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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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6,"style":1942},[1941],[1805,2778,2780],{"className":2779},[1946,1947,1948,1949],[1805,2781,2456],{"className":2782},[1913,1953,1949],[1805,2784,2785,2788],{"style":2076},[1805,2786],{"className":2787,"style":1942},[1941],[1805,2789,2791],{"className":2790},[1946,1947,1948,1949],[1805,2792,2489],{"className":2793},[1913,1949],[1805,2795,1958],{"className":2796},[1957],[1805,2798,2800],{"className":2799},[1929],[1805,2801,2804],{"className":2802,"style":2803},[1933],"height:0.247em;",[1805,2805],{},[1805,2807,1878],{"className":2808},[2064],[1798,2810,2811,2812,2875],{},"stationarity requires ",[1805,2813,2815,2839],{"className":2814},[1812],[1805,2816,2818],{"className":2817},[1816],[1818,2819,2820],{"xmlns":1820},[1823,2821,2822,2836],{},[1826,2823,2824,2827,2829,2831,2834],{},[1832,2825,2826],{"mathvariant":1834},"∣",[1832,2828,2436],{},[1832,2830,2826],{"mathvariant":1834},[1840,2832,2833],{},"\u003C",[1884,2835,1886],{},[1893,2837,2838],{"encoding":1895},"|\\phi|\u003C1",[1805,2840,2842,2866],{"className":2841,"ariaHidden":1877},[1900],[1805,2843,2845,2848,2851,2854,2857,2860,2863],{"className":2844},[1904],[1805,2846],{"className":2847,"style":1986},[1908],[1805,2849,2826],{"className":2850},[1913],[1805,2852,2436],{"className":2853},[1913,1953],[1805,2855,2826],{"className":2856},[1913],[1805,2858],{"className":2859,"style":1972},[1971],[1805,2861,2833],{"className":2862},[1976],[1805,2864],{"className":2865,"style":1972},[1971],[1805,2867,2869,2872],{"className":2868},[1904],[1805,2870],{"className":2871,"style":2399},[1908],[1805,2873,1886],{"className":2874},[1913]," and",[1805,2877,2879],{"className":2878},[1808],[1805,2880,2882,2942],{"className":2881},[1812],[1805,2883,2885],{"className":2884},[1816],[1818,2886,2887],{"xmlns":1820,"display":1821},[1823,2888,2889,2939],{},[1826,2890,2891,2893,2895,2898,2900,2902,2925,2937],{},[1832,2892,1849],{},[1840,2894,1852],{"stretchy":1845},[1832,2896,2897],{},"h",[1840,2899,1864],{"stretchy":1845},[1840,2901,1842],{},[2903,2904,2905,2913],"mfrac",{},[1866,2906,2907,2909,2911],{},[1832,2908,2484],{},[1832,2910,2456],{},[1884,2912,2489],{},[1826,2914,2915,2917,2919],{},[1884,2916,1886],{},[1840,2918,1858],{},[2161,2920,2921,2923],{},[1832,2922,2436],{},[1884,2924,2489],{},[2161,2926,2927,2929],{},[1832,2928,2436],{},[1826,2930,2931,2933,2935],{},[1832,2932,2826],{"mathvariant":1834},[1832,2934,2897],{},[1832,2936,2826],{"mathvariant":1834},[1832,2938,1891],{"mathvariant":1834},[1893,2940,2941],{"encoding":1895},"\\gamma(h)=\\frac{\\sigma_\\varepsilon^2}{1-\\phi^2}\\phi^{|h|}.",[1805,2943,2945,2972],{"className":2944,"ariaHidden":1877},[1900],[1805,2946,2948,2951,2954,2957,2960,2963,2966,2969],{"className":2947},[1904],[1805,2949],{"className":2950,"style":1986},[1908],[1805,2952,1849],{"className":2953,"style":1994},[1913,1953],[1805,2955,1852],{"className":2956},[1990],[1805,2958,2897],{"className":2959},[1913,1953],[1805,2961,1864],{"className":2962},[2026],[1805,2964],{"className":2965,"style":1972},[1971],[1805,2967,1842],{"className":2968},[1976],[1805,2970],{"className":2971,"style":1972},[1971],[1805,2973,2975,2979,3140,3179],{"className":2974},[1904],[1805,2976],{"className":2977,"style":2978},[1908],"height:2.3715em;vertical-align:-0.8804em;",[1805,2980,2982,2986,3137],{"className":2981},[1913],[1805,2983],{"className":2984},[1990,2985],"nulldelimiter",[1805,2987,2989],{"className":2988},[2903],[1805,2990,2992,3128],{"className":2991},[1924,1925],[1805,2993,2995,3125],{"className":2994},[1929],[1805,2996,2999,3052,3063],{"className":2997,"style":2998},[1933],"height:1.4911em;",[1805,3000,3002,3006],{"style":3001},"top:-2.314em;",[1805,3003],{"className":3004,"style":3005},[1941],"height:3em;",[1805,3007,3009,3012,3015,3018,3021],{"className":3008},[1913],[1805,3010,1886],{"className":3011},[1913],[1805,3013],{"className":3014,"style":2004},[1971],[1805,3016,1858],{"className":3017},[2008],[1805,3019],{"className":3020,"style":2004},[1971],[1805,3022,3024,3027],{"className":3023},[1913],[1805,3025,2436],{"className":3026},[1913,1953],[1805,3028,3030],{"className":3029},[1920],[1805,3031,3033],{"className":3032},[1924],[1805,3034,3036],{"className":3035},[1929],[1805,3037,3040],{"className":3038,"style":3039},[1933],"height:0.7401em;",[1805,3041,3043,3046],{"style":3042},"top:-2.989em;margin-right:0.05em;",[1805,3044],{"className":3045,"style":1942},[1941],[1805,3047,3049],{"className":3048},[1946,1947,1948,1949],[1805,3050,2489],{"className":3051},[1913,1949],[1805,3053,3055,3058],{"style":3054},"top:-3.23em;",[1805,3056],{"className":3057,"style":3005},[1941],[1805,3059],{"className":3060,"style":3062},[3061],"frac-line","border-bottom-width:0.04em;",[1805,3064,3066,3069],{"style":3065},"top:-3.677em;",[1805,3067],{"className":3068,"style":3005},[1941],[1805,3070,3072],{"className":3071},[1913],[1805,3073,3075,3078],{"className":3074},[1913],[1805,3076,2484],{"className":3077,"style":2757},[1913,1953],[1805,3079,3081],{"className":3080},[1920],[1805,3082,3084,3117],{"className":3083},[1924,1925],[1805,3085,3087,3114],{"className":3086},[1929],[1805,3088,3091,3102],{"className":3089,"style":3090},[1933],"height:0.8141em;",[1805,3092,3093,3096],{"style":2773},[1805,3094],{"className":3095,"style":1942},[1941],[1805,3097,3099],{"className":3098},[1946,1947,1948,1949],[1805,3100,2456],{"className":3101},[1913,1953,1949],[1805,3103,3105,3108],{"style":3104},"top:-3.063em;margin-right:0.05em;",[1805,3106],{"className":3107,"style":1942},[1941],[1805,3109,3111],{"className":3110},[1946,1947,1