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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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策略全流程","\u002Fzh\u002Fquant\u002F09-case-study","zh\u002Fquant\u002F09-case-study",{"title":1779,"path":1780,"stem":1781},"量化投资文献与软件图谱","\u002Fzh\u002Fquant\u002F10-reading-software-map","zh\u002Fquant\u002F10-reading-software-map",null,{"id":1784,"title":959,"body":1785,"description":12234,"extension":12235,"features":1782,"hero":1782,"layout":1782,"locale":1782,"meta":12236,"navigation":1782,"path":960,"published":12239,"seo":12240,"stem":961,"__hash__":12241},"docs\u002Fen\u002Ftime-series\u002F01-stochastic-process\u002Findex.md",{"type":1786,"value":1787,"toc":12214},"minimark",[1788,1792,1797,1988,1992,1995,2020,2024,2027,2182,2325,2553,2557,2702,3019,3022,3522,3525,4503,4577,4794,4909,4915,5017,5021,5024,5392,5548,5553,5556,5770,5916,6217,6633,6754,6899,6903,6953,6956,6960,6963,7486,7545,7549,7681,7684,7688,7691,8092,8158,8441,8444,8836,8996,9296,9402,9406,9409,9720,9723,10058,10066,10070,10131,10135,11013,12202,12206],[1789,1790,959],"h1",{"id":1791},"module-1-processes-dependence-and-stationarity",[1793,1794,1796],"h2",{"id":1795},"core-question","Core 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",[1802,1803,1806,1853],"span",{"className":1804},[1805],"katex",[1802,1807,1810],{"className":1808},[1809],"katex-mathml",[1811,1812,1814],"math",{"xmlns":1813},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1815,1816,1817,1848],"semantics",{},[1818,1819,1820,1831,1836,1839,1841],"mrow",{},[1821,1822,1823,1827],"msub",{},[1824,1825,1826],"mi",{},"x",[1828,1829,1830],"mn",{},"1",[1832,1833,1835],"mo",{"separator":1834},"true",",",[1832,1837,1838],{},"…",[1832,1840,1835],{"separator":1834},[1821,1842,1843,1845],{},[1824,1844,1826],{},[1824,1846,1847],{},"n",[1849,1850,1852],"annotation",{"encoding":1851},"application\u002Fx-tex","x_1,\\ldots,x_n",[1802,1854,1857],{"className":1855,"ariaHidden":1834},[1856],"katex-html",[1802,1858,1861,1866,1924,1928,1933,1937,1940,1943,1946],{"className":1859},[1860],"base",[1802,1862],{"className":1863,"style":1865},[1864],"strut","height:0.625em;vertical-align:-0.1944em;",[1802,1867,1870,1874],{"className":1868},[1869],"mord",[1802,1871,1826],{"className":1872},[1869,1873],"mathnormal",[1802,1875,1878],{"className":1876},[1877],"msupsub",[1802,1879,1883,1915],{"className":1880},[1881,1882],"vlist-t","vlist-t2",[1802,1884,1887,1910],{"className":1885},[1886],"vlist-r",[1802,1888,1892],{"className":1889,"style":1891},[1890],"vlist","height:0.3011em;",[1802,1893,1895,1900],{"style":1894},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1802,1896],{"className":1897,"style":1899},[1898],"pstrut","height:2.7em;",[1802,1901,1907],{"className":1902},[1903,1904,1905,1906],"sizing","reset-size6","size3","mtight",[1802,1908,1830],{"className":1909},[1869,1906],[1802,1911,1914],{"className":1912},[1913],"vlist-s","​",[1802,1916,1918],{"className":1917},[1886],[1802,1919,1922],{"className":1920,"style":1921},[1890],"height:0.15em;",[1802,1923],{},[1802,1925,1835],{"className":1926},[1927],"mpunct",[1802,1929],{"className":1930,"style":1932},[1931],"mspace","margin-right:0.1667em;",[1802,1934,1838],{"className":1935},[1936],"minner",[1802,1938],{"className":1939,"style":1932},[1931],[1802,1941,1835],{"className":1942},[1927],[1802,1944],{"className":1945,"style":1932},[1931],[1802,1947,1949,1952],{"className":1948},[1869],[1802,1950,1826],{"className":1951},[1869,1873],[1802,1953,1955],{"className":1954},[1877],[1802,1956,1958,1979],{"className":1957},[1881,1882],[1802,1959,1961,1976],{"className":1960},[1886],[1802,1962,1965],{"className":1963,"style":1964},[1890],"height:0.1514em;",[1802,1966,1967,1970],{"style":1894},[1802,1968],{"className":1969,"style":1899},[1898],[1802,1971,1973],{"className":1972},[1903,1904,1905,1906],[1802,1974,1847],{"className":1975},[1869,1873,1906],[1802,1977,1914],{"className":1978},[1913],[1802,1980,1982],{"className":1981},[1886],[1802,1983,1985],{"className":1984,"style":1921},[1890],[1802,1986],{},". What assumptions let that one path reveal a population mean, variance, and dependence structure?",[1793,1989,1991],{"id":1990},"learning-outcomes","Learning outcomes",[1798,1993,1994],{},"You will be able to:",[1996,1997,1998,2002,2005,2008,2011,2014,2017],"ul",{},[1999,2000,2001],"li",{},"distinguish a stochastic process, a random variable, and one sample path;",[1999,2003,2004],{},"construct the mean vector and Toeplitz covariance matrix of a stationary finite block;",[1999,2006,2007],{},"calculate and interpret autocovariance and autocorrelation;",[1999,2009,2010],{},"separate strict stationarity, weak stationarity, and ergodicity;",[1999,2012,2013],{},"distinguish white noise from independence and Gaussianity;",[1999,2015,2016],{},"derive the dependence and shock persistence of a stationary AR(1);",[1999,2018,2019],{},"state exactly what the Wold representation does and does not promise.",[1793,2021,2023],{"id":2022},"_1-process-versus-path","1. Process versus path",[1798,2025,2026],{},"A stochastic process is a collection of random variables indexed by time:",[1802,2028,2031],{"className":2029},[2030],"katex-display",[1802,2032,2034,2078],{"className":2033},[1805],[1802,2035,2037],{"className":2036},[1809],[1811,2038,2040],{"xmlns":1813,"display":2039},"block",[1815,2041,2042,2075],{},[1818,2043,2044,2048,2056,2059,2061,2064,2068,2071],{},[1832,2045,2047],{"stretchy":2046},"false","{",[1821,2049,2050,2053],{},[1824,2051,2052],{},"X",[1824,2054,2055],{},"t",[1832,2057,2058],{},":",[1824,2060,2055],{},[1832,2062,2063],{},"∈",[1824,2065,2067],{"mathvariant":2066},"double-struck","Z",[1832,2069,2070],{"stretchy":2046},"}",[1824,2072,2074],{"mathvariant":2073},"normal",".",[1849,2076,2077],{"encoding":1851},"\\{X_t:t\\in\\mathbb Z\\}.",[1802,2079,2081,2146,2165],{"className":2080,"ariaHidden":1834},[1856],[1802,2082,2084,2088,2092,2135,2139,2143],{"className":2083},[1860],[1802,2085],{"className":2086,"style":2087},[1864],"height:1em;vertical-align:-0.25em;",[1802,2089,2047],{"className":2090},[2091],"mopen",[1802,2093,2095,2099],{"className":2094},[1869],[1802,2096,2052],{"className":2097,"style":2098},[1869,1873],"margin-right:0.0785em;",[1802,2100,2102],{"className":2101},[1877],[1802,2103,2105,2127],{"className":2104},[1881,1882],[1802,2106,2108,2124],{"className":2107},[1886],[1802,2109,2112],{"className":2110,"style":2111},[1890],"height:0.2806em;",[1802,2113,2115,2118],{"style":2114},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1802,2116],{"className":2117,"style":1899},[1898],[1802,2119,2121],{"className":2120},[1903,1904,1905,1906],[1802,2122,2055],{"className":2123},[1869,1873,1906],[1802,2125,1914],{"className":2126},[1913],[1802,2128,2130],{"className":2129},[1886],[1802,2131,2133],{"className":2132,"style":1921},[1890],[1802,2134],{},[1802,2136],{"className":2137,"style":2138},[1931],"margin-right:0.2778em;",[1802,2140,2058],{"className":2141},[2142],"mrel",[1802,2144],{"className":2145,"style":2138},[1931],[1802,2147,2149,2153,2156,2159,2162],{"className":2148},[1860],[1802,2150],{"className":2151,"style":2152},[1864],"height:0.6542em;vertical-align:-0.0391em;",[1802,2154,2055],{"className":2155},[1869,1873],[1802,2157],{"className":2158,"style":2138},[1931],[1802,2160,2063],{"className":2161},[2142],[1802,2163],{"className":2164,"style":2138},[1931],[1802,2166,2168,2171,2175,2179],{"className":2167},[1860],[1802,2169],{"className":2170,"style":2087},[1864],[1802,2172,2067],{"className":2173},[1869,2174],"mathbb",[1802,2176,2070],{"className":2177},[2178],"mclose",[1802,2180,2074],{"className":2181},[1869],[1798,2183,2184,2185,2324],{},"The capital letters describe the population mechanism. The data ",[1802,2186,2188,2217],{"className":2187},[1805],[1802,2189,2191],{"className":2190},[1809],[1811,2192,2193],{"xmlns":1813},[1815,2194,2195,2215],{},[1818,2196,2197,2203,2205,2207,2209],{},[1821,2198,2199,2201],{},[1824,2200,1826],{},[1828,2202,1830],{},[1832,2204,1835],{"separator":1834},[1832,2206,1838],{},[1832,2208,1835],{"separator":1834},[1821,2210,2211,2213],{},[1824,2212,1826],{},[1824,2214,1847],{},[1849,2216,1852],{"encoding":1851},[1802,2218,2220],{"className":2219,"ariaHidden":1834},[1856],[1802,2221,2223,2226,2266,2269,2272,2275,2278,2281,2284],{"className":2222},[1860],[1802,2224],{"className":2225,"style":1865},[1864],[1802,2227,2229,2232],{"className":2228},[1869],[1802,2230,1826],{"className":2231},[1869,1873],[1802,2233,2235],{"className":2234},[1877],[1802,2236,2238,2258],{"className":2237},[1881,1882],[1802,2239,2241,2255],{"className":2240},[1886],[1802,2242,2244],{"className":2243,"style":1891},[1890],[1802,2245,2246,2249],{"style":1894},[1802,2247],{"className":2248,"style":1899},[1898],[1802,2250,2252],{"className":2251},[1903,1904,1905,1906],[1802,2253,1830],{"className":2254},[1869,1906],[1802,2256,1914],{"className":2257},[1913],[1802,2259,2261],{"className":2260},[1886],[1802,2262,2264],{"className":2263,"style":1921},[1890],[1802,2265],{},[1802,2267,1835],{"className":2268},[1927],[1802,2270],{"className":2271,"style":1932},[1931],[1802,2273,1838],{"className":2274},[1936],[1802,2276],{"className":2277,"style":1932},[1931],[1802,2279,1835],{"className":2280},[1927],[1802,2282],{"className":2283,"style":1932},[1931],[1802,2285,2287,2290],{"className":2286},[1869],[1802,2288,1826],{"className":2289},[1869,1873],[1802,2291,2293],{"className":2292},[1877],[1802,2294,2296,2316],{"className":2295},[1881,1882],[1802,2297,2299,2313],{"className":2298},[1886],[1802,2300,2302],{"className":2301,"style":1964},[1890],[1802,2303,2304,2307],{"style":1894},[1802,2305],{"className":2306,"style":1899},[1898],[1802,2308,2310],{"className":2309},[1903,1904,1905,1906],[1802,2311,1847],{"className":2312},[1869,1873,1906],[1802,2314,1914],{"className":2315},[1913],[1802,2317,2319],{"className":2318},[1886],[1802,2320,2322],{"className":2321,"style":1921},[1890],[1802,2323],{}," are one realised path. Observing 120 months is not the same as observing 120 independent copies: adjacent values may share shocks.",[2326,2327,2328,2344],"table",{},[2329,2330,2331],"thead",{},[2332,2333,2334,2338,2341],"tr",{},[2335,2336,2337],"th",{},"Object",[2335,2339,2340],{},"Example",[2335,2342,2343],{},"Question",[2345,2346,2347,2448,2539],"tbody",{},[2332,2348,2349,2353,2445],{},[2350,2351,2352],"td",{},"random variable",[2350,2354,2355,2356],{},"next month's demand ",[1802,2357,2359,2384],{"className":2358},[1805],[1802,2360,2362],{"className":2361},[1809],[1811,2363,2364],{"xmlns":1813},[1815,2365,2366,2381],{},[1818,2367,2368],{},[1821,2369,2370,2372],{},[1824,2371,2052],{},[1818,2373,2374,2376,2379],{},[1824,2375,2055],{},[1832,2377,2378],{},"+",[1828,2380,1830],{},[1849,2382,2383],{"encoding":1851},"X_{t+1}",[1802,2385,2387],{"className":2386,"ariaHidden":1834},[1856],[1802,2388,2390,2394],{"className":2389},[1860],[1802,2391],{"className":2392,"style":2393},[1864],"height:0.8917em;vertical-align:-0.2083em;",[1802,2395,2397,2400],{"className":2396},[1869],[1802,2398,2052],{"className":2399,"style":2098},[1869,1873],[1802,2401,2403],{"className":2402},[1877],[1802,2404,2406,2436],{"className":2405},[1881,1882],[1802,2407,2409,2433],{"className":2408},[1886],[1802,2410,2412],{"className":2411,"style":1891},[1890],[1802,2413,2414,2417],{"style":2114},[1802,2415],{"className":2416,"style":1899},[1898],[1802,2418,2420],{"className":2419},[1903,1904,1905,1906],[1802,2421,2423,2426,2430],{"className":2422},[1869,1906],[1802,2424,2055],{"className":2425},[1869,1873,1906],[1802,2427,2378],{"className":2428},[2429,1906],"mbin",[1802,2431,1830],{"className":2432},[1869,1906],[1802,2434,1914],{"className":2435},[1913],[1802,2437,2439],{"className":2438},[1886],[1802,2440,2443],{"className":2441,"style":2442},[1890],"height:0.2083em;",[1802,2444],{},[2350,2446,2447],{},"what