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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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1 — Predictor weights and Schur complement",[1798,2832,2833,2834,2948,2949,3041],{},"Use an AR(2) with ",[1805,2835,2837,2872],{"className":2836},[1808],[1805,2838,2840],{"className":2839},[1812],[1814,2841,2842],{"xmlns":1816},[1818,2843,2844,2869],{},[1821,2845,2846,2849,2851,2853,2856,2858,2860,2863],{},[1824,2847,2848],{},"ϕ",[1908,2850,1948],{},[1908,2852,1915],{"stretchy":1932},[2601,2854,2855],{},"0.65",[1908,2857,2378],{"separator":1837},[1908,2859,2567],{},[2601,2861,2862],{},"0.20",[1980,2864,2865,2867],{},[1908,2866,1945],{"stretchy":1932},[1824,2868,1986],{"mathvariant":1905},[1829,2870,2871],{"encoding":1831},"\\phi=(0.65,-0.20)^\\top",[1805,2873,2875,2894],{"className":2874,"ariaHidden":1837},[1836],[1805,2876,2878,2882,2885,2888,2891],{"className":2877},[1841],[1805,2879],{"className":2880,"style":2881},[1845],"height:0.8889em;vertical-align:-0.1944em;",[1805,2883,2848],{"className":2884},[1850,1883],[1805,2886],{"className":2887,"style":2120},[2029],[1805,2889,1948],{"className":2890},[2124],[1805,2892],{"className":2893,"style":2120},[2029],[1805,2895,2897,2901,2904,2907,2910,2913,2916,2919],{"className":2896},[1841],[1805,2898],{"className":2899,"style":2900},[1845],"height:1.0991em;vertical-align:-0.25em;",[1805,2902,1915],{"className":2903},[2038],[1805,2905,2855],{"className":2906},[1850],[1805,2908,2378],{"className":2909},[2503],[1805,2911],{"className":2912,"style":2030},[2029],[1805,2914,2567],{"className":2915},[1850],[1805,2917,2862],{"className":2918},[1850],[1805,2920,2922,2925],{"className":2921},[2113],[1805,2923,1945],{"className":2924},[2113],[1805,2926,2928],{"className":2927},[2194],[1805,2929,2931],{"className":2930},[2059],[1805,2932,2934],{"className":2933},[2064],[1805,2935,2937],{"className":2936,"style":2204},[2068],[1805,2938,2939,2942],{"style":2207},[1805,2940],{"className":2941,"style":2211},[2076],[1805,2943,2945],{"className":2944},[2215,2216,2045,2217],[1805,2946,1986],{"className":2947},[1850,2217]," and innovation variance 1.5. Predict ",[1805,2950,2952,2978],{"className":2951},[1808],[1805,2953,2955],{"className":2954},[1812],[1814,2956,2957],{"xmlns":1816},[1818,2958,2959,2975],{},[1821,2960,2961],{},[1992,2962,2963,2965],{},[1824,2964,1827],{},[1821,2966,2967,2970,2973],{},[1824,2968,2969],{},"t",[1908,2971,2972],{},"+",[2601,2974,2603],{},[1829,2976,2977],{"encoding":1831},"X_{t+1}",[1805,2979,2981],{"className":2980,"ariaHidden":1837},[1836],[1805,2982,2984,2988],{"className":2983},[1841],[1805,2985],{"className":2986,"style":2987},[1845],"height:0.8917em;vertical-align:-0.2083em;",[1805,2989,2991,2995],{"className":2990},[1850],[1805,2992,1827],{"className":2993,"style":2994},[1850,1883],"margin-right:0.0785em;",[1805,2996,2998],{"className":2997},[2194],[1805,2999,3001,3032],{"className":3000},[2059,2060],[1805,3002,3004,3029],{"className":3003},[2064],[1805,3005,3008],{"className":3006,"style":3007},[2068],"height:0.3011em;",[1805,3009,3011,3014],{"style":3010},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1805,3012],{"className":3013,"style":2211},[2076],[1805,3015,3017],{"className":3016},[2215,2216,2045,2217],[1805,3018,3020,3023,3026],{"className":3019},[1850,2217],[1805,3021,2969],{"className":3022},[1850,1883,2217],[1805,3024,2972],{"className":3025},[2639,2217],[1805,3027,2603],{"className":3028},[1850,2217],[1805,3030,2100],{"className":3031},[2099],[1805,3033,3035],{"className":3034},[2064],[1805,3036,3039],{"className":3037,"style":3038},[2068],"height:0.2083em;",[1805,3040],{}," from",[1805,3043,3045],{"className":3044},[1889],[1805,3046,3048,3109],{"className":3047},[1808],[1805,3049,3051],{"className":3050},[1812],[1814,3052,3053],{"xmlns":1816,"display":1898},[1818,3054,3055,3106],{},[1821,3056,3057,3059,3061,3063,3069,3071,3083,3085,3098,3104],{},[1824,3058,1827],{"mathvariant":1826},[1908,3060,1948],{},[1908,3062,1915],{"stretchy":1932},[1992,3064,3065,3067],{},[1824,3066,1827],{},[1824,3068,2969],{},[1908,3070,2378],{"separator":1837},[1992,3072,3073,3075],{},[1824,3074,1827],{},[1821,3076,3077,3079,3081],{},[1824,3078,2969],{},[1908,3080,2567],{},[2601,3082,2603],{},[1908,3084,2378],{"separator":1837},[1992,3086,3087,3089],{},[1824,3088,1827],{},[1821,3090,3091,3093,3095],{},[1824,3092,2969],{},[1908,3094,2567],{},[2601,3096,3097],{},"2",[1980,3099,3100,3102],{},[1908,3101,1945],{"stretchy":1932},[1824,3103,1986],{"mathvariant":1905},[1824,3105,2004],{"mathvariant":1905},[1829,3107,3108],{"encoding":1831},"\\mathbf