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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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question",[1798,1799,1800],"p",{},"What changes when variables forecast one another, seasonal positions have their own dynamics, or non-stationary levels remain tied by a stable long-run relation?",[1793,1802,1804],{"id":1803},"learning-outcomes","Learning outcomes",[1798,1806,1807],{},"You will be able to:",[1809,1810,1811,1815,1818,1821,1824,1827,1830],"ul",{},[1812,1813,1814],"li",{},"formulate and check the stability of a VAR;",[1812,1816,1817],{},"solve a stationary VAR covariance with a Kronecker-product equation;",[1812,1819,1820],{},"distinguish Granger predictability from causal effect;",[1812,1822,1823],{},"compute and qualify an impulse response;",[1812,1825,1826],{},"choose among seasonal indicators, seasonal differencing, and SARIMA terms;",[1812,1828,1829],{},"recognise cointegration and interpret an error-correction model;",[1812,1831,1832],{},"explain why forecasts across products or time aggregates must be reconciled.",[1793,1834,1836],{"id":1835},"_1-var-every-variable-can-use-every-lag","1. VAR: every variable can use every lag",[1798,1838,1839,1840,1887,1888,1917],{},"For a ",[1841,1842,1845,1867],"span",{"className":1843},[1844],"katex",[1841,1846,1849],{"className":1847},[1848],"katex-mathml",[1850,1851,1853],"math",{"xmlns":1852},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1854,1855,1856,1863],"semantics",{},[1857,1858,1859],"mrow",{},[1860,1861,1862],"mi",{},"k",[1864,1865,1862],"annotation",{"encoding":1866},"application\u002Fx-tex",[1841,1868,1872],{"className":1869,"ariaHidden":1871},[1870],"katex-html","true",[1841,1873,1876,1881],{"className":1874},[1875],"base",[1841,1877],{"className":1878,"style":1880},[1879],"strut","height:0.6944em;",[1841,1882,1862],{"className":1883,"style":1886},[1884,1885],"mord","mathnormal","margin-right:0.0315em;","-variable VAR(",[1841,1889,1891,1904],{"className":1890},[1844],[1841,1892,1894],{"className":1893},[1848],[1850,1895,1896],{"xmlns":1852},[1854,1897,1898,1902],{},[1857,1899,1900],{},[1860,1901,1798],{},[1864,1903,1798],{"encoding":1866},[1841,1905,1907],{"className":1906,"ariaHidden":1871},[1870],[1841,1908,1910,1914],{"className":1909},[1875],[1841,1911],{"className":1912,"style":1913},[1879],"height:0.625em;vertical-align:-0.1944em;",[1841,1915,1798],{"className":1916},[1884,1885],"),",[1841,1919,1922],{"className":1920},[1921],"katex-display",[1841,1923,1925,2019],{"className":1924},[1844],[1841,1926,1928],{"className":1927},[1848],[1850,1929,1931],{"xmlns":1852,"display":1930},"block",[1854,1932,1933,2016],{},[1857,1934,1935,1945,1949,1952,1955,1964,1977,1979,1982,1984,1990,2002,2004,2012],{},[1936,1937,1938,1942],"msub",{},[1860,1939,1941],{"mathvariant":1940},"bold","y",[1860,1943,1944],{},"t",[1946,1947,1948],"mo",{},"=",[1860,1950,1951],{"mathvariant":1940},"c",[1946,1953,1954],{},"+",[1936,1956,1957,1960],{},[1860,1958,1959],{},"A",[1961,1962,1963],"mn",{},"1",[1936,1965,1966,1968],{},[1860,1967,1941],{"mathvariant":1940},[1857,1969,1970,1972,1975],{},[1860,1971,1944],{},[1946,1973,1974],{},"−",[1961,1976,1963],{},[1946,1978,1954],{},[1946,1980,1981],{},"⋯",[1946,1983,1954],{},[1936,1985,1986,1988],{},[1860,1987,1959],{},[1860,1989,1798],{},[1936,1991,1992,1994],{},[1860,1993,1941],{"mathvariant":1940},[1857,1995,1996,1998,2000],{},[1860,1997,1944],{},[1946,1999,1974],{},[1860,2001,1798],{},[1946,2003,1954],{},[1936,2005,2006,2010],{},[1860,2007,2009],{"mathvariant":2008},"bold-italic","ε",[1860,2011,1944],{},[1860,2013,2015],{"mathvariant":2014},"normal",".",[1864,2017,2018],{"encoding":1866},"\\mathbf y_t=\\mathbf c+A_1\\mathbf y_{t-1}+\\cdots+A_p\\mathbf y_{t-p}+\\boldsymbol\\varepsilon_t.",[1841,2020,2022,2099,2120,2228,2247,2354],{"className":2021,"ariaHidden":1871},[1870],[1841,2023,2025,2029,2087,2092,2096],{"className":2024},[1875],[1841,2026],{"className":2027,"style":2028},[1879],"height:0.6389em;vertical-align:-0.1944em;",[1841,2030,2032,2037],{"className":2031},[1884],[1841,2033,1941],{"className":2034,"style":2036},[1884,2035],"mathbf","margin-right:0.016em;",[1841,2038,2041],{"className":2039},[2040],"msupsub",[1841,2042,2046,2078],{"className":2043},[2044,2045],"vlist-t","vlist-t2",[1841,2047,2050,2073],{"className":2048},[2049],"vlist-r",[1841,2051,2055],{"className":2052,"style":2054},[2053],"vlist","height:0.2806em;",[1841,2056,2058,2063],{"style":2057},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[1841,2059],{"className":2060,"style":2062},[2061],"pstrut","height:2.7em;",[1841,2064,2070],{"className":2065},[2066,2067,2068,2069],"sizing","reset-size6","size3","mtight",[1841,2071,1944],{"className":2072},[1884,1885,2069],[1841,2074,2077],{"className":2075},[2076],"vlist-s","​",[1841,2079,2081],{"className":2080},[2049],[1841,2082,2085],{"className":2083,"style":2084},[2053],"height:0.15em;",[1841,2086],{},[1841,2088],{"className":2089,"style":2091},[2090],"mspace","margin-right:0.2778em;",[1841,2093,1948],{"className