,11370,11373],{"className":11252},[1871],[1838,11254],{"className":11255,"style":4756},[1875],[1838,11257,11259,11262,11345,11348,11351,11354,11357,11360,11364,11367],{"className":11258},[2175],[1838,11260,2479],{"className":11261,"style":7249},[2814,7248],[1838,11263,11265,11310],{"className":11264},[1880],[1838,11266,11268],{"className":11267},[1880,4105],[1838,11269,11271,11302],{"className":11270},[2000,2001],[1838,11272,11274,11299],{"className":11273},[2005],[1838,11275,11278,11286],{"className":11276,"style":11277},[2009],"height:0.6706em;",[1838,11279,11280,11283],{"style":4118},[1838,11281],{"className":11282,"style":4122},[2017],[1838,11284,4691],{"className":11285,"style":4760},[1880,1881],[1838,11287,11290,11293],{"className":11288,"style":11289},[4135],"top:-3.4306em;",[1838,11291],{"className":11292,"style":4122},[2017],[1838,11294,11295],{"style":4142},[2841,11296,11297],{"xmlns":2843,"width":4145,"height":4146,"viewBox":4147,"preserveAspectRatio":4148},[2848,11298],{"d":4151},[1838,11300,2033],{"className":11301},[2032],[1838,11303,11305],{"className":11304},[2005],[1838,11306,11308],{"className":11307,"style":4161},[2009],[1838,11309],{},[1838,11311,11313],{"className":11312},[1996],[1838,11314,11316,11337],{"className":11315},[2000,2001],[1838,11317,11319,11334],{"className":11318},[2005],[1838,11320,11322],{"className":11321,"style":2210},[2009],[1838,11323,11325,11328],{"style":11324},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[1838,11326],{"className":11327,"style":2018},[2017],[1838,11329,11331],{"className":11330},[2022,2023,2024,2025],[1838,11332,11127],{"className":11333},[1880,1881,2025],[1838,11335,2033],{"className":11336},[2032],[1838,11338,11340],{"className":11339},[2005],[1838,11341,11343],{"className":11342,"style":2040},[2009],[1838,11344],{},[1838,11346,3989],{"className":11347},[2814],[1838,11349,8432],{"className":11350},[1880,1881],[1838,11352],{"className":11353,"style":2158},[2046],[1838,11355,1932],{"className":11356},[2139],[1838,11358],{"className":11359,"style":2158},[2046],[1838,11361,11152],{"className":11362,"style":11363},[1880,1881],"margin-right:0.0572em;",[1838,11365,4004],{"className":11366},[3084],[1838,11368,2563],{"className":11369,"style":7249},[3084,7248],[1838,11371],{"className":11372,"style":2437},[2046],[1838,11374,2773],{"className":11375},[1880],[1798,11377,11378],{},"A well-specified linear mean model should leave off-diagonal entries small relative to sampling variation. Also check:",[1809,11380,11381,11384,11387,11390,11393],{},[1812,11382,11383],{},"changing variance over time;",[1812,11385,11386],{},"outliers and heavy tails;",[1812,11388,11389],{},"parameter sensitivity to endpoints;",[1812,11391,11392],{},"roots close to the unit circle;",[1812,11394,11395],{},"near cancellation of AR and MA polynomials.",[1798,11397,11398],{},"The Ljung–Box statistic compresses several sample residual correlations:",[1838,11400,11402],{"className":11401},[1886],[1838,11403,11405,11490],{"className":11404},[1841],[1838,11406,11408],{"className":11407},[1845],[1847,11409,11410],{"xmlns":1849,"display":1895},[1851,11411,11412,11487],{},[1854,11413,11414,11417,11419,11422,11424,11426,11428,11430,11432,11434,11436,11438,11452,11485],{},[1857,11415,11416],{},"Q",[1910,11418,3989],{"stretchy":2496},[1857,11420,11421],{},"m",[1910,11423,4004],{"stretchy":2496},[1910,11425,1912],{},[1857,11427,2376],{},[1910,11429,3989],{"stretchy":2496},[1857,11431,2376],{},[1910,11433,1937],{},[1919,11435,2528],{},[1910,11437,4004],{"stretchy":2496},[8986,11439,11440,11442,11450],{},[1910,11441,8429],{},[1854,11443,11444,11446,11448],{},[1857,11445,4555],{},[1910,11447,1912],{},[1919,11449,1921],{},[1857,11451,11421],{},[7032,11453,11454,11477],{},[1854,11455,11456,11467,11469,11471],{},[1901,11457,11458,11465],{},[3967,11459,11460,11463],{"accent":1867},[1857,11461,11462],{},"ρ",[1910,11464,3973],{"stretchy":1867},[1857,11466,11127],{},[1910,11468,3989],{"stretchy":2496},[1857,11470,4555],{},[3991,11472,11473,11475],{},[1910,11474,4004],{"stretchy":2496},[1919,11476,2528],{},[1854,11478,11479,11481,11483],{},[1857,11480,2376],{},[1910,11482,1932],{},[1857,11484,4555],{},[1857,11486,2773],{"mathvariant":2539},[1860,11488,11489],{"encoding":1862},"Q(m)=n(n+2)\\sum_{h=1}^m\n\\frac{\\widehat\\rho_e(h)^2}{n-h}.",[1838,11491,11493,11520,11544],{"className":11492,"ariaHidden":1867},[1866],[1838,11494,11496,11499,11502,11505,11508,11511,11514,11517],{"className":11495},[1871],[1838,11497],{"className":11498,"style":4756},[1875],[1838,11500,11416],{"className":11501},[1880,1881],[1838,11503,3989],{"className":11504},[2814],[1838,11506,11421],{"className":11507},[1880,1881],[1838,11509,4004],{"className":11510},[3084],[1838,11512],{"className":11513,"style":2047},[2046],[1838,11515,1912],{"className":11516},[2051],[1838,11518],{"className":11519,"style":2047},[2046],[1838,11521,11523,11526,11529,11532,11535,11538,11541],{"className":11522},[1871],[1838,11524],{"className":11525,"style":4756},[1875],[1838,11527,2376],{"className":11528},[1880,1881],[1838,11530,3989],{"className":11531},[2814],[1838,11533,2376],{"className":11534},[1880,1881],[1838,11536],{"className":11537,"style":2158},[2046],[1838,11539,1937],{"className":11540},[2139],[1838,11542],{"className":11543,"style":2158},[2046],[1838,11545,11547,11551,11554,11557,11560,11626,11629,11817],{"className":11546},[1871],[1838,11548],{"className":11549,"style":11550},[1875],"height:2.9535em;vertical-align:-1.3021em;",[1838,11552,2528],{"className":11553},[1880],[1838,11555,4004],{"className":11556},[3084],[1838,11558],{"className":11559,"style":2437},[2046],[1838,11561,11563],{"className":11562},[7262,8498],[1838,11564,11566,11617],{"className":11565},[2000,2001],[1838,11567,11569,11614],{"className":11568},[2005],[1838,11570,11572,11593,11603],{"className":11571,"style":9240},[2009],[1838,11573,11575,11578],{"style":11574},"top:-1.8479em;margin-left:0em;",[1838,11576],{"className":11577,"style":8515},[2017],[1838,11579,11581],{"className":11580},[2022,2023,2024,2025],[1838,11582,11584,11587,11590],{"className":11583},[1880,2025],[1838,11585,4555],{"className":11586},[1880,1881,2025],[1838,11588,1912],{"className":11589},[2051,2025],[1838,11591,1921],{"className":11592},[1880,2025],[1838,11594,11595,11598],{"style":8524},[1838,11596],{"className":11597,"style":8515},[2017],[1838,11599,11600],{},[1838,11601,8429],{"className":11602},[7262,8533,8534],[1838,11604,11605,11608],{"style":9274},[1838,11606],{"className":11607,"style":8515},[2017],[1838,11609,11611],{"className":11610},[2022,2023,2024,2025],[1838,11612,11421],{"className":11613},[1880,1881,2025],[1838,11615,2033],{"className":11616},[2032],[1838,11618,11620],{"className":11619},[2005],[1838,11621,11624],{"className":11622,"style":11623},[2009],"height:1.3021em;",[1838,11625],{},[1838,11627],{"className":11628,"style":2437},[2046],[1838,11630,11632,11635,11814],{"className":11631},[1880],[1838,11633],{"className":11634},[2814,7175],[1838,11636,11638],{"className":11637},[7032],[1838,11639,11641,11805],{"className":11640},[2000,2001],[1838,11642,11644,11802],{"className":11643},[2005],[1838,11645,11647,11670,11678],{"className":11646,"style":7939},[2009],[1838,11648,11649,11652],{"style":7191},[1838,11650],{"className":11651,"style":4122},[2017],[1838,11653,11655,11658,11661,11664,11667],{"className":11654},[1880],[1838,11656,2376],{"className":11657},[1880,1881],[1838,11659],{"className":11660,"style":2158},[2046],[1838,11662,1932],{"className":11663},[2139],[1838,11665],{"className":11666,"style":2158},[2046],[1838,11668,4555],{"className":11669},[1880,1881],[1838,11671,11672,11675],{"style":7203},[1838,11673],{"className":11674,"style":4122},[2017],[1838,11676],{"className":11677,"style":7211},[7210],[1838,11679,11680,11683],{"style":7214},[1838,11681],{"className":11682,"style":4122},[2017],[1838,11684,11686,11767,11770,11773],{"className":11685},[1880],[1838,11687,11689,11733],{"className":11688},[1880],[1838,11690,11692],{"className":11691},[1880,4105],[1838,11693,11695,11725],{"className":11694},[2000,2001],[1838,11696,11698,11722],{"className":11697},[2005],[1838,11699,11701,11709],{"className":11700,"style":11277},[2009],[1838,11702,11703,11706],{"style":4118},[1838,11704],{"className":11705,"style":4122},[2017],[1838,11707,11462],{"className":11708},[1880,1881],[1838,11710,11713,11716],{"className":11711,"style":11712},[4135],"width:calc(100% - 