X=(X_t,X_{t-1},X_{t-2})^\\top.",[1805,3110,3112,3130],{"className":3111,"ariaHidden":1837},[1836],[1805,3113,3115,3118,3121,3124,3127],{"className":3114},[1841],[1805,3116],{"className":3117,"style":1846},[1845],[1805,3119,1827],{"className":3120},[1850,1851],[1805,3122],{"className":3123,"style":2120},[2029],[1805,3125,1948],{"className":3126},[2124],[1805,3128],{"className":3129,"style":2120},[2029],[1805,3131,3133,3137,3140,3181,3184,3187,3236,3239,3242,3291,3320],{"className":3132},[1841],[1805,3134],{"className":3135,"style":3136},[1845],"height:1.1491em;vertical-align:-0.25em;",[1805,3138,1915],{"className":3139},[2038],[1805,3141,3143,3146],{"className":3142},[1850],[1805,3144,1827],{"className":3145,"style":2994},[1850,1883],[1805,3147,3149],{"className":3148},[2194],[1805,3150,3152,3173],{"className":3151},[2059,2060],[1805,3153,3155,3170],{"className":3154},[2064],[1805,3156,3159],{"className":3157,"style":3158},[2068],"height:0.2806em;",[1805,3160,3161,3164],{"style":3010},[1805,3162],{"className":3163,"style":2211},[2076],[1805,3165,3167],{"className":3166},[2215,2216,2045,2217],[1805,3168,2969],{"className":3169},[1850,1883,2217],[1805,3171,2100],{"className":3172},[2099],[1805,3174,3176],{"className":3175},[2064],[1805,3177,3179],{"className":3178,"style":2313},[2068],[1805,3180],{},[1805,3182,2378],{"className":3183},[2503],[1805,3185],{"className":3186,"style":2030},[2029],[1805,3188,3190,3193],{"className":3189},[1850],[1805,3191,1827],{"className":3192,"style":2994},[1850,1883],[1805,3194,3196],{"className":3195},[2194],[1805,3197,3199,3228],{"className":3198},[2059,2060],[1805,3200,3202,3225],{"className":3201},[2064],[1805,3203,3205],{"className":3204,"style":3007},[2068],[1805,3206,3207,3210],{"style":3010},[1805,3208],{"className":3209,"style":2211},[2076],[1805,3211,3213],{"className":3212},[2215,2216,2045,2217],[1805,3214,3216,3219,3222],{"className":3215},[1850,2217],[1805,3217,2969],{"className":3218},[1850,1883,2217],[1805,3220,2567],{"className":3221},[2639,2217],[1805,3223,2603],{"className":3224},[1850,2217],[1805,3226,2100],{"className":3227},[2099],[1805,3229,3231],{"className":3230},[2064],[1805,3232,3234],{"className":3233,"style":3038},[2068],[1805,3235],{},[1805,3237,2378],{"className":3238},[2503],[1805,3240],{"className":3241,"style":2030},[2029],[1805,3243,3245,3248],{"className":3244},[1850],[1805,3246,1827],{"className":3247,"style":2994},[1850,1883],[1805,3249,3251],{"className":3250},[2194],[1805,3252,3254,3283],{"className":3253},[2059,2060],[1805,3255,3257,3280],{"className":3256},[2064],[1805,3258,3260],{"className":3259,"style":3007},[2068],[1805,3261,3262,3265],{"style":3010},[1805,3263],{"className":3264,"style":2211},[2076],[1805,3266,3268],{"className":3267},[2215,2216,2045,2217],[1805,3269,3271,3274,3277],{"className":3270},[1850,2217],[1805,3272,2969],{"className":3273},[1850,1883,2217],[1805,3275,2567],{"className":3276},[2639,2217],[1805,3278,3097],{"className":3279},[1850,2217],[1805,3281,2100],{"className":3282},[2099],[1805,3284,3286],{"className":3285},[2064],[1805,3287,3289],{"className":3288,"style":3038},[2068],[1805,3290],{},[1805,3292,3294,3297],{"className":3293},[2113],[1805,3295,1945],{"className":3296},[2113],[1805,3298,3300],{"className":3299},[2194],[1805,3301,3303],{"className":3302},[2059],[1805,3304,3306],{"className":3305},[2064],[1805,3307,3309],{"className":3308,"style":2484},[2068],[1805,3310,3311,3314],{"style":2487},[1805,3312],{"className":3313,"style":2211},[2076],[1805,3315,3317],{"className":3316},[2215,2216,2045,2217],[1805,3318,1986],{"className":3319},[1850,2217],[1805,3321,2004],{"className":3322},[1850],[1798,3324,3325],{},"The model says that only the first two coordinates should retain non-zero weights.",[3327,3328],"web-r",{"code64":3329,"layout":3330,"locale":7,"title":3331},"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","vertical","Solve and audit the best linear predictor",[1798,3333,3334],{},"The third weight should be zero up to numerical precision, and the Schur complement should reproduce the innovation variance. Both are model-implied identities.",[1793,3336,3338],{"id":3337},"example-2-three-estimators-answer-three-finite-sample-problems","Example 2 — Three estimators answer three finite-sample problems",[1798,3340,3341],{},"For a centred