":2094},[2095],"mrel",[1841,2097],{"className":2098,"style":2091},[2090],[1841,2100,2102,2106,2109,2113,2117],{"className":2101},[1875],[1841,2103],{"className":2104,"style":2105},[1879],"height:0.6667em;vertical-align:-0.0833em;",[1841,2107,1951],{"className":2108},[1884,2035],[1841,2110],{"className":2111,"style":2112},[2090],"margin-right:0.2222em;",[1841,2114,1954],{"className":2115},[2116],"mbin",[1841,2118],{"className":2119,"style":2112},[2090],[1841,2121,2123,2127,2169,2219,2222,2225],{"className":2122},[1875],[1841,2124],{"className":2125,"style":2126},[1879],"height:0.8917em;vertical-align:-0.2083em;",[1841,2128,2130,2133],{"className":2129},[1884],[1841,2131,1959],{"className":2132},[1884,1885],[1841,2134,2136],{"className":2135},[2040],[1841,2137,2139,2161],{"className":2138},[2044,2045],[1841,2140,2142,2158],{"className":2141},[2049],[1841,2143,2146],{"className":2144,"style":2145},[2053],"height:0.3011em;",[1841,2147,2149,2152],{"style":2148},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1841,2150],{"className":2151,"style":2062},[2061],[1841,2153,2155],{"className":2154},[2066,2067,2068,2069],[1841,2156,1963],{"className":2157},[1884,2069],[1841,2159,2077],{"className":2160},[2076],[1841,2162,2164],{"className":2163},[2049],[1841,2165,2167],{"className":2166,"style":2084},[2053],[1841,2168],{},[1841,2170,2172,2175],{"className":2171},[1884],[1841,2173,1941],{"className":2174,"style":2036},[1884,2035],[1841,2176,2178],{"className":2177},[2040],[1841,2179,2181,2210],{"className":2180},[2044,2045],[1841,2182,2184,2207],{"className":2183},[2049],[1841,2185,2187],{"className":2186,"style":2145},[2053],[1841,2188,2189,2192],{"style":2057},[1841,2190],{"className":2191,"style":2062},[2061],[1841,2193,2195],{"className":2194},[2066,2067,2068,2069],[1841,2196,2198,2201,2204],{"className":2197},[1884,2069],[1841,2199,1944],{"className":2200},[1884,1885,2069],[1841,2202,1974],{"className":2203},[2116,2069],[1841,2205,1963],{"className":2206},[1884,2069],[1841,2208,2077],{"className":2209},[2076],[1841,2211,2213],{"className":2212},[2049],[1841,2214,2217],{"className":2215,"style":2216},[2053],"height:0.2083em;",[1841,2218],{},[1841,2220],{"className":2221,"style":2112},[2090],[1841,2223,1954],{"className":2224},[2116],[1841,2226],{"className":2227,"style":2112},[2090],[1841,2229,2231,2234,2238,2241,2244],{"className":2230},[1875],[1841,2232],{"className":2233,"style":2105},[1879],[1841,2235,1981],{"className":2236},[2237],"minner",[1841,2239],{"className":2240,"style":2112},[2090],[1841,2242,1954],{"className":2243},[2116],[1841,2245],{"className":2246,"style":2112},[2090],[1841,2248,2250,2254,2296,2345,2348,2351],{"className":2249},[1875],[1841,2251],{"className":2252,"style":2253},[1879],"height:0.9694em;vertical-align:-0.2861em;",[1841,2255,2257,2260],{"className":2256},[1884],[1841,2258,1959],{"className":2259},[1884,1885],[1841,2261,2263],{"className":2262},[2040],[1841,2264,2266,2287],{"className":2265},[2044,2045],[1841,2267,2269,2284],{"className":2268},[2049],[1841,2270,2273],{"className":2271,"style":2272},[2053],"height:0.1514em;",[1841,2274,2275,2278],{"style":2148},[1841,2276],{"className":2277,"style":2062},[2061],[1841,2279,2281],{"className":2280},[2066,2067,2068,2069],[1841,2282,1798],{"className":2283},[1884,1885,2069],[1841,2285,2077],{"className":2286},[2076],[1841,2288,2290],{"className":2289},[2049],[1841,2291,2294],{"className":2292,"style":2293},[2053],"height:0.2861em;",[1841,2295],{},[1841,2297,2299,2302],{"className":2298},[1884],[1841,2300,1941],{"className":2301,"style":2036},[1884,2035],[1841,2303,2305],{"className":2304},[2040],[1841,2306,2308,2337],{"className":2307},[2044,2045],[1841,2309,2311,2334],{"className":2310},[2049],[1841,2312,2314],{"className":2313,"style":2054},[2053],[1841,2315,2316,2319],{"style":2057},[1841,2317],{"className":2318,"style":2062},[2061],[1841,2320,2322],{"className":2321},[2066,2067,2068,2069],[1841,2323,2325,2328,2331],{"className":2324},[1884,2069],[1841,2326,1944],{"className":2327},[1884,1885,2069],[1841,2329,1974],{"className":2330},[2116,2069],[1841,2332,1798],{"className":2333},[1884,1885,2069],[1841,2335,2077],{"className":2336},[2076],[1841,2338,2340],{"className":2339},[2049],[1841,2341,2343],{"className":2342,"style":2293},[2053],[1841,2344],{},[1841,2346],{"className":2347,"style":2112},[2090],[1841,2349,1954],{"className":2350},[2116],[1841,2352],{"className":2353,"style":2112},[2090],[1841,2355,2357,2361,2409],{"className":2356},[1875],[1841,2358],{"className":2359,"style":2360},[1879],"height:0.5944em;vertical-align:-0.15em;",[1841,2362,2364,2374],{"className":2363},[1884],[1841,2365,2367],{"className":2366},[1884],[1841,2368,2370],{"className":2369},[1884],[1841,2371,2009],{"className":2372},[1884,2373],"boldsymbol",[1841,2375,2377],{"className":2376},[2040],[1841,2378,2380,2401],{"className":2379},[2044,2045],[1841,2381,2383,2398],{"className":2382},[2049],[1841,2384,2386],{"className":2385,"style":2054},[2053],[1841,2387,2389,2392],{"style":2388},"top:-2.55em;margin-right:0.05em;",[1841,2390],{"className":2391,"style":2062},[2061],[1841,2393,2395],{"className":2394},[2066,2067,2068,2069],[1841,2396,1944],{"className":2397},[1884,1885,2069],[1841,2399,2077],{"className":2400},[2076],[1841,2402,2404],{"className":2403},[2049],[1841,2405,2407],{"className":2406,"style":2084},[2053],[1841,2408],{},[1841,2410,2015],{"className":2411},[1884],[1798,2413,2414,2415,2486,2487,2515],{},"A