values and probabilities are possible?",[2332,2449,2450,2453,2536],{},[2350,2451,2452],{},"process",[2350,2454,2455,2535],{},[1802,2456,2458,2480],{"className":2457},[1805],[1802,2459,2461],{"className":2460},[1809],[1811,2462,2463],{"xmlns":1813},[1815,2464,2465,2477],{},[1818,2466,2467,2469,2475],{},[1832,2468,2047],{"stretchy":2046},[1821,2470,2471,2473],{},[1824,2472,2052],{},[1824,2474,2055],{},[1832,2476,2070],{"stretchy":2046},[1849,2478,2479],{"encoding":1851},"\\{X_t\\}",[1802,2481,2483],{"className":2482,"ariaHidden":1834},[1856],[1802,2484,2486,2489,2492,2532],{"className":2485},[1860],[1802,2487],{"className":2488,"style":2087},[1864],[1802,2490,2047],{"className":2491},[2091],[1802,2493,2495,2498],{"className":2494},[1869],[1802,2496,2052],{"className":2497,"style":2098},[1869,1873],[1802,2499,2501],{"className":2500},[1877],[1802,2502,2504,2524],{"className":2503},[1881,1882],[1802,2505,2507,2521],{"className":2506},[1886],[1802,2508,2510],{"className":2509,"style":2111},[1890],[1802,2511,2512,2515],{"style":2114},[1802,2513],{"className":2514,"style":1899},[1898],[1802,2516,2518],{"className":2517},[1903,1904,1905,1906],[1802,2519,2055],{"className":2520},[1869,1873,1906],[1802,2522,1914],{"className":2523},[1913],[1802,2525,2527],{"className":2526},[1886],[1802,2528,2530],{"className":2529,"style":1921},[1890],[1802,2531],{},[1802,2533,2070],{"className":2534},[2178]," over all months",[2350,2537,2538],{},"how does the joint distribution change with time?",[2332,2540,2541,2544,2550],{},[2350,2542,2543],{},"sample path",[2350,2545,2546],{},[2547,2548,2549],"code",{},"102, 108, 105, ...",[2350,2551,2552],{},"what can this one history reveal?",[1793,2554,2556],{"id":2555},"_2-finite-blocks-expose-the-matrix-structure","2. Finite blocks expose the matrix structure",[1798,2558,2559,2560,2701],{},"For any selected times 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is symmetric Toeplitz: each diagonal is constant. It must also be positive semidefinite because",[1802,4578,4580],{"className":4579},[2030],[1802,4581,4583,4633],{"className":4582},[1805],[1802,4584,4586],{"className":4585},[1809],[1811,4587,4588],{"xmlns":1813,"display":2039},[1815,4589,4590,4630],{},[1818,4591,4592,4599,4605,4607,4609,4611,4613,4615,4621,4623,4625,4628],{},[2753,4593,4594,4597],{},[1824,4595,4596],{"mathvariant":2719},"a",[1824,4598,2760],{"mathvariant":2073},[1821,4600,4601,4603],{},[1824,4602,3073],{"mathvariant":2073},[1824,4604,1847],{},[1824,4606,4596],{"mathvariant":2719},[1832,4608,2722],{},[1824,4610,3059],{"mathvariant":2073},[1832,4612,3062],{},[1832,4614,2725],{"stretchy":2046},[2753,4616,4617,4619],{},[1824,4618,4596],{"mathvariant":2719},[1824,4620,2760],{"mathvariant":2073},[1824,4622,2052],{"mathvariant":2719},[1832,4624,2757],{"stretchy":2046},[1832,4626,4627],{},"≥",[1828,4629,3583],{},[1849,4631,4632],{"encoding":1851},"\\mathbf a^\\top\\Gamma_n\\mathbf a\n=\\operatorname{Var}(\\mathbf a^\\top\\mathbf X)\\ge0",[1802,4634,4636,4724,4784],{"className":4635,"ariaHidden":1834},[1856],[1802,4637,4639,4643,4672,4712,4715,4718,4721],{"className":4638},[1860],[1802,4640],{"className":4641,"style":4642},[1864],"height:1.0491em;vertical-align:-0.15em;",[1802,4644,4646,4649],{"className":4645},[1869],[1802,4647,4596],{"className":4648},[1869,2779],[1802,4650,4652],{"className":4651},[1877],[1802,4653,4655],{"className":4654},[1881],[1802,4656,4658],{"className":4657},[1886],[1802,4659,4661],{"className":4660,"style":3003},[1890],[1802,4662,4663,4666],{"style":3006},[1802,4664],{"className":4665,"style":1899},[1898],[1802,4667,4669],{"className":4668},[1903,1904,1905,1906],[1802,4670,2760],{"className":4671},[1869,1906],[1802,4673,4675,4678],{"className":4674},[1869],[1802,4676,3073],{"className":4677},[1869],[1802,4679,4681],{"className":4680},[1877],[1802,4682,4684,4704],{"className":4683},[1881,1882],[1802,4685,4687,4701],{"className":4686},[1886],[1802,4688,4690],{"className":4689,"style":1964},[1890],[1802,4691,4692,4695],{"style":1894},[1802,4693],{"className":4694,"style":1899},[1898],[1802,4696,4698],{"className":4697},[1903,1904,1905,1906],[1802,4699,1847],{"className":4700},[1869,1873,1906],[1802,4702,1914],{"className":4703},[1913],[1802,4705,4707],{"className":4706},[1886],[1802,4708,4710],{"className":4709,"style":1921},[1890],[1802,4711],{},[1802,4713,4596],{"className":4714},[1869,2779],[1802,4716],{"className":4717,"style":2138},[1931],[1802,4719,2722],{"className":4720},[2142],[1802,4722],{"className":4723,"style":2138},[1931],[1802,4725,4727,4731,4737,4740,4769,4772,4775,4778,4781],{"className":4726},[1860],[1802,4728],{"className":4729,"style":4730},[1864],"height:1.1491em;vertical-align:-0.25em;",[1802,4732,4734],{"className":4733},[3199],[1802,4735,3059],{"className":4736},[1869,3203],[1802,4738,2725],{"className":4739},[2091],[1802,4741,4743,4746],{"className":4742},[1869],[1802,4744,4596],{"className":4745},[1869,2779],[1802,4747,4749],{"className":4748},[1877],[1802,4750,4752],{"className":4751},[1881],[1802,4753,4755],{"className":4754},[1886],[1802,4756,4758],{"className":4757,"style":3003},[1890],[1802,4759,4760,4763],{"style":3006},[1802,4761],{"className":4762,"style":1899},[1898],[1802,4764,4766],{"className":4765},[1903,1904,1905,1906],[1802,4767,2760],{"className":4768},[1869,1906],[1802,4770,2052],{"className":4771},[1869,2779],[1802,4773,2757],{"className":4774},[2178],[1802,4776],{"className":4777,"style":2138},[1931],[1802,4779,4627],{"className":4780},[2142],[1802,4782],{"className":4783,"style":2138},[1931],[1802,4785,4787,4791],{"className":4786},[1860],[1802,4788],{"className":4789,"style":4790},[1864],"height:0.6444em;",[1802,4792,3583],{"className":4793},[1869],[1798,4795,4796,4797,4827,4828,4908],{},"for every ",[1802,4798,4800,4814],{"className":4799},[1805],[1802,4801,4803],{"className":4802},[1809],[1811,4804,4805],{"xmlns":1813},[1815,4806,4807,4811],{},[1818,4808,4809],{},[1824,4810,4596],{"mathvariant":2719},[1849,4812,4813],{"encoding":1851},"\\mathbf a",[1802,4815,4817],{"className":4816,"ariaHidden":1834},[1856],[1802,4818,4820,4824],{"className":4819},[1860],[1802,4821],{"className":4822,"style":4823},[1864],"height:0.4444em;",[1802,4825,4596],{"className":4826},[1869,2779],". The bounds ",[1802,4829,4831,4863],{"className":4830},[1805],[1802,4832,4834],{"className":4833},[1809],[1811,4835,4836],{"xmlns":1813},[1815,4837,4838,4860],{},[1818,4839,4840,4843,4846,4848,4851,4853,4855,4858],{},[1824,4841,4842],{"mathvariant":2073},"∣",[1824,4844,4845],{},"ρ",[1832,4847,2725],{"stretchy":2046},[1824,4849,4850],{},"h",[1832,4852,2757],{"stretchy":2046},[1824,4854,4842],{"mathvariant":2073},[1832,4856,4857],{},"≤",[1828,4859,1830],{},[1849,4861,4862],{"encoding":1851},"|\\rho(h)|\\le1",[1802,4864,4866,4899],{"className":4865,"ariaHidden":1834},[1856],[1802,4867,4869,4872,4875,4878,4881,4884,4887,4890,4893,4896],{"className":4868},[1860],[1802,4870],{"className":4871,"style":2087},[1864],[1802,4873,4842],{"className":4874},[1869],[1802,4876,4845],{"className":4877},[1869,1873],[1802,4879,2725],{"className":4880},[2091],[1802,4882,4850],{"className":4883},[1869,1873],[1802,4885,2757],{"className":4886},[2178],[1802,4888,4842],{"className":4889},[1869],[1802,4891],{"className":4892,"style":2138},[1931],[1802,4894,4857],{"className":4895},[2142],[1802,4897],{"className":4898,"style":2138},[1931],[1802,4900,4902,4905],{"className":4901},[1860],[1802,4903],{"className":4904,"style":4790},[1864],[1802,4906,1830],{"className":4907},[1869]," are necessary but not sufficient; all lags must fit together into valid covariance matrices.",[4910,4911],"web-r",{"code64":4912,"layout":4913,"locale":7,"title":4914},"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","vertical","Construct and verify a stationary covariance block",[1798,4916,4917,4918,4949,4950,4986,4987,5016],{},"Change ",[1802,4919,4921,4936],{"className":4920},[1805],[1802,4922,4924],{"className":4923},[1809],[1811,4925,4926],{"xmlns":1813},[1815,4927,4928,4933],{},[1818,4929,4930],{},[1824,4931,4932],{},"ϕ",[1849,4934,4935],{"encoding":1851},"\\phi",[1802,4937,4939],{"className":4938,"ariaHidden":1834},[1856],[1802,4940,4942,4946],{"className":4941},[1860],[1802,4943],{"className":4944,"style":4945},[1864],"height:0.8889em;vertical-align:-0.1944em;",[1802,4947,4932],{"className":4948},[1869,1873]," to ",[1802,4951,4953,4970],{"className":4952},[1805],[1802,4954,4956],{"className":4955},[1809],[1811,4957,4958],{"xmlns":1813},[1815,4959,4960,4967],{},[1818,4961,4962,4964],{},[1832,4963,3631],{},[1828,4965,4966],{},"0.7",[1849,4968,4969],{"encoding":1851},"-0.7",[1802,4971,4973],{"className":4972,"ariaHidden":1834},[1856],[1802,4974,4976,4980,4983],{"className":4975},[1860],[1802,4977],{"className":4978,"style":4979},[1864],"height:0.7278em;vertical-align:-0.0833em;",[1802,4981,3631],{"className":4982},[1869],[1802,4984,4966],{"className":4985},[1869],": alternating covariances appear, but validity remains. Move it to ",[1802,4988,4990,5004],{"className":4989},[1805],[1802,4991,4993],{"className":4992},[1809],[1811,4994,4995],{"xmlns":1813},[1815,4996,4997,5002],{},[1818,4998,4999],{},[1828,5000,5001],{},"0.98",[1849,5003,5001],{"encoding":1851},[1802,5005,5007],{"className":5006,"ariaHidden":1834},[1856],[1802,5008,5010,5013],{"className":5009},[1860],[1802,5011],{"className":5012,"style":4790},[1864],[1802,5014,5001],{"className":5015},[1869],": the smallest eigenvalue and condition number reveal near-collinearity even before estimation.",[1793,5018,5020],{"id":5019},"_3-dependence-is-indexed-by-lag","3. Dependence is indexed by lag",[1798,5022,5023],{},"For a weakly stationary