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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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h84z",[1805,3634,1971],{"className":3635},[1970],[1805,3637,3639],{"className":3638},[1943],[1805,3640,3642],{"className":3641,"style":3174},[1947],[1805,3643],{},[1805,3645],{"className":3646,"style":2488},[1984],[1805,3648,2320],{"className":3649},[2484],[1805,3651],{"className":3652,"style":3653},[1984],"margin-right:2em;",[1805,3655],{"className":3656,"style":2488},[1984],[1805,3658,2691],{"className":3659},[1925,1929],[1805,3661],{"className":3662,"style":1985},[1984],[1805,3664,1841],{"className":3665},[1989],[1805,3667],{"className":3668,"style":1985},[1984],[1805,3670,3672,3676,3679,3682,3685,3688,3691,3694,3697,3700,3703,3706,3709,3712,3741],{"className":3671},[1916],[1805,3673],{"className":3674,"style":3675},[1920],"height:1.1491em;vertical-align:-0.25em;",[1805,3677,2311],{"className":3678},[2440],[1805,3680,1850],{"className":3681},[1925],[1805,3683,2320],{"className":3684},[2484],[1805,3686],{"className":3687,"style":2488},[1984],[1805,3689,2925],{"className":3690},[1925],[1805,3692,2320],{"className":3693},[2484],[1805,3695],{"className":3696,"style":2488},[1984],[1805,3698,2337],{"className":3699},[2113],[1805,3701],{"className":3702,"style":2488},[1984],[1805,3704,2320],{"className":3705},[2484],[1805,3707],{"className":3708,"style":2488},[1984],[1805,3710,2925],{"className":3711},[1925],[1805,3713,3715,3718],{"className":3714},[2614],[1805,3716,2361],{"className":3717},[2614],[1805,3719,3721],{"className":3720},[1934],[1805,3722,3724],{"className":3723},[1938],[1805,3725,3727],{"className":3726},[1943],[1805,3728,3730],{"className":3729,"style":2630},[1947],[1805,3731,3732,3735],{"style":2633},[1805,3733],{"className":3734,"style":1956},[1955],[1805,3736,3738],{"className":3737},[1960,1961,1962,1963],[1805,3739,2364],{"className":3740},[1925,1963],[1805,3742,1902],{"className":3743},[1925],[1793,3745,3747],{"id":3746},"example-1-roots-eigenvalues-and-shock-propagation","Example 1 — Roots, eigenvalues, and shock propagation",[1798,3749,3750],{},"For",[1805,3752,3754],{"className":3753},[1808],[1805,3755,3757,3819],{"className":3756},[1812],[1805,3758,3760],{"className":3759},[1816],[1818,3761,3762],{"xmlns":1820,"display":1821},[1823,3763,3764,3816],{},[1826,3765,3766,3772,3774,3777,3789,3791,3794,3806,3808,3814],{},[1829,3767,3768,3770],{},[1832,3769,1834],{},[1832,3771,1837],{},[1839,3773,1841],{},[1848,3775,3776],{},"0.75",[1829,3778,3779,3781],{},[1832,3780,1834],{},[1826,3782,3783,3785,3787],{},[1832,3784,1837],{},[1839,3786,1861],{},[1848,3788,1850],{},[1839,3790,1861],{},[1848,3792,3793],{},"0.125",[1829,3795,3796,3798],{},[1832,3797,1834],{},[1826,3799,3800,3802,3804],{},[1832,3801,1837],{},[1839,3803,1861],{},[1848,3805,2942],{},[1839,3807,1866],{},[1829,3809,3810,3812],{},[1832,3811,1896],{},[1832,3813,1837],{},[1839,3815,2320],{"separator":1912},[1904,3817,3818],{"encoding":1906},"X_t=0.75X_{t-1}-0.125X_{t-2}+\\varepsilon_t,",[1805,3820,3822,3877,3944,4011],{"className":3821,"ariaHidden":1912},[1911],[1805,3823,3825,3828,3868,3871,3874],{"className":3824},[1916],[1805,3826],{"className":3827,"style":1921},[1920],[1805,3829,3831,3834],{"className":3830},[1925],[1805,3832,1834],{"className":3833,"style":1930},[1925,1929],[1805,3835,3837],{"className":3836},[1934],[1805,3838,3840,3860],{"className":3839},[1938,1939],[1805,3841,3843,3857],{"className":3842},[1943],[1805,3844,3846],{"className":3845,"style":1948},[1947],[1805,3847,3848,3851],{"style":1951},[1805,3849],{"className":3850,"style":1956},[1955],[1805,3852,3854],{"className":3853},[1960,1961,1962,1963],[1805,3855,1837],{"className":3856},[1925,1929,1963],[1805,3858,1971],{"className":3859},[1970],[1805,3861,3863],{"className":3862},[1943],[1805,3864,3866],{"className":3865,"style":1978},[1947],[1805,3867],{},[1805,3869],{"className":3870,"style":1985},[1984],[1805,3872,1841],{"className":3873},[1989],[1805,3875],{"className":3876,"style":1985},[1984],[1805,3878,3880,3883,3886,3935,3938,3941],{"className":3879},[1916],[1805,3881],{"className":3882,"style":2767},[1920],[1805,3884,3776],{"className":3885},[1925],[1805,3887,3889,3892],{"className":3888},[1925],[1805,3890,1834],{"className":3891,"style":1930},[1925,1929],[1805,3893,3895],{"className":3894},[1934],[1805,3896,3898,3927],{"className":3897},[1938,1939],[1805,3899,3901,3924],{"className":3900},[1943],[1805,3902,3904],{"className":3903,"style":2018},[1947],[1805,3905,3906,3909],{"style":1951},[1805,3907],{"className":3908,"style":1956},[1955],[1805,3910,3912],{"className":3911},[1960,1961,1962,1963],[1805,3913,3915,3918,3921],{"className":3914},[1925,1963],[1805,3916,1837],{"className":3917},[1925,1929,1963],[1805,3919,1861],{"className":3920},[2077,1963],[1805,3922,1850],{"className":3923},[1925,1963],[1805,3925,1971],{"className":3926},[1970],[1805,3928,3930],{"className":3929},[1943],[1805,3931,3933],{"className":3932,"style":2090},[1947],[1805,3934],{},[1805,3936],{"className":3937,"style":2096},[1984],[1805,3939,1861],{"className":3940},[2077],[1805,3942],{"className":3943,"style":2096},[1984],[1805,3945,3947,3950,3953,4002,4005,4008],{"className":3946},[1916],[1805,3948],{"className":3949,"style":2767},[1920],[1805,3951,3793],{"className":3952},[1925],[1805,3954,3956,3959],{"className":3955},[1925],[1805,3957,1834],{"className":3958,"style":1930},[1925,1929],[1805,3960,3962],{"className":3961},[1934],[1805,3963,3965,3994],{"className":3964},[1938,1939],[1805,3966,3968,3991],{"className":3967},[1943],[1805,3969,3971],{"className":3970,"style":2018},[1947],[1805,3972,3973,3976],{"style":1951},[1805,3974],{"className":3975,"style":1956},[1955],[1805,3977,3979],{"className":3978},[1960,1961,1962,1963],[1805,3980,3982,3985,3988],{"className":3981},[1925,1963],[1805,3983,1837],{"className":3984},[1925,1929,1963],[1805,3986,1861],{"className":3987},[2077,1963],[1805,3989,2942],{"className":3990},[1925,1963],[1805,3992,1971],{"className":3993},[1970],[1805,3995,3997],{"className":3996},[1943],[1805,3998,4000],{"className":3999,"style":2090},[1947],[1805,4001],{},[1805,4003],{"className":4004,"style":2096},[1984],[1805,4006,1866],{"className":4007},[2077],[1805,4009],{"className":4010,"style":2096},[1984],[1805,4012,4014,4018,4058],{"className":4013},[1916],[1805,4015],{"className":4016,"style":4017},[1920],"height:0.625em;vertical-align:-0.1944em;",[1805,4019,4021,4024],{"className":4020},[1925],[1805,4022,1896],{"className":4023},[1925,1929],[1805,4025,4027],{"className":4026},[1934],[1805,4028,4030,4050],{"className":4029},[1938,1939],[1805,4031,4033,4047],{"className":4032},[1943],[1805,4034,4036],{"className":4035,"style":1948},[1947],[1805,4037,4038,4041],{"style":2021},[1805,4039],{"className":4040,"style":1956},[1955],[1805,4042,4044],{"className":4043},[1960,1961,1962,1963],[1805,4045,1837],{"className":4046},[1925,1929,1963],[1805,4048,1971],{"className":4049},[1970],[1805,4051,4053],{"className":4052},[1943],[1805,4054,4056],{"className":4055,"style":1978},[1947],[1805,4057],{},[1805,4059,2320],{"className":4060},[2484],[1798,4062,4063],{},"the