VAR(1) is stationary when every eigenvalue of ",[1841,2416,2418,2436],{"className":2417},[1844],[1841,2419,2421],{"className":2420},[1848],[1850,2422,2423],{"xmlns":1852},[1854,2424,2425,2433],{},[1857,2426,2427],{},[1936,2428,2429,2431],{},[1860,2430,1959],{},[1961,2432,1963],{},[1864,2434,2435],{"encoding":1866},"A_1",[1841,2437,2439],{"className":2438,"ariaHidden":1871},[1870],[1841,2440,2442,2446],{"className":2441},[1875],[1841,2443],{"className":2444,"style":2445},[1879],"height:0.8333em;vertical-align:-0.15em;",[1841,2447,2449,2452],{"className":2448},[1884],[1841,2450,1959],{"className":2451},[1884,1885],[1841,2453,2455],{"className":2454},[2040],[1841,2456,2458,2478],{"className":2457},[2044,2045],[1841,2459,2461,2475],{"className":2460},[2049],[1841,2462,2464],{"className":2463,"style":2145},[2053],[1841,2465,2466,2469],{"style":2148},[1841,2467],{"className":2468,"style":2062},[2061],[1841,2470,2472],{"className":2471},[2066,2067,2068,2069],[1841,2473,1963],{"className":2474},[1884,2069],[1841,2476,2077],{"className":2477},[2076],[1841,2479,2481],{"className":2480},[2049],[1841,2482,2484],{"className":2483,"style":2084},[2053],[1841,2485],{}," has modulus below one. For VAR(",[1841,2488,2490,2503],{"className":2489},[1844],[1841,2491,2493],{"className":2492},[1848],[1850,2494,2495],{"xmlns":1852},[1854,2496,2497,2501],{},[1857,2498,2499],{},[1860,2500,1798],{},[1864,2502,1798],{"encoding":1866},[1841,2504,2506],{"className":2505,"ariaHidden":1871},[1870],[1841,2507,2509,2512],{"className":2508},[1875],[1841,2510],{"className":2511,"style":1913},[1879],[1841,2513,1798],{"className":2514},[1884,1885],"), use the equivalent companion matrix.",[2517,2518,2520],"h3",{"id":2519},"parameter-growth-is-the-first-constraint","Parameter growth is the first constraint",[1798,2522,2523,2524,2601,2602,2655,2656,2708],{},"With an intercept, a VAR has ",[1841,2525,2527,2556],{"className":2526},[1844],[1841,2528,2530],{"className":2529},[1848],[1850,2531,2532],{"xmlns":1852},[1854,2533,2534,2553],{},[1857,2535,2536,2538,2542,2544,2546,2548,2550],{},[1860,2537,1862],{},[1946,2539,2541],{"stretchy":2540},"false","(",[1860,2543,1862],{},[1860,2545,1798],{},[1946,2547,1954],{},[1961,2549,1963],{},[1946,2551,2552],{"stretchy":2540},")",[1864,2554,2555],{"encoding":1866},"k(kp+1)",[1841,2557,2559,2588],{"className":2558,"ariaHidden":1871},[1870],[1841,2560,2562,2566,2569,2573,2576,2579,2582,2585],{"className":2561},[1875],[1841,2563],{"className":2564,"style":2565},[1879],"height:1em;vertical-align:-0.25em;",[1841,2567,1862],{"className":2568,"style":1886},[1884,1885],[1841,2570,2541],{"className":2571},[2572],"mopen",[1841,2574,1862],{"className":2575,"style":1886},[1884,1885],[1841,2577,1798],{"className":2578},[1884,1885],[1841,2580],{"className":2581,"style":2112},[2090],[1841,2583,1954],{"className":2584},[2116],[1841,2586],{"className":2587,"style":2112},[2090],[1841,2589,2591,2594,2597],{"className":2590},[1875],[1841,2592],{"className":2593,"style":2565},[1879],[1841,2595,1963],{"className":2596},[1884],[1841,2598,2552],{"className":2599},[2600],"mclose"," regression coefficients. For ",[1841,2603,2605,2624],{"className":2604},[1844],[1841,2606,2608],{"className":2607},[1848],[1850,2609,2610],{"xmlns":1852},[1854,2611,2612,2621],{},[1857,2613,2614,2616,2618],{},[1860,2615,1862],{},[1946,2617,1948],{},[1961,2619,2620],{},"5",[1864,2622,2623],{"encoding":1866},"k=5",[1841,2625,2627,2645],{"className":2626,"ariaHidden":1871},[1870],[1841,2628,2630,2633,2636,2639,2642],{"className":2629},[1875],[1841,2631],{"className":2632,"style":1880},[1879],[1841,2634,1862],{"className":2635,"style":1886},[1884,1885],[1841,2637],{"className":2638,"style":2091},[2090],[1841,2640,1948],{"className":2641},[2095],[1841,2643],{"className":2644,"style":2091},[2090],[1841,2646,2648,2652],{"className":2647},[1875],[1841,2649],{"className":2650,"style":2651},[1879],"height:0.6444em;",[1841,2653,2620],{"className":2654},[1884]," and ",[1841,2657,2659,2678],{"className":2658},[1844],[1841,2660,2662],{"className":2661},[1848],[1850,2663,2664],{"xmlns":1852},[1854,2665,2666,2675],{},[1857,2667,2668,2670,2672],{},[1860,2669,1798],{},[1946,2671,1948],{},[1961,2673,2674],{},"4",[1864,2676,2677],{"encoding":1866},"p=4",[1841,2679,2681,2699],{"className":2680,"ariaHidden":1871},[1870],[1841,2682,2684,2687,2690,2693,2696],{"className":2683},[1875],[1841,2685],{"className":2686,"style":1913},[1879],[1841,2688,1798],{"className":2689},[1884,1885],[1841,2691],{"className":2692,"style":2091},[2090],[1841,2694,1948],{"className":2695},[2095],[1841,2697],{"className":2698,"style":2091},[2090],[1841,2700,2702,2705],{"className":2701},[1875],[1841,2703],{"className":2704,"style":2651},[1879],[1841,2706,2674],{"className":2707},[1884],", that