948,1949],[1805,3112,2489],{"className":3113},[1913,1949],[1805,3115,1958],{"className":3116},[1957],[1805,3118,3120],{"className":3119},[1929],[1805,3121,3123],{"className":3122,"style":2803},[1933],[1805,3124],{},[1805,3126,1958],{"className":3127},[1957],[1805,3129,3131],{"className":3130},[1929],[1805,3132,3135],{"className":3133,"style":3134},[1933],"height:0.8804em;",[1805,3136],{},[1805,3138],{"className":3139},[2026,2985],[1805,3141,3143,3146],{"className":3142},[1913],[1805,3144,2436],{"className":3145},[1913,1953],[1805,3147,3149],{"className":3148},[1920],[1805,3150,3152],{"className":3151},[1924],[1805,3153,3155],{"className":3154},[1929],[1805,3156,3159],{"className":3157,"style":3158},[1933],"height:0.938em;",[1805,3160,3161,3164],{"style":2076},[1805,3162],{"className":3163,"style":1942},[1941],[1805,3165,3167],{"className":3166},[1946,1947,1948,1949],[1805,3168,3170,3173,3176],{"className":3169},[1913,1949],[1805,3171,2826],{"className":3172},[1913,1949],[1805,3174,2897],{"className":3175},[1913,1953,1949],[1805,3177,2826],{"className":3178},[1913,1949],[1805,3180,1891],{"className":3181},[1913],[1798,3183,3184,3185,3256],{},"The first column of ",[1805,3186,3188,3207],{"className":3187},[1812],[1805,3189,3191],{"className":3190},[1816],[1818,3192,3193],{"xmlns":1820},[1823,3194,3195,3204],{},[1826,3196,3197],{},[1829,3198,3199,3201],{},[1832,3200,1835],{"mathvariant":1834},[1884,3202,3203],{},"6",[1893,3205,3206],{"encoding":1895},"\\Gamma_6",[1805,3208,3210],{"className":3209,"ariaHidden":1877},[1900],[1805,3211,3213,3216],{"className":3212},[1904],[1805,3214],{"className":3215,"style":1909},[1908],[1805,3217,3219,3222],{"className":3218},[1913],[1805,3220,1835],{"className":3221},[1913],[1805,3223,3225],{"className":3224},[1920],[1805,3226,3228,3248],{"className":3227},[1924,1925],[1805,3229,3231,3245],{"className":3230},[1929],[1805,3232,3234],{"className":3233,"style":2584},[1933],[1805,3235,3236,3239],{"style":1937},[1805,3237],{"className":3238,"style":1942},[1941],[1805,3240,3242],{"className":3241},[1946,1947,1948,1949],[1805,3243,3203],{"className":3244},[1913,1949],[1805,3246,1958],{"className":3247},[1957],[1805,3249,3251],{"className":3250},[1929],[1805,3252,3254],{"className":3253,"style":1965},[1933],[1805,3255],{}," is therefore",[1805,3258,3260],{"className":3259},[1808],[1805,3261,3263,3322],{"className":3262},[1812],[1805,3264,3266],{"className":3265},[1816],[1818,3267,3268],{"xmlns":1820,"display":1821},[1823,3269,3270,3319],{},[1826,3271,3272,3274,3276,3279,3281,3283,3285,3287,3289,3291,3297,3299,3302,3304,3311,3317],{},[1832,3273,1849],{},[1840,3275,1852],{"stretchy":1845},[1884,3277,3278],{},"0",[1840,3280,1864],{"stretchy":1845},[1840,3282,1852],{"stretchy":1845},[1884,3284,1886],{},[1840,3286,1878],{"separator":1877},[1832,3288,2436],{},[1840,3290,1878],{"separator":1877},[2161,3292,3293,3295],{},[1832,3294,2436],{},[1884,3296,2489],{},[1840,3298,1878],{"separator":1877},[1840,3300,3301],{},"…",[1840,3303,1878],{"separator":1877},[2161,3305,3306,3308],{},[1832,3307,2436],{},[1884,3309,3310],{},"5",[2161,3312,3313,3315],{},[1840,3314,1864],{"stretchy":1845},[1832,3316,2167],{"mathvariant":1834},[1832,3318,1891],{"mathvariant":1834},[1893,3320,3321],{"encoding":1895},"\\gamma(0)(1,\\phi,\\phi^2,\\ldots,\\phi^5)^\\top.",[1805,3323,3325],{"className":3324,"ariaHidden":1877},[1900],[1805,3326,3328,3331,3334,3337,3340,3343,3346,3349,3352,3355,3358,3361,3364,3393,3396,3399,3403,3406,3409,3412,3441,3470],{"className":3327},[1904],[1805,3329],{"className":3330,"style":2211},[1908],[1805,3332,1849],{"className":3333,"style":1994},[1913,1953],[1805,3335,1852],{"className":3336},[1990],[1805,3338,3278],{"className":3339},[1913],[1805,3341,1864],{"className":3342},[2026],[1805,3344,1852],{"className":3345},[1990],[1805,3347,1886],{"className":3348},[1913],[1805,3350,1878],{"className":3351},[2064],[1805,3353],{"className":3354,"style":2682},[1971],[1805,3356,2436],{"className":3357},[1913,1953],[1805,3359,1878],{"className":3360},[2064],[1805,3362],{"className":3363,"style":2682},[1971],[1805,3365,3367,3370],{"className":3366},[1913],[1805,3368,2436],{"className":3369},[1913,1953],[1805,3371,3373],{"className":3372},[1920],[1805,3374,3376],{"className":3375},[1924],[1805,3377,3379],{"className":3378},[1929],[1805,3380,3382],{"className":3381,"style":2770},[1933],[1805,3383,3384,3387],{"style":2076},[1805,3385],{"className":3386,"style":1942},[1941],[1805,3388,3390],{"className":3389},[1946,1947,1948,1949],[1805,3391,2489],{"className":3392},[1913,1949],[1805,3394,1878],{"className":3395},[2064],[1805,3397],{"className":3398,"style":2682},[1971],[1805,3400,3301],{"className":3401},[3402],"minner",[1805,3404],{"className":3405,"style":2682},[1971],[1805,3407,1878],{"className":3408},[2064],[1805,3410],{"className":3411,"style":2682},[1971],[1805,3413,3415,3418],{"className":3414},[1913],[1805,3416,2436],{"className":3417},[1913,1953],[1805,3419,3421],{"className":3420},[1920],[1805,3422,3424],{"className":3423},[1924],[1805,3425,3427],{"className":3426},[1929],[1805,3428,3430],{"className":3429,"style":2770},[1933],[1805,3431,3432,3435],{"style":2076},[1805,3433],{"className":3434,"style":1942},[1941],[1805,3436,3438],{"className":3437},[1946,1947,1948,1949],[1805,3439,3310],{"className":3440},[1913,1949],[1805,3442,3444,3447],{"className":3443},[2026],[1805,3445,1864],{"className":3446},[2026],[1805,3448,3450],{"className":3449},[1920],[1805,3451,3453],{"className":3452},[1924],[1805,3454,3456],{"className":3455},[1929],[1805,3457,3459],{"className":3458,"style":2241},[1933],[1805,3460,3461,3464],{"style":2076},[1805,3462],{"className":3463,"style":1942},[1941],[1805,3465,3467],{"className":3466},[1946,1947,1948,1949],[1805,3468,2167],{"className":3469},[1913,1949],[1805,3471,1891],{"className":3472},[1913],[3474,3475],"web-r",{"code64":3476,"layout":3477,"locale":7,"title":3478},"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","vertical","Build and audit an AR(1) covariance matrix",[3480,3481,3483],"h3",{"id":3482},"read-the-output","Read the output",[3485,3486,3487,3491,3494,3497],"ul",{},[3488,3489,3490],"li",{},"Equal diagonals confirm Toeplitz structure.",[3488,3492,3493],{},"Positive eigenvalues establish positive definiteness for this finite block.",[3488,3495,3496],{},"The Cholesky error checks the factorisation numerically.",[3488,3498,3499],{},"The quadratic-form check connects a matrix calculation to a familiar variance.",[1798,3501,3502,3503,3533,3534,3570,3571,3600],{},"Change ",[1805,3504,3506,3520],{"className":3505},[1812],[1805,3507,3509],{"className":3508},[1816],[1818,3510,3511],{"xmlns":1820},[1823,3512,3513,3517],{},[1826,3514,3515],{},[1832,3516,2436],{},[1893,3518,3519],{"encoding":1895},"\\phi",[1805,3521,3523],{"className":3522,"ariaHidden":1877},[1900],[1805,3524,3526,3530],{"className":3525},[1904],[1805,3527],{"className":3528,"style":3529},[1908],"height:0.8889em;vertical-align:-0.1944em;",[1805,3531,2436],{"className":3532},[1913,1953]," to ",[1805,3535,3537,3554],{"className":3536},[1812],[1805,3538,3540],{"className":3539},[1816],[1818,3541,3542],{"xmlns":1820},[1823,3543,3544,3551],{},[1826,3545,3546,3548],{},[1840,3547,1858],{},[1884,3549,3550],{},"0.6",[1893,3552,3553],{"encoding":1895},"-0.6",[1805,3555,3557],{"className":3556,"ariaHidden":1877},[1900],[1805,3558,3560,3564,3567],{"className":3559},[1904],[1805,3561],{"className":3562,"style":3563},[1908],"height:0.7278em;vertical-align:-0.0833em;",[1805,3565,1858],{"className":3566},[1913],[1805,3568,3550],{"className":3569},[1913],". The first off-diagonal changes sign, but the matrix remains positive definite. Change it to ",[1805,3572,3574,3588],{"className":3573},[1812],[1805,3575,3577],{"className":3576},[1816],[1818,3578,3579],{"xmlns":1820},[1823,3580,3581,3586],{},[1826,3582,3583],{},[1884,3584,3585],{},"0.95",[1893,3587,3585],{"encoding":1895},[1805,3589,3591],{"className":3590,"ariaHidden":1877},[1900],[1805,3592,3594,3597],{"className":3593},[1904],[1805,3595],{"className":3596,"style":2399},[1908],[1805,3598,3585],{"className":3599},[1913],": the process is still stationary, but the condition number rises because neighbouring variables become nearly redundant.",[1793,3602,3604],{"id":3603},"example-2-gaussian-simulation-by-a-matrix-square-root","Example 2 — Gaussian simulation by a matrix square root",[1798,3606,3607,3608,3733,3734,3764,3765,3893],{},"Let ",[1805,3609,3611,3647],{"className":3610},[1812],[1805,3612,3614],{"className":3613},[1816],[1818,3615,3616],{"xmlns":1820},[1823,3617,3618,3644],{},[1826,3619,3620,3623,3626,3629,3631,3633,3635,3642],{},[1832,3621,3622],{"mathvariant":2118},"z",[1840,3624,3625],{},"∼",[1832,3627,3628],{},"N",[1840,3630,1852],{"stretchy":1845},[1884,3632,3278],{"mathvariant":2118},[1840,3634,1878],{"separator":1877},[1829,3636,3637,3640],{},[1832,3638,3639],{},"I",[1832,3641,1838],{},[1840,3643,1864],{"stretchy":1845},[1893,3645,3646],{"encoding":1895},"\\mathbf z\\sim N(\\mathbf0,I_n)",[1805,3648,3650,3668],{"className":3649,"ariaHidden":1877},[1900],[1805,3651,3653,3656,3659,3662,3665],{"className":3652},[1904],[1805,3654],{"className":3655,"style":2132},[1908],[1805,3657,3622],{"className":3658},[1913,2136],[1805,3660],{"className":3661,"style":1972},[1971],[1805,3663,3625],{"className":3664},[1976],[1805,3666],{"className":3667,"style":1972},[1971],[1805,3669,3671,3674,3678,3681,3684,3687,3690,3730],{"className":3670},[1904],[1805,3672],{"className":3673,"style":1986},[1908],[1805,3675,3628],{"className":3676,"style":3677},[1913,1953],"margin-right:0.109em;",[1805,3679,1852],{"className":3680},[1990],[1805,3682,3278],{"className":3683},[1913,2136],[1805,3685,1878],{"className":3686},[2064],[1805,3688],{"className":3689,"style":2682},[1971],[1805,3691,3693,3696],{"className":3692},[1913],[1805,3694,3639],{"className":3695,"style":2510},[1913,1953],[1805,3697,3699],{"className":3698},[1920],[1805,3700,3702,3722],{"className":3701},[1924,1925],[1805,3703,3705,3719],{"className":3704},[1929],[1805,3706,3708],{"className":3707,"style":1934},[1933],[1805,3709,3710,3713],{"style":2526},[1805,3711],{"className":3712,"style":1942},[1941],[1805,3714,3716],{"className":3715},[1946,1947,1948,1949],[1805,3717,1838],{"className":3718},[1913,1953,1949],[1805,3720,1958],{"className":3721},[1957],[1805,3723,3725],{"className":3724},[1929],[1805,3726,3728],{"className":3727,"style":1965},[1933],[1805,3729],{},[1805,3731,1864],{"className":3732},[2026]," and choose ",[1805,3735,3737,3751],{"className":3736},[1812],[1805,3738,3740],{"className":3739},[1816],[1818,3741,3742],{"xmlns":1820},[1823,3743,3744,3749],{},[1826,3745,3746],{},[1832,3747,3748],{},"L",[1893,3750,3748],{"encoding":1895},[1805,3752,3754],{"className":3753,"ariaHidden":1877},[1900],[1805,3755,3757,3761],{"className":3756},[1904],[1805,3758],{"className":3759,"style":3760},[1908],"height:0.6833em;",[1805,3762,3748],{"className":3763},[1913,1953]," with ",[1805,3766,3768,3796],{"className":3767},[1812],[1805,3769,3771],{"className":3770},[1816],[1818,3772,3773],{"xmlns":1820},[1823,3774,3775,3793],{},[1826,3776,3777,3779,3785,3787],{},[1832,3778,3748],{},[2161,3780,3781,3783],{},[1832,3782,3748],{},[1832,3784,2167],{"mathvariant":1834},[1840,3786,1842],{},[1829,3788,3789,3791],{},[1832,3790,1835],{"mathvariant":1834},[1832,3792,1838],{},[1893,3794,3795],{"encoding":1895},"LL^\\top=\\Gamma_n",[1805,3797,3799,3847],{"className":3798,"ariaHidden":1877},[1900],[1805,3800,3802,3806,3809,3838,3841,3844],{"className":3801},[1904],[1805,3803],{"className":3804,"style":3805},[1908],"height:0.8491em;",[1805,3807,3748],{"className":3808},[1913,1953],[1805,3810,3812,3815],{"className":3811},[1913],[1805,3813,3748],{"className":3814},[1913,1953],[1805,3816,3818],{"className":3817},[1920],[1805,3819,3821],{"className":3820},[1924],[1805,3822,3824],{"className":3823},[1929],[1805,3825,3827],{"className":3826,"style":3805},[1933],[1805,3828,3829,3832],{"style":3104},[1805,3830],{"className":3831,"style":1942},[1941],[1805,3833,3835],{"className":3834},[1946,1947,1948,1949],[1805,3836,2167],{"className":3837},[1913,1949],[1805,3839],{"className":3840,"style":1972},[1971],[1805,3842,1842],{"className":3843},[1976],[1805,3845],{"className":3846,"style":1972},[1971],[1805,3848,3850,3853],{"className":3849},[1904],[1805,3851],{"className":3852,"style":1909},[1908],[1805,3854,3856,3859],{"className":3855},[1913],[1805,3857,1835],{"className":3858},[1913],[1805,3860,3862],{"className":3861},[1920],[1805,3863,3865,3885],{"className":3864},[1924,1925],[1805,3866,3868,3882],{"className":3867},[1929],[1805,3869,3871],{"className":3870,"style":1934},[1933],[1805,3872,3873,3876],{"style":1937},[1805,3874],{"className":3875,"style":1942},[1941],[1805,3877,3879],{"className":3878},[1946,1947,1948,1949],[1805,3880,1838],{"className":3881},[1913,1953,1949],[1805,3883,1958],{"className":3884},[1957],[1805,3886,3888],{"className":3887},[1929],[1805,3889,3891],{"className":3890,"style":1965},[1933],[1805,3892],{},". Then",[1805,3895,3897],{"className":3896},[1808],[1805,3898,3900,3930],{"className":3899},[1812],[1805,3901,3903],{"className":3902},[1816],[1818,3904,3905],{"xmlns":1820,"display":1821},[1823,3906,3907,3927],{},[1826,3908,3909,3915,3917,3921,3923,3925],{},[1829,3910,3911,3913],{},[1832,3912,2172],{"mathvariant":2118},[1832,3914,1838],{},[1840,3916,1842],{},[1832,3918,3920],{"mathvariant":3919},"bold-italic","μ",[1840,3922,2451],{},[1832,3924,3748],{},[1832,3926,3622],{"mathvariant":2118},[1893,3928,3929],{"encoding":1895},"\\mathbf X_n=\\boldsymbol\\mu+L\\mathbf z",[1805,3931,3933,3989,4015],{"className":3932,"ariaHidden":1877},[1900],[1805,3934,3936,3940,3980,3983,3986],{"className":3935},[1904],[1805,3937],{"className":3938,"style":3939},[1908],"height:0.8361em;vertical-align:-0.15em;",[1805,3941,3943,3946],{"className":3942},[1913],[1805,3944,2172],{"className":3945},[1913,2136],[1805,3947,3949],{"className":3948},[1920],[1805,3950,3952,3972],{"className":3951},[1924,1925],[1805,3953,3955,3969],{"className":3954},[1929],[1805,3956,3958],{"className":3957,"style":1934},[1933],[1805,3959,3960,3963],{"style":1937},[1805,3961],{"className":3962,"style":1942},[1941],[1805,3964,3966],{"className":3965},[1946,1947,1948,1949],[1805,3967,1838],{"className":3968},[1913,1953,1949],[1805,3970,1958],{"className":3971},[1957],[1805,3973,3975],{"className":3974},[1929],[1805,3976,3978],{"className":3977,"style":1965},[1933],[1805,3979],{},[1805,3981],{"className":3982,"style":1972},[1971],[1805,3984,1842],{"className":3985},[1976],[1805,3987],{"className":3988,"style":1972},[1971],[1805,3990,3992,3996,4006,4009,4012],{"className":3991},[1904],[1805,3993],{"className":3994,"style":3995},[1908],"height:0.7778em;vertical-align:-0.1944em;",[1805,3997,3999],{"className":3998},[1913],[1805,4000,4002],{"className":4001},[1913],[1805,4003,3920],{"className":4004},[1913,4005],"boldsymbol",[1805,4007],{"className":4008,"style":2004},[1971],[1805,4010,2451],{"className":4011},[2008],[1805,4013],{"className":4014,"style":2004},[1971],[1805,4016,4018,4021,4024],{"className":4017},[1904],[1805,4019],{"className":4020,"style":3760},[1908],[1805,4022,3748],{"className":4023},[1913,1953],[1805,4025,3622],{"className":4026},[1913,2136],[1798,4028,4029,4030,4066,4067,1891],{},"has mean ",[1805,4031,4033,4047],{"className":4032},[1812],[1805,4034,4036],{"className":4035},[1816],[1818,4037,4038],{"xmlns":1820},[1823,4039,4040,4044],{},[1826,4041,4042],{},[1832,4043,3920],{"mathvariant":3919},[1893,4045,4046],{"encoding":1895},"\\boldsymbol\\mu",[1805,4048,4050],{"className":4049,"ariaHidden":1877},[1900],[1805,4051,4053,4057],{"className":4052},[1904],[1805,4054],{"className":4055,"style":4056},[1908],"height:0.6389em;vertical-align:-0.1944em;",[1805,4058,4060],{"className":4059},[1913],[1805,4061,4063],{"className":4062},[1913],[1805,4064,3920],{"className":4065},[1913,4005]," and covariance ",[1805,4068,4070,4088],{"className":4069},[1812],[1805,4071,4073],{"className":4072},[1816],[1818,4074,4075],{"xmlns":1820},[1823,4076,4077,4085],{},[1826,4078,4079],{},[1829,4080,4081,4083],{},[1832,4082,1835],{"mathvariant":1834},[1832,4084,1838],{},[1893,4086,4087],{"encoding":1895},"\\Gamma_n",[1805,4089,4091],{"className":4090,"ariaHidden":1877},[1900],[1805,4092,4094,4097],{"className":4093},[1904],[1805,4095],{"className":4096,"style":1909},[1908],[1805,4098,4100,4103],{"className":4099},[1913],[1805,4101,1835],{"className":4102},[1913],[1805,4104,4106],{"className":4105},[1920],[1805,4107,4109,4129],{"className":4108},[1924,1925],[1805,4110,4112,4126],{"className":4111},[1929],[1805,4113,4115],{"className":4114,"style":1934},[1933],[1805,4116,4117,4120],{"style":1937},[1805,4118],{"className":4119,"style":1942},[1941],[1805,4121,4123],{"className":4122},[1946,1947,1948,1949],[1805,4124,1838],{"className":4125},[1913,1953,1949],[1805,4127,1958],{"className":4128},[1957],[1805,4130,4132],{"className":4131},[1929],[1805,4133,4135],{"className":4134,"style":1965},[1933],[1805,4136],{},[1798,4138,4139,4140,3764,4254,4381,4382,1891],{},"R