AR polynomial is",[1805,4065,4067],{"className":4066},[1808],[1805,4068,4070,4113],{"className":4069},[1812],[1805,4071,4073],{"className":4072},[1816],[1818,4074,4075],{"xmlns":1820,"display":1821},[1823,4076,4077,4110],{},[1826,4078,4079,4081,4083,4086,4088,4090,4092,4094,4096,4098,4100,4102,4108],{},[1832,4080,1846],{},[1839,4082,2311],{"stretchy":2310},[1832,4084,4085],{},"z",[1839,4087,2361],{"stretchy":2310},[1839,4089,1841],{},[1848,4091,1850],{},[1839,4093,1861],{},[1848,4095,3776],{},[1832,4097,4085],{},[1839,4099,1866],{},[1848,4101,3793],{},[2357,4103,4104,4106],{},[1832,4105,4085],{},[1848,4107,2942],{},[1832,4109,1902],{"mathvariant":1901},[1904,4111,4112],{"encoding":1906},"\\phi(z)=1-0.75z+0.125z^2.",[1805,4114,4116,4145,4164,4185],{"className":4115,"ariaHidden":1912},[1911],[1805,4117,4119,4123,4126,4129,4133,4136,4139,4142],{"className":4118},[1916],[1805,4120],{"className":4121,"style":4122},[1920],"height:1em;vertical-align:-0.25em;",[1805,4124,1846],{"className":4125},[1925,1929],[1805,4127,2311],{"className":4128},[2440],[1805,4130,4085],{"className":4131,"style":4132},[1925,1929],"margin-right:0.044em;",[1805,4134,2361],{"className":4135},[2614],[1805,4137],{"className":4138,"style":1985},[1984],[1805,4140,1841],{"className":4141},[1989],[1805,4143],{"className":4144,"style":1985},[1984],[1805,4146,4148,4152,4155,4158,4161],{"className":4147},[1916],[1805,4149],{"className":4150,"style":4151},[1920],"height:0.7278em;vertical-align:-0.0833em;",[1805,4153,1850],{"className":4154},[1925],[1805,4156],{"className":4157,"style":2096},[1984],[1805,4159,1861],{"className":4160},[2077],[1805,4162],{"className":4163,"style":2096},[1984],[1805,4165,4167,4170,4173,4176,4179,4182],{"className":4166},[1916],[1805,4168],{"className":4169,"style":4151},[1920],[1805,4171,3776],{"className":4172},[1925],[1805,4174,4085],{"className":4175,"style":4132},[1925,1929],[1805,4177],{"className":4178,"style":2096},[1984],[1805,4180,1866],{"className":4181},[2077],[1805,4183],{"className":4184,"style":2096},[1984],[1805,4186,4188,4192,4195,4224],{"className":4187},[1916],[1805,4189],{"className":4190,"style":4191},[1920],"height:0.8641em;",[1805,4193,3793],{"className":4194},[1925],[1805,4196,4198,4201],{"className":4197},[1925],[1805,4199,4085],{"className":4200,"style":4132},[1925,1929],[1805,4202,4204],{"className":4203},[1934],[1805,4205,4207],{"className":4206},[1938],[1805,4208,4210],{"className":4209},[1943],[1805,4211,4213],{"className":4212,"style":4191},[1947],[1805,4214,4215,4218],{"style":2633},[1805,4216],{"className":4217,"style":1956},[1955],[1805,4219,4221],{"className":4220},[1960,1961,1962,1963],[1805,4222,2942],{"className":4223},[1925,1963],[1805,4225,1902],{"className":4226},[1925],[1798,4228,4229],{},"Its roots must lie outside the unit circle. The companion eigenvalues are their reciprocals and must lie inside.",[4231,4232],"web-r",{"code64":4233,"layout":4234,"locale":7,"title":4235},"cGhpIDwtIGMoMC43NSwgLTAuMTI1KQpwIDwtIGxlbmd0aChwaGkpCgpGIDwtIHJiaW5kKHBoaSwgY2JpbmQoZGlhZyhwIC0gMSksIG1hdHJpeCgwLCBwIC0gMSwgMSkpKQpHIDwtIGMoMSwgcmVwKDAsIHAgLSAxKSkKYXJfcm9vdHMgPC0gcG9seXJvb3QoYygxLCAtcGhpKSkKY29tcGFuaW9uX2VpZ2VudmFsdWVzIDwtIGVpZ2VuKEYsIG9ubHkudmFsdWVzID0gVFJVRSkkdmFsdWVzCgpob3Jpem9uIDwtIDEwCnJlc3BvbnNlIDwtIG51bWVyaWMoaG9yaXpvbiArIDEpCkZfcG93ZXIgPC0gZGlhZyhwKQpmb3IgKGggaW4gMDpob3Jpem9uKSB7CiAgcmVzcG9uc2VbaCArIDFdIDwtIGRyb3AoYygxLCByZXAoMCwgcCAtIDEpKSAlKiUgRl9wb3dlciAlKiUgRykKICBGX3Bvd2VyIDwtIEZfcG93ZXIgJSolIEYKfQoKY2F0KCJDb21wYW5pb24gbWF0cml4IEY6XG4iKQpwcmludChGKQpjYXQoIlxuQVIgcG9seW5vbWlhbCByb290czpcbiIpCnByaW50KHJvdW5kKGFyX3Jvb3RzLCA2KSkKY2F0KCJSb290IG1vZHVsaTpcbiIpCnByaW50KHJvdW5kKE1vZChhcl9yb290cyksIDYpKQpjYXQoIlxuQ29tcGFuaW9uIGVpZ2VudmFsdWVzOlxuIikKcHJpbnQocm91bmQoY29tcGFuaW9uX2VpZ2VudmFsdWVzLCA2KSkKY2F0KCJTcGVjdHJhbCByYWRpdXM6Iiwgcm91bmQobWF4KE1vZChjb21wYW5pb25fZWlnZW52YWx1ZXMpKSwgNiksICJcbiIpCmNhdCgiXG5JbXB1bHNlLXJlc3BvbnNlIGNvZWZmaWNpZW50cyBwc2lfMCwuLi4scHNpXzEwOlxuIikKcHJpbnQocm91bmQocmVzcG9uc2UsIDYpKQoKcmVjdXJzaW9uX2Vycm9yIDwtIG1heChhYnMoCiAgcmVzcG9uc2VbMzpsZW5ndGgocmVzcG9uc2UpXSAtCiAgICBwaGlbMV0gKiByZXNwb25zZVsyOihsZW5ndGgocmVzcG9uc2UpIC0gMSldIC0KICAgIHBoaVsyXSAqIHJlc3BvbnNlWzE6KGxlbmd0aChyZXNwb25zZSkgLSAyKV0KKSkKY2F0KCJNYXhpbXVtIGltcHVsc2UtcmVjdXJzaW9uIGVycm9yOiIsCiAgICBmb3JtYXQocmVjdXJzaW9uX2Vycm9yLCBzY2llbnRpZmljID0gVFJVRSksICJcbiIp","vertical","Compare AR roots, companion eigenvalues, and impulse responses",[1798,4237,4238,4239,4306,4307,4382,4383,4414],{},"The first output checks admissibility. The second shows dynamics: ",[1805,4240,4242,4263],{"className":4241},[1812],[1805,4243,4245],{"className":4244},[1816],[1818,4246,4247],{"xmlns":1820},[1823,4248,4249,4260],{},[1826,4250,4251,4258],{},[2357,4252,4253,4255],{},[1832,4254,2674],{},[1832,4256,4257],{},"h",[1832,4259,2691],{},[1904,4261,4262],{"encoding":1906},"F^hG",[1805,4264,4266],{"className":4265,"ariaHidden":1912},[1911],[1805,4267,4269,4273,4303],{"className":4268},[1916],[1805,4270],{"className":4271,"style":4272},[1920],"height:0.8491em;",[1805,4274,4276,4279],{"className":4275},[1925],[1805,4277,2674],{"className":4278,"style":2771},[1925,1929],[1805,4280,4282],{"className":4281},[1934],[1805,4283,4285],{"className":4284},[1938],[1805,4286,4288],{"className":4287},[1943],[1805,4289,4291],{"className":4290,"style":4272},[1947],[1805,4292,4294,4297],{"style":4293},"top:-3.063em;margin-right:0.05em;",[1805,4295],{"className":4296,"style":1956},[1955],[1805,4298,4300],{"className":4299},[1960,1961,1962,1963],[1805,4301,4257],{"className":4302},[1925,1929,1963],[1805,4304,2691],{"className":4305},[1925,1929]," is the state response to one innovation, and its first component is ",[1805,4308,4310,4329],{"className":4309},[1812],[1805,4311,4313],{"className":4312},[1816],[1818,4314,4315],{"xmlns":1820},[1823,4316,4317,4326],{},[1826,4318,4319],{},[1829,4320,4321,4324],{},[1832,4322,4323],{},"ψ",[1832,4325,4257],{},[1904,4327,4328],{"encoding":1906},"\\psi_h",[1805,4330,4332],{"className":4331,"ariaHidden":1912},[1911],[1805,4333,4335,4339],{"className":4334},[1916],[1805,4336],{"className":4337,"style":4338},[1920],"height:0.8889em;vertical-align:-0.1944em;",[1805,4340,4342,4346],{"className":4341},[1925],[1805,4343,4323],{"className":4344,"style":4345},[1925,1929],"margin-right:0.0359em;",[1805,4347,4349],{"className":4348},[1934],[1805,4350,4352,4374],{"className":4351},[1938,1939],[1805,4353,4355,4371],{"className":4354},[1943],[1805,4356,4359],{"className":4357,"style":4358},[1947],"height:0.3361em;",[1805,4360,4362,4365],{"style":4361},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1805,4363],{"className":4364,"style":1956},[1955],[1805,4366,4368],{"className":4367},[1960,1961,1962,1963],[1805,4369,4257],{"className":4370},[1925,1929,1963],[1805,4372,1971],{"className":4373},[1970],[1805,4375,4377],{"className":4376},[1943],[1805,4378,4380],{"className":4379,"style":1978},[1947],[1805,4381],{}," in the MA(",[1805,4384,4386,4401],{"className":4385},[1812],[1805,4387,4389],{"className":4388},[1816],[1818,4390,4391],{"xmlns":1820},[1823,4392,4393,4398],{},[1826,4394,4395],{},[1832,4396,4397],{"mathvariant":1901},"∞",[1904,4399,4400],{"encoding":1906},"\\infty",[1805,4402,4404],{"className":4403,"ariaHidden":1912},[1911],[1805,4405,4407,4411],{"className":4406},[1916],[1805,4408],{"className":4409,"style":4410},[1920],"height:0.4306em;",[1805,4412,4397],{"className":4413},[1925],") representation.",[1793,4416,4418],{"id":4417},"example-2-yulewalker-as-a-toeplitz-solve","Example 2 — Yule–Walker as a Toeplitz solve",[1798,4420,4421,4422,4450],{},"For a stationary 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innovation variance follows from the lag-zero equation:",[1805,5019,5021],{"className":5020},[1808],[1805,5022,5024,5072],{"className":5023},[1812],[1805,5025,5027],{"className":5026},[1816],[1818,5028,5029],{"xmlns":1820,"display":1821},[1823,5030,5031,5069],{},[1826,5032,5033,5043,5045,5047,5049,5051,5053,5055,5061,5067],{},[5034,5035,5036,5039,5041],"msubsup",{},[1832,5037,5038],{},"σ",[1832,5040,1896],{},[1848,5042,2942],{},[1839,5044,1841],{},[1832,5046,4482],{},[1839,5048,2311],{"stretchy":2310},[1848,5050,2925],{},[1839,5052,2361],{"stretchy":2310},[1839,5054,1861],{},[2357,5056,5057,5059],{},[1832,5058,1846],{"mathvariant":4475},[1832,5060,2364],{"mathvariant":1901},[1829,5062,5063,5065],{},[1832,5064,4482],{"mathvariant":4475},[1832,5066,1798],{},[1832,5068,1902],{"mathvariant":1901},[1904,5070,5071],{"encoding":1906},"\\sigma_\\varepsilon^2\n=\\gamma(0)-\\boldsymbol\\phi^\\top\\boldsymbol\\gamma_p.",[1805,5073,5075,5144,5171],{"className":5074,"ariaHidden":1912},[1911],[1805,5076,5078,5082,5135,5138,5141],{"className":5077},[1916],[1805,5079],{"className":5080,"style":5081},[1920],"height:1.1111em;vertical-align:-0.247em;",[1805,5083,5085,5088],{"className":5084},[1925],[1805,5086,5038],{"className":5087,"style":4345},[1925,1929],[1805,5089,5091],{"className":5090},[1934],[1805,5092,5094,5126],{"className":5093},[1938,1939],[1805,5095,5097,5123],{"className":5096},[1943],[1805,5098,5100,5112],{"className":5099,"style":4191},[1947],[1805,5101,5103,5106],{"style":5102},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[1805,5104],{"className":5105,"style":1956},[1955],[1805,5107,5109],{"className":5108},[1960,1961,1962,1963],[1805,5110,1896],{"className":5111},[1925,1929,1963],[1805,5113,5114,5117],{"style":2633},[1805,5115],{"className":5116,"style":1956},[1955],[1805,5118,5120],{"className":5119},[1960,1961,1962,1963],[1805,5121,2942],{"className":5122},[1925,1963],[1805,5124,1971],{"className":5125},[1970],[1805,5127,5129],{"className":5128},[1943],[1805,5130,5133],{"className":5131,"style":5132},[1947],"height:0.247em;",[1805,5134],{},[1805,5136],{"className":5137,"style":1985},[1984],[1805,5139,1841],{"className":5140},[1989],[1805,5142],{"className":5143,"style":1985},[1984],[1805,5145,5147,5150,5153,5156,5159,5162,5165,5168],{"className":5146},[1916],[1805,5148],{"className":5149,"style":4122},[1920],[1805,5151,4482],{"className":5152,"style":4808},[1925,1929],[1805,5154,2311],{"className":5155},[2440],[1805,5157,2925],{"className":5158},[1925],[1805,5160,2361],{"className":5161},[2614],[1805,5163],{"className":5164,"style":2096},[1984],[1805,5166,1861],{"className":5167},[2077],[1805,5169],{"className":5170,"style":2096},[1984],[1805,5172,5174,5178,5215,5261],{"className":5173},[1916],[1805,5175],{"className":5176,"style":5177},[1920],"height:1.3137em;vertical-align:-0.3802em;",[1805,5179,5181,5190],{"className":5180},[1925],[1805,5182,5184],{"className":5183},[1925],[1805,5185,5187],{"className":5186},[1925],[1805,5188,1846],{"className":5189},[1925,4548],[1805,5191,5193],{"className":5192},[1934],[1805,5194,5196],{"className":5195},[1938],[1805,5197,5199],{"className":5198},[1943],[1805,5200,5203],{"className":5201,"style":5202},[1947],"height:0.9334em;",[1805,5204,5206,5209],{"style":5205},"top:-3.1473em;margin-right:0.05em;",[1805,5207],{"className":5208,"style":1956},[1955],[1805,5210,5212],{"className":5211},[1960,1961,1962,1963],[1805,5213,2364],{"className":5214},[1925,1963],[1805,5216,5218,5227],{"className":5217},[1925],[1805,5219,5221],{"className":5220},[1925],[1805,5222,5224],{"className":5223},[1925],[1805,5225,4482],{"className":5226,"style":4577},[1925,4548],[1805,5228,5230],{"className":5229},[1934],[1805,5231,5233,5253],{"className":5232},[1938,1939],[1805,5234,5236,5250],{"className":5235},[1943],[1805,5237,5239],{"className":5238,"style":4590},[1947],[1805,5240,5241,5244],{"style":4593},[1805,5242],{"className":5243,"style":1956},[1955],[1805,5245,5247],{"className":5246},[1960,1961,1962,1963],[1805,5248,1798],{"className":5249},[1925,1929,1963],[1805,5251,1971],{"className":5252},[1970],[1805,5254,5256],{"className":5255},[1943],[1805,5257,5259],{"className":5258,"style":4612},[1947],[1805,5260],{},[1805,5262,1902],{"className":5263},[1925],[4231,5265],{"code64":5266,"layout":4234,"locale":7,"title":5267},"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AR coefficients from theoretical autocovariances",[1798,5269,5270],{},"This cell deliberately starts from population autocovariances. In estimation, replacing them by sample autocovariances