AR(1):",[1805,3343,3345],{"className":3344},[1889],[1805,3346,3348,3393],{"className":3347},[1808],[1805,3349,3351],{"className":3350},[1812],[1814,3352,3353],{"xmlns":1816,"display":1898},[1818,3354,3355,3390],{},[1821,3356,3357,3363,3365,3367,3379,3381,3388],{},[1992,3358,3359,3361],{},[1824,3360,1827],{},[1824,3362,2969],{},[1908,3364,1948],{},[1824,3366,2848],{},[1992,3368,3369,3371],{},[1824,3370,1827],{},[1821,3372,3373,3375,3377],{},[1824,3374,2969],{},[1908,3376,2567],{},[2601,3378,2603],{},[1908,3380,2972],{},[1992,3382,3383,3386],{},[1824,3384,3385],{},"ε",[1824,3387,2969],{},[1824,3389,2004],{"mathvariant":1905},[1829,3391,3392],{"encoding":1831},"X_t=\\phi X_{t-1}+\\varepsilon_t.",[1805,3394,3396,3452,3520],{"className":3395,"ariaHidden":1837},[1836],[1805,3397,3399,3403,3443,3446,3449],{"className":3398},[1841],[1805,3400],{"className":3401,"style":3402},[1845],"height:0.8333em;vertical-align:-0.15em;",[1805,3404,3406,3409],{"className":3405},[1850],[1805,3407,1827],{"className":3408,"style":2994},[1850,1883],[1805,3410,3412],{"className":3411},[2194],[1805,3413,3415,3435],{"className":3414},[2059,2060],[1805,3416,3418,3432],{"className":3417},[2064],[1805,3419,3421],{"className":3420,"style":3158},[2068],[1805,3422,3423,3426],{"style":3010},[1805,3424],{"className":3425,"style":2211},[2076],[1805,3427,3429],{"className":3428},[2215,2216,2045,2217],[1805,3430,2969],{"className":3431},[1850,1883,2217],[1805,3433,2100],{"className":3434},[2099],[1805,3436,3438],{"className":3437},[2064],[1805,3439,3441],{"className":3440,"style":2313},[2068],[1805,3442],{},[1805,3444],{"className":3445,"style":2120},[2029],[1805,3447,1948],{"className":3448},[2124],[1805,3450],{"className":3451,"style":2120},[2029],[1805,3453,3455,3459,3462,3511,3514,3517],{"className":3454},[1841],[1805,3456],{"className":3457,"style":3458},[1845],"height:0.9028em;vertical-align:-0.2083em;",[1805,3460,2848],{"className":3461},[1850,1883],[1805,3463,3465,3468],{"className":3464},[1850],[1805,3466,1827],{"className":3467,"style":2994},[1850,1883],[1805,3469,3471],{"className":3470},[2194],[1805,3472,3474,3503],{"className":3473},[2059,2060],[1805,3475,3477,3500],{"className":3476},[2064],[1805,3478,3480],{"className":3479,"style":3007},[2068],[1805,3481,3482,3485],{"style":3010},[1805,3483],{"className":3484,"style":2211},[2076],[1805,3486,3488],{"className":3487},[2215,2216,2045,2217],[1805,3489,3491,3494,3497],{"className":3490},[1850,2217],[1805,3492,2969],{"className":3493},[1850,1883,2217],[1805,3495,2567],{"className":3496},[2639,2217],[1805,3498,2603],{"className":3499},[1850,2217],[1805,3501,2100],{"className":3502},[2099],[1805,3504,3506],{"className":3505},[2064],[1805,3507,3509],{"className":3508,"style":3038},[2068],[1805,3510],{},[1805,3512],{"className":3513,"style":1884},[2029],[1805,3515,2972],{"className":3516},[2639],[1805,3518],{"className":3519,"style":1884},[2029],[1805,3521,3523,3527,3568],{"className":3522},[1841],[1805,3524],{"className":3525,"style":3526},[1845],"height:0.5806em;vertical-align:-0.15em;",[1805,3528,3530,3533],{"className":3529},[1850],[1805,3531,3385],{"className":3532},[1850,1883],[1805,3534,3536],{"className":3535},[2194],[1805,3537,3539,3560],{"className":3538},[2059,2060],[1805,3540,3542,3557],{"className":3541},[2064],[1805,3543,3545],{"className":3544,"style":3158},[2068],[1805,3546,3548,3551],{"style":3547},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1805,3549],{"className":3550,"style":2211},[2076],[1805,3552,3554],{"className":3553},[2215,2216,2045,2217],[1805,3555,2969],{"className":3556},[1850,1883,2217],[1805,3558,2100],{"className":3559},[2099],[1805,3561,3563],{"className":3562},[2064],[1805,3564,3566],{"className":3565,"style":2313},[2068],[1805,3567],{},[1805,3569,2004],{"className":3570},[1850],[1798,3572,3573],{},"Compare:",[1805,3575,3577],{"className":3576},[1889],[1805,3578,3580,3738],{"className":3579},[1808],[1805,3581,3583],{"className":3582},[1812],[1814,3584,3585],{"xmlns":1816,"display":1898},[1818,3586,3587,3735],{},[1821,3588,3589,3608,3610,3682,3684,3686,3701,3703,3733],{},[1992,3590,3591,3597],{},[2358,3592,3593,3595],{"accent":1837},[1824,3594,2848],{},[1908,3596,2364],{"stretchy":1837},[1821,3598,3599,3602,3605],{},[1824,3600,3601],{},"C",[1824,3603,3604],{},"L",[1824,3606,3607],{},"S",[1908,3609,1948],{},[3611,3612,3613,3651],"mfrac",{},[1821,3614,3615,3632,3639],{},[3616,3617,3618,3621,3629],"munderover",{},[1908,3619,3620],{},"∑",[1821,3622,3623,3625,3627],{},[1824,3624,2969],{},[1908,3626,1948],{},[2601,3628,3097],{},[1824,3630,3631],{},"n",[1992,3633,3634,3637],{},[1824,3635,3636],{},"x",[1824,3638,2969],{},[1992,3640,3641,3643],{},[1824,3642,3636],{},[1821,3644,3645,3647,3649],{},[1824,3646,2969],{},[1908,3648,2567],{},[2601,3650,2603],{},[1821,3652,3653,3667],{},[3616,3654,3655,3657,3665],{},[1908,3656,3620],{},[1821,3658,3659,3661,3663],{},[1824,3660