returns ",[1805,4141,4143,4175],{"className":4142},[1812],[1805,4144,4146],{"className":4145},[1816],[1818,4147,4148],{"xmlns":1820},[1823,4149,4150,4172],{},[1826,4151,4152,4155,4157,4160,4162,4164,4170],{},[1832,4153,4154],{},"U",[1840,4156,1842],{},[1832,4158,4159],{"mathvariant":1834},"chol",[1840,4161,2157],{},[1840,4163,1852],{"stretchy":1845},[1829,4165,4166,4168],{},[1832,4167,1835],{"mathvariant":1834},[1832,4169,1838],{},[1840,4171,1864],{"stretchy":1845},[1893,4173,4174],{"encoding":1895},"U=\\operatorname{chol}(\\Gamma_n)",[1805,4176,4178,4196],{"className":4177,"ariaHidden":1877},[1900],[1805,4179,4181,4184,4187,4190,4193],{"className":4180},[1904],[1805,4182],{"className":4183,"style":3760},[1908],[1805,4185,4154],{"className":4186,"style":3677},[1913,1953],[1805,4188],{"className":4189,"style":1972},[1971],[1805,4191,1842],{"className":4192},[1976],[1805,4194],{"className":4195,"style":1972},[1971],[1805,4197,4199,4202,4208,4211,4251],{"className":4198},[1904],[1805,4200],{"className":4201,"style":1986},[1908],[1805,4203,4205],{"className":4204},[2215],[1805,4206,4159],{"className":4207},[1913,2219],[1805,4209,1852],{"className":4210},[1990],[1805,4212,4214,4217],{"className":4213},[1913],[1805,4215,1835],{"className":4216},[1913],[1805,4218,4220],{"className":4219},[1920],[1805,4221,4223,4243],{"className":4222},[1924,1925],[1805,4224,4226,4240],{"className":4225},[1929],[1805,4227,4229],{"className":4228,"style":1934},[1933],[1805,4230,4231,4234],{"style":1937},[1805,4232],{"className":4233,"style":1942},[1941],[1805,4235,4237],{"className":4236},[1946,1947,1948,1949],[1805,4238,1838],{"className":4239},[1913,1953,1949],[1805,4241,1958],{"className":4242},[1957],[1805,4244,4246],{"className":4245},[1929],[1805,4247,4249],{"className":4248,"style":1965},[1933],[1805,4250],{},[1805,4252,1864],{"className":4253},[2026],[1805,4255,4257,4285],{"className":4256},[1812],[1805,4258,4260],{"className":4259},[1816],[1818,4261,4262],{"xmlns":1820},[1823,4263,4264,4282],{},[1826,4265,4266,4272,4274,4276],{},[2161,4267,4268,4270],{},[1832,4269,4154],{},[1832,4271,2167],{"mathvariant":1834},[1832,4273,4154],{},[1840,4275,1842],{},[1829,4277,4278,4280],{},[1832,4279,1835],{"mathvariant":1834},[1832,4281,1838],{},[1893,4283,4284],{"encoding":1895},"U^\\top U=\\Gamma_n",[1805,4286,4288,4335],{"className":4287,"ariaHidden":1877},[1900],[1805,4289,4291,4294,4323,4326,4329,4332],{"className":4290},[1904],[1805,4292],{"className":4293,"style":3805},[1908],[1805,4295,4297,4300],{"className":4296},[1913],[1805,4298,4154],{"className":4299,"style":3677},[1913,1953],[1805,4301,4303],{"className":4302},[1920],[1805,4304,4306],{"className":4305},[1924],[1805,4307,4309],{"className":4308},[1929],[1805,4310,4312],{"className":4311,"style":3805},[1933],[1805,4313,4314,4317],{"style":3104},[1805,4315],{"className":4316,"style":1942},[1941],[1805,4318,4320],{"className":4319},[1946,1947,1948,1949],[1805,4321,2167],{"className":4322},[1913,1949],[1805,4324,4154],{"className":4325,"style":3677},[1913,1953],[1805,4327],{"className":4328,"style":1972},[1971],[1805,4330,1842],{"className":4331},[1976],[1805,4333],{"className":4334,"style":1972},[1971],[1805,4336,4338,4341],{"className":4337},[1904],[1805,4339],{"className":4340,"style":1909},[1908],[1805,4342,4344,4347],{"className":4343},[1913],[1805,4345,1835],{"className":4346},[1913],[1805,4348,4350],{"className":4349},[1920],[1805,4351,4353,4373],{"className":4352},[1924,1925],[1805,4354,4356,4370],{"className":4355},[1929],[1805,4357,4359],{"className":4358,"style":1934},[1933],[1805,4360,4361,4364],{"style":1937},[1805,4362],{"className":4363,"style":1942},[1941],[1805,4365,4367],{"className":4366},[1946,1947,1948,1949],[1805,4368,1838],{"className":4369},[1913,1953,1949],[1805,4371,1958],{"className":4372},[1957],[1805,4374,4376],{"className":4375},[1929],[1805,4377,4379],{"className":4378,"style":1965},[1933],[1805,4380],{},", so use ",[1805,4383,4385,4407],{"className":4384},[1812],[1805,4386,4388],{"className":4387},[1816],[1818,4389,4390],{"xmlns":1820},[1823,4391,4392,4404],{},[1826,4393,4394,4396,4398],{},[1832,4395,3748],{},[1840,4397,1842],{},[2161,4399,4400,4402],{},[1832,4401,4154],{},[1832,4403,2167],{"mathvariant":1834},[1893,4405,4406],{"encoding":1895},"L=U^\\top",[1805,4408,4410,4428],{"className":4409,"ariaHidden":1877},[1900],[1805,4411,4413,4416,4419,4422,4425],{"className":4412},[1904],[1805,4414],{"className":4415,"style":3760},[1908],[1805,4417,3748],{"className":4418},[1913,1953],[1805,4420],{"className":4421,"style":1972},[1971],[1805,4423,1842],{"className":4424},[1976],[1805,4426],{"className":4427,"style":1972},[1971],[1805,4429,4431,4434],{"className":4430},[1904],[1805,4432],{"className":4433,"style":3805},[1908],[1805,4435,4437,4440],{"className":4436},[1913],[1805,4438,4154],{"className":4439,"style":3677},[1913,1953],[1805,4441,4443],{"className":4442},[1920],[1805,4444,4446],{"className":4445},[1924],[1805,4447,4449],{"className":4448},[1929],[1805,4450,4452],{"className":4451,"style":3805},[1933],[1805,4453,4454,4457],{"style":3104},[1805,4455],{"className":4456,"style":1942},[1941],[1805,4458,4460],{"className":4459},[1946,1947,1948,1949],[1805,4461,2167],{"className":4462},[1913,1949],[3474,4464],{"code64":4465,"layout":3477,"locale":7,"title":4466},"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","Simulate Gaussian blocks with a Cholesky factor",[1798,4468,4469,4470,4474],{},"The empirical error will not be exactly zero because only finitely many blocks were simulated. Increase ",[4471,4472,4473],"code",{},"replications"," by a factor of four: Monte Carlo error should shrink by roughly a factor of