introduces sampling error; the linear algebra remains the same.",[1793,5272,5274],{"id":5273},"one-model-four-equivalent-views","One model, four equivalent views",[5276,5277,5278,5294],"table",{},[5279,5280,5281],"thead",{},[5282,5283,5284,5288,5291],"tr",{},[5285,5286,5287],"th",{},"View",[5285,5289,5290],{},"Object",[5285,5292,5293],{},"Stability or identification check",[5295,5296,5297,5463,5517,5777],"tbody",{},[5282,5298,5299,5303,5460],{},[5300,5301,5302],"td",{},"difference equation",[5300,5304,5305],{},[1805,5306,5308,5343],{"className":5307},[1812],[1805,5309,5311],{"className":5310},[1816],[1818,5312,5313],{"xmlns":1820},[1823,5314,5315,5340],{},[1826,5316,5317,5319,5321,5324,5326,5332,5334],{},[1832,5318,1846],{},[1839,5320,2311],{"stretchy":2310},[1832,5322,5323],{},"B",[1839,5325,2361],{"stretchy":2310},[1829,5327,5328,5330],{},[1832,5329,1834],{},[1832,5331,1837],{},[1839,5333,1841],{},[1829,5335,5336,5338],{},[1832,5337,1896],{},[1832,5339,1837],{},[1904,5341,5342],{"encoding":1906},"\\phi(B)X_t=\\varepsilon_t",[1805,5344,5346,5414],{"className":5345,"ariaHidden":1912},[1911],[1805,5347,5349,5352,5355,5358,5362,5365,5405,5408,5411],{"className":5348},[1916],[1805,5350],{"className":5351,"style":4122},[1920],[1805,5353,1846],{"className":5354},[1925,1929],[1805,5356,2311],{"className":5357},[2440],[1805,5359,5323],{"className":5360,"style":5361},[1925,1929],"margin-right:0.0502em;",[1805,5363,2361],{"className":5364},[2614],[1805,5366,5368,5371],{"className":5367},[1925],[1805,5369,1834],{"className":5370,"style":1930},[1925,1929],[1805,5372,5374],{"className":5373},[1934],[1805,5375,5377,5397],{"className":5376},[1938,1939],[1805,5378,5380,5394],{"className":5379},[1943],[1805,5381,5383],{"className":5382,"style":1948},[1947],[1805,5384,5385,5388],{"style":1951},[1805,5386],{"className":5387,"style":1956},[1955],[1805,5389,5391],{"className":5390},[1960,1961,1962,1963],[1805,5392,1837],{"className":5393},[1925,1929,1963],[1805,5395,1971],{"className":5396},[1970],[1805,5398,5400],{"className":5399},[1943],[1805,5401,5403],{"className":5402,"style":1978},[1947],[1805,5404],{},[1805,5406],{"className":5407,"style":1985},[1984],[1805,5409,1841],{"className":5410},[1989],[1805,5412],{"className":5413,"style":1985},[1984],[1805,5415,5417,5420],{"className":5416},[1916],[1805,5418],{"className":5419,"style":2236},[1920],[1805,5421,5423,5426],{"className":5422},[1925],[1805,5424,1896],{"className":5425},[1925,1929],[1805,5427,5429],{"className":5428},[1934],[1805,5430,5432,5452],{"className":5431},[1938,1939],[1805,5433,5435,5449],{"className":5434},[1943],[1805,5436,5438],{"className":5437,"style":1948},[1947],[1805,5439,5440,5443],{"style":2021},[1805,5441],{"className":5442,"style":1956},[1955],[1805,5444,5446],{"className":5445},[1960,1961,1962,1963],[1805,5447,1837],{"className":5448},[1925,1929,1963],[1805,5450,1971],{"className":5451},[1970],[1805,5453,5455],{"className":5454},[1943],[1805,5456,5458],{"className":5457,"style":1978},[1947],[1805,5459],{},[5300,5461,5462],{},"coefficients 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s_{t-1}+G\\varepsilon_t",[1805,5572,5574,5629,5696],{"className":5573,"ariaHidden":1912},[1911],[1805,5575,5577,5580,5620,5623,5626],{"className":5576},[1916],[1805,5578],{"className":5579,"style":2379},[1920],[1805,5581,5583,5586],{"className":5582},[1925],[1805,5584,2303],{"className":5585},[1925,2386],[1805,5587,5589],{"className":5588},[1934],[1805,5590,5592,5612],{"className":5591},[1938,1939],[1805,5593,5595,5609],{"className":5594},[1943],[1805,5596,5598],{"className":5597,"style":1948},[1947],[1805,5599,5600,5603],{"style":2021},[1805,5601],{"className":5602,"style":1956},[1955],[1805,5604,5606],{"className":5605},[1960,1961,1962,1963],[1805,5607,1837],{"className":5608},[1925,1929,1963],[1805,5610,1971],{"className":5611},[1970],[1805,5613,5615],{"className":5614},[1943],[1805,5616,5618],{"className":5617,"style":1978},[1947],[1805,5619],{},[1805,5621],{"className":5622,"style":1985},[1984],[1805,5624,1841],{"className":5625},[1989],[1805,5627],{"className":5628,"style":1985},[1984],[1805,5630,5632,5635,5638,5687,5690,5693],{"className":5631},[1916],[1805,5633],{"className":5634,"style":2767},[1920],[1805,5636,2674],{"className":5637,"style":2771},[1925,1929],[1805,5639,5641,5644],{"className":5640},[1925],[1805,5642,2303],{"className":5643},[1925,2386],[1805,5645,5647],{"className":5646},[1934],[1805,5648,5650,5679],{"className":5649},[1938,1939],[1805,5651,5653,5676],{"className":5652},[1943],[1805,5654,5656],{"className":5655,"style":2018},[1947],[1805,5657,5658,5661],{"style":2021},[1805,5659],{"className":5660,"style":1956},[1955],[1805,5662,5664],{"className":5663},[1960,1961,1962,1963],[1805,5665,5667,5670,5673],{"className":5666},[1925,1963],[1805,5668,1837],{"className":5669},[1925,1929,1963],[1805,5671,1861],{"className":5672},[2077,1963],[1805,5674,1850],{"className":5675},[1925,1963],[1805,5677,1971],{"className":5678},[1970],[1805,5680,5682],{"className":5681},[1943],[1805,5683,5685],{"className":5684,"style":2090},[1947],[1805,5686],{},[1805,5688],{"className":5689,"style":2096},[1984],[1805,5691,1866],{"className":5692},[2077],[1805,5694],{"className":5695,"style":2096},[1984],[1805,5697,5699,5702,5705],{"className":5698},[1916],[1805,5700],{"className":5701,"style":1921},[1920],[1805,5703,2691],{"className":5704},[1925,1929],[1805,5706,5708,5711],{"className":5707},[1925],[1805,5709,1896],{"className":5710},[1925,1929],[1805,5712,5714],{"className":5713},[1934],[1805,5715,5717,5737],{"className":5716},[1938,1939],[1805,5718,5720,5734],{"className":5719},[1943],[1805,5721,5723],{"className":5722,"style":1948},[1947],[1805,5724,5725,5728],{"style":2021},[1805,5726],{"className":5727,"style":1956},[1955],[1805,5729,5731],{"className":5730},[1960,1961,1962,1963],[1805,5732,1837],{"className":5733},[1925,1929,1963],[1805,5735,1971],{"className":5736},[1970],[1805,5738,5740],{"className":5739},[1943],[1805,5741,5743],{"className":5742,"style":1978},[1947],[1805,5744],{},[5300,5746,5747,5748,5776],{},"eigenvalues of ",[1805,5749,5751,5764],{"className":5750},[1812],[1805,5752,5754],{"className":5753},[1816],[1818,5755,5756],{"xmlns":1820},[1823,5757,5758,5762],{},[1826,5759,5760],{},[1832,5761,2674],{},[1904,5763,2674],{"encoding":1906},[1805,5765,5767],{"className":5766,"ariaHidden":1912},[1911],[1805,5768,5770,5773],{"className":5769},[1916],[1805,5771],{"className":5772,"style":3103},[1920],[1805,5774,2674],{"className":5775,"style":2771},[1925,1929]," inside