is 105 coefficients before the 15 distinct innovation-covariance terms. A rich system estimated from 80 quarters is not automatically informative; shrinkage, fewer variables\u002Flags, or theory-based restrictions may be necessary.",[2517,2710,2712],{"id":2711},"stacked-regression-form","Stacked regression form",[1798,2714,2715,2716,2897,2898,2928,2929,2897,3052,3085],{},"For VAR(1), collect rows ",[1841,2717,2719,2759],{"className":2718},[1844],[1841,2720,2722],{"className":2721},[1848],[1850,2723,2724],{"xmlns":1852},[1854,2725,2726,2756],{},[1857,2727,2728,2739,2742,2745,2747],{},[2729,2730,2731,2733,2736],"msubsup",{},[1860,2732,1941],{"mathvariant":1940},[1961,2734,2735],{},"2",[1860,2737,2738],{"mathvariant":2014},"⊤",[1946,2740,2741],{"separator":1871},",",[1946,2743,2744],{},"…",[1946,2746,2741],{"separator":1871},[2729,2748,2749,2751,2754],{},[1860,2750,1941],{"mathvariant":1940},[1860,2752,2753],{},"n",[1860,2755,2738],{"mathvariant":2014},[1864,2757,2758],{"encoding":1866},"\\mathbf y_2^\\top,\\ldots,\\mathbf y_n^\\top",[1841,2760,2762],{"className":2761,"ariaHidden":1871},[1870],[1841,2763,2765,2769,2824,2828,2832,2835,2838,2841,2844],{"className":2764},[1875],[1841,2766],{"className":2767,"style":2768},[1879],"height:1.0972em;vertical-align:-0.2481em;",[1841,2770,2772,2775],{"className":2771},[1884],[1841,2773,1941],{"className":2774,"style":2036},[1884,2035],[1841,2776,2778],{"className":2777},[2040],[1841,2779,2781,2815],{"className":2780},[2044,2045],[1841,2782,2784,2812],{"className":2783},[2049],[1841,2785,2788,2800],{"className":2786,"style":2787},[2053],"height:0.8491em;",[1841,2789,2791,2794],{"style":2790},"top:-2.4519em;margin-left:-0.016em;margin-right:0.05em;",[1841,2792],{"className":2793,"style":2062},[2061],[1841,2795,2797],{"className":2796},[2066,2067,2068,2069],[1841,2798,2735],{"className":2799},[1884,2069],[1841,2801,2803,2806],{"style":2802},"top:-3.063em;margin-right:0.05em;",[1841,2804],{"className":2805,"style":2062},[2061],[1841,2807,2809],{"className":2808},[2066,2067,2068,2069],[1841,2810,2738],{"className":2811},[1884,2069],[1841,2813,2077],{"className":2814},[2076],[1841,2816,2818],{"className":2817},[2049],[1841,2819,2822],{"className":2820,"style":2821},[2053],"height:0.2481em;",[1841,2823],{},[1841,2825,2741],{"className":2826},[2827],"mpunct",[1841,2829],{"className":2830,"style":2831},[2090],"margin-right:0.1667em;",[1841,2833,2744],{"className":2834},[2237],[1841,2836],{"className":2837,"style":2831},[2090],[1841,2839,2741],{"className":2840},[2827],[1841,2842],{"className":2843,"style":2831},[2090],[1841,2845,2847,2850],{"className":2846},[1884],[1841,2848,1941],{"className":2849,"style":2036},[1884,2035],[1841,2851,2853],{"className":2852},[2040],[1841,2854,2856,2888],{"className":2855},[2044,2045],[1841,2857,2859,2885],{"className":2858},[2049],[1841,2860,2862,2874],{"className":2861,"style":2787},[2053],[1841,2863,2865,2868],{"style":2864},"top:-2.453em;margin-left:-0.016em;margin-right:0.05em;",[1841,2866],{"className":2867,"style":2062},[2061],[1841,2869,2871],{"className":2870},[2066,2067,2068,2069],[1841,2872,2753],{"className":2873},[1884,1885,2069],[1841,2875,2876,2879],{"style":2802},[1841,2877],{"className":2878,"style":2062},[2061],[1841,2880,2882],{"className":2881},[2066,2067,2068,2069],[1841,2883,2738],{"className":2884},[1884,2069],[1841,2886,2077],{"className":2887},[2076],[1841,2889,2891],{"className":2890},[2049],[1841,2892,2895],{"className":2893,"style":2894},[2053],"height:0.247em;",[1841,2896],{}," in ",[1841,2899,2901,2915],{"className":2900},[1844],[1841,2902,2904],{"className":2903},[1848],[1850,2905,2906],{"xmlns":1852},[1854,2907,2908,2913],{},[1857,2909,2910],{},[1860,2911,2912],{},"Y",[1864,2914,2912],{"encoding":1866},[1841,2916,2918],{"className":2917,"ariaHidden":1871},[1870],[1841,2919,2921,2925],{"className":2920},[1875],[1841,2922],{"className":2923,"style":2924},[1879],"height:0.6833em;",[1841,2926,2912],{"className":2927,"style":2112},[1884,1885]," and rows 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y_{t-1}^\\top)",[1841,2967,2969],{"className":2968,"ariaHidden":1871},[1870],[1841,2970,2972,2976,2979,2982,2985,2988,3049],{"className":2971},[1875],[1841,2973],{"className":2974,"style":2975},[1879],"height:1.1555em;vertical-align:-0.3064em;",[1841,2977,2541],{"className":2978},[2572],[1841,2980,1963],{"className":2981},[1884],[1841,2983,2741],{"className":2984},[2827],[1841,2986],{"className":2987,"style":2831},[2090],[1841,2989,2991,2994],{"className":2990},[1884],[1841,2992,1941],{"className":2993,"style":2036},[1884,2035],[1841,2995,2997],{"className":2996},[2040],[1841,2998,3000,3040],{"className":2999},[2044,2045],[1841,3001,3003,3037],{"className":3002},[2049],[1841,3004,3006,3026],{"className":3005,"style":2787},[2053],[1841,3007,3008,3011],{"style":2790},[1841,3009],{"className":3010,"style":2062},[2061],[1841,3012,3014],{"className":3013},[2066,2067,2068,2069