,2969],{},[1908,3662,1948],{},[2601,3664,3097],{},[1824,3666,3631],{},[3668,3669,3670,3672,3680],"msubsup",{},[1824,3671,3636],{},[1821,3673,3674,3676,3678],{},[1824,3675,2969],{},[1908,3677,2567],{},[2601,3679,2603],{},[2601,3681,3097],{},[1908,3683,2378],{"separator":1837},[2029,3685],{"width":2381},[1992,3687,3688,3694],{},[2358,3689,3690,3692],{"accent":1837},[1824,3691,2848],{},[1908,3693,2364],{"stretchy":1837},[1821,3695,3696,3698],{},[1824,3697,1867],{},[1824,3699,3700],{},"W",[1908,3702,1948],{},[3611,3704,3705,3719],{},[1821,3706,3707,3713,3715,3717],{},[2358,3708,3709,3711],{"accent":1837},[1824,3710,1996],{},[1908,3712,2364],{"stretchy":1837},[1908,3714,1915],{"stretchy":1932},[2601,3716,2603],{},[1908,3718,1945],{"stretchy":1932},[1821,3720,3721,3727,3729,3731],{},[2358,3722,3723,3725],{"accent":1837},[1824,3724,1996],{},[1908,3726,2364],{"stretchy":1837},[1908,3728,1915],{"stretchy":1932},[2601,3730,1931],{},[1908,3732,1945],{"stretchy":1932},[1908,3734,2378],{"separator":1837},[1829,3736,3737],{"encoding":1831},"\\widehat\\phi_{CLS}\n=\\frac{\\sum_{t=2}^n x_tx_{t-1}}\n{\\sum_{t=2}^n x_{t-1}^2},\n\\qquad\n\\widehat\\phi_{YW}\n=\\frac{\\widehat\\gamma(1)}{\\widehat\\gamma(0)},",[1805,3739,3741,3851,4313],{"className":3740,"ariaHidden":1837},[1836],[1805,3742,3744,3748,3842,3845,3848],{"className":3743},[1841],[1805,3745],{"className":3746,"style":3747},[1845],"height:1.1289em;vertical-align:-0.1944em;",[1805,3749,3751,3797],{"className":3750},[1850],[1805,3752,3754],{"className":3753},[1850,2408],[1805,3755,3757,3788],{"className":3756},[2059,2060],[1805,3758,3760,3785],{"className":3759},[2064],[1805,3761,3764,3772],{"className":3762,"style":3763},[2068],"height:0.9344em;",[1805,3765,3766,3769],{"style":2420},[1805,3767],{"className":3768,"style":2077},[2076],[1805,3770,2848],{"className":3771},[1850,1883],[1805,3773,3776,3779],{"className":3774,"style":3775},[2430],"width:calc(100% - 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conditional, Yule-Walker, and exact Gaussian estimates",[1798,4882,4883],{},"The estimates are not expected to match exactly:",[4885,4886,4887,4903],"table",{},[4888,4889,4890],"thead",{},[4891,4892,4893,4897,4900],"tr",{},[4894,4895,4896],"th",{},"Estimator",[4894,4898,4899],{},"Initial-condition treatment",[4894,4901,4902],{},"Main sample summary",[4904,4905,4906,4988,4999],"tbody",{},[4891,4907,4908,4912,4985],{},[4909,4910,4911],"td",{},"conditional LS",[4909,4913,4914,4915],{},"conditions on ",[1805,4916,4918,4936],{"className":4917},[1808],[1805,4919,4921],{"className":4920},[1812],[1814,4922,4923],{"xmlns":1816},[1818,4924,4925,4933],{},[1821,4926,4927],{},[1992,4928,4929,4931],{},[1824,4930,3636],{},[2601,4932,2603],{},[1829,4934,4935],{"encoding":1831},"x_1",[1805,4937,4939],{"className":4938,"ariaHidden":1837},[1836],[1805,4940,4942,4945],{"className":4941},[1841],[1805,4943],{"className":4944,"style":3526},[1845],[1805,4946,4948,4951],{"className":4947},[1850],[1805,4949,3636],{"className":4950},[1850,1883],[1805,4952,4954],{"className":4953},[2194],[1805,4955,4957,4977],{"className":4956},[2059,2060],[1805,4958,4960,4974],{"className":4959},[2064],[1805,4961,4963],{"className":4962,"style":3007},[2068],[1805,4964,4965,4968],{"style":3547},[1805,4966],{"className":4967,"style":2211},[2076],[1805,4969,4971],{"className":4970},[2215,2216,2045,2217],[1805,4972,2603],{"className":4973},[1850,2217],[1805,4975,2100],{"className":4976},[2099],[1805,4978,4980],{"className":4979},[2064],[1805,4981,4983],{"className":4982,"style":2313},[2068],[1805,4984],{},[4909,4986,4987],{},"transition residual sum of squares",[4891,4989,4990,4993,4996],{},[4909,4991,4992],{},"Yule–Walker",[4909,4994,4995],{},"replaces population moments",[4909,4997,4998],{},"sample autocovariances",[4891,5000,5001,5004,5007],{},[4909,5002,5003],{},"exact Gaussian ML",[4909,5005,5006],{},"models stationary distribution of the full vector",[4909,5008,5009],{},"determinant and quadratic form",[1798,5011,5012,5013,5017],{},"Repeat with ",[5014,5015,5016],"code",{},"n \u003C- 500",". Differences usually shrink, illustrating asymptotic agreement without pretending finite-sample identity.",[1793,5019,5021],{"id":5020},"numerical-discipline","Numerical discipline",[1798,5023,5024],{},"Do not write code that evaluates",[5026,5027,5031],"pre",{"className":5028,"code":5029,"language":5030,"meta":10,"style":10},"language-r shiki shiki-themes github-light-high-contrast github-light-high-contrast