unit circle",[5282,5778,5779,5782,5934],{},[5300,5780,5781],{},"moment equations",[5300,5783,5784],{},[1805,5785,5787,5815],{"className":5786},[1812],[1805,5788,5790],{"className":5789},[1816],[1818,5791,5792],{"xmlns":1820},[1823,5793,5794,5812],{},[1826,5795,5796,5802,5804,5806],{},[1829,5797,5798,5800],{},[1832,5799,4470],{"mathvariant":1901},[1832,5801,1798],{},[1832,5803,1846],{"mathvariant":4475},[1839,5805,1841],{},[1829,5807,5808,5810],{},[1832,5809,4482],{"mathvariant":4475},[1832,5811,1798],{},[1904,5813,5814],{"encoding":1906},"\\Gamma_p\\boldsymbol\\phi=\\boldsymbol\\gamma_p",[1805,5816,5818,5882],{"className":5817,"ariaHidden":1912},[1911],[1805,5819,5821,5824,5864,5873,5876,5879],{"className":5820},[1916],[1805,5822],{"className":5823,"style":2129},[1920],[1805,5825,5827,5830],{"className":5826},[1925],[1805,5828,4470],{"className":5829},[1925],[1805,5831,5833],{"className":5832},[1934],[1805,5834,5836,5856],{"className":5835},[1938,1939],[1805,5837,5839,5853],{"className":5838},[1943],[1805,5840,5842],{"className":5841,"style":2148},[1947],[1805,5843,5844,5847],{"style":2021},[1805,5845],{"className":5846,"style":1956},[1955],[1805,5848,5850],{"className":5849},[1960,1961,1962,1963],[1805,5851,1798],{"className":5852},[1925,1929,1963],[1805,5854,1971],{"className":5855},[1970],[1805,5857,5859],{"className":5858},[1943],[1805,5860,5862],{"className":5861,"style":2169},[1947],[1805,5863],{},[1805,5865,5867],{"className":5866},[1925],[1805,5868,5870],{"className":5869},[1925],[1805,5871,1846],{"className":5872},[1925,4548],[1805,5874],{"className":5875,"style":1985},[1984],[1805,5877,1841],{"className":5878},[1989],[1805,5880],{"className":5881,"style":1985},[1984],[1805,5883,5885,5888],{"className":5884},[1916],[1805,5886],{"className":5887,"style":4564},[1920],[1805,5889,5891,5900],{"className":5890},[1925],[1805,5892,5894],{"className":5893},[1925],[1805,5895,5897],{"className":5896},[1925],[1805,5898,4482],{"className":5899,"style":4577},[1925,4548],[1805,5901,5903],{"className":5902},[1934],[1805,5904,5906,5926],{"className":5905},[1938,1939],[1805,5907,5909,5923],{"className":5908},[1943],[1805,5910,5912],{"className":5911,"style":4590},[1947],[1805,5913,5914,5917],{"style":4593},[1805,5915],{"className":5916,"style":1956},[1955],[1805,5918,5920],{"className":5919},[1960,1961,1962,1963],[1805,5921,1798],{"className":5922},[1925,1929,1963],[1805,5924,1971],{"className":5925},[1970],[1805,5927,5929],{"className":5928},[1943],[1805,5930,5932],{"className":5931,"style":4612},[1947],[1805,5933],{},[5300,5935,5936,6006],{},[1805,5937,5939,5957],{"className":5938},[1812],[1805,5940,5942],{"className":5941},[1816],[1818,5943,5944],{"xmlns":1820},[1823,5945,5946,5954],{},[1826,5947,5948],{},[1829,5949,5950,5952],{},[1832,5951,4470],{"mathvariant":1901},[1832,5953,1798],{},[1904,5955,5956],{"encoding":1906},"\\Gamma_p",[1805,5958,5960],{"className":5959,"ariaHidden":1912},[1911],[1805,5961,5963,5966],{"className":5962},[1916],[1805,5964],{"className":5965,"style":4738},[1920],[1805,5967,5969,5972],{"className":5968},[1925],[1805,5970,4470],{"className":5971},[1925],[1805,5973,5975],{"className":5974},[1934],[1805,5976,5978,5998],{"className":5977},[1938,1939],[1805,5979,5981,5995],{"className":5980},[1943],[1805,5982,5984],{"className":5983,"style":2148},[1947],[1805,5985,5986,5989],{"style":2021},[1805,5987],{"className":5988,"style":1956},[1955],[1805,5990,5992],{"className":5991},[1960,1961,1962,1963],[1805,5993,1798],{"className":5994},[1925,1929,1963],[1805,5996,1971],{"className":5997},[1970],[1805,5999,6001],{"className":6000},[1943],[1805,6002,6004],{"className":6003,"style":2169},[1947],[1805,6005],{}," invertible",[1798,6008,6009],{},"The views are equivalent under the model, but each makes a different task easier.",[1793,6011,6013],{"id":6012},"exercises","Exercises",[6015,6016,6017,6094,6097,6213,6308,6516],"ol",{},[6018,6019,6020,6021,1902],"li",{},"Build the companion matrix for AR(3) coefficients ",[1805,6022,6024,6055],{"className":6023},[1812],[1805,6025,6027],{"className":6026},[1816],[1818,6028,6029],{"xmlns":1820},[1823,6030,6031,6052],{},[1826,6032,6033,6035,6038,6040,6042,6045,6047,6050],{},[1839,6034,2311],{"stretchy":2310},[1848,6036,6037],{},"0.5",[1839,6039,2320],{"separator":1912},[1839,6041,1861],{},[1848,6043,6044],{},"0.2",[1839,6046,2320],{"separator":1912},[1848,6048,6049],{},"0.1",[1839,6051,2361],{"stretchy":2310},[1904,6053,6054],{"encoding":1906},"(0.5,-0.2,0.1)",[1805,6056,6058],{"className":6057,"ariaHidden":1912},[1911],[1805,6059,6061,6064,6067,6070,6073,6076,6079,6082,6085,6088,6091],{"className":6060},[1916],[1805,6062],{"className":6063,"style":4122},[1920],[1805,6065,2311],{"className":6066},[2440],[1805,6068,6037],{"className":6069},[1925],[1805,6071,2320],{"className":6072},[2484],[1805,6074],{"className":6075,"style":2488},[1984],[1805,6077,1861],{"className":6078},[1925],[1805,6080,6044],{"className":6081},[1925],[1805,6083,2320],{"className":6084},[2484],[1805,6086],{"className":6087,"style":2488},[1984],[1805,6089,6049],{"className":6090},[1925],[1805,6092,2361],{"className":6093},[2614],[6018,6095,6096],{},"Verify numerically that each companion eigenvalue is the reciprocal of one AR-polynomial root.",[6018,6098,6099,6100,6212],{},"Set ",[1805,6101,6103,6137],{"className":6102},[1812],[1805,6104,6106],{"className":6105},[1816],[1818,6107,6108],{"xmlns":1820},[1823,6109,6110,6134],{},[1826,6111,6112,6114,6116,6118,6121,6123,6125,6128],{},[1832,6113,1846],{},[1839,6115,1841],{},[1839,6117,2311],{"stretchy":2310},[1848,6119,6120],{},"1.2",[1839,6122,2320],{"separator":1912},[1839,6124,1861],{},[1848,6126,6127],{},"0.32",[2357,6129,6130,6132],{},[1839,6131,2361],{"stretchy":2310},[1832,6133,2364],{"mathvariant":1901},[1904,6135,6136],{"encoding":1906},"\\phi=(1.2,-0.32)^\\top",[1805,6138,6140,6158],{"className":6139,"ariaHidden":1912},[1911],[1805,6141,6143,6146,6149,6152,6155],{"className":6142},[1916],[1805,6144],{"className":6145,"style":4338},[1920],[1805,6147,1846],{"className":6148},[1925,1929],[1805,6150],{"className":6151,"style":1985},[1984],[1805,6153,1841],{"className":6154},[1989],[1805,6156],{"className":6157,"style":1985},[1984],[1805,6159,6161,6165,6168,6171,6174,6177,6180,6183],{"className":6160},[1916],[1805,6162],{"className":6163,"style":6164},[1920],"height:1.0991em;vertical-align:-0.25em;",[1805,6166,2311],{"className":6167},[2440],[1805,6169,6120],{"className":6170},[1925],[1805,6172,2320],{"className":6173},[2484],[1805,6175],{"className":6176,"style":2488},[1984],[1805,6178,1861],{"className":6179},[1925],[1805,6181,6127],{"className":6182},[1925],[1805,6184,6186,6189],{"className":6185},[2614],[1805,6187,2361],{"className":6188},[2614],[1805,6190,6192],{"className":6191},[1934],[1805,6193,6195],{"className":6194},[1938],[1805,6196,6198],{"className":6197},[1943],[1805,6199,6201],{"className":6200,"style":4272},[1947],[1805,6202,6203,6206],{"style":4293},[1805,6204],{"className":6205,"style":1956},[1955],[1805,6207,6209],{"className":6208},[1960,1961,1962,1963],[1805,6210,2364],{"className":6211},[1925,1963],". Check