],[1841,3015,3017,3020,3023],{"className":3016},[1884,2069],[1841,3018,1944],{"className":3019},[1884,1885,2069],[1841,3021,1974],{"className":3022},[2116,2069],[1841,3024,1963],{"className":3025},[1884,2069],[1841,3027,3028,3031],{"style":2802},[1841,3029],{"className":3030,"style":2062},[2061],[1841,3032,3034],{"className":3033},[2066,2067,2068,2069],[1841,3035,2738],{"className":3036},[1884,2069],[1841,3038,2077],{"className":3039},[2076],[1841,3041,3043],{"className":3042},[2049],[1841,3044,3047],{"className":3045,"style":3046},[2053],"height:0.3064em;",[1841,3048],{},[1841,3050,2552],{"className":3051},[2600],[1841,3053,3055,3071],{"className":3054},[1844],[1841,3056,3058],{"className":3057},[1848],[1850,3059,3060],{"xmlns":1852},[1854,3061,3062,3068],{},[1857,3063,3064],{},[1860,3065,3067],{"mathvariant":3066},"script","X",[1864,3069,3070],{"encoding":1866},"\\mathcal 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Then",[1841,3087,3089],{"className":3088},[1921],[1841,3090,3092,3177],{"className":3091},[1844],[1841,3093,3095],{"className":3094},[1848],[1850,3096,3097],{"xmlns":1852,"display":1930},[1854,3098,3099,3174],{},[1857,3100,3101,3103,3105,3107,3110,3112,3115,3117,3120,3123,3126,3128,3130,3132,3134,3136,3143,3146,3148,3150,3152,3154,3156,3158,3160,3162,3164,3166,3168,3170,3172],{},[1860,3102,2912],{},[1946,3104,1948],{},[1860,3106,3067],{"mathvariant":3066},[1860,3108,3109],{},"B",[1946,3111,1954],{},[1860,3113,3114],{},"E",[1946,3116,2741],{"separator":1871},[2090,3118],{"width":3119},"2em",[1860,3121,3122],{"mathvariant":2014},"vec",[1946,3124,3125],{},"⁡",[1946,3127,2541],{"stretchy":2540},[1860,3129,2912],{},[1946,3131,2552],{"stretchy":2540},[1946,3133,1948],{},[1946,3135,2541],{"stretchy":2540},[1936,3137,3138,3141],{},[1860,3139,3140],{},"I",[1860,3142,1862],{},[1946,3144,3145],{},"⊗",[1860,3147,3067],{"mathvariant":3066},[1946,3149,2552],{"stretchy":2540},[1860,3151,3122],{"mathvariant":2014},[1946,3153,3125],{},[1946,3155,2541],{"stretchy":2540},[1860,3157,3109],{},[1946,3159,2552],{"stretchy":2540},[1946,3161,1954],{},[1860,3163,3122],{"mathvariant":2014},[1946,3165,3125],{},[1946,3167,2541],{"stretchy":2540},[1860,3169,3114],{},[1946,3171,2552],{"stretchy":2540},[1860,3173,2015],{"mathvariant":2014},[1864,3175,3176],{"encoding":1866},"Y=\\mathcal XB+E,\n\\qquad\n\\operatorname{vec}(Y)\n=(I_k\\otimes\\mathcal X)\\operatorname{vec}(B)+\\operatorname{vec}(E).",[1841,3178,3180,3198,3221,3267,3328,3367],{"className":3179,"ariaHidden":1871},[1870],[1841,3181,3183,3186,3189,3192,3195],{"className":3182},[1875],[1841,3184],{"className":3185,"style":2924},[1879],[1841,3187,2912],{"className":3188,"style":2112},[1884,1885],[1841,3190],{"className":3191,"style":2091},[2090],[1841,3193,1948],{"className":3194},[2095],[1841,3196],{"className":3197,"style":2091},[2090],[1841,3199,3201,3205,3208,3212,3215,3218],{"className":3200},[1875],[1841,3202],{"className":3203,"style":3204},[1879],"height:0.7667em;vertical-align:-0.0833em;",[1841,3206,3067],{"className":3207,"style":3084},[1884,3083],[1841,3209,3109],{"className":3210,"style":3211},[1884,1885],"margin-right:0.0502em;",[1841,3213],{"className":3214,"style":2112},[2090],[1841,3216,1954],{"className":3217},[2116],[1841,3219],{"className":3220,"style":2112},[2090],[1841,3222,3224,3227,3231,3234,3238,3241,3249,3252,3255,3258,3261,3264],{"className":3223},[1875],[1841,3225],{"className":3226,"style":2565},[1879],[1841,3228,3114],{"className":3229,"style":3230},[1884,1885],"margin-right:0.0576em;",[1841,3232,2741],{"className":3233},[2827],[1841,3235],{"className":3236,"style":3237},[2090],"margin-right:2em;",[1841,3239],{"className":3240,"style":2831},[2090],[1841,3242,3245],{"className":3243},[3244],"mop",[1841,3246,3122],{"className":3247},[1884,3248],"mathrm",[1841,3250,2541],{"className":3251},[2572],[1841,3253,2912],{"className":3254,"style":2112},[1884,1885],[1841,3256,2552],{"className":3257},[2600],[1841,3259],{"className":3260,"style":2091},[2090],[1841,3262,1948],{"className":3263},[2095],[1841,3265],{"className":3266,"style":2091},[2090],[1841,3268,3270,3273,3276,3319,3322,3325],{"className":3269},[1875],[1841,3271],{"className":3272,"style":2565},[1879],[1841,3274,2541],{"className":3275},[2572],[1841,3277,3279,3283],{"className":3278},[1884],[1841,3280,3140],{"className":3281,"style":3282},[1884,1885],"margin-right:0.0785em;",[1841,3284,3286],{"className":3285},[2040],[1841,3287,3289,3311],{"className":3288},[2044,2045],[1841,3290,3292,3308],{"className":3291},[2049],[1841,3293,3296],{"className":3294,"style":3295},[2053],"height:0.3361em;",[1841,3297,3299,3302],{"style":3298},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1841,3300],{"className":3301,"style":2062},[2061],[1841,3303,3305],{"className":3304},[2066,2067,2068,2069],[1841,3306,1862],{"className":3307,"style":1886},[1884,1885,2069],[1841,3309,2077],{"className":3310},[2076],[1841,3312,3314],{"className":3313},[2049],[1841,3315,3317],{"className":3316,"style":2084},[2053],[1841,3318],{},[1841,3320],{"className":3321,"style":2112},[2