process,",[1802,5025,5027],{"className":5026},[2030],[1802,5028,5030,5119],{"className":5029},[1805],[1802,5031,5033],{"className":5032},[1809],[1811,5034,5035],{"xmlns":1813,"display":2039},[1815,5036,5037,5116],{},[1818,5038,5039,5041,5043,5045,5047,5049,5051,5053,5055,5061,5063,5075,5077,5079,5081,5083,5085,5087,5089,5091,5114],{},[1824,5040,3588],{},[1832,5042,2725],{"stretchy":2046},[1824,5044,4850],{},[1832,5046,2757],{"stretchy":2046},[1832,5048,2722],{},[1824,5050,3086],{"mathvariant":2073},[1832,5052,3062],{},[1832,5054,2725],{"stretchy":2046},[1821,5056,5057,5059],{},[1824,5058,2052],{},[1824,5060,2055],{},[1832,5062,1835],{"separator":1834},[1821,5064,5065,5067],{},[1824,5066,2052],{},[1818,5068,5069,5071,5073],{},[1824,5070,2055],{},[1832,5072,3631],{},[1824,5074,4850],{},[1832,5076,2757],{"stretchy":2046},[1832,5078,1835],{"separator":1834},[1931,5080],{"width":3056},[1824,5082,4845],{},[1832,5084,2725],{"stretchy":2046},[1824,5086,4850],{},[1832,5088,2757],{"stretchy":2046},[1832,5090,2722],{},[5092,5093,5094,5104],"mfrac",{},[1818,5095,5096,5098,5100,5102],{},[1824,5097,3588],{},[1832,5099,2725],{"stretchy":2046},[1824,5101,4850],{},[1832,5103,2757],{"stretchy":2046},[1818,5105,5106,5108,5110,5112],{},[1824,5107,3588],{},[1832,5109,2725],{"stretchy":2046},[1828,5111,3583],{},[1832,5113,2757],{"stretchy":2046},[1824,5115,2074],{"mathvariant":2073},[1849,5117,5118],{"encoding":1851},"\\gamma(h)=\\operatorname{Cov}(X_t,X_{t-h}),\n\\qquad\n\\rho(h)=\\frac{\\gamma(h)}{\\gamma(0)}.",[1802,5120,5122,5149,5293],{"className":5121,"ariaHidden":1834},[1856],[1802,5123,5125,5128,5131,5134,5137,5140,5143,5146],{"className":5124},[1860],[1802,5126],{"className":5127,"style":2087},[1864],[1802,5129,3588],{"className":5130,"style":4034},[1869,1873],[1802,5132,2725],{"className":5133},[2091],[1802,5135,4850],{"className":5136},[1869,1873],[1802,5138,2757],{"className":5139},[2178],[1802,5141],{"className":5142,"style":2138},[1931],[1802,5144,2722],{"className":5145},[2142],[1802,5147],{"className":5148,"style":2138},[1931],[1802,5150,5152,5155,5161,5164,5204,5207,5210,5260,5263,5266,5269,5272,5275,5278,5281,5284,5287,5290],{"className":5151},[1860],[1802,5153],{"className":5154,"style":2087},[1864],[1802,5156,5158],{"className":5157},[3199],[1802,5159,3086],{"className":5160,"style":3269},[1869,3203],[1802,5162,2725],{"className":5163},[2091],[1802,5165,5167,5170],{"className":5166},[1869],[1802,5168,2052],{"className":5169,"style":2098},[1869,1873],[1802,5171,5173],{"className":5172},[1877],[1802,5174,5176,5196],{"className":5175},[1881,1882],[1802,5177,5179,5193],{"className":5178},[1886],[1802,5180,5182],{"className":5181,"style":2111},[1890],[1802,5183,5184,5187],{"style":2114},[1802,5185],{"className":5186,"style":1899},[1898],[1802,5188,5190],{"className":5189},[1903,1904,1905,1906],[1802,5191,2055],{"className":5192},[1869,1873,1906],[1802,5194,1914],{"className":5195},[1913],[1802,5197,5199],{"className":5198},[1886],[1802,5200,5202],{"className":5201,"style":1921},[1890],[1802,5203],{},[1802,5205,1835],{"className":5206},[1927],[1802,5208],{"className":5209,"style":1932},[1931],[1802,5211,5213,5216],{"className":5212},[1869],[1802,5214,2052],{"className":5215,"style":2098},[1869,1873],[1802,5217,5219],{"className":5218},[1877],[1802,5220,5222,5252],{"className":5221},[1881,1882],[1802,5223,5225,5249],{"className":5224},[1886],[1802,5226,5229],{"className":5227,"style":5228},[1890],"height:0.3361em;",[1802,5230,5231,5234],{"style":2114},[1802,5232],{"className":5233,"style":1899},[1898],[1802,5235,5237],{"className":5236},[1903,1904,1905,1906],[1802,5238,5240,5243,5246],{"className":5239},[1869,1906],[1802,5241,2055],{"className":5242},[1869,1873,1906],[1802,5244,3631],{"className":5245},[2429,1906],[1802,5247,4850],{"className":5248},[1869,1873,1906],[1802,5250,1914],{"className":5251},[1913],[1802,5253,5255],{"className":5254},[1886],[1802,5256,5258],{"className":5257,"style":2442},[1890],[1802,5259],{},[1802,5261,2757],{"className":5262},[2178],[1802,5264,1835],{"className":5265},[1927],[1802,5267],{"className":5268,"style":3192},[1931],[1802,5270],{"className":5271,"style":1932},[1931],[1802,5273,4845],{"className":5274},[1869,1873],[1802,5276,2725],{"className":5277},[2091],[1802,5279,4850],{"className":5280},[1869,1873],[1802,5282,2757],{"className":5283},[2178],[1802,5285],{"className":5286,"style":2138},[1931],[1802,5288,2722],{"className":5289},[2142],[1802,5291],{"className":5292,"style":2138},[1931],[1802,5294,5296,5300,5389],{"className":5295},[1860],[1802,5297],{"className":5298,"style":5299},[1864],"height:2.363em;vertical-align:-0.936em;",[1802,5301,5303,5307,5386],{"className":5302},[1869],[1802,5304],{"className":5305},[2091,5306],"nulldelimiter",[1802,5308,5310],{"className":5309},[5092],[1802,5311,5313,5377],{"className":5312},[1881,1882],[1802,5314,5316,5374],{"className":5315},[1886],[1802,5317,5320,5342,5353],{"className":5318,"style":5319},[1890],"height:1.427em;",[1802,5321,5323,5327],{"style":5322},"top:-2.314em;",[1802,5324],{"className":5325,"style":5326},[1898],"height:3em;",[1802,5328,5330,5333,5336,5339],{"className":5329},[1869],[1802,5331,3588],{"className":5332,"style":4034},[1869,1873],[1802,5334,2725],{"className":5335},[2091],[1802,5337,3583],{"className":5338},[1869],[1802,5340,2757],{"className":5341},[2178],[1802,5343,5345,5348],{"style":5344},"top:-3.23em;",[1802,5346],{"className":5347,"style":5326},[1898],[1802,5349],{"className":5350,"style":5352},[5351],"frac-line","border-bottom-width:0.04em;",[1802,5354,5356,5359],{"style":5355},"top:-3.677em;",[1802,5357],{"className":5358,"style":5326},[1898],[1802,5360,5362,5365,5368,5371],{"className":5361},[1869],[1802,5363,3588],{"className":5364,"style":4034},[1869,1873],[1802,5366,2725],{"className":5367},[2091],[1802,5369,4850],{"className":5370},[1869,1873],[1802,5372,2757],{"className":5373},[2178],[1802,5375,1914],{"className":5376},[1913],[1802,5378,5380],{"className":5379},[1886],[1802,5381,5384],{"className":5382,"style":5383},[1890],"height:0.936em;",[1802,5385],{},[1802,5387],{"className":5388},[2178,5306],[1802,5390,2074],{"className":5391},[1869],[1798,5393,5394,5438,5439,5483,5484,5518,5519,5547],{},[1802,5395,5397,5417],{"className":5396},[1805],[1802,5398,5400],{"className":5399},[1809],[1811,5401,5402],{"xmlns":1813},[1815,5403,5404,5414],{},[1818,5405,5406,5408,5410,5412],{},[1824,5407,3588],{},[1832,5409,2725],{"stretchy":2046},[1824,5411,4850],{},[1832,5413,2757],{"stretchy":2046},[1849,5415,5416],{"encoding":1851},"\\gamma(h)",[1802,5418,5420],{"className":5419,"ariaHidden":1834},[1856],[1802,5421,5423,5426,5429,5432,5435],{"className":5422},[1860],[1802,5424],{"className":5425,"style":2087},[1864],[1802,5427,3588],{"className":5428,"style":4034},[1869,1873],[1802,5430,2725],{"className":5431},[2091],[1802,5433,4850],{"className":5434},[1869,1873],[1802,5436,2757],{"className":5437},[2178]," retains the unit squared; ",[1802,5440,5442,5462],{"className":5441},[1805],[1802,5443,5445],{"className":5444},[1809],[1811,5446,5447],{"xmlns":1813},[1815,5448,5449,5459],{},[1818,5450,5451,5453,5455,5457],{},[1824,5452,4845],{},[1832,5454,2725],{"stretchy":2046},[1824,5456,4850],{},[1832,5458,2757],{"stretchy":2046},[1849,5460,5461],{"encoding":1851},"\\rho(h)",[1802,5463,5465],{"className":5464,"ariaHidden":1834},[1856],[1802,5466,5468,5471,5474,5477,5480],{"className":5467},[1860],[1802,5469],{"className":5470,"style":2087},[1864],[1802,5472,4845],{"className":5473},[1869,1873],[1802,5475,2725],{"className":5476},[2091],[1802,5478,4850],{"className":5479},[1869,1873],[1802,5481,2757],{"className":5482},[2178]," is unit-free and lies between ",[1802,5485,5487,5503],{"className":5486},[1805],[1802,5488,5490],{"className":5489},[1809],[1811,5491,5492],{"xmlns":1813},[1815,5493,5494,5500],{},[1818,5495,5496,5498],{},[1832,5497,3631],{},[1828,5499,1830],{},[1849,5501,5502],{"encoding":1851},"-1",[1802,5504,5506],{"className":5505,"ariaHidden":1834},[1856],[1802,5507,5509,5512,5515],{"className":5508},[1860],[1802,5510],{"className":5511,"style":4979},[1864],[1802,5513,3631],{"className":5514},[1869],[1802,5516,1830],{"className":5517},[1869]," and ",[1802,5520,5522,5535],{"className":5521},[1805],[1802,5523,5525],{"className":5524},[1809],[1811,5526,5527],{"xmlns":1813},[1815,5528,5529,5533],{},[1818,5530,5531],{},[1828,5532,1830],{},[1849,5534,1830],{"encoding":1851},[1802,5536,5538],{"className":5537,"ariaHidden":1834},[1856],[1802,5539,5541,5544],{"className":5540},[1860],[1802,5542],{"className":5543,"style":4790},[1864],[1802,5545,1830],{"className":5546},[1869],". Neither measures nonlinear dependence completely.",[5549,5550,5552],"h3",{"id":5551},"worked-sample-calculation","Worked sample calculation",[1798,5554,5555],{},"Take one short 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(-1)+(-1)(1)+(1)(0)+(0)(2)}{6}=-0.167,",[1802,6339,6341,6408,6618],{"className":6340,"ariaHidden":1834},[1856],[1802,6342,6344,6347,6390,6393,6396,6399,6402,6405],{"className":6343},[1860],[1802,6345],{"className":6346,"style":2087},[1864],[1802,6348,6350],{"className":6349},[1869,5719],[1802,6351,6353,6382],{"className":6352},[1881,1882],[1802,6354,6356,6379],{"className":6355},[1886],[1802,6357,6359,6367],{"className":6358,"style":6005},[1890],[1802,6360,6361,6364],{"style":5732},[1802,6362],{"className":6363,"style":5326},[1898],[1802,6365,3588],{"className":6366,"style":4034},[1869,1873],[1802,6368,6370,6373],{"className":6369,"style":6018},[6017],[1802,6371],{"className":6372,"style":5326},[1898],[1802,6374,6375],{"style":6024},[3979,6376,6377],{"xmlns":3981,"width":6027,"height":6028,"viewBox":6029,"preserveAspectRatio":6030},[3986,6378],{"d":6033},[1802,6380,1914],{"className":6381},[1913],[1802,6383,6385],{"className":6384},[1886],[1802,6386,6388],{"className":6387,"style":6043},[1890],[1802,6389],{},[1802,6391,2725],{"className":6392},[2091],[1802,6394,1830],{"className":6395},[1869],[1802,6397,2757],{"className":6398},[2178],[1802,6400],{"className":6401,"style":2138},[1931],[1802,6403,2722],{"className":6404},[2142],[1802,6406],{"className":6407,"style":2138},[1931],[1802,6409,6411,6415,6609,6612,6615],{"className":6410},[1860],[1802,6412],{"className":6413,"style":6414},[1864],"height:2.113em;vertical-align:-0.686em;",[1802,6416,6418,6421,6606],{"className":6417},[1869],[1802,6419],{"className":6420},[2091,5306],[1802,6422,6424],{"className":6423},[5092],[1802,6425,6427,6598],{"className":6426},[1881,1882],[1802,6428,6430,6595],{"className":6429},[1886],[1802,6431,6433,6444,6452],{"className":6432,"style":5319},[1890],[1802,6434,6435,6438],{"style":5322},[1802,6436],{"className":6437,"style":5326},[1898],[1802,6439,6441],{"className":6440},[1869],[1802,6442,5602],{"className":6443},[1869],[1802,6445,6446,6449],{"style":5344},[1802,6447],{"className":6448,"style":5326},[1898],[1802,6450],{"className":6451,"style":5352},[5351],[1802,6453,6454,6457],{"style":5355},[1802,6455],{"className":6456,"style":5326},[1898],[1802,6458,6460,6463,6466,6469,6472,6475,6478,6481,6484,6487,6490,6493,6496,6499,6502,6505,6508,6511,6514,6517,6520,6523,6526,6529,6532,6535,6538,6541,6544,6547,6550,6553,6556,6559,6562,6565,6568,6571,6574,6577,6580,6583,6586,6589,6592],{"className":6459},[1869],[1802,6461,2725],{"className":6462},[2091],[1802,6464,3631],{"className":6465},[1869],[1802,6467,3688],{"className":6468},[1869],[1802,6470,2757],{"className":6471},[2178],[1802,6473,2725],{"className":6474},[2091],[1802,6476,3583],{"className":6477},[1869],[1802,6479,2757],{"className":6480},[2178],[1802,6482],{"className":6483,"style":4106},[1931],[1802,6485,2378],{"className":6486},[2429],[1802,6488],{"className":6489,"style":4106},[1931],[1802,6491,2725],{"className":6492},[2091],[1802,6494,3583],{"className":6495},[1869],[1802,6497,2757],{"className":6498},[2178],[1802,6500,2725],{"className":6501},[2091],[1802,6503,3631],{"className":6504},[1869],[1802,6506,1830],{"className":6507},[1869],[1802,6509,2757],{"className":6510},[2178],[1802,6512],{"className":6513,"style":4106},[1931],[1802,6515,2378],{"className":6516},[2429],[1802,6518],{"className":6519,"style":4106},[1931],[1802,6521,2725],{"className":6522},[2091],[1802,6524,3631],{"className":6525},[1869],[1802,6527,1830],{"className":6528},[1869],[1802,6530,2757],{"className":6531},[2178],[1802,6533,2725],{"className":6534},[2091],[1802,6536,1830],{"className":6537},[1869],[1802,6539,2757],{"className":6540},[2178],[1802,6542],{"className":6543,"style":4106},[1931],[1802,6545,2378],{"className":6546},[2429],[1802,6548],{"className":6549,"style":4106},[1931],[1802,6551,2725],{"className":6552},[2091],[1802,6554,1830],{"className":6555},[1869],[1802,6557,2757],{"className":6558},[2178],[1802,6560,2725],{"className":6561},[2091],[1802,6563,3583],{"className":6564},[1869],[1802,6566,2757],{"className":6567},[2178],[1802,6569],{"className":6570,"style":4106},[1931],[1802,6572,2378],{"className":6573},[2429],[1802,6575],{"className":6576,"style":4106},[1931],[1802,6578,2725],{"className":6579},[2091],[1802,6581,3583],{"className":6582},[1869],[1802,6584,2757],{"className":6585},[2178],[1802,6587,2725],{"className":6588},[2091],[1802,6590,3688],{"className":6591},[1869],[1802,6593,2757],{"className":6594},[2178],[1802,6596,1914],{"className":6597},[1913],[1802,6599,6601],{"className":6600},[1886],[1802,6602,6604],{"className":6603,"style":6189},[1890],[1802,6605],{},[1802,6607],{"className":6608},[2178,5306],[1802,6610],{"className":6611,"style":2138},[1931],[1802,6613,2722],{"className":6614},[2142],[1802,6616],{"className":6617,"style":2138},[1931],[1802,6619,6621,6624,6627,6630],{"className":6620},[1860],[1802,6622],{"className":6623,"style":6210},[1864],[1802,6625,3631],{"className":6626},[1869],[1802,6628,6332],{"className":6629},[1869],[1802,6631,1835],{"className":6632},[1927],[1798,6634,6635,6636,6753],{},"so ",[1802,6637,6639,6670],{"className":6638},[1805],[1802,6640,6642],{"className":6641},[1809],[1811,6643,6644],{"xmlns":1813},[1815,6645,6646,6667],{},[1818,6647,6648,6654,6656,6658,6660,6662,6664],{},[5610,6649,6650,6652],{"accent":1834},[1824,6651,4845],{},[1832,6653,5937],{"stretchy":1834},[1832,6655,2725],{"stretchy":2046},[1828,6657,1830],{},[1832,6659,2757],{"stretchy":2046},[1832,6661,2722],{},[1832,6663,3631],{},[1828,6665,6666],{},"0.10",[1849,6668,6669],{"encoding":1851},"\\widehat\\rho(1)=-0.10",[1802,6671,6673,6741],{"className":6672,"ariaHidden":1834},[1856],[1802,6674,6676,6679,6723,6726,6729,6732,6735,6738],{"className":6675},[1860],[1802,6677],{"className":6678,"style":2087},[1864],[1802,6680,6682],{"className":6681},[1869,5719],[1802,6683,6685,6715],{"className":6684},[1881,1882],[1802,6686,6688,6712],{"className":6687},[1886],[1802,6689,6691,6699],{"className":6690,"style":6005},[1890],[1802,6692,6693,6696],{"style":5732},[1802,6694],{"className":6695,"style":5326},[1898],[1802,6697,4845],{"className":6698},[1869,1873],[1802,6700,6703,6706],{"className":6701,"style":6702},[6017],"width:calc(100% - 0.1667em);margin-left:0.1667em;top:-3.4306em;",[1802,6704],{"className":6705,"style":5326},[1898],[1802,6707,6708],{"style":6024},[3979,6709,6710],{"xmlns":3981,"width":6027,"height":6028,"viewBox":6029,"preserveAspectRatio":6030},[3986,6711],{"d":6033},[1802,6713,1914],{"className":6714},[1913],[1802,6716,6718],{"className":6717},[1886],[1802,6719,6721],{"className":6720,"style":6043},[1890],[1802,6722],{},[1802,6724,2725],{"className":6725},[2091],[1802,6727,1830],{"className":6728},[1869],[1802,6730,2757],{"className":6731},[2178],[1802,6733],{"className":6734,"style":2138},[1931],[1802,6736,2722],{"className":6737},[2142],[1802,6739],{"className":6740,"style":2138},[1931],[1802,6742,6744,6747,6750],{"className":6743},[1860],[1802,6745],{"className":6746,"style":4979},[1864],[1802,6748,3631],{"className":6749},[1869],[1802,6751,6666],{"className":6752},[1869],". The sample is far too short for a strong conclusion; the exercise teaches what each ACF bar aggregates.",[6755,6756,6758,6759,6788,6789,6841,6842,6870,6871,2074],"tip",{"title":6757},"Estimator convention","Some software divides lag-",[1802,6760,6762,6775],{"className":6761},[1805],[1802,6763,6765],{"className":6764},[1809],[1811,6766,6767],{"xmlns":1813},[1815,6768,6769,6773],{},[1818,6770,6771],{},[1824,6772,4850],{},[1849,6774,4850],{"encoding":1851},[1802,6776,6778],{"className":6777,"ariaHidden":1834},[1856],[1802,6779,6781,6785],{"className":6780},[1860],[1802,6782],{"className":6783,"style":6784},[1864],"height:0.6944em;",[1802,6786,4850],{"className":6787},[1869,1873]," covariance by ",[1802,6790,6792,6810],{"className":6791},[1805],[1802,6793,6795],{"className":6794},[1809],[1811,6796,6797],{"xmlns":1813},[1815,6798,6799,6807],{},[1818,6800,6801,6803,6805],{},[1824,6802,1847],{},[1832,6804,3631],{},[1824,6806,4850],{},[1849,6808,6809],{"encoding":1851},"n-h",[1802,6811,6813,6832],{"className":6812,"ariaHidden":1834},[1856],[1802,6814,6816,6820,6823,6826,6829],{"className":6815},[1860],[1802,6817],{"className":6818,"style":6819},[1864],"height:0.6667em;vertical-align:-0.0833em;",[1802,6821,1847],{"className":6822},[1869,1873],[1802,6824],{"className":6825,"style":4106},[1931],[1802,6827,3631],{"className":6828},[2429],[1802,6830],{"className":6831,"style":4106},[1931],[1802,6833,6835,6838],{"className":6834},[1860],[1802,6836],{"className":6837,"style":6784},[1864],[1802,6839,4850],{"className":6840},[1869,1873]," instead of ",[1802,6843,6845,6858],{"className":6844},[1805],[1802,6846,6848],{"className":6847},[1809],[1811,6849,6850],{"xmlns":1813},[1815,6851,6852,6856],{},[1818,6853,6854],{},[1824,6855,1847],{},[1849,6857,1847],{"encoding":1851},[1802,6859,6861],{"className":6860,"ariaHidden":1834},[1856],[1802,6862,6864,6867],{"className":6863},[1860],[1802,6865],{"className":6866,"style":5634},[1864],[1802,6868,1847],{"className":6869},[1869,1873],". Both conventions occur. State the convention when reproducing an exact number; in large samples the practical difference shrinks for fixed ",[1802,6872,6874,6887],{"className":6873},[1805],[1802,6875,6877],{"className":6876},[1809],[1811,6878,6879],{"xmlns":1813},[1815,6880,6881,6885],{},[1818,6882,6883],{},[1824,6884,4850],{},[1849,6886,4850],{"encoding":1851},[1802,6888,6890],{"className":6889,"ariaHidden":1834},[1856],[1802,6891,6893,6896],{"className":6892},[1860],[1802,6894],{"className":6895,"style":6784},[1864],[1802,6897,4850],{"className":6898},[1869,1873],[1793,6900,6902],{"id":6901},"_4-three-kinds-of-stability","4. Three kinds of stability",[2326,6904,6905,6918],{},[2329,6906,6907],{},[2332,6908,6909,6912,6915],{},[2335,6910,6911],{},"Concept",[2335,6913,6914],{},"Requirement",[2335,6916,6917],{},"Why it matters",[2345,6919,6920,6931,6942],{},[2332,6921,6922,6925,6928],{},[2350,6923,6924],{},"strict stationarity",[2350,6926,6927],{},"every finite joint distribution is unchanged by a time shift",[2350,6929,6930],{},"complete distributional stability",[2332,6932,6933,6936,6939],{},[2350,6934,6935],{},"weak stationarity",[2350,6937,6938],{},"constant finite mean\u002Fvariance; covariance depends only on lag",[2350,6940,6941],{},"ARMA, linear prediction, spectra",[2332,6943,6944,6947,6950],{},[2350,6945,6946],{},"ergodicity",[2350,6948,6949],{},"time averages converge to population quantities",[2350,6951,6952],{},"learning from one long path",[1798,6954,6955],{},"Strict stationarity implies weak stationarity only when second moments exist. For a Gaussian process, weak stationarity is enough to determine shift-invariant finite-dimensional distributions. Stationarity alone does not guarantee that one path explores the population adequately; that is the role of ergodic conditions.",[5549,6957,6959],{"id":6958},"why-persistence-reduces-effective-information","Why persistence reduces effective information",[1798,6961,6962],{},"For a stationary mean-zero series,",[1802,6964,6966],{"className":6965},[2030],[1802,6967,6969,7073],{"className":6968},[1805],[1802,6970,6972],{"className":6971},[1809],[1811,6973,6974],{"xmlns":1813,"display":2039},[1815,6975,6976,7070],{},[1818,6977,6978,6980,6982,6984,6994,6996,6998,7004,7068],{},[1824,6979,3059],{"mathvariant":2073},[1832,6981,3062],{},[1832,6983,2725],{"stretchy":2046},[1821,6985,6986,6992],{},[5610,6987,6988,6990],{"accent":1834},[1824,6989,2052],{},[1832,6991,5616],{},[1824,6993,1847],{},[1832,6995,2757],{"stretchy":2046},[1832,6997,2722],{},[5092,6999,7000,7002],{},[1828,7001,1830],{},[1824,7003,1847],{},[1818,7005,7006,7008,7010,7012,7014,7016,7018,7020,7042,7058,7060,7062,7064,7066],{},[1832,7007,3083],{"fence":1834},[1824,7009,3588],{},[1832,7011,2725],{"stretchy":2046},[1828,7013,3583],{},[1832,7015,2757],{"stretchy":2046},[1832,7017,2378],{},[1828,7019,3688],{},[7021,7022,7023,7026,7034],"munderover",{},[1832,7024,7025],{},"∑",[1818,7027,7028,7030,7032],{},[1824,7029,4850],{},[1832,7031,2722],{},[1828,7033,1830],{},[1818,7035,7036,7038,7040],{},[1824,7037,1847],{},[1832,7039,3631],{},[1828,7041,1830],{},[1818,7043,7044,7046,7048,7050,7056],{},[1832,7045,2725],{"fence":1834},[1828,7047,1830],{},[1832,7049,3631],{},[5092,7051,7052,7054],{},[1824,7053,4850],{},[1824,7055,1847],{},[1832,7057,2757],{"fence":1834},[1824,7059,3588],{},[1832,7061,2725],{"stretchy":2046},[1824,7063,4850],{},[1832,7065,2757],{"stretchy":2046},[1832,7067,3119],{"fence":1834},[1824,7069,2074],{"mathvariant":2073},[1849,7071,7072],{"encoding":1851},"\\operatorname{Var}(\\bar X_n)\n=\\frac{1}{n}\\left[\\gamma(0)+2\\sum_{h=1}^{n-1}\\left(1-\\frac{h}{n}\\right)\\gamma(h)\\right].",[1802,7074,7076,7175],{"className":7075,"ariaHidden":1834},[1856],[1802,7077,7079,7083,7089,7092,7163,7166,7169,7172],{"className":7078},[1860],[1802,7080],{"className":7081,"style":7082},[1864],"height:1.0701em;vertical-align:-0.25em;",[1802,7084,7086],{"className":7085},[3199],[1802,7087,3059],{"className":7088},[1869,3203],[1802,7090,2725],{"className":7091},[2091],[1802,7093,7095,7129],{"className":7094},[1869],[1802,7096,7098],{"className":7097},[1869,5719],[1802,7099,7101],{"className":7100},[1881],[1802,7102,7104],{"className":7103},[1886],[1802,7105,7108,7116],{"className":7106,"style":7107},[1890],"height:0.8201em;",[1802,7109,7110,7113],{"style":5732},[1802,7111],{"className":7112,"style":5326},[1898],[1802,7114,2052],{"className":7115,"style":2098},[1869,1873],[1802,7117,7119,7122],{"style":7118},"top:-3.2523em;",[1802,7120],{"className":7121,"style":5326},[1898],[1802,7123,7126],{"className":7124,"style":7125},[5747],"left:-0.1667em;",[1802,7127,5616],{"className":7128},[1869],[1802,7130,7132],{"className":7131},[1877],[1802,7133,7135,7155],{"className":7134},[1881,1882],[1802,7136,7138,7152],{"className":7137},[1886],[1802,7139,7141],{"className":7140,"style":1964},[1890],[1802,7142,7143,7146],{"style":2114},[1802,7144],{"className":7145,"style":1899},[1898],[1802,7147,7149],{"className":7148},[1903,1904,1905,1906],[1802,7150,1847],{"className":7151},[1869,1873,1906],[1802,7153,1914],{"className":7154},[1913],[1802,7156,7158],{"className":7157},[1886],[1802,7159,7161],{"className":7160,"style":1921},[1890],[1802,7162],{},[1802,7164,2757],{"className":7165},[2178],[1802,7167],{"className":7168,"style":2138},[1931],[1802,7170,2722],{"className":7171},[2142],[1802,7173],{"className":7174,"style":2138},[1931],[1802,7176,7178,7182,7244,7247,7480,7483],{"className":7177},[1860],[1802,7179],{"className":7180,"style":7181},[1864],"height:3.1032em;vertical-align:-1.3021em;",[1802,7183,7185,7188,7241],{"className":7184},[1869],[1802,7186],{"className":7187},[2091,5306],[1802,7189,7191],{"className":7190},[5092],[1802,7192,7194,7233],{"className":7193},[1881,1882],[1802,7195,7197,7230],{"className":7196},[1886],[1802,7198,7200,7211,7219],{"className":7199,"style":6089},[1890],[1802,7201,7202,7205],{"style":5322},[1802,7203],{"className":7204,"style":5326},[1898],[1802,7206,7208],{"className":7207},[1869],[1802,7209,1847],{"className":7210},[1869,1873],[1802,7212,7213,7216],{"style":5344},[1802,7214],{"className":7215,"style":5326},[1898],[1802,7217],{"className":7218,"style":5352},[5351],[1802,7220,7221,7224],{"style":5355},[1802,7222],{"className":7223,"style":5326},[1898],[1802,7225,7227],{"className":7226},[1869],[1802,7228,1830],{"className":7229},[1869],[1802,7231,1914],{"className":7232},[1913],[1802,7234,7236],{"className":7235},[1886],[1802,7237,7239],{"className":7238,"style":6189},[1890],[1802,7240],{},[1802,7242],{"className":7243},[2178,5306],[1802,7245],{"className":7246,"style":1932},[1931],[1802,7248,7250,7257,7260,7263