github-dark-high-contrast","det(Sigma)\nsolve(Sigma)\n","r",[5014,5032,5033,5040],{"__ignoreMap":10},[1805,5034,5037],{"class":5035,"line":5036},"line",1,[1805,5038,5039],{},"det(Sigma)\n",[1805,5041,5043],{"class":5035,"line":5042},2,[1805,5044,5045],{},"solve(Sigma)\n",[1798,5047,5048],{},"inside an optimiser. Determinants can underflow or overflow, and a full inverse performs unnecessary work. Use",[5026,5050,5052],{"className":5028,"code":5051,"language":5030,"meta":10,"style":10},"U \u003C- chol(Sigma)\nlogdet \u003C- 2 * sum(log(diag(U)))\nz \u003C- backsolve(U, x, transpose = TRUE)\nquadratic \u003C- sum(z^2)\n",[5014,5053,5054,5059,5064,5070],{"__ignoreMap":10},[1805,5055,5056],{"class":5035,"line":5036},[1805,5057,5058],{},"U \u003C- chol(Sigma)\n",[1805,5060,5061],{"class":5035,"line":5042},[1805,5062,5063],{},"logdet \u003C- 2 * sum(log(diag(U)))\n",[1805,5065,5067],{"class":5035,"line":5066},3,[1805,5068,5069],{},"z \u003C- backsolve(U, x, transpose = TRUE)\n",[1805,5071,5073],{"class":5035,"line":5072},4,[1805,5074,5075],{},"quadratic \u003C- sum(z^2)\n",[1798,5077,5078,5079,5108],{},"The code then mirrors the mathematics and exposes failure when ",[1805,5080,5082,5096],{"className":5081},[1808],[1805,5083,5085],{"className":5084},[1812],[1814,5086,5087],{"xmlns":1816},[1818,5088,5089,5093],{},[1821,5090,5091],{},[1824,5092,4509],{"mathvariant":1905},[1829,5094,5095],{"encoding":1831},"\\Sigma",[1805,5097,5099],{"className":5098,"ariaHidden":1837},[1836],[1805,5100,5102,5105],{"className":5101},[1841],[1805,5103],{"className":5104,"style":1879},[1845],[1805,5106,4509],{"className":5107},[1850]," is not positive definite.",[1793,5110,5112],{"id":5111},"exercises","Exercises",[5114,5115,5116,5309,5398,5504,5629],"ol",{},[5117,5118,5119,5120,5204,5205,5308],"li",{},"Predict 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for state ",[1805,5206,5208,5243],{"className":5207},[1808],[1805,5209,5211],{"className":5210},[1812],[1814,5212,5213],{"xmlns":1816},[1818,5214,5215,5240],{},[1821,5216,5217,5219,5222,5224,5227,5229,5231,5234],{},[1908,5218,1915],{"stretchy":1932},[2601,5220,5221],{},"1.2",[1908,5223,2378],{"separator":1837},[2601,5225,5226],{},"0.4",[1908,5228,2378],{"separator":1837},[1908,5230,2567],{},[2601,5232,5233],{},"0.5",[1980,5235,5236,5238],{},[1908,5237,1945],{"stretchy":1932},[1824,5239,1986],{"mathvariant":1905},[1829,5241,5242],{"encoding":1831},"(1.2,0.4,-0.5)^\\top",[1805,5244,5246],{"className":5245,"ariaHidden":1837},[1836],[1805,5247,5249,5252,5255,5258,5261,5264,5267,5270,5273,5276,5279],{"className":5248},[1841],[1805,5250],{"className":5251,"style":2900},[1845],[1805,5253,1915],{"className":5254},[2038],[1805,5256,5221],{"className":5257},[1850],[1805,5259,2378],{"className":5260},[2503],[1805,5262],{"className":5263,"style":2030},[2029],[1805,5265,5226],{"className":5266},[1850],[1805,5268,2378],{"className":5269},[2503],[1805,5271],{"className":5272,"style":2030},[2029],[1805,5274,2567],{"className":5275},[1850],[1805,5277,5233],{"className":5278},[1850],[1805,5280,5282,5285],{"className":5281},[2113],[1805,5283,1945],{"className":5284},[2113],[1805,5286,5288],{"className":5287},[2194],[1805,5289,5291],{"className":5290},[2059],[1805,5292,5294],{"className":5293},[2064],[1805,5295,5297],{"className":5296,"style":2204},[2068],[1805,5298,5299,5302],{"style":2207},[1805,5300],{"className":5301,"style":2211},[2076],[1805,5303,5305],{"className":5304},[2215,2216,2045,2217],[1805,5306,1986],{"className":5307},[1850,2217]," by the AR(2) recursion and compare with the cell.",[5117,5310,5311,5312,5397],{},"Remove ",[1805,5313,5315,5339],{"className":5314},[1808],[1805,5316,5318],{"className":5317},[1812],[1814,5319,5320],{"xmlns":1816},[1818,5321,5322,5336],{},[1821,5323,5324],{},[1992,5325,5326,5328],{},[1824,5327,1827],{},[1821,5329,5330,5332,5334],{},[1824,5331,2969],{},[1908,5333,2567],{},[2601,5335,3097],{},[1829,5337,5338],{"encoding":1831},"X_{t-2}",[1805,5340,5342],{"className":5341,"ariaHidden":1837},[1836],[1805,5343,5345,5348],{"className":5344},[1841],[1805,5346],{"className":5347,"style":2987},[1845],[1805,5349,5351,5354],{"className":5350},[1850],[1805,5352,1827],{"className":5353,"style":2994},[1850,1883],[1805,5355,5357],{"className":5356},[2194],[1805,5358,5360,5389],{"className":5359},[2059,2060],[1805,5361,5363,5386],{"className":5362},[2064],[1805,5364,5366],{"className":5365,"style":3007},[2068],[1805,5367,5368,5371],{"style":3010},[1805,5369],{"className":5370,"style":2211},[2076],[1805,5372,5374],{"className":5373},[2215,2216,2045,2217],[1805,5375,5377,5380,5383],{"className":5376},[1850,2217],[1805,5378,2969],{"className":5379},[1850,1883,2217],[1805,5381,2567],{"className":5382},[2639,2217],[1805,5384,3097],{"className":5385},[1850,2217],[1805,5387,2100],{"className":5388},[2099],[1805,5390,5392],{"className":5391},[2064],[1805,5393,5395],{"className":5394,"style":3038},[2068],[1805,5396],{}," from