090],[1841,3323,3145],{"className":3324},[2116],[1841,3326],{"className":3327,"style":2112},[2090],[1841,3329,3331,3334,3337,3340,3343,3349,3352,3355,3358,3361,3364],{"className":3330},[1875],[1841,3332],{"className":3333,"style":2565},[1879],[1841,3335,3067],{"className":3336,"style":3084},[1884,3083],[1841,3338,2552],{"className":3339},[2600],[1841,3341],{"className":3342,"style":2831},[2090],[1841,3344,3346],{"className":3345},[3244],[1841,3347,3122],{"className":3348},[1884,3248],[1841,3350,2541],{"className":3351},[2572],[1841,3353,3109],{"className":3354,"style":3211},[1884,1885],[1841,3356,2552],{"className":3357},[2600],[1841,3359],{"className":3360,"style":2112},[2090],[1841,3362,1954],{"className":3363},[2116],[1841,3365],{"className":3366,"style":2112},[2090],[1841,3368,3370,3373,3379,3382,3385,3388],{"className":3369},[1875],[1841,3371],{"className":3372,"style":2565},[1879],[1841,3374,3376],{"className":3375},[3244],[1841,3377,3122],{"className":3378},[1884,3248],[1841,3380,2541],{"className":3381},[2572],[1841,3383,3114],{"className":3384,"style":3230},[1884,1885],[1841,3386,2552],{"className":3387},[2600],[1841,3389,2015],{"className":3390},[1884],[1798,3392,3393],{},"Equation-by-equation OLS and multivariate least squares give the same coefficient estimates when every equation has the same regressors. The innovation covariance still matters for joint inference.",[1793,3395,3397],{"id":3396},"_2-a-two-variable-shock-calculation","2. 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",[1841,6310,6312,6346],{"className":6311},[1844],[1841,6313,6315],{"className":6314},[1848],[1850,6316,6317],{"xmlns":1852},[1854,6318,6319,6343],{},[1857,6320,6321,6327,6329,6331,6333,6339,6341],{},[1936,6322,6323,6325],{},[1860,6324,5938],{},[1961,6326,1963],{},[1946,6328,1948],{},[1946,6330,1981],{},[1946,6332,1948],{},[1936,6334,6335,6337],{},[1860,6336,5938],{},[1860,6338,1798],{},[1946,6340,1948],{},[1961,6342,3444],{},[1864,6344,6345],{"encoding":1866},"b_1=\\cdots=b_p=0",[1841,6347,6349,6405,6424,6480],{"className":6348,"ariaHidden":1871},[1870],[1841,6350,6352,6356,6396,6399,6402],{"className":6351},[1875],[1841,6353],{"className":6354,"style":6355},[1879],"height:0.8444em;vertical-align:-0.15em;",[1841,6357,6359,6362],{"className":6358},[1884],[1841,6360,5938],{"className":6361},[1884,1885],[1841,6363,6365],{"className":6364},[2040],[1841,6366,6368,6388],{"className":6367},[2044,2045],[1841,6369,6371,6385],{"className":6370},[2049],[1841,6372,6374],{"className":6373,"style":2145},[2053],[1841,6375,6376,6379],{"style":2148},[1841,6377],{"className":6378,"style":2062},[2061],[1841,6380,6382],{"className":6381},[2066,2067,2068,2069],[1841,6383,1963],{"className":6384},[1884,2069],[1841,6386,2077],{"className":6387},[2076],[1841,6389,6391],{"className":6390},[2049],[1841,6392,6394],{"className":6393,"style":2084},[2053],[1841,6395],{},[1841,6397],{"className":6398,"style":2091},[2090],[1841,6400,1948],{"className":6401},[2095],[1841,6403],{"className":6404,"style":2091},[2090],[1841,6406,6408,6412,6415,6418,6421],{"className":6407},[1875],[1841,6409],{"className":6410,"style":6411},[1879],"height:0.3669em;",[1841,6413,1981],{"className":6414},[2237],[1841,6416],{"className":6417,"style":2091},[2090],[1841,6419,1948],{"className":6420},[2095],[1841,6422],{"className":6423,"style":2091},[2090],[1841,6425,6427,6431,6471,6474,6477],{"className":6426},[1875],[1841,6428],{"className":6429,"style":6430},[1879],"height:0.9805em;vertical-align:-0.2861em;",[1841,6432,6434,6437],{"className":6433},[1884],[1841,6435,5938],{"className":6436},[1884,1885],[1841,6438,6440],{"className":6439},[2040],[1841,6441,6443,6463],{"className":6442},[2044,2045],[1841,6444,6446,6460],{"className":6445},[2049],[1841,6447,6449],{"className":6448,"style":2272},[2053],[1841,6450,6451,6454],{"style":2148},[1841,6452],{"className":6453,"style":2062},[2061],[1841,6455,6457],{"className":6456},[2066,2067,2068,2069],[1841,6458,1798],{"className":6459},[1884,1885,2069],[1841,6461,2077],{"className":6462},[2076],[1841,6464,6466],{"className":6465},[2049],[1841,6467,6469],{"className":6468,"style":2293},[2053],[1841,6470],{},[1841,6472],{"className":6473,"style":2091},[2090],[1841,6475,1948],{"className":6476},[2095],[1841,6478],{"className":6479,"style":2091},[2090],[1841,6481,6483,6486],{"className":6482},[1875],[1841,6484],{"className":6485,"style":2651},[1879],[1841,6487,3444],{"className":6488},[1884],[4162,6490,6491,6504],{},[4165,6492,6493],{},[4168,6494,6495,6498,6501],{},[4171,6496,6497],{},"Result",[4171,6499,6500],{},"Supported claim",[4171,6502,6503],{},"Not supported without more design",[4209,6505,6506,6602],{},[4168,6507,6508,6511,6542],{},[4214,6509,6510],{},"reject restrictions",[4214,6512,6513,6541],{},[1841,6514,6516,6529],{"className":6515},[1844],[1841,6517,6519],{"className":6518},[1848],[1850,6520,6521],{"xmlns":1852},[1854,6522,6523,6527],{},[1857,6524,6525],{},[1860,6526,5474],{},[1864,6528,5474],{"encoding":1866},[1841,6530,6532],{"className":6531,"ariaHidden":1871},[1870],[1841,6533,6535,6538],{"className":6534},[1875],[1841,6536],{"className":6537,"style":5486},[1879],[1841,6539,5474],{"className":6540},[1884,1885]," has