0.1667em);margin-left:0.1667em;top:-3.4306em;",[1838,11714],{"className":11715,"style":4122},[2017],[1838,11717,11718],{"style":4142},[2841,11719,11720],{"xmlns":2843,"width":4145,"height":4146,"viewBox":4147,"preserveAspectRatio":4148},[2848,11721],{"d":4151},[1838,11723,2033],{"className":11724},[2032],[1838,11726,11728],{"className":11727},[2005],[1838,11729,11731],{"className":11730,"style":4161},[2009],[1838,11732],{},[1838,11734,11736],{"className":11735},[1996],[1838,11737,11739,11759],{"className":11738},[2000,2001],[1838,11740,11742,11756],{"className":11741},[2005],[1838,11743,11745],{"className":11744,"style":2210},[2009],[1838,11746,11747,11750],{"style":2083},[1838,11748],{"className":11749,"style":2018},[2017],[1838,11751,11753],{"className":11752},[2022,2023,2024,2025],[1838,11754,11127],{"className":11755},[1880,1881,2025],[1838,11757,2033],{"className":11758},[2032],[1838,11760,11762],{"className":11761},[2005],[1838,11763,11765],{"className":11764,"style":2040},[2009],[1838,11766],{},[1838,11768,3989],{"className":11769},[2814],[1838,11771,4555],{"className":11772},[1880,1881],[1838,11774,11776,11779],{"className":11775},[3084],[1838,11777,4004],{"className":11778},[3084],[1838,11780,11782],{"className":11781},[1996],[1838,11783,11785],{"className":11784},[2000],[1838,11786,11788],{"className":11787},[2005],[1838,11789,11791],{"className":11790,"style":4477},[2009],[1838,11792,11793,11796],{"style":4446},[1838,11794],{"className":11795,"style":2018},[2017],[1838,11797,11799],{"className":11798},[2022,2023,2024,2025],[1838,11800,2528],{"className":11801},[1880,2025],[1838,11803,2033],{"className":11804},[2032],[1838,11806,11808],{"className":11807},[2005],[1838,11809,11812],{"className":11810,"style":11811},[2009],"height:0.7693em;",[1838,11813],{},[1838,11815],{"className":11816},[3084,7175],[1838,11818,2773],{"className":11819},[1880],[1798,11821,11822],{},"A large p-value says that this test did not detect residual autocorrelation at the chosen lags. It does not establish independence, normality, or parameter constancy.",[1793,11824,11826],{"id":11825},"_8-model-comparison-remains-conditional","8. Model comparison remains conditional",[1798,11828,11829,11830,11860,11861,11891],{},"For maximised log likelihood ",[1838,11831,11833,11847],{"className":11832},[1841],[1838,11834,11836],{"className":11835},[1845],[1847,11837,11838],{"xmlns":1849},[1851,11839,11840,11844],{},[1854,11841,11842],{},[1857,11843,7020],{"mathvariant":2539},[1860,11845,11846],{"encoding":1862},"\\ell",[1838,11848,11850],{"className":11849,"ariaHidden":1867},[1866],[1838,11851,11853,11857],{"className":11852},[1871],[1838,11854],{"className":11855,"style":11856},[1875],"height:0.6944em;",[1838,11858,7020],{"className":11859},[1880]," and ",[1838,11862,11864,11878],{"className":11863},[1841],[1838,11865,11867],{"className":11866},[1845],[1847,11868,11869],{"xmlns":1849},[1851,11870,11871,11876],{},[1854,11872,11873],{},[1857,11874,11875],{},"k",[1860,11877,11875],{"encoding":1862},[1838,11879,11881],{"className":11880,"ariaHidden":1867},[1866],[1838,11882,11884,11887],{"className":11883},[1871],[1838,11885],{"className":11886,"style":11856},[1875],[1838,11888,11875],{"className":11889,"style":11890},[1880,1881],"margin-right:0.0315em;"," estimated parameters,",[1838,11893,11895],{"className":11894},[1886],[1838,11896,11898,11966],{"className":11897},[1841],[1838,11899,11901],{"className":11900},[1845],[1847,11902,11903],{"xmlns":1849,"display":1895},[1851,11904,11905,11963],{},[1854,11906,11907,11916,11918,11920,11922,11924,11926,11928,11930,11932,11934,11943,11945,11947,11949,11951,11953,11955,11957,11959,11961],{},[1854,11908,11909,11912,11914],{},[1857,11910,11911],{"mathvariant":2539},"A",[1857,11913,10288],{"mathvariant":2539},[1857,11915,3978],{"mathvariant":2539},[1910,11917,1912],{},[1910,11919,1932],{},[1919,11921,2528],{},[1857,11923,7020],{"mathvariant":2539},[1910,11925,1937],{},[1919,11927,2528],{},[1857,11929,11875],{},[1910,11931,1972],{"separator":1867},[2046,11933],{"width":2568},[1854,11935,11936,11939,11941],{},[1857,11937,11938],{"mathvariant":2539},"B",[1857,11940,10288],{"mathvariant":2539},[1857,11942,3978],{"mathvariant":2539},[1910,11944,1912],{},[1910,11946,1932],{},[1919,11948,2528],{},[1857,11950,7020],{"mathvariant":2539},[1910,11952,1937],{},[1857,11954,11875],{},[1857,11956,7046],{},[1910,11958,7049],{},[1857,11960,2376],{},[1857,11962,2773],{"mathvariant":2539},[1860,11964,11965],{"encoding":1862},"\\mathrm{AIC}=-2\\ell+2k,\\qquad\n\\mathrm{BIC}=-2\\ell+k\\log