,7266,7269,7272,7275,7278,7281,7284,7366,7369,7459,7462,7465,7468,7471,7474],{"className":7249},[1936],[1802,7251,7253],{"className":7252,"style":3258},[2091,3257],[1802,7254,3083],{"className":7255},[3262,7256],"size4",[1802,7258,3588],{"className":7259,"style":4034},[1869,1873],[1802,7261,2725],{"className":7262},[2091],[1802,7264,3583],{"className":7265},[1869],[1802,7267,2757],{"className":7268},[2178],[1802,7270],{"className":7271,"style":4106},[1931],[1802,7273,2378],{"className":7274},[2429],[1802,7276],{"className":7277,"style":4106},[1931],[1802,7279,3688],{"className":7280},[1869],[1802,7282],{"className":7283,"style":1932},[1931],[1802,7285,7288],{"className":7286},[3199,7287],"op-limits",[1802,7289,7291,7357],{"className":7290},[1881,1882],[1802,7292,7294,7354],{"className":7293},[1886],[1802,7295,7298,7320,7333],{"className":7296,"style":7297},[1890],"height:1.8011em;",[1802,7299,7301,7305],{"style":7300},"top:-1.8479em;margin-left:0em;",[1802,7302],{"className":7303,"style":7304},[1898],"height:3.05em;",[1802,7306,7308],{"className":7307},[1903,1904,1905,1906],[1802,7309,7311,7314,7317],{"className":7310},[1869,1906],[1802,7312,4850],{"className":7313},[1869,1873,1906],[1802,7315,2722],{"className":7316},[2142,1906],[1802,7318,1830],{"className":7319},[1869,1906],[1802,7321,7323,7326],{"style":7322},"top:-3.05em;",[1802,7324],{"className":7325,"style":7304},[1898],[1802,7327,7328],{},[1802,7329,7025],{"className":7330},[3199,7331,7332],"op-symbol","large-op",[1802,7334,7336,7339],{"style":7335},"top:-4.3em;margin-left:0em;",[1802,7337],{"className":7338,"style":7304},[1898],[1802,7340,7342],{"className":7341},[1903,1904,1905,1906],[1802,7343,7345,7348,7351],{"className":7344},[1869,1906],[1802,7346,1847],{"className":7347},[1869,1873,1906],[1802,7349,3631],{"className":7350},[2429,1906],[1802,7352,1830],{"className":7353},[1869,1906],[1802,7355,1914],{"className":7356},[1913],[1802,7358,7360],{"className":7359},[1886],[1802,7361,7364],{"className":7362,"style":7363},[1890],"height:1.3021em;",[1802,7365],{},[1802,7367],{"className":7368,"style":1932},[1931],[1802,7370,7372,7378,7381,7384,7387,7390,7453],{"className":7371},[1936],[1802,7373,7375],{"className":7374,"style":3258},[2091,3257],[1802,7376,2725],{"className":7377},[3262,1905],[1802,7379,1830],{"className":7380},[1869],[1802,7382],{"className":7383,"style":4106},[1931],[1802,7385,3631],{"className":7386},[2429],[1802,7388],{"className":7389,"style":4106},[1931],[1802,7391,7393,7396,7450],{"className":7392},[1869],[1802,7394],{"className":7395},[2091,5306],[1802,7397,7399],{"className":7398},[5092],[1802,7400,7402,7442],{"className":7401},[1881,1882],[1802,7403,7405,7439],{"className":7404},[1886],[1802,7406,7409,7420,7428],{"className":7407,"style":7408},[1890],"height:1.3714em;",[1802,7410,7411,7414],{"style":5322},[1802,7412],{"className":7413,"style":5326},[1898],[1802,7415,7417],{"className":7416},[1869],[1802,7418,1847],{"className":7419},[1869,1873],[1802,7421,7422,7425],{"style":5344},[1802,7423],{"className":7424,"style":5326},[1898],[1802,7426],{"className":7427,"style":5352},[5351],[1802,7429,7430,7433],{"style":5355},[1802,7431],{"className":7432,"style":5326},[1898],[1802,7434,7436],{"className":7435},[1869],[1802,7437,4850],{"className":7438},[1869,1873],[1802,7440,1914],{"className":7441},[1913],[1802,7443,7445],{"className":7444},[1886],[1802,7446,7448],{"className":7447,"style":6189},[1890],[1802,7449],{},[1802,7451],{"className":7452},[2178,5306],[1802,7454,7456],{"className":7455,"style":3258},[2178,3257],[1802,7457,2757],{"className":7458},[3262,1905],[1802,7460],{"className":7461,"style":1932},[1931],[1802,7463,3588],{"className":7464,"style":4034},[1869,1873],[1802,7466,2725],{"className":7467},[2091],[1802,7469,4850],{"className":7470},[1869,1873],[1802,7472,2757],{"className":7473},[2178],[1802,7475,7477],{"className":7476,"style":3258},[2178,3257],[1802,7478,3119],{"className":7479},[3262,7256],[1802,7481],{"className":7482,"style":1932},[1931],[1802,7484,2074],{"className":7485},[1869],[1798,7487,7488,7489,7544],{},"Positive autocovariances make the sample mean noisier than the iid formula ",[1802,7490,7492,7517],{"className":7491},[1805],[1802,7493,7495],{"className":7494},[1809],[1811,7496,7497],{"xmlns":1813},[1815,7498,7499,7514],{},[1818,7500,7501,7503,7505,7507,7509,7512],{},[1824,7502,3588],{},[1832,7504,2725],{"stretchy":2046},[1828,7506,3583],{},[1832,7508,2757],{"stretchy":2046},[1824,7510,7511],{"mathvariant":2073},"\u002F",[1824,7513,1847],{},[1849,7515,7516],{"encoding":1851},"\\gamma(0)\u002Fn",[1802,7518,7520],{"className":7519,"ariaHidden":1834},[1856],[1802,7521,7523,7526,7529,7532,7535,7538,7541],{"className":7522},[1860],[1802,7524],{"className":7525,"style":2087},[1864],[1802,7527,3588],{"className":7528,"style":4034},[1869,1873],[1802,7530,2725],{"className":7531},[2091],[1802,7533,3583],{"className":7534},[1869],[1802,7536,2757],{"className":7537},[2178],[1802,7539,7511],{"className":7540},[1869],[1802,7542,1847],{"className":7543},[1869,1873],". One hundred highly persistent months need not contain one hundred months' worth of independent information.",[1793,7546,7548],{"id":7547},"_5-white-noise-is-not-one-assumption","5. White noise is not one assumption",[2326,7550,7551,7561],{},[2329,7552,7553],{},[2332,7554,7555,7558],{},[2335,7556,7557],{},"Label",[2335,7559,7560],{},"Required claim",[2345,7562,7563,7571,7579],{},[2332,7564,7565,7568],{},[2350,7566,7567],{},"white noise",[2350,7569,7570],{},"zero mean, constant variance, zero autocovariance at non-zero lags",[2332,7572,7573,7576],{},[2350,7574,7575],{},"independent white noise",[2350,7577,7578],{},"white noise observations are also independent",[2332,7580,7581,7584],{},[2350,7582,7583],{},"Gaussian white noise",[2350,7585,7586,7587],{},"independent observations follow ",[1802,7588,7590,7620],{"className":7589},[1805],[1802,7591,7593],{"className":7592},[1809],[1811,7594,7595],{"xmlns":1813},[1815,7596,7597,7617],{},[1818,7598,7599,7602,7604,7606,7608,7615],{},[1824,7600,7601],{},"N",[1832,7603,2725],{"stretchy":2046},[1828,7605,3583],{},[1832,7607,1835],{"separator":1834},[2753,7609,7610,7613],{},[1824,7611,7612],{},"σ",[1828,7614,3688],{},[1832,7616,2757],{"stretchy":2046},[1849,7618,7619],{"encoding":1851},"N(0,\\sigma^2)",[1802,7621,7623],{"className":7622,"ariaHidden":1834},[1856],[1802,7624,7626,7630,7634,7637,7640,7643,7646,7678],{"className":7625},[1860],[1802,7627],{"className":7628,"style":7629},[1864],"height:1.0641em;vertical-align:-0.25em;",[1802,7631,7601],{"className":7632,"style":7633},[1869,1873],"margin-right:0.109em;",[1802,7635,2725],{"className":7636},[2091],[1802,7638,3583],{"className":7639},[1869],[1802,7641,1835],{"className":7642},[1927],[1802,7644],{"className":7645,"style":1932},[1931],[1802,7647,7649,7653],{"className":7648},[1869],[1802,7650,7612],{"className":7651,"style":7652},[1869,1873],"margin-right:0.0359em;",[1802,7654,7656],{"className":7655},[1877],[1802,7657,7659],{"className":7658},[1881],[1802,7660,7662],{"className":7661},[1886],[1802,7663,7666],{"className":7664,"style":7665},[1890],"height:0.8141em;",[1802,7667,7669,7672],{"style":7668},"top:-3.063em;margin-right:0.05em;",[1802,7670],{"className":7671,"style":1899},[1898],[1802,7673,7675],{"className":7674},[1903,1904,1905,1906],[1802,7676,3688],{"className":7677},[1869,1906],[1802,7679,2757],{"className":7680},[2178],[1798,7682,7683],{},"Uncorrelated does not imply independent outside special families such as jointly Gaussian variables. A squared-noise series can be predictable in variance even when the level has zero autocorrelation.",[1793,7685,7687],{"id":7686},"_6-the-ar1-as-a-complete-dependence-example","6. The AR(1) as a complete dependence example",[1798,7689,7690],{},"Let",[1802,7692,7694],{"className":7693},[2030],[1802,7695,7697,7776],{"className":7696},[1805],[1802,7698,7700],{"className":7699},[1809],[1811,7701,7702],{"xmlns":1813,"display":2039},[1815,7703,7704,7773],{},[1818,7705,7706,7712,7714,7716,7728,7730,7737,7739,7741,7747,7750,7753,7755,7757,7759,7761,7769,7771],{},[1821,7707,7708,7710],{},[1824,7709,2052],{},[1824,7711,2055],{},[1832,7713,2722],{},[1824,7715,4932],{},[1821,7717,7718,7720],{},[1824,7719,2052],{},[1818,7721,7722,7724,7726],{},[1824,7723,2055],{},[1832,7725,3631],{},[1828,7727,1830],{},[1832,7729,2378],{},[1821,7731,7732,7735],{},[1824,7733,7734],{},"ε",[1824,7736,2055],{},[1832,7738,1835],{"separator":1834},[1931,7740],{"width":3056},[1821,7742,7743,7745],{},[1824,7744,7734],{},[1824,7746,2055],{},[1832,7748,7749],{},"∼",[1824,7751,7752],{},"W",[1824,7754,7601],{},[1832,7756,2725],{"stretchy":2046},[1828,7758,3583],{},[1832,7760,1835],{"separator":1834},[3077,7762,7763,7765,7767],{},[1824,7764,7612],{},[1824,7766,7734],{},[1828,7768,3688],{},[1832,7770,2757],{"stretchy":2046},[1824,7772,2074],{"mathvariant":2073},[1849,7774,7775],{"encoding":1851},"X_t=\\phi X_{t-1}+\\varepsilon_t,\n\\qquad \\varepsilon_t\\sim 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unit shock has about half its effect after ",[1802,9189,9191,9232],{"className":9190},[1805],[1802,9192,9194],{"className":9193},[1809],[1811,9195,9196],{"xmlns":1813},[1815,9197,9198,9229],{},[1818,9199,9200,9203,9205,9207,9210,9212,9214,9216,9218,9220,9222,9224,9226],{},[1824,9201,9202],{},"log",[1832,9204,3062],{},[1832,9206,2725],{"stretchy":2046},[1828,9208,9209],{},"0.5",[1832,9211,2757],{"stretchy":2046},[1824,9213,7511],{"mathvariant":2073},[1824,9215,9202],{},[1832,9217,3062],{},[1832,9219,2725],{"stretchy":2046},[1828,9221,4966],{},[1832,9223,2757],{"stretchy":2046},[1832,9225,2722],{},[1828,9227,9228],{},"1.94",[1849,9230,9231],{"encoding":1851},"\\log(0.5)\u002F\\log(0.7)=1.94",[1802,9233,9235,9286],{"className":9234,"ariaHidden":1834},[1856],[1802,9236,9238,9241,9248,9251,9254,9257,9260,9263,9268,9271,9274,9277,9280,9283],{"className":9237},[1860],[1802,9239],{"className":9240,"style":2087},[1864],[1802,9242,9244,9245],{"className":9243},[3199],"lo",[1802,9246,9247],{"style":3269},"g",[1802,9249,2725],{"className":9250},[2091],[1802,9252,9209],{"className":9253},[1869],[1802,9255,2757],{"className":9256},[2178],[1802,9258,7511],{"className":9259},[1869],[1802,9261],{"className":9262,"style":1932},[1931],[1802,9264,9244,9266],{"className":9265},[3199],[1802,9267,9247],{"style":3269},[1802,9269,2725],{"className":9270},[2091],[1802,9272,4966],{"className":9273},[1869],[1802,9275,2757],{"className":9276},[2178],[1802,9278],{"className":9279,"style":2138},[1931],[1802,9281,2722],{"className":9282},[2142],[1802,9284],{"className":9285,"style":2138},[1931],[1802,9287,9289,9292],{"className":9288},[1860],[1802,9290],{"className":9291,"style":4790},[1864],[1802,9293,9228],{"className":9294},[1869]," periods.",[1798,9297,8094,9298,9349,9350,9401],{},[1802,9299,9301,9319],{"className":9300},[1805],[1802,9302,9304],{"className":9303},[1809],[1811,9305,9306],{"xmlns":1813},[1815,9307,9308,9316],{},[1818,9309,9310,9312,9314],{},[1824,9311,4932],{},[1832,9313,8115],{},[1828,9315,3583],{},[1849,9317,9318],{"encoding":1851},"\\phi\u003C0",[1802,9320,9322,9340],{"className":9321,"ariaHidden":1834},[1856],[1802,9323,9325,9328,9331,9334,9337],{"className":9324},[1860],[1802,9326],{"className":9327,"style":4945},[1864],[1802,9329,4932],{"className":9330},[1869,1873],[1802,9332],{"className":9333,"style":2138},[1931],[1802,9335,8115],{"className":9336},[2142],[1802,9338],{"className":9339,"style":2138},[1931],[1802,9341,9343,9346],{"className":9342},[1860],[1802,9344],{"className":9345,"style":4790},[1864],[1802,9347,3583],{"className":9348},[1869],", signs