the predictor vector. Why does the MSPE remain unchanged under the true AR(2)?",[5117,5399,5400,5401,5503],{},"Set ",[1805,5402,5404,5428],{"className":5403},[1808],[1805,5405,5407],{"className":5406},[1812],[1814,5408,5409],{"xmlns":1816},[1818,5410,5411,5425],{},[1821,5412,5413,5420,5422],{},[1992,5414,5415,5417],{},[1824,5416,2848],{},[5418,5419,1837],"mtext",{},[1908,5421,1948],{},[2601,5423,5424],{},"0.95",[1829,5426,5427],{"encoding":1831},"\\phi_{\\text{true}}=0.95",[1805,5429,5431,5493],{"className":5430,"ariaHidden":1837},[1836],[1805,5432,5434,5437,5484,5487,5490],{"className":5433},[1841],[1805,5435],{"className":5436,"style":2881},[1845],[1805,5438,5440,5443],{"className":5439},[1850],[1805,5441,2848],{"className":5442},[1850,1883],[1805,5444,5446],{"className":5445},[2194],[1805,5447,5449,5476],{"className":5448},[2059,2060],[1805,5450,5452,5473],{"className":5451},[2064],[1805,5453,5455],{"className":5454,"style":3158},[2068],[1805,5456,5457,5460],{"style":3547},[1805,5458],{"className":5459,"style":2211},[2076],[1805,5461,5463],{"className":5462},[2215,2216,2045,2217],[1805,5464,5466],{"className":5465},[1850,2217],[1805,5467,5470],{"className":5468},[1850,5469,2217],"text",[1805,5471,1837],{"className":5472},[1850,2217],[1805,5474,2100],{"className":5475},[2099],[1805,5477,5479],{"className":5478},[2064],[1805,5480,5482],{"className":5481,"style":2313},[2068],[1805,5483],{},[1805,5485],{"className":5486,"style":2120},[2029],[1805,5488,1948],{"className":5489},[2124],[1805,5491],{"className":5492,"style":2120},[2029],[1805,5494,5496,5500],{"className":5495},[1841],[1805,5497],{"className":5498,"style":5499},[1845],"height:0.6444em;",[1805,5501,5424],{"className":5502},[1850]," and repeat the estimator comparison. Record convergence, condition number, and estimate dispersion across seeds.",[5117,5505,5506,5507,5538,5539,5569,5570,5628],{},"Add an unknown mean ",[1805,5508,5510,5525],{"className":5509},[1808],[1805,5511,5513],{"className":5512},[1812],[1814,5514,5515],{"xmlns":1816},[1818,5516,5517,5522],{},[1821,5518,5519],{},[1824,5520,5521],{},"μ",[1829,5523,5524],{"encoding":1831},"\\mu",[1805,5526,5528],{"className":5527,"ariaHidden":1837},[1836],[1805,5529,5531,5535],{"className":5530},[1841],[1805,5532],{"className":5533,"style":5534},[1845],"height:0.625em;vertical-align:-0.1944em;",[1805,5536,5521],{"className":5537},[1850,1883]," to the likelihood and replace ",[1805,5540,5542,5556],{"className":5541},[1808],[1805,5543,5545],{"className":5544},[1812],[1814,5546,5547],{"xmlns":1816},[1818,5548,5549,5553],{},[1821,5550,5551],{},[1824,5552,3636],{"mathvariant":1826},[1829,5554,5555],{"encoding":1831},"\\mathbf x",[1805,5557,5559],{"className":5558,"ariaHidden":1837},[1836],[1805,5560,5562,5566],{"className":5561},[1841],[1805,5563],{"className":5564,"style":5565},[1845],"height:0.4444em;",[1805,5567,3636],{"className":5568},[1850,1851]," by ",[1805,5571,5573,5593],{"className":5572},[1808],[1805,5574,5576],{"className":5575},[1812],[1814,5577,5578],{"xmlns":1816},[1818,5579,5580,5590],{},[1821,5581,5582,5584,5586,5588],{},[1824,5583,3636],{"mathvariant":1826},[1908,5585,2567],{},[1824,5587,5521],{},[2601,5589,2603],{"mathvariant":1826},[1829,5591,5592],{"encoding":1831},"\\mathbf x-\\mu\\mathbf1",[1805,5594,5596,5615],{"className":5595,"ariaHidden":1837},[1836],[1805,5597,5599,5603,5606,5609,5612],{"className":5598},[1841],[1805,5600],{"className":5601,"style":5602},[1845],"height:0.6667em;vertical-align:-0.0833em;",[1805,5604,3636],{"className":5605},[1850,1851],[1805,5607],{"className":5608,"style":1884},[2029],[1805,5610,2567],{"className":5611},[2639],[1805,5613],{"className":5614,"style":1884},[2029],[1805,5616,5618,5622,5625],{"className":5617},[1841],[1805,5619],{"className":5620,"style":5621},[1845],"height:0.8389em;vertical-align:-0.1944em;",[1805,5623,5521],{"className":5624},[1850,1883],[1805,5626,2603],{"className":5627},[1850,1851],". Which parameter correlations do you expect?",[5117,5630,5631,5632,5747,5748,2004],{},"Graduate extension: use the delta method to transform the Hessian covariance from ",[1805,5633,5635,5667],{"className":5634},[1808],[1805,5636,5638],{"className":5637},[1812],[1814,5639,5640],{"xmlns":1816},[1818,5641,5642,5664],{},[1821,5643,5644,5651,5653,5656,5658,5660,5662],{},[1992,5645,5646,5649],{},[1824,5647,5648],{},"η",[2601,5650,2603],{},[1908,5652,1948],{},[1824,5654,5655],{"mathvariant":1905},"atanh",[1908,5657,1910],{},[1908,5659,1915],{"stretchy":1932},[1824,5661,2848],{},[1908,5663,1945],{"stretchy":1932},[1829,5665,5666],{"encoding":1831},"\\eta_1=\\operatorname{atanh}(\\phi)",[1805,5668,5670,5726],{"className":5669,"ariaHidden":1837},[1836],[1805,5671,5673,5676,5717,5720,5723],{"className":5672},[1841],[1805,5674],{"className":5675,"style":5534},[1845],[1805,5677,5679,5682],{"className":5678},[1850],[1805,5680,5648],{"className":5681,"style":4617},[1850,1883],[1805,5683,5685],{"className":5684},[2194],[1805,5686,5688,5709],{"className":5687},[2059,2060],[1805,5689,5691,5706],{"className":5690},[2064],[1805,5692,5694],{"className":5693,"style":3007},[2068],[1805,5695,5697,5700],{"style":5696},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1805,5698],{"className":5699,"style":2211},[2076],[1805,5701,5703],{"className":5702},[2215,2216,2045,2217],[1805,5704,2603],{"className":5705},[1850,2217],[1805,5707,2100],{"className":5708},[2099],[1805,5710,5712],{"className":5711},[2064],[1805,5713,5715],{"className":5714,"style":2313},[2068],[1805,5716],{},[1805,5718],{"className":5719,"style":2120},[2029],[1805,5721,1948],{"className":5722},[2124],[1805,5724],{"className":5725,"style":2120},[2029],[1805,5727,5729,5732,5738,5741,5744],{"className":5728},[1841],[1805,5730],{"className":5731,"style":2620},[1845],[1805,5733,5735],{"className":5734},[2021],[1805,5736,5655],{"className":5737},[1850,2025],[1805,5739,1915],{"className":5740},[2038],[1805,5742,2848],{"className":5743},[1850,1883],[1805,5745,1945],{"className":5746},[2113]," to ",[1805,5749,5751,5765],{"className":5750},[1808],[1805,5752,5754],{"className":5753},[1812],[1814,5755,5756],{"xmlns":1816},[1818,5757,5758,5762],{},[1821,5759,5760],{},[1824,5761,2848],{},[1829,5763,5764],{"encoding":1831},"\\phi",[1805,5766,5768],{"className":5767,"ariaHidden":1837},[1836],[1805,5769,5771,5774],{"className":5770},[1841],[1805,5772],{"className":5773,"style":2881},[1845],[1805,5775,2848],{"className":5776},[1850,1883],[5778,5779,5781],"legacy-details",{"title":5780},"Checkpoints",[5114,5782,5783,5890,6043,6046,6049],{},[5117,5784,5785,5889],{},[1805,5786,5788,5823],{"className":5787},[1808],[1805,5789,5791],{"className":5790},[1812],[1814,5792,5793],{"xmlns":1816},[1818,5794,5795,5820],{},[1821,5796,5797,5799,5801,5803,5805,5807,5809,5811,5813,5815,5817],{},[2601,5798,2855],{},[1908,5800,1915],{"stretchy":1932},[2601,5802,5221],{},[1908,5804,1945],{"stretchy":1932},[1908,5806,2567],{},[2601,5808,2862],{},[1908,5810,1915],{"stretchy":1932},[2601,5812,5226],{},[1908,5814,1945],{"stretchy":1932},[1908,5816,1948],{},[2601,5818,5819],{},"0.70",[1829,5821,5822],{"encoding":1831},"0.65(1.2)-0.20(0.4)=0.70",[1805,5824,5826,5853,5880],{"className":5825,"ariaHidden":1837},[1836],[1805,5827,5829,5832,5835,5838,5841,5844,5847,5850],{"className":5828},[1841],[1805,5830],{"className":5831,"style":2620},[1845],[1805,5833,2855],{"className":5834},[1850],[1805,5836,1915],{"className":5837},[2038],[1805,5839,5221],{"className":5840},[1850],[1805,5842,1945],{"className":5843},[2113],[1805,5845],{"className":5846,"style":1884},[2029],[1805,5848,2567],{"className":5849},[2639],[1805,5851],{"className":5852,"style":1884},[2029],[1805,5854,5856,5859,5862,5865,5868,5871,5874,5877],{"className":5855},[1841],[1805,5857],{"className":5858,"style":2620},[1845],[1805,5860,2862],{"className":5861},[1850],[1805,5863,1915],{"className":5864},[2038],[1805,5866,5226],{"className":5867},[1850],[1805,5869,1945],{"className":5870},[2113],[1805,5872],{"className":5873,"style":2120},[2029],[1805,5875,1948],{"className":5876},[2124],[1805,5878],{"className":5879,"style":2120},[2029],[1805,5881,5883,5886],{"className":5882},[1841],[1805,5884],{"className":5885,"style":5499},[1845],[1805,5887,5819],{"className":5888},[1850],"; the third coordinate has no direct coefficient.",[5117,5891,5892,5893,6042],{},"The Markov state already consists of ",[1805,5894,5896,5932],{"className":5895},[1808],[1805,5897,5899],{"className":5898},[1812],[1814,5900,5901],{"xmlns":1816},[1818,5902,5903,5929],{},[1821,5904,5905,5907,5913,5915,5927],{},[1908,5906,1915],{"stretchy":1932},[1992,5908,5909,5911],{},[1824,5910,1827],{},[1824,5912,2969],{},[1908,5914,2378],{"separator":1837},[1992,5916,5917,5919],{},[1824,5918,1827],{},[1821,5920,5921,5923,5925],{},[1824,5922,2969],{},[1908,5924,2567],{},[2601,5926,2603],{},[1908,5928,1945],{"stretchy":1932},[1829,5930,5931],{"encoding":1831},"(X_t,X_{t-1})",[1805,5933,5935],{"className":5934,"ariaHidden":1837},[1836],[1805,5936,5938,5941,5944,5984,5987,5990,6039],{"className":5937},[1841],[1805,5939],{"className":5940,"style":2620},[1845],[1805,5942,1915],{"className":5943},[2038],[1805,5945,5947,5950],{"className":5946},[1850],[1805,5948,1827],{"className":5949,"style":2994},[1850,1883],[1805,5951,5953],{"className":5952},[2194],[1805,5954,5956,5976],{"className":5955},[2059,2060],[1805,5957,5959,5973],{"className":5958},[2064],[1805,5960,5962],{"className":5961,"style":3158},[2068],[1805,5963,5964,5967],{"style":3010},[1805,5965],{"className":5966,"style":2211},[2076],[1805,5968,5970],{"className":5969},[2215,2216,2045,2217],[1805,5971,2969],{"className":5972},[1850,1883,2217],[1805,5974,2100],{"className":5975},[2099],[1805,5977,5979],{"className":5978},[2064],[1805,5980,5982],{"className":5981,"style":2313},[2068],[1805,5983],{},[1805,5985,2378],{"className":5986},[2503],[1805,5988],{"className":5989,"style":2030},[2029],[1805,5991,5993,5996],{"className":5992},[1850],[1805,5994,1827],{"className":5995,"style":2994},[1850,1883],[1805,5997,5999],{"className":5998},[2194],[1805,6000,6002,6031],{"className":6001},[2059,2060],[1805,6003,6005,6028],{"className":6004},[2064],[1805,6006,6008],{"className":6007,"style":3007},[2068],[1805,6009,6010,6013],{"style":3010},[1805,6011],{"className":6012,"style":2211},[2076],[1805,6014,6016],{"className":6015},[2215,2216,2045,2217],[1805,6017,6019,6022,6025],{"className":6018},[1850,2217],[1805,6020,2969],{"className":6021},[1850,1883,2217],[1805,6023,2567],{"className":6024},[2639,2217],[1805,6026,2603],{"className":6027},[1850,2217],[1805,6029,2100],{"className":6030},[2099],[1805,6032,6034],{"className":6033},[2064],[1805,6035,6037],{"className":6036,"style":3038},[2068],[1805,6038],{},[1805,6040,1945],{"className":6041},[2113],"; the older value adds no linear information conditional on that state.",[5117,6044,6045],{},"Near the boundary, covariance matrices become ill-conditioned and normal approximations become less reliable.",[5117,6047,6048],{},"The mean and persistence can be strongly confounded in short, persistent series.",[5117,6050,6051,6052,6174],{},"Multiply the eta-scale variance by ",[1805,6053,6055,6085],{"className":6054},[1808],[1805,6056,6058],{"className":6057},[1812],[1814,6059,6060],{"xmlns":1816},[1818,6061,6062,6082],{},[1821,6063,6064,6066,6068,6070,6076],{},[1908,6065,1915],{"stretchy":1932},[2601,6067,2603],{},[1908,6069,2567],{},[1980,6071,6072,6074],{},[1824,6073,2848],{},[2601,6075,3097],{},[1980,6077,6078,6080],{},[1908,6079,1945],{"stretchy":1932},[2601,6081,3097],{},[1829,6083,6084],{"encoding":1831},"(1-\\phi^2)^2",[1805,6086,6088,6109],{"className":6087,"ariaHidden":1837},[1836],[1805,6089,6091,6094,6097,6100,6103,6106],{"className":6090},[1841],[1805,6092],{"className":6093,"style":2620},[1845],[1805,6095,1915],{"className":6096},[2038],[1805,6098,2603],{"className":6099},[1850],[1805,6101],{"className":6102,"style":1884},[2029],[1805,6104,2567],{"className":6105},[2639],[1805,6107],{"className":6108,"style":1884},[2029],[1805,6110,6112,6116,6145],{"className":6111},[1841],[1805,6113],{"className":6114,"style":6115},[1845],"height:1.0641em;vertical-align:-0.25em;",[1805,6117,6119,6122],{"className":6118},[1850],[1805,6120,2848],{"className":6121},[1850,1883],[1805,6123,6125],{"className":6124},[2194],[1805,6126,6128],{"className":6127},[2059],[1805,6129,6131],{"className":6130},[2064],[1805,6132,6134],{"className":6133,"style":4764},[2068],[1805,6135,6136,6139],{"style":2207},[1805,6137],{"className":6138,"style":2211},[2076],[1805,6140,6142],{"className":6141},[2215,2216,2045,2217],[1805,6143,3097],{"className":6144},[1850,2217],[1805,6146,6148,6151],{"className":6147},[2113],[1805,6149,1945],{"className":6150},[2113],[1805,6152,6154],{"className":6153},[2194],[1805,6155,6157],{"className":6156},[2059],[1805,6158,6160],{"className":6159},[2064],[1805,6161,6163],{"className":6162,"style":4764},[2068],[1805,6164,6165,6168],{"style":2207},[1805,6166],{"className":6167,"style":2211},[2076],[1805,6169,6171],{"className":6170},[2215,2216,2045,2217],[1805,6172,3097],{"className":6173},[1850,2217]," at the estimate.",[1793,6176,6178],{"id":6177},"completion-standard","Completion standard",[1798,6180,6181],{},"You should be able to identify the covariance block used for prediction, verify its normal equations, and explain why conditional, moment, and exact likelihood estimates differ in finite samples.",[1798,6183,6184,6185,2004],{},"Continue to ",[2371,6186,6188],{"href":6187},".\u002F04-state-space","Lab 4 — State Space",[6190,6191,6192],"style",{},"html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: 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