two.",[1793,4476,4478],{"id":4477},"why-pairwise-correlations-are-not-enough","Why pairwise correlations are not enough",[1798,4480,4481,4482,4560],{},"Values satisfying ",[1805,4483,4485,4515],{"className":4484},[1812],[1805,4486,4488],{"className":4487},[1816],[1818,4489,4490],{"xmlns":1820},[1823,4491,4492,4512],{},[1826,4493,4494,4496,4499,4501,4503,4505,4507,4510],{},[1832,4495,2826],{"mathvariant":1834},[1832,4497,4498],{},"ρ",[1840,4500,1852],{"stretchy":1845},[1832,4502,2897],{},[1840,4504,1864],{"stretchy":1845},[1832,4506,2826],{"mathvariant":1834},[1840,4508,4509],{},"≤",[1884,4511,1886],{},[1893,4513,4514],{"encoding":1895},"|\\rho(h)|\\le1",[1805,4516,4518,4551],{"className":4517,"ariaHidden":1877},[1900],[1805,4519,4521,4524,4527,4530,4533,4536,4539,4542,4545,4548],{"className":4520},[1904],[1805,4522],{"className":4523,"style":1986},[1908],[1805,4525,2826],{"className":4526},[1913],[1805,4528,4498],{"className":4529},[1913,1953],[1805,4531,1852],{"className":4532},[1990],[1805,4534,2897],{"className":4535},[1913,1953],[1805,4537,1864],{"className":4538},[2026],[1805,4540,2826],{"className":4541},[1913],[1805,4543],{"className":4544,"style":1972},[1971],[1805,4546,4509],{"className":4547},[1976],[1805,4549],{"className":4550,"style":1972},[1971],[1805,4552,4554,4557],{"className":4553},[1904],[1805,4555],{"className":4556,"style":2399},[1908],[1805,4558,1886],{"className":4559},[1913]," do not automatically form a valid autocorrelation sequence. All finite Toeplitz blocks must be positive semidefinite. A proposed sequence can pass every pairwise bound and still produce a negative eigenvalue.",[1798,4562,4563],{},"This is the matrix reason that an ACF cannot be invented one lag at a time.",[1793,4565,4567],{"id":4566},"exercises","Exercises",[4569,4570,4571,4716,4731,4989,5132],"ol",{},[3488,4572,4573,4574,4625,4626,4670,4671,4715],{},"With ",[1805,4575,4577,4595],{"className":4576},[1812],[1805,4578,4580],{"className":4579},[1816],[1818,4581,4582],{"xmlns":1820},[1823,4583,4584,4592],{},[1826,4585,4586,4588,4590],{},[1832,4587,2436],{},[1840,4589,1842],{},[1884,4591,3550],{},[1893,4593,4594],{"encoding":1895},"\\phi=0.6",[1805,4596,4598,4616],{"className":4597,"ariaHidden":1877},[1900],[1805,4599,4601,4604,4607,4610,4613],{"className":4600},[1904],[1805,4602],{"className":4603,"style":3529},[1908],[1805,4605,2436],{"className":4606},[1913,1953],[1805,4608],{"className":4609,"style":1972},[1971],[1805,4611,1842],{"className":4612},[1976],[1805,4614],{"className":4615,"style":1972},[1971],[1805,4617,4619,4622],{"className":4618},[1904],[1805,4620],{"className":4621,"style":2399},[1908],[1805,4623,3550],{"className":4624},[1913]," and innovation variance 2, calculate ",[1805,4627,4629,4649],{"className":4628},[1812],[1805,4630,4632],{"className":4631},[1816],[1818,4633,4634],{"xmlns":1820},[1823,4635,4636,4646],{},[1826,4637,4638,4640,4642,4644],{},[1832,4639,1849],{},[1840,4641,1852],{"stretchy":1845},[1884,4643,3278],{},[1840,4645,1864],{"stretchy":1845},[1893,4647,4648],{"encoding":1895},"\\gamma(0)",[1805,4650,4652],{"className":4651,"ariaHidden":1877},[1900],[1805,4653,4655,4658,4661,4664,4667],{"className":4654},[1904],[1805,4656],{"className":4657,"style":1986},[1908],[1805,4659,1849],{"className":4660,"style":1994},[1913,1953],[1805,4662,1852],{"className":4663},[1990],[1805,4665,3278],{"className":4666},[1913],[1805,4668,1864],{"className":4669},[2026]," and ",[1805,4672,4674,4694],{"className":4673},[1812],[1805,4675,4677],{"className":4676},[1816],[1818,4678,4679],{"xmlns":1820},[1823,4680,4681,4691],{},[1826,4682,4683,4685,4687,4689],{},[1832,4684,1849],{},[1840,4686,1852],{"stretchy":1845},[1884,4688,1886],{},[1840,4690,1864],{"stretchy":1845},[1893,4692,4693],{"encoding":1895},"\\gamma(1)",[1805,4695,4697],{"className":4696,"ariaHidden":1877},[1900],[1805,4698,4700,4703,4706,4709,4712],{"className":4699},[1904],[1805,4701],{"className":4702,"style":1986},[1908],[1805,4704,1849],{"className":4705,"style":1994},[1913,1953],[1805,4707,1852],{"className":4708},[1990],[1805,4710,1886],{"className":4711},[1913],[1805,4713,1864],{"className":4714},[2026]," by hand.",[3488,4717,4718,4719,4722,4723,4726,4727,4730],{},"Replace ",[4471,4720,4721],{},"gamma"," by ",[4471,4724,4725],{},"c(1, 0.9, -0.9)"," and inspect the eigenvalues of ",[4471,4728,4729],{},"toeplitz(gamma)",". Is it a valid three-lag covariance block?",[3488,4732,4733,4734,1891],{},"Show algebraically that ",[1805,4735,4737,4803],{"className":4736},[1812],[1805,4738,4740],{"className":4739},[1816],[1818,4741,4742],{"xmlns":1820},[1823,4743,4744,4800],{},[1826,4745,4746,4748,4750,4752,4758,4760,4772,4774,4776,4778,4780,4782,4784,4786,4788,4790,4792,4794,4796,4798],{},[1832,4747,2154],{"mathvariant":1834},[1840,4749,2157],{},[1840,4751,1852],{"stretchy":1845},[1829,4753,4754,4756],{},[1832,4755,2172],{},[1832,4757,2431],{},[1840,4759,1858],{},[1829,4761,4762,4764],{},[1832,4763,2172],{},[1826,4765,4766,4768,4770],{},[1832,4767,2431],{},[1840,4769,1858],{},[1884,4771,1886],{},[1840,4773,1864],{"stretchy":1845},[1840,4775,1842],{},[1884,4777,2489],{},[1840,4779,1846],{"stretchy":1845},[1832,4781,1849],{},[1840,4783,1852],{"stretchy":1845},[1884,4785,3278],{},[1840,4787,1864],{"stretchy":1845},[1840,4789,1858],{},[1832,4791,1849],{},[1840,4793,1852],{"stretchy":1845},[1884,4795,1886],{},[1840,4797,1864],{"stretchy":1845},[1840,4799,1870],{"stretchy":1845},[1893,4801,4802],{"encoding":1895},"\\operatorname{Var}(X_t-X_{t-1})=2[\\gamma(0)-\\gamma(1)]",[1805,4804,4806,4870,4937,4970],{"className":4805,"ariaHidden":1877},[1900],[1805,4807,4809,4812,4818,4821,4861,4864,4867],{"className":4808},[1904],[1805,4810],{"className":4811,"style":1986},[1908