roots and eigenvalues before simulating.",[6018,6214,6215,6216,1902],{},"For AR(1), reduce the matrix Yule–Walker equation to ",[1805,6217,6219,6254],{"className":6218},[1812],[1805,6220,6222],{"className":6221},[1816],[1818,6223,6224],{"xmlns":1820},[1823,6225,6226,6251],{},[1826,6227,6228,6230,6232,6234,6236,6238,6240,6243,6245,6247,6249],{},[1832,6229,1846],{},[1839,6231,1841],{},[1832,6233,4482],{},[1839,6235,2311],{"stretchy":2310},[1848,6237,1850],{},[1839,6239,2361],{"stretchy":2310},[1832,6241,6242],{"mathvariant":1901},"\u002F",[1832,6244,4482],{},[1839,6246,2311],{"stretchy":2310},[1848,6248,2925],{},[1839,6250,2361],{"stretchy":2310},[1904,6252,6253],{"encoding":1906},"\\phi=\\gamma(1)\u002F\\gamma(0)",[1805,6255,6257,6275],{"className":6256,"ariaHidden":1912},[1911],[1805,6258,6260,6263,6266,6269,6272],{"className":6259},[1916],[1805,6261],{"className":6262,"style":4338},[1920],[1805,6264,1846],{"className":6265},[1925,1929],[1805,6267],{"className":6268,"style":1985},[1984],[1805,6270,1841],{"className":6271},[1989],[1805,6273],{"className":6274,"style":1985},[1984],[1805,6276,6278,6281,6284,6287,6290,6293,6296,6299,6302,6305],{"className":6277},[1916],[1805,6279],{"className":6280,"style":4122},[1920],[1805,6282,4482],{"className":6283,"style":4808},[1925,1929],[1805,6285,2311],{"className":6286},[2440],[1805,6288,1850],{"className":6289},[1925],[1805,6291,2361],{"className":6292},[2614],[1805,6294,6242],{"className":6295},[1925],[1805,6297,4482],{"className":6298,"style":4808},[1925,1929],[1805,6300,2311],{"className":6301},[2440],[1805,6303,2925],{"className":6304},[1925],[1805,6306,2361],{"className":6307},[2614],[6018,6309,6099,6310,6416,6417,6486,6487,6515],{},[1805,6311,6313,6345],{"className":6312},[1812],[1805,6314,6316],{"className":6315},[1816],[1818,6317,6318],{"xmlns":1820},[1823,6319,6320,6342],{},[1826,6321,6322,6324,6326,6328,6331,6333,6336],{},[1832,6323,1846],{},[1839,6325,1841],{},[1839,6327,2311],{"stretchy":2310},[1848,6329,6330],{},"0.9",[1839,6332,2320],{"separator":1912},[1848,6334,6335],{},"0.09",[2357,6337,6338,6340],{},[1839,6339,2361],{"stretchy":2310},[1832,6341,2364],{"mathvariant":1901},[1904,6343,6344],{"encoding":1906},"\\phi=(0.9,0.09)^\\top",[1805,6346,6348,6366],{"className":6347,"ariaHidden":1912},[1911],[1805,6349,6351,6354,6357,6360,6363],{"className":6350},[1916],[1805,6352],{"className":6353,"style":4338},[1920],[1805,6355,1846],{"className":6356},[1925,1929],[1805,6358],{"className":6359,"style":1985},[1984],[1805,6361,1841],{"className":6362},[1989],[1805,6364],{"className":6365,"style":1985},[1984],[1805,6367,6369,6372,6375,6378,6381,6384,6387],{"className":6368},[1916],[1805,6370],{"className":6371,"style":6164},[1920],[1805,6373,2311],{"className":6374},[2440],[1805,6376,6330],{"className":6377},[1925],[1805,6379,2320],{"className":6380},[2484],[1805,6382],{"className":6383,"style":2488},[1984],[1805,6385,6335],{"className":6386},[1925],[1805,6388,6390,6393],{"className":6389},[2614],[1805,6391,2361],{"className":6392},[2614],[1805,6394,6396],{"className":6395},[1934],[1805,6397,6399],{"className":6398},[1938],[1805,6400,6402],{"className":6401},[1943],[1805,6403,6405],{"className":6404,"style":4272},[1947],[1805,6406,6407,6410],{"style":4293},[1805,6408],{"className":6409,"style":1956},[1955],[1805,6411,6413],{"className":6412},[1960,1961,1962,1963],[1805,6414,2364],{"className":6415},[1925,1963],". Inspect the spectral radius and the decay of ",[1805,6418,6420,6437],{"className":6419},[1812],[1805,6421,6423],{"className":6422},[1816],[1818,6424,6425],{"xmlns":1820},[1823,6426,6427,6435],{},[1826,6428,6429],{},[1829,6430,6431,6433],{},[1832,6432,4323],{},[1832,6434,4257],{},[1904,6436,4328],{"encoding":1906},[1805,6438,6440],{"className":6439,"ariaHidden":1912},[1911],[1805,6441,6443,6446],{"className":6442},[1916],[1805,6444],{"className":6445,"style":4338},[1920],[1805,6447,6449,6452],{"className":6448},[1925],[1805,6450,4323],{"className":6451,"style":4345},[1925,1929],[1805,6453,6455],{"className":6454},[1934],[1805,6456,6458,6478],{"className":6457},[1938,1939],[1805,6459,6461,6475],{"className":6460},[1943],[1805,6462,6464],{"className":6463,"style":4358},[1947],[1805,6465,6466,6469],{"style":4361},[1805,6467],{"className":6468,"style":1956},[1955],[1805,6470,6472],{"className":6471},[1960,1961,1962,1963],[1805,6473,4257],{"className":6474},[1925,1929,1963],[1805,6476,1971],{"className":6477},[1970],[1805,6479,6481],{"className":6480},[1943],[1805,6482,6484],{"className":6483,"style":1978},[1947],[1805,6485],{},". Explain why “coefficients sum to less than one” is not the general AR(",[1805,6488,6490,6503],{"className":6489},[1812],[1805,6491,6493],{"className":6492},[1816],[1818,6494,6495],{"xmlns":1820},[1823,6496,6497,6501],{},[1826,6498,6499],{},[1832,6500,1798],{},[1904,6502,1798],{"encoding":1906},[1805,6504,6506],{"className":6505,"ariaHidden":1912},[1911],[1805,6507,6509,6512],{"className":6508},[1916],[1805,6510],{"className":6511,"style":4017},[1920],[1805,6513,1798],{"className":6514},[1925,1929],") stability test.",[6018,6517,6518,6519,6788,6789,6864,6865,1902],{},"Graduate extension: solve\n",[1805,6520,6522,6594],{"className":6521},[1812],[1805,6523,6525],{"className":6524},[1816],[1818,6526,6527],{"xmlns":1820},[1823,6528,6529,6591],{},[1826,6530,6531,6534,6536,6538,6541,6543,6545,6547,6550,6552,6554,6557,6559,6569,6571,6573,6575,6581,6583,6589],{},[1832,6532,6533],{"mathvariant":1901},"vec",[1839,6535,4649],{},[1839,6537,2311],{"stretchy":2310},[1832,6539,6540],{},"P",[1839,6542,2361],{"stretchy":2310},[1839,6544,1841],{},[1839,6546,2311],{"stretchy":2310},[1832,6548,6549],{},"I",[1839,6551,1861],{},[1832,6553,2674],{},[1839,6555,6556],{},"⊗",[1832,6558,2674],{},[2357,6560,6561,6563],{},[1839,6562,2361],{"stretchy":2310},[1826,6564,6565,6567],{},[1839,6566,1861],{},[1848,6568,1850],{},[1832,6570,6533],{"mathvariant":1901},[1839,6572,4649],{},[1839,6574,2311],{"stretchy":2310},[2357,6576,6577,6579],{},[1832,6578,5038],{},[1848,6580,2942],{},[1832,6582,2691],{},[2357,6584,6585,6587],{},[1832,6586,2691],{},[1832,6588,2364],{"mathvariant":1901},[1839,6590,2361],{"stretchy":2310},[1904,6592,6593],{"encoding":1906},"\\operatorname{vec}(P)=(I-F\\otimes