alternate; if ",[1802,9351,9353,9371],{"className":9352},[1805],[1802,9354,9356],{"className":9355},[1809],[1811,9357,9358],{"xmlns":1813},[1815,9359,9360,9368],{},[1818,9361,9362,9364,9366],{},[1824,9363,4932],{},[1832,9365,2722],{},[1828,9367,1830],{},[1849,9369,9370],{"encoding":1851},"\\phi=1",[1802,9372,9374,9392],{"className":9373,"ariaHidden":1834},[1856],[1802,9375,9377,9380,9383,9386,9389],{"className":9376},[1860],[1802,9378],{"className":9379,"style":4945},[1864],[1802,9381,4932],{"className":9382},[1869,1873],[1802,9384],{"className":9385,"style":2138},[1931],[1802,9387,2722],{"className":9388},[2142],[1802,9390],{"className":9391,"style":2138},[1931],[1802,9393,9395,9398],{"className":9394},[1860],[1802,9396],{"className":9397,"style":4790},[1864],[1802,9399,1830],{"className":9400},[1869],", shocks do not decay and the process becomes a random walk.",[1793,9403,9405],{"id":9404},"_7-linear-processes-and-wolds-result","7. Linear processes and Wold's result",[1798,9407,9408],{},"A causal linear process has the form",[1802,9410,9412],{"className":9411},[2030],[1802,9413,9415,9474],{"className":9414},[1805],[1802,9416,9418],{"className":9417},[1809],[1811,9419,9420],{"xmlns":1813,"display":2039},[1815,9421,9422,9471],{},[1818,9423,9424,9430,9432,9434,9436,9450,9457,9469],{},[1821,9425,9426,9428],{},[1824,9427,2052],{},[1824,9429,2055],{},[1832,9431,2722],{},[1824,9433,3051],{},[1832,9435,2378],{},[7021,9437,9438,9440,9448],{},[1832,9439,7025],{},[1818,9441,9442,9444,9446],{},[1824,9443,3114],{},[1832,9445,2722],{},[1828,9447,3583],{},[1824,9449,8195],{"mathvariant":2073},[1821,9451,9452,9455],{},[1824,9453,9454],{},"ψ",[1824,9456,3114],{},[1821,9458,9459,9461],{},[1824,9460,7734],{},[1818,9462,9463,9465,9467],{},[1824,9464,2055],{},[1832,9466,3631],{},[1824,9468,3114],{},[1832,9470,1835],{"separator":1834},[1849,9472,9473],{"encoding":1851},"X_t=\\mu+\\sum_{j=0}^{\\infty}\\psi_j\\varepsilon_{t-j},",[1802,9475,9477,9532,9551],{"className":9476,"ariaHidden":1834},[1856],[1802,9478,9480,9483,9523,9526,9529],{"className":9479},[1860],[1802,9481],{"className":9482,"style":4535},[1864],[1802,9484,9486,9489],{"className":9485},[1869],[1802,9487,2052],{"className":9488,"style":2098},[1869,1873],[1802,9490,9492],{"className":9491},[1877],[1802,9493,9495,9515],{"className":9494},[1881,1882],[1802,9496,9498,9512],{"className":9497},[1886],[1802,9499,9501],{"className":9500,"style":2111},[1890],[1802,9502,9503,9506],{"style":2114},[1802,9504],{"className":9505,"style":1899},[1898],[1802,9507,9509],{"className":9508},[1903,1904,1905,1906],[1802,9510,2055],{"className":9511},[1869,1873,1906],[1802,9513,1914],{"className":9514},[1913],[1802,9516,9518],{"className":9517},[1886],[1802,9519,9521],{"className":9520,"style":1921},[1890],[1802,9522],{},[1802,9524],{"className":9525,"style":2138},[1931],[1802,9527,2722],{"className":9528},[2142],[1802,9530],{"className":9531,"style":2138},[1931],[1802,9533,9535,9539,9542,9545,9548],{"className":9534},[1860],[1802,9536],{"className":9537,"style":9538},[1864],"height:0.7778em;vertical-align:-0.1944em;",[1802,9540,3051],{"className":9541},[1869,1873],[1802,9543],{"className":9544,"style":4106},[1931],[1802,9546,2378],{"className":9547},[2429],[1802,9549],{"className":9550,"style":4106},[1931],[1802,9552,9554,9557,9624,9627,9668,9717],{"className":9553},[1860],[1802,9555],{"className":9556,"style":8283},[1864],[1802,9558,9560],{"className":9559},[3199,7287],[1802,9561,9563,9616],{"className":9562},[1881,1882],[1802,9564,9566,9613],{"className":9565},[1886],[1802,9567,9569,9589,9599],{"className":9568,"style":8296},[1890],[1802,9570,9571,9574],{"style":8299},[1802,9572],{"className":9573,"style":7304},[1898],[1802,9575,9577],{"className":9576},[1903,1904,1905,1906],[1802,9578,9580,9583,9586],{"className":9579},[1869,1906],[1802,9581,3114],{"className":9582,"style":3418},[1869,1873,1906],[1802,9584,2722],{"className":9585},[2142,1906],[1802,9587,3583],{"className":9588},[1869,1906],[1802,9590,9591,9594],{"style":7322},[1802,9592],{"className":9593,"style":7304},[1898],[1802,9595,9596],{},[1802,9597,7025],{"className":9598},[3199,7331,7332],[1802,9600,9601,9604],{"style":7335},[1802,9602],{"className":9603,"style":7304},[1898],[1802,9605,9607],{"className":9606},[1903,1904,1905,1906],[1802,9608,9610],{"className":9609},[1869,1906],[1802,9611,8195],{"className":9612},[1869,1906],[1802,9614,1914],{"className":9615},[1913],[1802,9617,9619],{"className":9618},[1886],[1802,9620,9622],{"className":9621,"style":8351},[1890],[1802,9623],{},[1802,9625],{"className":9626,"style":1932},[1931],[1802,9628,9630,9633],{"className":9629},[1869],[1802,9631,9454],{"className":9632,"style":7652},[1869,1873],[1802,9634,9636],{"className":9635},[1877],[1802,9637,9639,9660],{"className":9638},[1881,1882],[1802,9640,9642,9657],{"className":9641},[1886],[1802,9643,9645],{"className":9644,"style":8405},[1890],[1802,9646,9648,9651],{"style":9647},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1802,9649],{"className":9650,"style":1899},[1898],[1802,9652,9654],{"className":9653},[1903,1904,1905,1906],[1802,9655,3114],{"className":9656,"style":3418},[1869,1873,1906],[1802,9658,1914],{"className":9659},[1913],[1802,9661,9663],{"className":9662},[1886],[1802,9664,9666],{"className":9665,"style":8435},[1890],[1802,9667],{},[1802,9669,9671,9674],{"className":9670},[1869],[1802,9672,7734],{"className":9673},[1869,1873],[1802,9675,9677],{"className":9676},[1877],[1802,9678,9680,9709],{"className":9679},[1881,1882],[1802,9681,9683,9706],{"className":9682},[1886],[1802,9684,9686],{"className":9685,"style":8405},[1890],[1802,9687,9688,9691],{"style":1894},[1802,9689],{"className":9690,"style":1899},[1898],[1802,9692,9694],{"className":9693},[1903,1904,1905,1906],[1802,9695,9697,9700,9703],{"className":9696},[1869,1906],[1802,9698,2055],{"className":9699},[1869,1873,1906],[1802,9701,3631],{"className":9702},[2429,1906],[1802,9704,3114],{"className":9705,"style":3418},[1869,1873,1906],[1802,9707,1914],{"className":9708},[1913],[1802,9710,9712],{"className":9711},[1886],[1802,9713,9715],{"className":9714,"style":8435},[1890],[1802,9716],{},[1802,9718,1835],{"className":9719},[1927],[1798,9721,9722],{},"with coefficients that decay sufficiently for the sum to exist. Its autocovariance is",[1802,9724,9726],{"className":9725},[2030],[1802,9727,9729,9797],{"className":9728},[1805],[1802,9730,9732],{"className":9731},[1809],[1811,9733,9734],{"xmlns":1813,"display":2039},[1815,9735,9736,9794],{},[1818,9737,9738,9740,9742,9744,9746,9748,9756,9770,9776,9792],{},[1824,9739,3588],{},[1832,9741,2725],{"stretchy":2046},[1824,9743,4850],{},[1832,9745,2757],{"stretchy":2046},[1832,9747,2722],{},[3077,9749,9750,9752,9754],{},[1824,9751,7612],{},[1824,9753,7734],{},[1828,9755,3688],{},[7021,9757,9758,9760,9768],{},[1832,9759,7025],{},[1818,9761,9762,9764,9766],{},[1824,9763,3114],{},[1832,9765,2722],{},[1828,9767,3583],{},[1824,9769,8195],{"mathvariant":2073},[1821,9771,9772,9774],{},[1824,9773,9454],{},[1824,9775,3114],{},[1821,9777,9778,9780],{},[1824,9779,9454],{},[1818,9781,9782,9784,9786,9788,9790],{},[1824,9783,3114],{},[1832,9785,2378],{},[1824,9787,4842],{"mathvariant":2073},[1824,9789,4850],{},[1824,9791,4842],{"mathvariant":2073},[1824,9793,2074],{"mathvariant":2073},[1849,9795,9796],{"encoding":1851},"\\gamma(h)=\\sigma_\\varepsilon^2\\sum_{j=0}^{\\infty}\\psi_j\\psi_{j+|h|}.",[1802,9798,9800,9827],{"className":9799,"ariaHidden":1834},[1856],[1802,9801,9803,9806,9809,9812,9815,9818,9821,9824],{"className":9802},[1860],[1802,9804],{"className":9805,"style":2087},[1864],[1802,9807,3588],{"className":9808,"style":4034},[1869,1873],[1802,9810,2725],{"className":9811},[2091],[1802,9813,4850],{"className":9814},[1869,1873],[1802,9816,2757],{"className":9817},[2178],[1802,9819],{"className":9820,"style":2138},[1931],[1802,9822,2722],{"className":9823},[2142],[1802,9825],{"className":9826,"style":2138},[1931],[1802,9828,9830,9833,9884,9887,9954,9957,9997,10055],{"className":9829},[1860],[1802,9831],{"className":9832,"style":8283},[1864],[1802,9834,9836,9839],{"className":9835},[1869],[1802,9837,7612],{"className":9838,"style":7652},[1869,1873],[1802,9840,9842],{"className":9841},[1877],[1802,9843,9845,9876],{"className":9844},[1881,1882],[1802,9846,9848,9873],{"className":9847},[1886],[1802,9849,9851,9862],{"className":9850,"style":8050},[1890],[1802,9852,9853,9856],{"style":8053},[1802,9854],{"className":9855,"style":1899},[1898],[1802,9857,9859],{"className":9858},[1903,1904,1905,1906],[1802,9860,7734],{"className":9861},[1869,1873,1906],[1802,9863,9864,9867],{"style":3006},[1802,9865],{"className":9866,"style":1899},[1898],[1802,9868,9870],{"className":9869},[1903,1904,1905,1906],[1802,9871,3688],{"className":9872},[1869,1906],[1802,9874,1914],{"className":9875},[1913],[1802,9877,9879],{"className":9878},[1886],[1802,9880,9882],{"className":9881,"style":8083},[1890],[1802,9883],{},[1802,9885],{"className":9886,"style":1932},[1931],[1802,9888,9890],{"className":9889},[3199,7287],[1802,9891,9893,9946],{"className":9892},[1881,1882],[1802,9894,9896,9943],{"className":9895},[1886],[1802,9897,9899,9919,9929],{"className":9898,"style":8296},[1890],[1802,9900,9901,9904],{"style":8299},[1802,9902],{"className":9903,"style":7304},[1898],[1802,9905,9907],{"className":9906},[1903,1904,1905,1906],[1802,9908,9910,9913,9916],{"className":9909},[1869,1906],[1802,9911,3114],{"className":9912,"style":3418},[1869,1873,1906],[1802,9914,2722],{"className":9915},[2142,1906],[1802,9917,3583],{"className":9918},[1869,1906],[1802,9920,9921,9924],{"style":7322},[1802,9922],{"className":9923,"style":7304},[1898],[1802,9925,9926],{},[1802,9927,7025],{"className":9928},[3199,7331,7332],[1802,9930,9931,9934],{"style":7335},[1802,9932],{"className":9933,"style":7304},[1898],[1802,9935,9937],{"className":9936},[1903,1904,1905,1906],[1802,9938,9940],{"className":9939},[1869,1906],[1802,9941,8195],{"className":9942},[1869,1906],[1802,9944,1914],{"className":9945},[1913],[1802,9947,9949],{"className":9948},[1886],[1802,9950,9952],{"className":9951,"style":8351},[1890],[1802,9953],{},[1802,9955],{"className":9956,"style":1932},[1931],[1802,9958,9960,9963],{"className":9959},[1869],[1802,9961,9454],{"className":9962,"style":7652},[1869,1873],[1802,9964,9966],{"className":9965},[1877],[1802,9967,9969,9989],{"className":9968},[1881,1882],[1802,9970,9972,9986],{"className":9971},[1886],[1802,9973,9975],{"className":9974,"style":8405},[1890],[1802,9976,9977,9980],{"style":9647},[1802,9978],{"className":9979,"style":1899},[1898],[1802,9981,9983],{"className":9982},[1903,1904,1905,1906],[1802,9984,3114],{"className":9985,"style":3418},[1869,1873,1906],[1802,9987,1914],{"className":9988},[1913],[1802,9990,9992],{"className":9991},[1886],[1802,9993,9995],{"className":9994,"style":8435},[1890],[1802,9996],{},[1802,9998,10000,10003],{"className":9999},[1869],[1802,10001,9454],{"className":10002,"style":7652},[1869,1873],[1802,10004,10006],{"className":10005},[1877],[1802,10007,10009,10046],{"className":10008},[1881,1882],[1802,10010,10012,10043],{"className":10011},[1886],[1802,10013,10016],{"className":10014,"style":10015},[1890],"height:0.3448em;",[1802,10017,10019,10022],{"style":10018},"top:-2.5198em;margin-left:-0.0359em;margin-right:0.05em;",[1802,10020],{"className":10021,"style":1899},[1898],[1802,10023,10025],{"className":10024},[1903,1904,1905,1906],[1802,10026,10028,10031,10034,10037,10040],{"className":10027},[1869,1906],[1802,10029,3114],{"className":10030,"style":3418},[1869,1873,1906],[1802,10032,2378],{"className":10033},[2429,1906],[1802,10035,4842],{"className":10036},[1869,1906],[1802,10038,4850],{"className":10039},[1869,1873,1906],[1802,10041,4842],{"className":10042},[1869,1906],[1802,10044,1914],{"className":10045},[1913],[1802,10047,10049],{"className":10048},[1886],[1802,10050,10053],{"className":10051,"style":10052},[1890],"height:0.3552em;",[1802,10054],{},[1802,10056,2074],{"className":10057},[1869],[1798,10059,10060,10061,10065],{},"Wold's