],[1805,4813,4815],{"className":4814},[2215],[1805,4816,2154],{"className":4817},[1913,2219],[1805,4819,1852],{"className":4820},[1990],[1805,4822,4824,4827],{"className":4823},[1913],[1805,4825,2172],{"className":4826,"style":2510},[1913,1953],[1805,4828,4830],{"className":4829},[1920],[1805,4831,4833,4853],{"className":4832},[1924,1925],[1805,4834,4836,4850],{"className":4835},[1929],[1805,4837,4839],{"className":4838,"style":2523},[1933],[1805,4840,4841,4844],{"style":2526},[1805,4842],{"className":4843,"style":1942},[1941],[1805,4845,4847],{"className":4846},[1946,1947,1948,1949],[1805,4848,2431],{"className":4849},[1913,1953,1949],[1805,4851,1958],{"className":4852},[1957],[1805,4854,4856],{"className":4855},[1929],[1805,4857,4859],{"className":4858,"style":1965},[1933],[1805,4860],{},[1805,4862],{"className":4863,"style":2004},[1971],[1805,4865,1858],{"className":4866},[2008],[1805,4868],{"className":4869,"style":2004},[1971],[1805,4871,4873,4876,4925,4928,4931,4934],{"className":4872},[1904],[1805,4874],{"className":4875,"style":1986},[1908],[1805,4877,4879,4882],{"className":4878},[1913],[1805,4880,2172],{"className":4881,"style":2510},[1913,1953],[1805,4883,4885],{"className":4884},[1920],[1805,4886,4888,4917],{"className":4887},[1924,1925],[1805,4889,4891,4914],{"className":4890},[1929],[1805,4892,4894],{"className":4893,"style":2584},[1933],[1805,4895,4896,4899],{"style":2526},[1805,4897],{"className":4898,"style":1942},[1941],[1805,4900,4902],{"className":4901},[1946,1947,1948,1949],[1805,4903,4905,4908,4911],{"className":4904},[1913,1949],[1805,4906,2431],{"className":4907},[1913,1953,1949],[1805,4909,1858],{"className":4910},[2008,1949],[1805,4912,1886],{"className":4913},[1913,1949],[1805,4915,1958],{"className":4916},[1957],[1805,4918,4920],{"className":4919},[1929],[1805,4921,4923],{"className":4922,"style":2614},[1933],[1805,4924],{},[1805,4926,1864],{"className":4927},[2026],[1805,4929],{"className":4930,"style":1972},[1971],[1805,4932,1842],{"className":4933},[1976],[1805,4935],{"className":4936,"style":1972},[1971],[1805,4938,4940,4943,4946,4949,4952,4955,4958,4961,4964,4967],{"className":4939},[1904],[1805,4941],{"className":4942,"style":1986},[1908],[1805,4944,2489],{"className":4945},[1913],[1805,4947,1846],{"className":4948},[1990],[1805,4950,1849],{"className":4951,"style":1994},[1913,1953],[1805,4953,1852],{"className":4954},[1990],[1805,4956,3278],{"className":4957},[1913],[1805,4959,1864],{"className":4960},[2026],[1805,4962],{"className":4963,"style":2004},[1971],[1805,4965,1858],{"className":4966},[2008],[1805,4968],{"className":4969,"style":2004},[1971],[1805,4971,4973,4976,4979,4982,4985],{"className":4972},[1904],[1805,4974],{"className":4975,"style":1986},[1908],[1805,4977,1849],{"className":4978,"style":1994},[1913,1953],[1805,4980,1852],{"className":4981},[1990],[1805,4983,1886],{"className":4984},[1913],[1805,4986,4988],{"className":4987},[2026],")]",[3488,4990,4991,4992,5043,5044,5043,5072,5101,5102,5131],{},"Compare the condition numbers for ",[1805,4993,4995,5013],{"className":4994},[1812],[1805,4996,4998],{"className":4997},[1816],[1818,4999,5000],{"xmlns":1820},[1823,5001,5002,5010],{},[1826,5003,5004,5006,5008],{},[1832,5005,2436],{},[1840,5007,1842],{},[1884,5009,3278],{},[1893,5011,5012],{"encoding":1895},"\\phi=0",[1805,5014,5016,5034],{"className":5015,"ariaHidden":1877},[1900],[1805,5017,5019,5022,5025,5028,5031],{"className":5018},[1904],[1805,5020],{"className":5021,"style":3529},[1908],[1805,5023,2436],{"className":5024},[1913,1953],[1805,5026],{"className":5027,"style":1972},[1971],[1805,5029,1842],{"className":5030},[1976],[1805,5032],{"className":5033,"style":1972},[1971],[1805,5035,5037,5040],{"className":5036},[1904],[1805,5038],{"className":5039,"style":2399},[1908],[1805,5041,3278],{"className":5042},[1913],", ",[1805,5045,5047,5060],{"className":5046},[1812],[1805,5048,5050],{"className":5049},[1816],[1818,5051,5052],{"xmlns":1820},[1823,5053,5054,5058],{},[1826,5055,5056],{},[1884,5057,3550],{},[1893,5059,3550],{"encoding":1895},[1805,5061,5063],{"className":5062,"ariaHidden":1877},[1900],[1805,5064,5066,5069],{"className":5065},[1904],[1805,5067],{"className":5068,"style":2399},[1908],[1805,5070,3550],{"className":5071},[1913],[1805,5073,5075,5089],{"className":5074},[1812],[1805,5076,5078],{"className":5077},[1816],[1818,5079,5080],{"xmlns":1820},[1823,5081,5082,5087],{},[1826,5083,5084],{},[1884,5085,5086],{},"0.9",[1893,5088,5086],{"encoding":1895},[1805,5090,5092],{"className":5091,"ariaHidden":1877},[1900],[1805,5093,5095,5098],{"className":5094},[1904],[1805,5096],{"className":5097,"style":2399},[1908],[1805,5099,5086],{"className":5100},[1913],", and ",[1805,5103,5105,5119],{"className":5104},[1812],[1805,5106,5108],{"className":5107},[1816],[1818,5109,5110],{"xmlns":1820},[1823,5111,5112,5117],{},[1826,5113,5114],{},[1884,5115,5116],{},"0.99",[1893,5118,5116],{"encoding":1895},[1805,5120,5122],{"className":5121,"ariaHidden":1877},[1900],[1805,5123,5125,5128],{"className":5124},[1904],[1805,5126],{"className":5127,"style":2399},[1908],[1805,5129,5116],{"className":5130},[1913],". Explain the statistical meaning.",[3488,5133,5134],{},"Graduate extension: prove that every principal submatrix of a positive semidefinite matrix is positive