incremental lagged predictive content in this specification",[4214,6543,6544,6545,6573,6574],{},"intervention on ",[1841,6546,6548,6561],{"className":6547},[1844],[1841,6549,6551],{"className":6550},[1848],[1850,6552,6553],{"xmlns":1852},[1854,6554,6555,6559],{},[1857,6556,6557],{},[1860,6558,5474],{},[1864,6560,5474],{"encoding":1866},[1841,6562,6564],{"className":6563,"ariaHidden":1871},[1870],[1841,6565,6567,6570],{"className":6566},[1875],[1841,6568],{"className":6569,"style":5486},[1879],[1841,6571,5474],{"className":6572},[1884,1885]," changes ",[1841,6575,6577,6590],{"className":6576},[1844],[1841,6578,6580],{"className":6579},[1848],[1850,6581,6582],{"xmlns":1852},[1854,6583,6584,6588],{},[1857,6585,6586],{},[1860,6587,1941],{},[1864,6589,1941],{"encoding":1866},[1841,6591,6593],{"className":6592,"ariaHidden":1871},[1870],[1841,6594,6596,6599],{"className":6595},[1875],[1841,6597],{"className":6598,"style":1913},[1879],[1841,6600,1941],{"className":6601,"style":5519},[1884,1885],[4168,6603,6604,6607,6610],{},[4214,6605,6606],{},"fail to reject",[4214,6608,6609],{},"sample does not show incremental linear content",[4214,6611,6612,6640],{},[1841,6613,6615,6628],{"className":6614},[1844],[1841,6616,6618],{"className":6617},[1848],[1850,6619,6620],{"xmlns":1852},[1854,6621,6622,6626],{},[1857,6623,6624],{},[1860,6625,5474],{},[1864,6627,5474],{"encoding":1866},[1841,6629,6631],{"className":6630,"ariaHidden":1871},[1870],[1841,6632,6634,6637],{"className":6633},[1875],[1841,6635],{"className":6636,"style":5486},[1879],[1841,6638,5474],{"className":6639},[1884,1885]," is irrelevant in all horizons\u002Fregimes",[1798,6642,6643],{},"Omitted common causes, measurement timing, anticipation, aggregation, and regime changes can all alter the result.",[1793,6645,6647],{"id":6646},"_5-seasonality-is-not-one-phenomenon","5. Seasonality is not one phenomenon",[1798,6649,6650,6651,6704],{},"For monthly data, ",[1841,6652,6654,6674],{"className":6653},[1844],[1841,6655,6657],{"className":6656},[1848],[1850,6658,6659],{"xmlns":1852},[1854,6660,6661,6671],{},[1857,6662,6663,6666,6668],{},[1860,6664,6665],{},"s",[1946,6667,1948],{},[1961,6669,6670],{},"12",[1864,6672,6673],{"encoding":1866},"s=12",[1841,6675,6677,6695],{"className":6676,"ariaHidden":1871},[1870],[1841,6678,6680,6683,6686,6689,6692],{"className":6679},[1875],[1841,6681],{"className":6682,"style":5486},[1879],[1841,6684,6665],{"className":6685},[1884,1885],[1841,6687],{"className":6688,"style":2091},[2090],[1841,6690,1948],{"className":6691},[2095],[1841,6693],{"className":6694,"style":2091},[2090],[1841,6696,6698,6701],{"className":6697},[1875],[1841,6699],{"className":6700,"style":2651},[1879],[1841,6702,6670],{"className":6703},[1884],". Choose the smallest assumption that matches the mechanism:",[4162,6706,6707,6720],{},[4165,6708,6709],{},[4168,6710,6711,6714,6717],{},[4171,6712,6713],{},"Mechanism",[4171,6715,6716],{},"Model device",[4171,6718,6719],{},"Interpretation",[4209,6721,6722,6733,6744,6896],{},[4168,6723,6724,6727,6730],{},[4214,6725,6726],{},"stable January effect",[4214,6728,6729],{},"11 month indicators",[4214,6731,6732],{},"deterministic seasonal mean",[4168,6734,6735,6738,6741],{},[4214,6736,6737],{},"smoothly changing recurring pattern",[4214,6739,6740],{},"Fourier sine\u002Fcosine terms",[4214,6742,6743],{},"parsimonious deterministic cycle",[4168,6745,6746,6749,6893],{},[4214,6747,6748],{},"seasonal shocks accumulate",[4214,6750,6751],{},[1841,6752,6754,6786],{"className":6753},[1844],[1841,6755,6757],{"className":6756},[1848],[1850,6758,6759],{"xmlns":1852},[1854,6760,6761,6783],{},[1857,6762,6763,6765,6767,6769,6775,6777],{},[1946,6764,2541],{"stretchy":2540},[1961,6766,1963],{},[1946,6768,1974],{},[3841,6770,6771,6773],{},[1860,6772,3109],{},[1961,6774,6670],{},[1946,6776,2552],{"stretchy":2540},[1936,6778,6779,6781],{},[1860,6780,3067],{},[1860,6782,1944],{},[1864,6784,6785],{"encoding":1866},"(1-B^{12})X_t",[1841,6787,6789,6810],{"className":6788,"ariaHidden":1871},[1870],[1841,6790,6792,6795,6798,6801,6804,6807],{"className":6791},[1875],[1841,6793],{"className":6794,"style":2565},[1879],[1841,6796,2541],{"className":6797},[2572],[1841,6799,1963],{"className":6800},[1884],[1841,6802],{"className":6803,"style":2112},[2090],[1841,6805,1974],{"className":6806},[2116],[1841,6808],{"className":6809,"style":2112},[2090],[1841,6811,6813,6817,6850,6853],{"className":6812},[1875],[1841,6814],{"className":6815,"style":6816},[1879],"height:1.0641em;vertical-align:-0.25em;",[1841,6818,6820,6823],{"className":6819},[1884],[1841,6821,3109],{"className":6822,"style":3211},[1884,1885],[1841,6824,6826],{"className":6825},[2040],[1841,6827,6829],{"className":6828},[2044],[1841,6830,6832],{"className":6831},[2049],[1841,6833,6836],{"className":6834,"style":6835},[2053],"height:0.8141em;",[1841,6837,6838,6841],{"style":2802},[1841,6839],{"className":6840,"style":2062},[2061],[1841