decomposition says that every covariance-stationary, purely non-deterministic process can be represented as an infinite moving average of uncorrelated innovations. It does ",[10062,10063,10064],"strong",{},"not"," say that a short finite ARMA model is true, that innovations are independent, or that the coefficients are easy to estimate.",[1793,10067,10069],{"id":10068},"_8-diagnostic-distinctions","8. Diagnostic distinctions",[2326,10071,10072,10085],{},[2329,10073,10074],{},[2332,10075,10076,10079,10082],{},[2335,10077,10078],{},"Observation",[2335,10080,10081],{},"Supported interpretation",[2335,10083,10084],{},"Unsupported leap",[2345,10086,10087,10098,10109,10120],{},[2332,10088,10089,10092,10095],{},[2350,10090,10091],{},"ACF decays slowly",[2350,10093,10094],{},"strong linear persistence or unremoved low-frequency structure",[2350,10096,10097],{},"“the process has a unit root”",[2332,10099,10100,10103,10106],{},[2350,10101,10102],{},"residual ACF is near zero",[2350,10104,10105],{},"little remaining linear serial correlation at inspected lags",[2350,10107,10108],{},"“residuals are iid Gaussian”",[2332,10110,10111,10114,10117],{},[2350,10112,10113],{},"sample mean stabilises",[2350,10115,10116],{},"evidence consistent with mean ergodicity",[2350,10118,10119],{},"proof of stationarity",[2332,10121,10122,10125,10128],{},[2350,10123,10124],{},"variance changes after an intervention",[2350,10126,10127],{},"possible break or volatility shift",[2350,10129,10130],{},"automatic need for first differencing",[1793,10132,10134],{"id":10133},"practice","Practice",[10136,10137,10138,10554,10557,10700,10703],"ol",{},[1999,10139,10140,10141,10363,10364,5518,10454,2074],{},"For ",[1802,10142,10144,10188],{"className":10143},[1805],[1802,10145,10147],{"className":10146},[1809],[1811,10148,10149],{"xmlns":1813},[1815,10150,10151,10185],{},[1818,10152,10153,10159,10161,10163,10165,10177,10179],{},[1821,10154,10155,10157],{},[1824,10156,2052],{},[1824,10158,2055],{},[1832,10160,2722],{},[1832,10162,3631],{},[1828,10164,9209],{},[1821,10166,10167,10169],{},[1824,10168,2052],{},[1818,10170,10171,10173,10175],{},[1824,10172,2055],{},[1832,10174,3631],{},[1828,10176,1830],{},[1832,10178,2378],{},[1821,10180,10181,10183],{},[1824,10182,7734],{},[1824,10184,2055],{},[1849,10186,10187],{"encoding":1851},"X_t=-0.5X_{t-1}+\\varepsilon_t",[1802,10189,10191,10246,10316],{"className":10190,"ariaHidden":1834},[1856],[1802,10192,10194,10197,10237,10240,10243],{"className":10193},[1860],[1802,10195],{"className":10196,"style":4535},[1864],[1802,10198,10200,10203],{"className":10199},[1869],[1802,10201,2052],{"className":10202,"style":2098},[1869,1873],[1802,10204,10206],{"className":10205},[1877],[1802,10207,10209,10229],{"className":10208},[1881,1882],[1802,10210,10212,10226],{"className":10211},[1886],[1802,10213,10215],{"className":10214,"style":2111},[1890],[1802,10216,10217,10220],{"style":2114},[1802,10218],{"className":10219,"style":1899},[1898],[1802,10221,10223],{"className":10222},[1903,1904,1905,1906],[1802,10224,2055],{"className":10225},[1869,1873,1906],[1802,10227,1914],{"className":10228},[1913],[1802,10230,10232],{"className":10231},[1886],[1802,10233,10235],{"className":10234,"style":1921},[1890],[1802,10236],{},[1802,10238],{"className":10239,"style":2138},[1931],[1802,10241,2722],{"className":10242},[2142],[1802,10244],{"className":10245,"style":2138},[1931],[1802,10247,10249,10252,10255,10258,10307,10310,10313],{"className":10248},[1860],[1802,10250],{"className":10251,"style":2393},[1864],[1802,10253,3631],{"className":10254},[1869],[1802,10256,9209],{"className":10257},[1869],[1802,10259,10261,10264],{"className":10260},[1869],[1802,10262,2052],{"className":10263,"style":2098},[1869,1873],[1802,10265,10267],{"className":10266},[1877],[1802,10268,10270,10299],{"className":10269},[1881,1882],[1802,10271,10273,10296],{"className":10272},[1886],[1802,10274,10276],{"className":10275,"style":1891},[1890],[1802,10277,10278,10281],{"style":2114},[1802,10279],{"className":10280,"style":1899},[1898],[1802,10282,10284],{"className":10283},[1903,1904,1905,1906],[1802,10285,10287,10290,10293],{"className":10286},[1869,1906],[1802,10288,2055],{"className":10289},[1869,1873,1906],[1802,10291,3631],{"className":10292},[2429,1906],[1802,10294,1830],{"className":10295},[1869,1906],[1802,10297,1914],{"className":10298},[1913],[1802,10300,10302],{"className":10301},[1886],[1802,10303,10305],{"className":10304,"style":2442},[1890],[1802,10306],{},[1802,10308],{"className":10309,"style":4106},[1931],[1802,10311,2378],{"className":10312},[2429],[1802,10314],{"className":10315,"style":4106},[1931],[1802,10317,10319,10323],{"className":10318},[1860],[1802,10320],{"className":10321,"style":10322},[1864],"height:0.5806em;vertical-align:-0.15em;",[1802,10324,10326,10329],{"className":10325},[1869],[1802,10327,7734],{"className":10328},[1869,1873],[1802,10330,10332],{"className":10331},[1877],[1802,10333,10335,10355],{"className":10334},[1881,1882],[1802,10336,10338,10352],{"className":10337},[1886],[1802,10339,10341],{"className":10340,"style":2111},[1890],[1802,10342,10343,10346],{"style":1894},[1802,10344],{"className":10345,"style":1899},[1898],[1802,10347,10349],{"className":10348},[1903,1904,1905,1906],[1802,10350,2055],{"className":10351},[1869,1873,1906],[1802,10353,1914],{"className":10354},[1913],[1802,10356,10358],{"className":10357},[1886],[1802,10359,10361],{"className":10360,"style":1921},[1890],[1802,10362],{}," with innovation variance 3, calculate ",[1802,10365,10367,10393],{"className":10366},[1805],[1802,10368,10370],{"className":10369},[1809],[1811,10371,10372],{"xmlns":1813},[1815,10373,10374,10390],{},[1818,10375,10376,10378,10380,10382,10388],{},[1824,10377,3059],{"mathvariant":2073},[1832,10379,3062],{},[1832,10381,2725],{"stretchy":2046},[1821,10383,10384,10386],{},[1824,10385,2052],{},[1824,10387,2055],{},[1832,10389,2757],{"stretchy":2046},[1849,10391,10392],{"encoding":1851},"\\operatorname{Var}(X_t)",[1802,10394,10396],{"className":10395,"ariaHidden":1834},[1856],[1802,10397,10399,10402,10408,10411,10451],{"className":10398},[1860],[1802,10400],{"className":10401,"style":2087},[1864],[1802,10403,10405],{"className":10404},[3199],[1802,10406,3059],{"className":10407},[1869,3203],[1802,10409,2725],{"className":10410},[2091],[1802,10412,10414,10417],{"className":10413},[1869],[1802,10415,2052],{"className":10416,"style":2098},[1869,1873],[1802,10418,10420],{"className":10419},[1877],[1802,10421,10423,10443],{"className":10422},[1881,1882],[1802,10424,10426,10440],{"className":10425},[1886],[1802,10427,10429],{"className":10428,"style":2111},[1890],[1802,10430,10431,10434],{"style":2114},[1802,10432],{"className":10433,"style":1899},[1898],[1802,10435,10437],{"className":10436},[1903,1904,1905,1906],[1802,10438,2055],{"className":10439},[1869,1873,1906],[1802,10441,1914],{"className":10442},[1913],[1802,10444,10446],{"className":10445},[1886],[1802,10447,10449],{"className":10448,"style":1921},[1890],[1802,10450],{},[1802,10452,2757],{"className":10453},[2178],[1802,10455,10457,10497],{"className":10456},[1805],[1802,10458,10460],{"className":10459},[1809],[1811,10461,10462],{"xmlns":1813},[1815,10463,10464,10494],{},[1818,10465,10466,10468,10470,10472,10474,10476,10478,10480,10482,10484,10486,10488,10490,10492],{},[1824,10467,4845],{},[1832,10469,2725],{"stretchy":2046},[1828,10471,1830],{},[1832,10473,2757],{"stretchy":2046},[1832,10475,1835],{"separator":1834},[1824,10477,4845],{},[1832,10479,2725],{"stretchy":2046},[1828,10481,3688],{},[1832,10483,2757],{"stretchy":2046},[1832,10485,1835],{"separator":1834},[1824,10487,4845],{},[1832,10489,2725],{"stretchy":2046},[1828,10491,5588],{},[1832,10493,2757],{"stretchy":2046},[1849,10495,10496],{"encoding":1851},"\\rho(1),\\rho(2),\\rho(3)",[1802,10498,10500],{"className":10499,"ariaHidden":1834},[1856],[1802,10501,10503,10506,10509,10512,10515,10518,10521,10524,10527,10530,10533,10536,10539,10542,10545,10548,10551],{"className":10502},[1860],[1802,10504],{"className":10505,"style":2087},[1864],[1802,10507,4845],{"className":10508},[1869,1873],[1802,10510,2725],{"className":10511},[2091],[1802,10513,1830],{"className":10514},[1869],[1802,10516,2757],{"className":10517},[2178],[1802,10519,1835],{"className":10520},[1927],[1802,10522],{"className":10523,"style":1932},[1931],[1802,10525,4845],{"className":10526},[1869,1873],[1802,10528,2725],{"className":10529},[2091],[1802,10531,3688],{"className":10532},[1869],[1802,10534,2757],{"className":10535},[2178],[1802,10537,1835],{"className":10538},[1927],[1802,10540],{"className":10541,"style":1932},[1931],[1802,10543,4845],{"className":10544},[1869,1873],[1802,10546,2725],{"className":10547},[2091],[1802,10549,5588],{"className":10550},[1869],[1802,10552,2757],{"className":10553},[2178],[1999,10555,10556],{},"Construct two dependent variables with zero correlation; explain why an ACF cannot detect their dependence.",[1999,10558,10559,10560,10648,10649,2074],{},"For the six-point path above, recompute ",[1802,10561,10563,10587],{"className":10562},[1805],[1802,10564,10566],{"className":10565},[1809],[1811,10567,10568],{"xmlns":1813},[1815,10569,10570,10584],{},[1818,10571,10572,10578,10580,10582],{},[5610,10573,10574,10576],{"accent":1834},[1824,10575,3588],{},[1832,10577,5937],{"stretchy":1834},[1832,10579,2725],{"stretchy":2046},[1828,10581,1830],{},[1832,10583,2757],{"stretchy":2046},[1849,10585,10586],{"encoding":1851},"\\widehat\\gamma(1)",[1802,10588,10590],{"className":10589,"ariaHidden":1834},[1856],[1802,10591,10593,10596,10639,10642,10645],{"className":10592},[1860],[1802,10594],{"className":10595,"style":2087},[1864],[1802,10597,10599],{"className":10598},[1869,5719],[1802,10600,10602,10631],{"className":10601},[1881,1882],[1802,10603,10605,10628],{"className":10604},[1886],[1802,10606,10608,10616],{"className":10607,"style":6005},[1890],[1802,10609,10610,10613],{"style":5732},[1802,10611],{"className":10612,"style":5326},[1898],[1802,10614,3588],{"className":10615,"style":4034},[1869,1873],[1802,10617,10619,10622],{"className":10618,"style":6018},[6017],[1802,10620],{"className":10621,"style":5326},[1898],[1802,10623,10624],{"style":6024},[3979,10625,10626],{"xmlns":3981,"width":6027,"height":6028,"viewBox":6029,"preserveAspectRatio":6030},[3986,10627],{"d":6033},[1802,10629,1914],{"className":10630},[1913],[1802,10632,10634],{"className":10633},[1886],[1802,10635,10637],{"className":10636,"style":6043},[1890],[1802,10638],{},[1802,10640,2725],{"className":10641},[2091],[1802,10643,1830],{"className":10644},[1869],[1802,10646,2757],{"className":10647},[2178]," using divisor ",[1802,10650,10652,10670],{"className":10651},[1805],[1802,10653,10655],{"className":10654},[1809],[1811,10656,10657],{"xmlns":1813},[1815,10658,10659,10667],{},[1818,10660,10661,10663,10665],{},[1824,10662,1847],{},[1832,10664,3631],{},[1828,10666,1830],{},[1849,10668,10669],{"encoding":1851},"n-1",[1802,10671,10673,10691],{"className":10672,"ariaHidden":1834},[1856],[1802,10674,10676,10679,10682,10685,10688],{"className":10675},[1860],[1802,10677],{"className":10678,"style":6819},[1864],[1802,10680,1847],{"className":10681},[1869,1873],[1802,10683],{"className":10684,"style":4106},[1931],[1802,10686,3631],{"className":10687},[2429],[1802,10689],{"className":10690,"style":4106},[1931],[1802,10692,10694,10697],{"className":10693},[1860],[1802,10695],{"className":10696,"style":4790},[1864],[1802,10698,1830],{"className":10699},[1869],[1999,10701,10702],{},"Explain why stationarity and ergodicity answer different questions.",[1999,10704,10705,10706,10776,10777,10828,10829,10858,10859,11012],{},"Construct ",[1802,10707,10709,10727],{"className":10708},[1805],[1802,10710,10712],{"className":10711},[1809],[1811,10713,10714],{"xmlns":1813},[1815,10715,10716,10724],{},[1818,10717,10718],{},[1821,10719,10720,10722],{},[1824,10721,3073],{"mathvariant":2073},[1828,10723,5588],{},[1849,10725,10726],{"encoding":1851},"\\Gamma_3",[1802,10728,10730],{"className":10729,"ariaHidden":1834},[1856],[1802,10731,10733,10736],{"className":10732},[1860],[1802,10734],{"className":10735,"style":4535},[1864],[1802,10737,10739,10742],{"className":10738},[1869],[1802,10740,3073],{"className":10741},[1869],[1802,10743,10745],{"className":10744},[1877],[1802,10746,10748,10768],{"className":10747},[1881,1882],[1802,10749,10751,10765],{"className":10750},[1886],[1802,10752,10754],{"className":10753,"style":1891},[1890],[1802,10755,10756,10759],{"style":1894},[1802,10757],{"className":10758,"style":1899},[1898],[1802,10760,10762],{"className":10761},[1903,1904,1905,1906],[1802,10763,5588],{"className":10764},[1869,1906],[1802,10766,1914],{"className":10767},[1913],[1802,10769,10771],{"className":10770},[1886],[1802,10772,10774],{"className":10773,"style":1921},[1890],[1802,10775],{}," for an AR(1) with ",[1802,10778,10780,10798],{"className":10779},[1805],[1802,10781,10783],{"className":10782},[1809],[1811,10784,10785],{"xmlns":1813},[1815,10786,10787,10795],{},[1818,10788,10789,10791,10793],{},[1824,10790,4932],{},[1832,10792,2722],{},[1828,10794,9209],{},[1849,10796,10797],{"encoding":1851},"\\phi=0.5",[1802,10799,10801,10819],{"className":10800,"ariaHidden":1834},[1856],[1802,10802,10804,10807,10810,10813,10816],{"className":10803},[1860],[1802,10805],{"className":10806,"style":4945},[1864],[1802,10808,4932],{"className":10809},[1869,1873],[1802,10811],{"className":10812,"style":2138},[1931],[1802,10814,2722],{"className":10815},[2142],[1802,10817],{"className":10818,"style":2138},[1931],[1802,10820,10822,10825],{"className":10821},[1860],[1802,10823],{"className":10824,"style":4790},[1864],[1802,10826,9209],{"className":10827},[1869]," and innovation variance ",[1802,10830,10832,10846],{"className":10831},[1805],[1802,10833,10835],{"className":10834},[1809],[1811,10836,10837],{"xmlns":1813},[1815,10838,10839,10844],{},[1818,10840,10841],{},[1828,10842,10843],{},"0.75",[1849,10845,10843],{"encoding":1851},[1802,10847,10849],{"className":10848,"ariaHidden":1834},[1856],[1802,10850,10852,10855],{"className":10851},[1860],[1802,10853],{"className":10854,"style":4790},[1864],[1802,10856,10843],{"className":10857},[1869],", then find ",[1802,10860,10862,10896],{"className":10861},[1805],[1802,10863,10865],{"className":10864},[1809],[1811,10866,10867],{"xmlns":1813},[1815,10868,10869,10893],{},[1818,10870,10871,10873,10875,10877,10883,10885,10891],{},[1824,10872,3059],{"mathvariant":2073},[1832,10874,3062],{},[1832,10876,2725],{"stretchy":2046},[1821,10878,10879,10881],{},[1824,10880,2052],{},[1828,10882,1830],{},[1832,10884,3631],{},[1821,10886,10887,10889],{},[1824,10888,2052],{},[1828,10890,3688],{},[1832,10892,2757],{"stretchy":2046},[1849,10894,10895],{"encoding":1851},"\\operatorname{Var}(X_1-X_2)",[1802,10897,10899,10963],{"className":10898,"ariaHidden":1834},[1856],[1802,10900,10902,10905,10911,10914,10954,10957,10960],{"className":10901},[1860],[1802,10903],{"className":10904,"style":2087},[1864],[1802,10906,10908],{"className":10907},[3199],[1802,10909,3059],{"className":10910},[1869,3203],[1802,10912,2725],{"className":10913},[2091],[1802,10915,10917,10920],{"className":10916},[1869],[1802,10918,2052],{"className":10919,"style":2098},[1869,1873],[1802,10921,10923],{"className":10922},[1877],[1802,10924,10926,10946],{"className":10925},[1881,1882],[1802,10927,10929,10943],{"className":10928},[1886],[1802,10930,10932],{"className":10931,"style":1891},[1890],[1802,10933,10934,10937],{"style":2114},[1802,10935],{"className":10936,"style":1899},[1898],[1802,10938,10940],{"className":10939},[1903,1904,1905,1906],[1802,10941,1830],{"className":10942},[1869,1906],[1802,10944,1914],{"className":10945},[1913],[1802,10947,10949],{"className":10948},[1886],[1802,10950,10952],{"className":10951,"style":1921},[1890],[1802,10953],{},[1802,10955],{"className":10956,"style":4106},[1931],[1802,10958,3631],{"className":10959},[2429],[1802,10961],{"className":10962,"style":4106},[1931],[1802,10964,10966,10969,11009],{"className":10965},[1860],[1802,10967],{"className":10968,"style":2087},[1864],[1802,10970,10972,10975],{"className":10971},[1869],[1802,10973,2052],{"className":10974,"style":2098},[1869,1873],[1802,10976,10978],{"className":10977},[1877],[1802,10979,10981,11001],{"className":10980},[1881,1882],[1802,10982,10984,10998],{"className":10983},[1886],[1802,10985,10987],{"className":10986,"style":1891},[1890],[1802,10988,10989,10992],{"style":2114},[1802,10990],{"className":10991,"style":1899},[1898],[1802,10993,10995],{"className":10994},[1903,1904,1905,1906],[1802,10996,3688],{"className":10997},[1869,1906],[1802,10999,1914],{"className":11000},[1913],[1802,11002,11004],{"className":11003},[1886],[1802,11005,11007],{"className":11006,"style":1921},[1890],[1802,11008],{},[1802,11010,2757],{"className":11011},[2178]," as a quadratic form.",[11014,11015,11017],"legacy-details",{"title":11016},"Answers",[10136,11018,11019,11196,11412,11481,11484],{},[1999,11020,11021,11022,11129,11130,2074],{},"Variance ",[1802,11023,11025,11058],{"className":11024},[1805],[1802,11026,11028],{"className":11027},[1809],[1811,11029,11030],{"xmlns":1813},[1815,11031,11032,11055],{},[1818,11033,11034,11036,11038,11040,11042,11044,11046,11049,11051,11053],{},[1832,11035,2722],{},[1828,11037,5588],{},[1824,11039,7511],{"mathvariant":2073},[1832,11041,2725],{"stretchy":2046},[1828,11043,1830],{},[1832,11045,3631],{},[1828,11047,11048],{},"0.25",[1832,11050,2757],{"stretchy":2046},[1832,11052,2722],{},[1828,11054,5583],{},[1849,11056,11057],{"encoding":1851},"=3\u002F(1-0.25)=4",[1802,11059,11061,11074,11099,11120],{"className":11060,"ariaHidden":1834},[1856],[1802,11062,11064,11068,11071],{"className":11063},[1860],[1802,11065],{"className":11066,"style":11067},[1864],"height:0.3669em;",[1802,11069,2722],{"className":11070},[2142],[1802,11072],{"className":11073,"style":2138},[1931],[1802,11075,11077,11080,11084,11087,11090,11093,11096],{"className":11076},[1860],[1802,11078],{"className":11079,"style":2087},[1864],[1802,11081,11083],{"className":11082},[1869],"3\u002F",[1802,11085,2725],{"className":11086},[2091],[1802,11088,1830],{"className":11089},[1869],[1802,11091],{"className":11092,"style":4106},[1931],[1802,11094,3631],{"className":11095},[2429],[1802,11097],{"className":11098,"style":4106},[1931],[1802,11100,11102,11105,11108,11111,11114,11117],{"className":11101},[1860],[1802,11103],{"className":11104,"style":2087},[1864],[1802,11106,11048],{"className":11107},[1869],[1802,11109,2757],{"className":11110},[2178],[1802,11112],{"className":11113,"style":2138},[1931],[1802,11115,2722],{"className":11116},[2142],[1802,11118],{"className":11119,"style":2138},[1931],[1802,11121,11123,11126],{"className":11122},[1860],[1802,11124],{"className":11125,"style":4790},[1864],[1802,11127,5583],{"className":11128},[1869],"; correlations are 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is symmetric around zero, 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are dependent but can have zero covariance because 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concerns invariance across calendar shifts; ergodicity concerns whether one long path recovers population quantities.",[1999,11485,11486,11552,11553,11985,11986,2074],{},[1802,11487,11489,11513],{"className":11488},[1805],[1802,11490,11492],{"className":11491},[1809],[1811,11493,11494],{"xmlns":1813},[1815,11495,11496,11510],{},[1818,11497,11498,11500,11502,11504,11506,11508],{},[1824,11499,3588],{},[1832,11501,2725],{"stretchy":2046},[1828,11503,3583],{},[1832,11505,2757],{"stretchy":2046},[1832,11507,2722],{},[1828,11509,1830],{},[1849,11511,11512],{"encoding":1851},"\\gamma(0)=1",[1802,11514,11516,11543],{"className":11515,"ariaHidden":1834},[1856],[1802,11517,11519,11522,11525,11528,11531,11534,11537,11540],{"className":11518},[1860],[1802,11520],{"className":11521,"style":2087},[1864],[1802,11523,3588],{"className":11524,"style":4034},[1869,1873],[1802,11526,2725],{"className":11527},[2091],[1802,11529,3583],{"className":11530},[1869],[1802,11532,2757],{"className":11533},[2178],[1802,11535],{"className":11536,"style":2138},[1931],[1802,11538,2722],{"className":11539},[2142],[1802,11541],{"className":11542,"style":2138},[1931],[1802,11544,11546,11549],{"className":11545},[1860],[1802,11547],{"className":11548,"style":4790},[1864],[1802,11550,1830],{"className":11551},[1869],", so\n",[1802,11554,11556,11645],{"className":11555},[1805],[1802,11557,11559],{"className":11558},[1809],[1811,11560,11561],{"xmlns":1813},[1815,11562,11563,11642],{},[1818,11564,11565,11571,11573],{},[1821,11566,11567,11569],{},[1824,11568,3073],{"mathvariant":2073},[1828,11570,5588],{},[1832,11572,2722],{},[1818,11574,11575,11577,11640],{},[1832,11576,3083],{"fence":1834},[3569,11578,11580,11600,11620],{"rowspacing":3571,"columnalign":11579,"columnspacing":3573},"center center 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Continue to ",[4596,12210,12212],{"href":12211},"..\u002F02-arma\u002F","ARMA and ARIMA"," to encode that stable dependence with lag polynomials.",{"title":10,"searchDepth":12215,"depth":12215,"links":12216},2,[12217,12218,12219,12220,12221,12225,12228,12229,12230,12231,12232,12233],{"id":1795,"depth":12215,"text":1796},{"id":1990,"depth":12215,"text":1991},{"id":2022,"depth":12215,"text":2023},{"id":2555,"depth":12215,"text":2556},{"id":5019,"depth":12215,"text":5020,"children":12222},[12223],{"id":5551,"depth":12224,"text":5552},3,{"id":6901,"depth":12215,"text":6902,"children":12226},[12227],{"id":6958,"depth":12224,"text":6959},{"id":7547,"depth":12215,"text":7548},{"id":7686,"depth":12215,"text":7687},{"id":9404,"depth":12215,"text":9405},{"id":10068,"depth":12215,"text":10069},{"id":10133,"depth":12215,"text":10134},{"id":12204,"depth":12215,"text":12205},"Represent a stochastic process through finite random vectors, Toeplitz covariance matrices, stationarity, and Wold innovations.","md",{"sidebar":12237},{"order":12238},1,true,{"title":959,"description":12234},"jXu0m6v8i3ONyfuBUhjIXl3O2lsuahUH8rsrte75tPI",[12243,12245],{"title":955,"path":956,"stem":957,"description":12244,"children":-1},"A compact diagnostic primer on stable dependence, trends, unit roots, seasonality, and breaks.",{"title":965,"path":966,"stem":967,"description":12246,"children":-1},"Build AR, MA, and ARIMA models from lag polynomials, roots, shock propagation, and finite-sample diagnostics.",1785754737714]