n.",[1838,11967,11969,11991,12016,12053,12077],{"className":11968,"ariaHidden":1867},[1866],[1838,11970,11972,11975,11982,11985,11988],{"className":11971},[1871],[1838,11973],{"className":11974,"style":8271},[1875],[1838,11976,11978],{"className":11977},[1880],[1838,11979,11981],{"className":11980},[1880,8084],"AIC",[1838,11983],{"className":11984,"style":2047},[2046],[1838,11986,1912],{"className":11987},[2051],[1838,11989],{"className":11990,"style":2047},[2046],[1838,11992,11994,11998,12001,12004,12007,12010,12013],{"className":11993},[1871],[1838,11995],{"className":11996,"style":11997},[1875],"height:0.7778em;vertical-align:-0.0833em;",[1838,11999,1932],{"className":12000},[1880],[1838,12002,2528],{"className":12003},[1880],[1838,12005,7020],{"className":12006},[1880],[1838,12008],{"className":12009,"style":2158},[2046],[1838,12011,1937],{"className":12012},[2139],[1838,12014],{"className":12015,"style":2158},[2046],[1838,12017,12019,12022,12025,12028,12031,12034,12037,12044,12047,12050],{"className":12018},[1871],[1838,12020],{"className":12021,"style":4051},[1875],[1838,12023,2528],{"className":12024},[1880],[1838,12026,11875],{"className":12027,"style":11890},[1880,1881],[1838,12029,1972],{"className":12030},[2341],[1838,12032],{"className":12033,"style":3129},[2046],[1838,12035],{"className":12036,"style":2437},[2046],[1838,12038,12040],{"className":12039},[1880],[1838,12041,12043],{"className":12042},[1880,8084],"BIC",[1838,12045],{"className":12046,"style":2047},[2046],[1838,12048,1912],{"className":12049},[2051],[1838,12051],{"className":12052,"style":2047},[2046],[1838,12054,12056,12059,12062,12065,12068,12071,12074],{"className":12055},[1871],[1838,12057],{"className":12058,"style":11997},[1875],[1838,12060,1932],{"className":12061},[1880],[1838,12063,2528],{"className":12064},[1880],[1838,12066,7020],{"className":12067},[1880],[1838,12069],{"className":12070,"style":2158},[2046],[1838,12072,1937],{"className":12073},[2139],[1838,12075],{"className":12076,"style":2158},[2046],[1838,12078,12080,12083,12086,12089,12094,12097,12100],{"className":12079},[1871],[1838,12081],{"className":12082,"style":4051},[1875],[1838,12084,11875],{"className":12085,"style":11890},[1880,1881],[1838,12087],{"className":12088,"style":2437},[2046],[1838,12090,7263,12092],{"className":12091},[7262],[1838,12093,7267],{"style":7266},[1838,12095],{"className":12096,"style":2437},[2046],[1838,12098,2376],{"className":12099},[1880,1881],[1838,12101,2773],{"className":12102},[1880],[1798,12104,12105],{},"Compare values only when models use the same response, likelihood convention, and effective sample. Before reading the smaller number, check admissible roots, identifiability, and residual covariance.",[1793,12107,12109],{"id":12108},"practice","Practice",[12111,12112,12113,12316,12319,12368,12480],"ol",{},[1812,12114,12115,12116,11860,12145,12174,12175,2773],{},"Write ",[1838,12117,12119,12133],{"className":12118},[1841],[1838,12120,12122],{"className":12121},[1845],[1847,12123,12124],{"xmlns":1849},[1851,12125,12126,12130],{},[1854,12127,12128],{},[1857,12129,1905],{"mathvariant":2571},[1860,12131,12132],{"encoding":1862},"\\mathcal X",[1838,12134,12136],{"className":12135,"ariaHidden":1867},[1866],[1838,12137,12139,12142],{"className":12138},[1871],[1838,12140],{"className":12141,"style":8271},[1875],[1838,12143,1905],{"className":12144,"style":3137},[1880,3136],[1838,12146,12148,12162],{"className":12147},[1841],[1838,12149,12151],{"className":12150},[1845],[1847,12152,12153],{"xmlns":1849},[1851,12154,12155,12159],{},[1854,12156,12157],{},[1857,12158,2472],{"mathvariant":2471},[1860,12160,12161],{"encoding":1862},"\\mathbf y",[1838,12163,12165],{"className":12164,"ariaHidden":1867},[1866],[1838,12166,12168,12171],{"className":12167},[1871],[1838,12169],{"className":12170,"style":2786},[1875],[1838,12172,2472],{"className":12173,"style":2791},[1880,2790]," for an AR(2) observed at ",[1838,12176,12178,12209],{"className":12177},[1841],[1838,12179,12181],{"className":12180},[1845],[1847,12182,12183],{"xmlns":1849},[1851,12184,12185,12206],{},[1854,12186,12187,12193,12195,12197,12199],{},[1901,12188,12189,12191],{},[1857,12190,2501],{},[1919,12192,1921],{},[1910,12194,1972],{"separator":1867},[1910,12196,2371],{},[1910,12198,1972],{"separator":1867},[1901,12200,12201,12203],{},[1857,12202,2501],{},[1919,12204,12205],{},"6",[1860,12207,12208],{"encoding":1862},"x_1,\\ldots,x_6",[1838,12210,12212],{"className":12211,"ariaHidden":1867},[1866],[1838,12213,12215,12218,12258,12261,12264,12267,12270,12273,12276],{"className":12214},[1871],[1838,12216],{"className":12217,"style":1876},[1875],[1838,12219,12221,12224],{"className":12220},[1880],[1838,12222,2501],{"className":12223},[1880,1881],[1838,12225,12227],{"className":12226},[1996],[1838,12228,12230,12250],{"className":12229},[2000,2001],[1838,12231,12233,12247],{"className":12232},[2005],[1838,12234,12236],{"className":12235,"style":2080},[2009],[1838,12237,12238,12241],{"style":2083},[1838,12239],{"className":12240,"style":2018},[2017],[1838,12242,12244],{"className":12243},[2022,2023,2024,2025],[1838,12245,1921],{"className":12246},[1880,2025],[1838,12248,2033],{"className":12249},[2032],[1838,12251,12253],{"className":12252},[2005],[1838,12254,12256],{"className":12255,"style":2040},[2009],[1838,12257],{},[1838,12259,1972],{"className":12260},[2341],[1838,12262],{"className":12263,"style":2437},[2046],[1838,12265,2371],{"className":12266},[2175],[1838,12268],{"className":12269,"style":2437},[2046],[1838,12271,1972],{"className":12272},[2341],[1838,12274],{"className":12275,"style":2437},[2046],[1838,12277,12279,12282],{"className":12278},[1880],[1838,12280,2501],{"className":12281},[1880,1881],[1838,12283,12285],{"className":12284},[1996],[1838,12286,12288,12308],{"className":12287},[2000,2001],[1838,12289,12291,12305],{"className":12290},[2005],[1838,12292,12294],{"className":12293,"style":2080},[2009],[1838,12295,12296,12299],{"style":2083},[1838,12297],{"className":12298,"style":2018},[2017],[1838,12300,12302],{"className":12301},[2022,2023,2024,2025],[1838,12303,12205],{"className":12304},[1880,2025],[1838,12306,2033],{"className":12307},[2032],[1838,12309,12311],{"className":12310},[2005],[1838,12312,12314],{"className":12313,"style":2040},[2009],[1838,12315],{},[1812,12317,12318],{},"Derive the AR(1) Yule–Walker estimator.",[1812,12320,12321,12322,12367],{},"Explain why ",[1838,12323,12325,12347],{"className":12324},[1841],[1838,12326,12328],{"className":12327},[1845],[1847,12329,12330],{"xmlns":1849},[1851,12331,12332,12344],{},[1854,12333,12334,12336,12338,12340,12342],{},[1857,12335,7046],{},[1910,12337,7049],{},[1857,12339,7067],{"mathvariant":2539},[1857,12341,6818],{"mathvariant":2539},[1857,12343,7067],{"mathvariant":2539},[1860,12345,12346],{"encoding":1862},"\\log|\\Sigma|",[1838,12348,12350],{"className":12349,"ariaHidden":1867},[1866],[1838,12351,12353,12356,12361,12364],{"className":12352},[1871],[1838,12354],{"className":12355,"style":4756},[1875],[1838,12357,7263,12359],{"className":12358},[7262],[1838,12360,7267],{"style":7266},[1838,12362],{"className":12363,"style":2437},[2046],[1838,12365,8472],{"className":12366},[1880]," cannot be dropped when comparing covariance parameters.",[1812,12369,12370,12371,2773],{},"Show how the Cholesky factor computes ",[1838,12372,12374,12404],{"className":12373},[1841],[1838,12375,12377],{"className":12376},[1845],[1847,12378,12379],{"xmlns":1849},[1851,12380,12381,12401],{},[1854,12382,12383,12389,12399],{},[3991,12384,12385,12387],{},[1857,12386,2501],{"mathvariant":2471},[1857,12388,3997],{"mathvariant":2539},[3991,12390,12391,12393],{},[1857,12392,6818],{"mathvariant":2539},[1854,12394,12395,12397],{},[1910,12396,1932],{},[1919,12398,1921],{},[1857,12400,2501],{"mathvariant":2471},[1860,12402,12403],{"encoding":1862},"\\mathbf x^\\top\\Sigma^{-1}\\mathbf