semidefinite.",[5136,5137,5139],"legacy-details",{"title":5138},"Checkpoints",[4569,5140,5141,5341,5420,5773],{},[3488,5142,5143,4670,5274,1891],{},[1805,5144,5146,5189],{"className":5145},[1812],[1805,5147,5149],{"className":5148},[1816],[1818,5150,5151],{"xmlns":1820},[1823,5152,5153,5186],{},[1826,5154,5155,5157,5159,5161,5163,5165,5167,5170,5172,5174,5176,5179,5181,5183],{},[1832,5156,1849],{},[1840,5158,1852],{"stretchy":1845},[1884,5160,3278],{},[1840,5162,1864],{"stretchy":1845},[1840,5164,1842],{},[1884,5166,2489],{},[1832,5168,5169],{"mathvariant":1834},"\u002F",[1840,5171,1852],{"stretchy":1845},[1884,5173,1886],{},[1840,5175,1858],{},[1884,5177,5178],{},"0.36",[1840,5180,1864],{"stretchy":1845},[1840,5182,1842],{},[1884,5184,5185],{},"3.125",[1893,5187,5188],{"encoding":1895},"\\gamma(0)=2\u002F(1-0.36)=3.125",[1805,5190,5192,5219,5244,5265],{"className":5191,"ariaHidden":1877},[1900],[1805,5193,5195,5198,5201,5204,5207,5210,5213,5216],{"className":5194},[1904],[1805,5196],{"className":5197,"style":1986},[1908],[1805,5199,1849],{"className":5200,"style":1994},[1913,1953],[1805,5202,1852],{"className":5203},[1990],[1805,5205,3278],{"className":5206},[1913],[1805,5208,1864],{"className":5209},[2026],[1805,5211],{"className":5212,"style":1972},[1971],[1805,5214,1842],{"className":5215},[1976],[1805,5217],{"className":5218,"style":1972},[1971],[1805,5220,5222,5225,5229,5232,5235,5238,5241],{"className":5221},[1904],[1805,5223],{"className":5224,"style":1986},[1908],[1805,5226,5228],{"className":5227},[1913],"2\u002F",[1805,5230,1852],{"className":5231},[1990],[1805,5233,1886],{"className":5234},[1913],[1805,5236],{"className":5237,"style":2004},[1971],[1805,5239,1858],{"className":5240},[2008],[1805,5242],{"className":5243,"style":2004},[1971],[1805,5245,5247,5250,5253,5256,5259,5262],{"className":5246},[1904],[1805,5248],{"className":5249,"style":1986},[1908],[1805,5251,5178],{"className":5252},[1913],[1805,5254,1864],{"className":5255},[2026],[1805,5257],{"className":5258,"style":1972},[1971],[1805,5260,1842],{"className":5261},[1976],[1805,5263],{"className":5264,"style":1972},[1971],[1805,5266,5268,5271],{"className":5267},[1904],[1805,5269],{"className":5270,"style":2399},[1908],[1805,5272,5185],{"className":5273},[1913],[1805,5275,5277,5302],{"className":5276},[1812],[1805,5278,5280],{"className":5279},[1816],[1818,5281,5282],{"xmlns":1820},[1823,5283,5284,5299],{},[1826,5285,5286,5288,5290,5292,5294,5296],{},[1832,5287,1849],{},[1840,5289,1852],{"stretchy":1845},[1884,5291,1886],{},[1840,5293,1864],{"stretchy":1845},[1840,5295,1842],{},[1884,5297,5298],{},"1.875",[1893,5300,5301],{"encoding":1895},"\\gamma(1)=1.875",[1805,5303,5305,5332],{"className":5304,"ariaHidden":1877},[1900],[1805,5306,5308,5311,5314,5317,5320,5323,5326,5329],{"className":5307},[1904],[1805,5309],{"className":5310,"style":1986},[1908],[1805,5312,1849],{"className":5313,"style":1994},[1913,1953],[1805,5315,1852],{"className":5316},[1990],[1805,5318,1886],{"className":5319},[1913],[1805,5321,1864],{"className":5322},[2026],[1805,5324],{"className":5325,"style":1972},[1971],[1805,5327,1842],{"className":5328},[1976],[1805,5330],{"className":5331,"style":1972},[1971],[1805,5333,5335,5338],{"className":5334},[1904],[1805,5336],{"className":5337,"style":2399},[1908],[1805,5339,5298],{"className":5340},[1913],[3488,5342,5343,5344,5419],{},"It has a negative eigenvalue, so satisfying the individual bounds ",[1805,5345,5347,5374],{"className":5346},[1812],[1805,5348,5350],{"className":5349},[1816],[1818,5351,5352],{"xmlns":1820},[1823,5353,5354,5372],{},[1826,5355,5356,5358,5360,5362,5364,5366,5368,5370],{},[1832,5357,2826],{"mathvariant":1834},[1832,5359,4498],{},[1840,5361,1852],{"stretchy":1845},[1832,5363,2897],{},[1840,5365,1864],{"stretchy":1845},[1832,5367,2826],{"mathvariant":1834},[1840,5369,4509],{},[1884,5371,1886],{},[1893,5373,4514],{"encoding":1895},[1805,5375,5377,5410],{"className":5376,"ariaHidden":1877},[1900],[1805,5378,5380,5383,5386,5389,5392,5395,5398,5401,5404,5407],{"className":5379},[1904],[1805,5381],{"className":5382,"style":1986},[1908],[1805,5384,2826],{"className":5385},[1913],[1805,5387,4498],{"className":5388},[1913,1953],[1805,5390,1852],{"className":5391},[1990],[1805,5393,2897],{"className":5394},[1913,1953],[1805,5396,1864],{"className":5397},[2026],[1805,5399,2826],{"className":5400},[1913],[1805,5402],{"className":5403,"style":1972},[1971],[1805,5405,4509],{"className":5406},[1976],[1805,5408],{"className":5409,"style":1972},[1971],[1805,5411,5413,5416],{"className":5412},[1904],[1805,5414],{"className":5415,"style":2399},[1908],[1805,5417,1886],{"className":5418},[1913]," was insufficient.",[3488,5421,5422,5423,1891],{},"Expand 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justify positive semidefiniteness with quadratic forms, and use Cholesky reconstruction as a reproducible numerical check.",[1798,5965,5966,5967,1891],{},"Continue to ",[2119,5968,5970],{"href":5969},".\u002F02-ar-recursions","Lab 2 — AR Recursions",{"title":10,"searchDepth":5972,"depth":5972,"links":5973},2,[5974,5975,5979,5980,5981,5982],{"id":1795,"depth":5972,"text":1796},{"id":2405,"depth":5972,"text":2406,"children":5976},[5977],{"id":3482,"depth":5978,"text":3483},3,{"id":3603,"depth":5972,"text":3604},{"id":4477,"depth":5972,"text":4478},{"id":4566,"depth":5972,"text":4567},{"id":5778,"depth":5972,"text":5779},"Construct stationary Toeplitz covariance matrices, test positive definiteness, factor them, and simulate Gaussian finite blocks.","md",{"sidebar":5986},{"order":5987},1,true,{"title":1001,"description":5983},"WESaYsoAzrYHI60jRhJY3Nm9gieWoPl3Xp8F2ZyaOa0",[5992,5994],{"title":995,"path":996,"stem":997,"description":5993,"children":-1},"Four browser-based base R laboratories for covariance matrices, AR recursions, linear prediction, likelihood, and state space.",{"title":1005,"path":1006,"stem":1007,"description":5995,"children":-1},"Convert AR models to companion form, compare polynomial roots with eigenvalues, propagate shocks, and recover coefficients from autocovariances.",1785754738595]