F)^{-1}\\operatorname{vec}(\\sigma^2GG^\\top)",[1805,6595,6597,6627,6648,6667],{"className":6596,"ariaHidden":1912},[1911],[1805,6598,6600,6603,6609,6612,6615,6618,6621,6624],{"className":6599},[1916],[1805,6601],{"className":6602,"style":4122},[1920],[1805,6604,6606],{"className":6605},[4797],[1805,6607,6533],{"className":6608},[1925,4801],[1805,6610,2311],{"className":6611},[2440],[1805,6613,6540],{"className":6614,"style":2771},[1925,1929],[1805,6616,2361],{"className":6617},[2614],[1805,6619],{"className":6620,"style":1985},[1984],[1805,6622,1841],{"className":6623},[1989],[1805,6625],{"className":6626,"style":1985},[1984],[1805,6628,6630,6633,6636,6639,6642,6645],{"className":6629},[1916],[1805,6631],{"className":6632,"style":4122},[1920],[1805,6634,2311],{"className":6635},[2440],[1805,6637,6549],{"className":6638,"style":1930},[1925,1929],[1805,6640],{"className":6641,"style":2096},[1984],[1805,6643,1861],{"className":6644},[2077],[1805,6646],{"className":6647,"style":2096},[1984],[1805,6649,6651,6655,6658,6661,6664],{"className":6650},[1916],[1805,6652],{"className":6653,"style":6654},[1920],"height:0.7667em;vertical-align:-0.0833em;",[1805,6656,2674],{"className":6657,"style":2771},[1925,1929],[1805,6659],{"className":6660,"style":2096},[1984],[1805,6662,6556],{"className":6663},[2077],[1805,6665],{"className":6666,"style":2096},[1984],[1805,6668,6670,6673,6676,6712,6715,6721,6724,6753,6756,6785],{"className":6669},[1916],[1805,6671],{"className":6672,"style":6164},[1920],[1805,6674,2674],{"className":6675,"style":2771},[1925,1929],[1805,6677,6679,6682],{"className":6678},[2614],[1805,6680,2361],{"className":6681},[2614],[1805,6683,6685],{"className":6684},[1934],[1805,6686,6688],{"className":6687},[1938],[1805,6689,6691],{"className":6690},[1943],[1805,6692,6695],{"className":6693,"style":6694},[1947],"height:0.8141em;",[1805,6696,6697,6700],{"style":4293},[1805,6698],{"className":6699,"style":1956},[1955],[1805,6701,6703],{"className":6702},[1960,1961,1962,1963],[1805,6704,6706,6709],{"className":6705},[1925,1963],[1805,6707,1861],{"className":6708},[1925,1963],[1805,6710,1850],{"className":6711},[1925,1963],[1805,6713],{"className":6714,"style":2488},[1984],[1805,6716,6718],{"className":6717},[4797],[1805,6719,6533],{"className":6720},[1925,4801],[1805,6722,2311],{"className":6723},[2440],[1805,6725,6727,6730],{"className":6726},[1925],[1805,6728,5038],{"className":6729,"style":4345},[1925,1929],[1805,6731,6733],{"className":6732},[1934],[1805,6734,6736],{"className":6735},[1938],[1805,6737,6739],{"className":6738},[1943],[1805,6740,6742],{"className":6741,"style":6694},[1947],[1805,6743,6744,6747],{"style":4293},[1805,6745],{"className":6746,"style":1956},[1955],[1805,6748,6750],{"className":6749},[1960,1961,1962,1963],[1805,6751,2942],{"className":6752},[1925,1963],[1805,6754,2691],{"className":6755},[1925,1929],[1805,6757,6759,6762],{"className":6758},[1925],[1805,6760,2691],{"className":6761},[1925,1929],[1805,6763,6765],{"className":6764},[1934],[1805,6766,6768],{"className":6767},[1938],[1805,6769,6771],{"className":6770},[1943],[1805,6772,6774],{"className":6773,"style":4272},[1947],[1805,6775,6776,6779],{"style":4293},[1805,6777],{"className":6778,"style":1956},[1955],[1805,6780,6782],{"className":6781},[1960,1961,1962,1963],[1805,6783,2364],{"className":6784},[1925,1963],[1805,6786,2361],{"className":6787},[2614],"\nand compare ",[1805,6790,6792,6811],{"className":6791},[1812],[1805,6793,6795],{"className":6794},[1816],[1818,6796,6797],{"xmlns":1820},[1823,6798,6799,6808],{},[1826,6800,6801],{},[1829,6802,6803,6805],{},[1832,6804,6540],{},[1848,6806,6807],{},"11",[1904,6809,6810],{"encoding":1906},"P_{11}",[1805,6812,6814],{"className":6813,"ariaHidden":1912},[1911],[1805,6815,6817,6820],{"className":6816},[1916],[1805,6818],{"className":6819,"style":1921},[1920],[1805,6821,6823,6826],{"className":6822},[1925],[1805,6824,6540],{"className":6825,"style":2771},[1925,1929],[1805,6827,6829],{"className":6828},[1934],[1805,6830,6832,6856],{"className":6831},[1938,1939],[1805,6833,6835,6853],{"className":6834},[1943],[1805,6836,6838],{"className":6837,"style":2018},[1947],[1805,6839,6841,6844],{"style":6840},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1805,6842],{"className":6843,"style":1956},[1955],[1805,6845,6847],{"className":6846},[1960,1961,1962,1963],[1805,6848,6850],{"className":6849},[1925,1963],[1805,6851,6807],{"className":6852},[1925,1963],[1805,6854,1971],{"className":6855},[1970],[1805,6857,6859],{"className":6858},[1943],[1805,6860,6862],{"className":6861,"style":1978},[1947],[1805,6863],{}," with ",[1805,6866,6868,6888],{"className":6867},[1812],[1805,6869,6871],{"className":6870},[1816],[1818,6872,6873],{"xmlns":1820},[1823,6874,6875,6885],{},[1826,6876,6877,6879,6881,6883],{},[1832,6878,4482],{},[1839,6880,2311],{"stretchy":2310},[1848,6882,2925],{},[1839,6884,2361],{"stretchy":2310},[1904,6886,6887],{"encoding":1906},"\\gamma(0)",[1805,6889,6891],{"className":6890,"ariaHidden":1912},[1911],[1805,6892,6894,6897,6900,6903,6906],{"className":6893},[1916],[1805,6895],{"className":6896,"style":4122},[1920],[1805,6898,4482],{"className":6899,"style":4808},[1925,1929],[1805,6901,2311],{"className":6902},[2440],[1805,6904,2925],{"className":6905},[1925],[1805,6907,2361],{"className":6908},[2614],[6910,6911,6913],"legacy-details",{"title":6912},"Checkpoints",[6015,6914,6915,6988,6991,6994,7235],{},[6018,6916,6917,6918,6987],{},"The first row is ",[1805,6919,6921,6948],{"className":6920},[1812],[1805,6922,6924],{"className":6923},[1816],[1818,6925,6926],{"xmlns":1820},[1823,6927,6928,6946],{},[1826,6929,6930,6932,6934,6936,6938,6940,6942,6944],{},[1839,6931,2311],{"stretchy":2310},[1848,6933,6037],{},[1839,6935,2320],{"separator":1912},[1839,6937,1861],{},[1848,6939,6044],{},[1839,6941,2320],{"separator":1912},[1848,6943,6049],{},[1839,6945,2361],{"stretchy":2310},[1904,6947,6054],{"encoding":1906},[1805,6949,6951],{"className":6950,"ariaHidden":1912},[1911],[1805,6952,6954,6957,6960,6963,6966,6969,6972,6975,6978,6981,6984],{"className":6953},[1916],[1805,6955],{"className":6956,"style":4122},[1920],[1805,6958,2311],{"className":6959},[2440],[1805,6961,6037],{"className":6962},[1925],[1805,6964,2320],{"className":6965},[2484],[1805,6967],{"className":6968,"style":2488},[1984],[1805,6970,1861],{"className":6971},[1925],[1805,6973,6044],{"className":6974},[1925],[1805,6976,2320],{"className":6977},[2484],[1805,6979],{"className":6980,"style":2488},[1984],[1805,6982,6049],{"className":6983},[1925],[1805,6985,2361],{"className":6986},[2614],"; rows two and three shift the state with ones on the first subdiagonal.",[6018,6989,6990],{},"Match the two unordered sets after taking reciprocals; allow floating-point tolerance.",[6018,6992,6993],{},"A coefficient larger than one does not decide stability by itself; the polynomial roots or companion eigenvalues do.",[6018,6995,6996,7111,7112,1902],{},[1805,6997,6999,7031],{"className":6998},[1812],[1805,7000,7002],{"className":7001},[1816],[1818,7003,7004],{"xmlns":1820},[1823,7005,7006,7028],{},[1826,7007,7008,7014,7016,7018,7020,7022,7024,7026],{},[1829,7009,7010,7012],{},[1832,7011,4470],{"mathvariant":1901},[1848,7013,1850],{},[1839,7015,1841],{},[1839,7017,2909],{"stretchy":2310},[1832,7019,4482],{},[1839,7021,2311],{"stretchy":2310},[1848,7023,2925],{},[1839,7025,2361],{"stretchy":2310},[1839,7027,3057],{"stretchy":2310},[1904,7029,7030],{"encoding":1906},"\\Gamma_1=[\\gamma(0)]",[1805,7032,7034,7089],{"className":7033,"ariaHidden":1912},[1911],[1805,7035,7037,7040,7080,7083,7086],{"className":7036},[1916],[1805,7038],{"className":7039,"style":1921},[1920],[1805,7041,7043,7046],{"className":7042},[1925],[1805,7044,4470],{"className":7045},[1925],[1805,7047,7049],{"className":7048},[1934],[1805,7050,7052,7072],{"className":7051},[1938,1939],[1805,7053,7055,7069],{"className":7054},[1943],[1805,7056,7058],{"className":7057,"style":2018},[1947],[1805,7059,7060,7063],{"style":2021},[1805,7061],{"className":7062,"style":1956},[1955],[1805,7064,7066],{"className":7065},[1960,1961,1962,1963],[1805,7067,1850],{"className":7068},[1925,1963],[1805,7070,1971],{"className":7071},[1970],[1805,7073,7075],{"className":7074},[1943],[1805,7076,7078],{"className":7077,"style":1978},[1947],[1805,7079],{},[1805,7081],{"className":7082,"style":1985},[1984],[1805,7084,1841],{"className":7085},[1989],[1805,7087],{"className":7088,"style":1985},[1984],[1805,7090,7092,7095,7098,7101,7104,7107],{"className":7091},[1916],[1805,7093],{"className":7094,"style":4122},[1920],[1805,7096,2909],{"className":7097},[2440],[1805,7099,4482],{"className":7100,"style":4808},[1925,1929],[1805,7102,2311],{"className":7103},[2440],[1805,7105,2925],{"className":7106},[1925],[1805,7108,7110],{"className":7109},[2614],")]"," and ",[1805,7113,7115,7147],{"className":7114},[1812],[1805,7116,7118],{"className":7117},[1816],[1818,7119,7120],{"xmlns":1820},[1823,7121,7122,7144],{},[1826,7123,7124,7130,7132,7134,7136,7138,7140,7142],{},[1829,7125,7126,7128],{},[1832,7127,4482],{"mathvariant":4475},[1848,7129,1850],{},[1839,7131,1841],{},[1839,7133,2909],{"stretchy":2310},[1832,7135,4482],{},[1839,7137,2311],{"stretchy":2310},[1848,7139,1850],{},[1839,7141,2361],{"stretchy":2310},[1839,7143,3057],{"stretchy":2310},[1904,7145,7146],{"encoding":1906},"\\boldsymbol\\gamma_1=[\\gamma(1)]",[1805,7148,7150,7214],{"className":7149,"ariaHidden":1912},[1911],[1805,7151,7153,7157,7205,7208,7211],{"className":7152},[1916],[1805,7154],{"className":7155,"style":7156},[1920],"height:0.6886em;vertical-align:-0.2441em;",[1805,7158,7160,7169],{"className":7159},[1925],[1805,7161,7163],{"className":7162},[1925],[1805,7164,7166],{"className":7165},[1925],[1805,7167,4482],{"className":7168,"style":4577},[1925,4548],[1805,7170,7172],{"className":7171},[1934],[1805,7173,7175,7196],{"className":7174},[1938,1939],[1805,7176,7178,7193],{"className":7177},[1943],[1805,7179,7182],{"className":7180,"style":7181},[1947],"height:0.207em;",[1805,7183,7184,7187],{"style":4593},[1805,7185],{"className":7186,"style":1956},[1955],[1805,7188,7190],{"className":7189},[1960,1961,1962,1963],[1805,7191,1850],{"className":7192},[1925,1963],[1805,7194,1971],{"className":7195},[1970],[1805,7197,7199],{"className":7198},[1943],[1805,7200,7203],{"className":7201,"style":7202},[1947],"height:0.2441em;",[1805,7204],{},[1805,7206],{"className":7207,"style":1985},[1984],[1805,7209,1841],{"className":7210},[1989],[1805,7212],{"className":7213,"style":1985},[1984],[1805,7215,7217,7220,7223,7226,7229,7232],{"className":7216},[1916],[1805,7218],{"className":7219,"style":4122},[1920],[1805,7221,2909],{"className":7222},[2440],[1805,7224,4482],{"className":7225,"style":4808},[1925,1929],[1805,7227,2311],{"className":7228},[2440],[1805,7230,1850],{"className":7231},[1925],[1805,7233,7110],{"className":7234},[2614],[6018,7236,7237],{},"Stability depends on all roots jointly, not on one coefficient or one sum.",[1793,7239,7241],{"id":7240},"completion-standard","Completion standard",[1798,7243,7244],{},"You should be able to move without ambiguity among lag-polynomial, companion-matrix, impulse-response, and Yule–Walker representations.",[1798,7246,7247,7248,1902],{},"Continue to ",[7249,7250,7252],"a",{"href":7251},".\u002F03-prediction-likelihood","Lab 3 — Prediction and Likelihood",{"title":10,"searchDepth":7254,"depth":7254,"links":7255},2,[7256,7257,7258,7259,7260,7261],{"id":1795,"depth":7254,"text":1796},{"id":3746,"depth":7254,"text":3747},{"id":4417,"depth":7254,"text":4418},{"id":5273,"depth":7254,"text":5274},{"id":6012,"depth":7254,"text":6013},{"id":7240,"depth":7254,"text":7241},"Convert AR models to companion form, compare polynomial roots with eigenvalues, propagate shocks, and recover coefficients from autocovariances.","md",{"sidebar":7265},{"order":7254},true,{"title":1005,"description":7262},"mDK-fcdu54vZyw3hrkgr7UIEUP3uvgV1EgCWYn1pfmo",[7270,7272],{"title":1001,"path":1002,"stem":1003,"description":7271,"children":-1},"Construct stationary Toeplitz covariance matrices, test positive definiteness, factor them, and simulate Gaussian finite blocks.",{"title":1009,"path":1010,"stem":1011,"description":7273,"children":-1},"Solve best-linear-prediction equations, verify Schur-complement uncertainty, and compare conditional, moment, and exact Gaussian AR estimates.",1785754738714]