x",[1838,12405,12407],{"className":12406,"ariaHidden":1867},[1866],[1838,12408,12410,12413,12442,12477],{"className":12409},[1871],[1838,12411],{"className":12412,"style":4443},[1875],[1838,12414,12416,12419],{"className":12415},[1880],[1838,12417,2501],{"className":12418},[1880,2790],[1838,12420,12422],{"className":12421},[1996],[1838,12423,12425],{"className":12424},[2000],[1838,12426,12428],{"className":12427},[2005],[1838,12429,12431],{"className":12430,"style":4443},[2009],[1838,12432,12433,12436],{"style":4446},[1838,12434],{"className":12435,"style":2018},[2017],[1838,12437,12439],{"className":12438},[2022,2023,2024,2025],[1838,12440,3997],{"className":12441},[1880,2025],[1838,12443,12445,12448],{"className":12444},[1880],[1838,12446,6818],{"className":12447},[1880],[1838,12449,12451],{"className":12450},[1996],[1838,12452,12454],{"className":12453},[2000],[1838,12455,12457],{"className":12456},[2005],[1838,12458,12460],{"className":12459,"style":4477},[2009],[1838,12461,12462,12465],{"style":4446},[1838,12463],{"className":12464,"style":2018},[2017],[1838,12466,12468],{"className":12467},[2022,2023,2024,2025],[1838,12469,12471,12474],{"className":12470},[1880,2025],[1838,12472,1932],{"className":12473},[1880,2025],[1838,12475,1921],{"className":12476},[1880,2025],[1838,12478,2501],{"className":12479},[1880,2790],[1812,12481,12482,12483,12557],{},"Why can Hessian standard errors be misleading when ",[1838,12484,12486,12504],{"className":12485},[1841],[1838,12487,12489],{"className":12488},[1845],[1847,12490,12491],{"xmlns":1849},[1851,12492,12493,12501],{},[1854,12494,12495],{},[3967,12496,12497,12499],{"accent":1867},[1857,12498,1917],{},[1910,12500,3973],{"stretchy":1867},[1860,12502,12503],{"encoding":1862},"\\widehat\\phi",[1838,12505,12507],{"className":12506,"ariaHidden":1867},[1866],[1838,12508,12510,12514],{"className":12509},[1871],[1838,12511],{"className":12512,"style":12513},[1875],"height:1.1289em;vertical-align:-0.1944em;",[1838,12515,12517],{"className":12516},[1880,4105],[1838,12518,12520,12549],{"className":12519},[2000,2001],[1838,12521,12523,12546],{"className":12522},[2005],[1838,12524,12526,12534],{"className":12525,"style":4115},[2009],[1838,12527,12528,12531],{"style":4118},[1838,12529],{"className":12530,"style":4122},[2017],[1838,12532,1917],{"className":12533},[1880,1881],[1838,12535,12537,12540],{"className":12536,"style":10766},[4135],[1838,12538],{"className":12539,"style":4122},[2017],[1838,12541,12542],{"style":4142},[2841,12543,12544],{"xmlns":2843,"width":4145,"height":4146,"viewBox":4147,"preserveAspectRatio":4148},[2848,12545],{"d":4151},[1838,12547,2033],{"className":12548},[2032],[1838,12550,12552],{"className":12551},[2005],[1838,12553,12555],{"className":12554,"style":4161},[2009],[1838,12556],{}," is close to one?",[12559,12560,12562],"legacy-details",{"title":12561},"Answers",[12111,12563,12564,13161,13387,13390,13546],{},[1812,12565,12566,12862,12863,12891,12892,13026,13027,2773],{},[1838,12567,12569,12625],{"className":12568},[1841],[1838,12570,12572],{"className":12571},[1845],[1847,12573,12574],{"xmlns":1849},[1851,12575,12576,12622],{},[1854,12577,12578,12580,12582,12584,12590,12592,12599,12601,12608,12610,12616],{},[1857,12579,2472],{"mathvariant":2471},[1910,12581,1912],{},[1910,12583,3989],{"stretchy":2496},[1901,12585,12586,12588],{},[1857,12587,2501],{},[1919,12589,10193],{},[1910,12591,1972],{"separator":1867},[1901,12593,12594,12596],{},[1857,12595,2501],{},[1919,12597,12598],{},"4",[1910,12600,1972],{"separator":1867},[1901,12602,12603,12605],{},[1857,12604,2501],{},[1919,12606,12607],{},"5",[1910,12609,1972],{"separator":1867},[1901,12611,12612,12614],{},[1857,12613,2501],{},[1919,12615,12205],{},[3991,12617,12618,12620],{},[1910,12619,4004],{"stretchy":2496},[1857,12621,3997],{"mathvariant":2539},[1860,12623,12624],{"encoding":1862},"\\mathbf y=(x_3,x_4,x_5,x_6)^\\top",[1838,12626,12628,12646],{"className":12627,"ariaHidden":1867},[1866],[1838,12629,12631,12634,12637,12640,12643],{"className":12630},[1871],[1838,12632],{"className":12633,"style":2786},[1875],[1838,12635,2472],{"className":12636,"style":2791},[1880,2790],[1838,12638],{"className":12639,"style":2047},[2046],[1838,12641,1912],{"className":12642},[2051],[1838,12644],{"className":12645,"style":2047},[2046],[1838,12647,12649,12652,12655,12695,12698,12701,12741,12744,12747,12787,12790,12793,12833],{"className":12648},[1871],[1838,12650],{"className":12651,"style":4421},[1875],[1838,12653,3989],{"className":12654},[2814],[1838,12656,12658,12661],{"className":12657},[1880],[1838,12659,2501],{"className":12660},[1880,1881],[1838,12662,12664],{"className":12663},[1996],[1838,12665,12667,12687],{"className":12666},[2000,2001],[1838,12668,12670,12684],{"className":12669},[2005],[1838,12671,12673],{"className":12672,"style":2080},[2009],[1838,12674,12675,12678],{"style":2083},[1838,12676],{"className":12677,"style":2018},[2017],[1838,12679,12681],{"className":12680},[2022,2023,2024,2025],[1838,12682,10193],{"className":12683},[1880,2025],[1838,12685,2033],{"className":12686},[2032],[1838,12688,12690],{"className":12689},[2005],[1838,12691,12693],{"className":12692,"style":2040},[2009],[1838,12694],{},[1838,12696,1972],{"className":12697},[2341],[1838,12699],{"className":12700,"style":2437},[2046],[1838,12702,12704,12707],{"className":12703},[1880],[1838,12705,2501],{"className":12706},[1880,1881],[1838,12708,12710],{"className":12709},[1996],[1838,12711,12713,12733],{"className":12712},[2000,2001],[1838,12714,12716,12730],{"className":12715},[2005],[1838,12717,12719],{"className":12718,"style":2080},[2009],[1838,12720,12721,12724],{"style":2083},[1838,12722],{"className":12723,"style":2018},[2017],[1838,12725,12727],{"className":12726},[2022,2023,2024,2025],[1838,12728,12598],{"className":12729},[1880,2025],[1838,12731,2033],{"className":12732},[2032],[1838,12734,12736],{"className":12735},[2005],[1838,12737,12739],{"className":12738,"style":2040},[2009],[1838,12740],{},[1838,12742,1972],{"className":12743},[2341],[1838,12745],{"className":12746,"style":2437},[2046],[1838,12748,12750,12753],{"className":12749},[1880],[1838,12751,2501],{"className":12752},[1880,1881],[1838,12754,12756],{"className":12755},[1996],[1838,12757,12759,12779],{"className":12758},[2000,2001],[1838,12760,12762,12776],{"className":12761},[2005],[1838,12763,12765],{"className":12764,"style":2080},[2009],[1838,12766,12767,12770],{"style":2083},[1838,12768],{"className":12769,"style":2018},[2017],[1838,12771,12773],{"className":12772},[2022,2023,2024,2025],[1838,12774,12607],{"className":12775},[1880,2025],[1838,12777,2033],{"className":12778},[2032],[1838,12780,12782],{"className":12781},[2005],[1838,12783,12785],{"className":12784,"style":2040},[2009],[1838,12786],{},[1838,12788,1972],{"className":12789},[2341],[1838,12791],{"className":12792,"style":2437},[2046],[1838,12794,12796,12799],{"className":12795},[1880],[1838,12797,2501],{"className":12798},[1880,1881],[1838,12800,12802],{"className":12801},[1996],[1838,12803,12805,12825],{"className":12804},[2000,2001],[1838,12806,12808,12822],{"className":12807},[2005],[1838,12809,12811],{"className":12810,"style":2080},[2009],[1838,12812,12813,12816],{"style":2083},[1838,12814],{"className":12815,"style":2018},[2017],[1838,12817,12819],{"className":12818},[2022,2023,2024,2025],[1838,12820,12205],{"className":12821},[1880,2025],[1838,12823,2033],{"className":12824},[2032],[1838,12826,12828],{"className":12827},[2005],[1838,12829,12831],{"className":12830,"style":2040},[2009],[1838,12832],{},[1838,12834,12836,12839],{"className":12835},[3084],[1838,12837,4004],{"className":12838},[3084],[1838,12840,12842],{"className":12841},[1996],[1838,12843,12845],{"className":12844},[2000],[1838,12846,12848],{"className":12847},[2005],[1838,12849,12851],{"className":12850,"style":4443},[2009],[1838,12852,12853,12856],{"style":4446},[1838,12854],{"className":12855,"style":2018},[2017],[1838,12857,12859],{"className":12858},[2022,2023,2024,2025],[1838,12860,3997],{"className":12861},[1880,2025]," and the rows of ",[1838,12864,12866,12879],{"className":12865},[1841],[1838,12867,12869],{"className":12868},[1845],[1847,12870,12871],{"xmlns":1849},[1851,12872,12873,12877],{},[1854,12874,12875],{},[1857,12876,1905],{"mathvariant":2571},[1860,12878,12132],{"encoding":1862},[1838,12880,12882],{"className":12881,"ariaHidden":1867},[1866],[1838,12883,12885,12888],{"className":12884},[1871],[1838,12886],{"className":12887,"style":8271},[1875],[1838,12889,1905],{"className":12890,"style":3137},[1880,3136]," are 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through 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penalises covariance matrices that assign broad volume; the quadratic form alone could favour inflated variance.",[1812,13391,13392,13393,13481,13482,2773],{},"Solve ",[1838,13394,13396,13421],{"className":13395},[1841],[1838,13397,13399],{"className":13398},[1845],[1847,13400,13401],{"xmlns":1849},[1851,13402,13403,13418],{},[1854,13404,13405,13411,13414,13416],{},[3991,13406,13407,13409],{},[1857,13408,8254],{},[1857,13410,3997],{"mathvariant":2539},[1857,13412,13413],{"mathvariant":2471},"z",[1910,13415,1912],{},[1857,13417,2501],{"mathvariant":2471},[1860,13419,13420],{"encoding":1862},"U^\\top\\mathbf z=\\mathbf x",[1838,13422,13424,13471],{"className":13423,"ariaHidden":1867},[1866],[1838,13425,13427,13430,13459,13462,13465,13468],{"className":13426},[1871],[1838,13428],{"className":13429,"style":4443},[1875],[1838,13431,13433,13436],{"className":13432},[1880],[1838,13434,8254],{"className":13435,"style":6896},[1880,1881],[1838,13437,13439],{"className":13438},[1996],[1838,13440,13442],{"className":13441},[2000],[1838,13443,13445],{"className":13444},[2005],[1838,13446,13448],{"className":13447,"style":4443},[2009],[1838,13449,13450,13453],{"style":4446},[1838,13451],{"className":13452,"style":2018},[2017],[1838,13454,13456],{"className":13455},[2022,2023,2024,2025],[1838,13457,3997],{"className":13458},[1880,2025],[1838,13460,13413],{"className":13461},[1880,2790],[1838,13463],{"className":13464,"style":2047},[2046],[1838,13466,1912],{"className":13467},[2051],[1838,13469],{"className":13470,"style":2047},[2046],[1838,13472,13474,13478],{"className":13473},[1871],[1838,13475],{"className":13476,"style":13477},[1875],"height:0.4444em;",[1838,13479,2501],{"className":13480},[1880,2790]," and compute ",[1838,13483,13485,13505],{"className":13484},[1841],[1838,13486,13488],{"className":13487},[1845],[1847,13489,13490],{"xmlns":1849},[1851,13491,13492,13502],{},[1854,13493,13494,13500],{},[3991,13495,13496,13498],{},[1857,13497,13413],{"mathvariant":2471},[1857,13499,3997],{"mathvariant":2539},[1857,13501,13413],{"mathvariant":2471},[1860,13503,13504],{"encoding":1862},"\\mathbf z^\\top\\mathbf z",[1838,13506,13508],{"className":13507,"ariaHidden":1867},[1866],[1838,13509,13511,13514,13543],{"className":13510},[1871],[1838,13512],{"className":13513,"style":4443},[1875],[1838,13515,13517,13520],{"className":13516},[1880],[1838,13518,13413],{"className":13519},[1880,2790],[1838,13521,13523],{"className":13522},[1996],[1838,13524,13526],{"className":13525},[2000],[1838,13527,13529],{"className":13528},[2005],[1838,13530,13532],{"className":13531,"style":4443},[2009],[1838,13533,13534,13537],{"style":4446},[1838,13535],{"className":13536,"style":2018},[2017],[1838,13538,13540],{"className":13539},[2022,2023,2024,2025],[1838,13541,3997],{"className":13542},[1880,2025],[1838,13544,13413],{"className":13545},[1880,2790],[1812,13547,13548],{},"The parameter lies near a non-regular boundary, the likelihood may be asymmetric, and the normal quadratic approximation can be poor.",[1793,13550,13552],{"id":13551},"takeaway","Takeaway",[1798,13554,13555,13556,2773],{},"Least squares, moments, and likelihood are different matrix summaries of the same dependent vector. Their assumptions, initial conditions, and numerical factorizations must be visible. Continue to ",[13557,13558,13560],"a",{"href":13559},"..\u002F05-multi-ar\u002F","Multivariate Systems",{"title":10,"searchDepth":13562,"depth":13562,"links":13563},2,[13564,13565,13566,13567,13568,13569,13573,13574,13575,13576,13577,13578],{"id":1795,"depth":13562,"text":1796},{"id":1803,"depth":13562,"text":1804},{"id":1832,"depth":13562,"text":1833},{"id":4497,"depth":13562,"text":4498},{"id":6770,"depth":13562,"text":6771},{"id":8229,"depth":13562,"text":8230,"children":13570},[13571],{"id":8910,"depth":13572,"text":8911},3,{"id":8927,"depth":13562,"text":8928},{"id":10264,"depth":13562,"text":10265},{"id":11028,"depth":13562,"text":11029},{"id":11825,"depth":13562,"text":11826},{"id":12108,"depth":13562,"text":12109},{"id":13551,"depth":13562,"text":13552},"Derive AR estimators from regression and moment equations, evaluate exact Gaussian likelihoods with covariance matrices, and interpret uncertainty.","md",{"sidebar":13582},{"order":13583},4,true,{"title":977,"description":13579},"9bPfrJkbxBYGGDV_8kmaXgx8S9Lhv86VWv94CXlBxWk",[13588,13590],{"title":971,"path":972,"stem":973,"description":13589,"children":-1},"Derive best linear predictors from partitioned covariance matrices, quantify error with Schur complements, and introduce Kalman recursions.",{"title":983,"path":984,"stem":985,"description":13591,"children":-1},"Express vector autoregressions, seasonal operators, cointegration, and forecast constraints with block matrices and stable linear systems.",1785754738194]