,6842,6844],{"className":6843},[2066,2067,2068,2069],[1841,6845,6847],{"className":6846},[1884,2069],[1841,6848,6670],{"className":6849},[1884,2069],[1841,6851,2552],{"className":6852},[2600],[1841,6854,6856,6859],{"className":6855},[1884],[1841,6857,3067],{"className":6858,"style":3282},[1884,1885],[1841,6860,6862],{"className":6861},[2040],[1841,6863,6865,6885],{"className":6864},[2044,2045],[1841,6866,6868,6882],{"className":6867},[2049],[1841,6869,6871],{"className":6870,"style":2054},[2053],[1841,6872,6873,6876],{"style":3298},[1841,6874],{"className":6875,"style":2062},[2061],[1841,6877,6879],{"className":6878},[2066,2067,2068,2069],[1841,6880,1944],{"className":6881},[1884,1885,2069],[1841,6883,2077],{"className":6884},[2076],[1841,6886,6888],{"className":6887},[2049],[1841,6889,6891],{"className":6890,"style":2084},[2053],[1841,6892],{},[4214,6894,6895],{},"seasonal stochastic trend",[4168,6897,6898,6901,6904],{},[4214,6899,6900],{},"dependence remains at seasonal lags",[4214,6902,6903],{},"seasonal AR\u002FMA terms",[4214,6905,6906],{},"shocks propagate across years",[1798,6908,6909],{},"A multiplicative SARIMA model is written",[1841,6911,6913],{"className":6912},[1921],[1841,6914,6916,7022],{"className":6915},[1844],[1841,6917,6919],{"className":6918},[1848],[1850,6920,6921],{"xmlns":1852,"display":1930},[1854,6922,6923,7019],{},[1857,6924,6925,6928,6930,6932,6934,6937,6939,6945,6947,6949,6951,6953,6955,6962,6964,6966,6968,6974,6981,6987,6989,6992,6994,6996,6998,7001,7003,7009,7011,7017],{},[1860,6926,6927],{},"ϕ",[1946,6929,2541],{"stretchy":2540},[1860,6931,3109],{},[1946,6933,2552],{"stretchy":2540},[1860,6935,6936],{"mathvariant":2014},"Φ",[1946,6938,2541],{"stretchy":2540},[3841,6940,6941,6943],{},[1860,6942,3109],{},[1961,6944,6670],{},[1946,6946,2552],{"stretchy":2540},[1946,6948,2541],{"stretchy":2540},[1961,6950,1963],{},[1946,6952,1974],{},[1860,6954,3109],{},[3841,6956,6957,6959],{},[1946,6958,2552],{"stretchy":2540},[1860,6960,6961],{},"d",[1946,6963,2541],{"stretchy":2540},[1961,6965,1963],{},[1946,6967,1974],{},[3841,6969,6970,6972],{},[1860,6971,3109],{},[1961,6973,6670],{},[3841,6975,6976,6978],{},[1946,6977,2552],{"stretchy":2540},[1860,6979,6980],{},"D",[1936,6982,6983,6985],{},[1860,6984,3067],{},[1860,6986,1944],{},[1946,6988,1948],{},[1860,6990,6991],{},"θ",[1946,6993,2541],{"stretchy":2540},[1860,6995,3109],{},[1946,6997,2552],{"stretchy":2540},[1860,6999,7000],{"mathvariant":2014},"Θ",[1946,7002,2541],{"stretchy":2540},[3841,7004,7005,7007],{},[1860,7006,3109],{},[1961,7008,6670],{},[1946,7010,2552],{"stretchy":2540},[1936,7012,7013,7015],{},[1860,7014,2009],{},[1860,7016,1944],{},[1860,7018,2015],{"mathvariant":2014},[1864,7020,7021],{"encoding":1866},"\\phi(B)\\Phi(B^{12})(1-B)^d(1-B^{12})^D X_t\n=\\theta(B)\\Theta(B^{12})\\varepsilon_t.",[1841,7023,7025,7100,7154,7273],{"className":7024,"ariaHidden":1871},[1870],[1841,7026,7028,7032,7035,7038,7041,7044,7047,7050,7082,7085,7088,7091,7094,7097],{"className":7027},[1875],[1841,7029],{"className":7030,"style":7031},[1879],"height:1.1141em;vertical-align:-0.25em;",[1841,7033,6927],{"className":7034},[1884,1885],[1841,7036,2541],{"className":7037},[2572],[1841,7039,3109],{"className":7040,"style":3211},[1884,1885],[1841,7042,2552],{"className":7043},[2600],[1841,7045,6936],{"className":7046},[1884],[1841,7048,2541],{"className":7049},[2572],[1841,7051,7053,7056],{"className":7052},[1884],[1841,7054,3109],{"className":7055,"style":3211},[1884,1885],[1841,7057,7059],{"className":7058},[2040],[1841,7060,7062],{"className":7061},[2044],[1841,7063,7065],{"className":7064},[2049],[1841,7066,7068],{"className":7067,"style":5382},[2053],[1841,7069,7070,7073],{"style":4867},[1841,7071],{"className":7072,"style":2062},[2061],[1841,7074,7076],{"className":7075},[2066,2067,2068,2069],[1841,7077,7079],{"className":7078},[1884,2069],[1841,7080,6670],{"className":7081},[1884,2069],[1841,7083,2552],{"className":7084},[2600],[1841,7086,2541],{"className":7087},[2572],[1841,7089,1963],{"className":7090},[1884],[1841,7092],{"className":7093,"style":2112},[2090],[1841,7095,1974],{"className":7096},[2116],[1841,7098],{"className":7099,"style":2112},[2090],[1841,7101,7103,7107,7110,7139,7142,7145,7148,7151],{"className":7102},[1875],[1841,7104],{"className":7105,"style":7106},[1879],"height:1.1491em;vertical-align:-0.25em;",[1841,7108,3109],{"className":7109,"style":3211},[1884,1885],[1841,7111,7113,7116],{"className":7112},[2600],[1841,7114,2552],{"className":7115},[2600],[1841,7117,7119],{"className":7118},[2040],[1841,7120,7122],{"className":7121},[2044],[1841,7123,7125],{"className":7124},[2049],[1841,7126,7128],{"className":7127,"style":4864},[2053],[1841,7129,7130,7133],{"style":4867},[1841,7131],{"className":7132,"style":2062},[2061],[1841,7134,7136],{"className":7135},[2066,2067,2068,2069],[1841,7137,6961],{"className":7138},[1884,1885,2069],[1841,7140,2541],{"className":7141},[2572],[1841,7143,1963],{"className":7144},[1884],[1841,7146],{"className":7147,"style":2112},[2090],[1841,7149,1974],{"className":7150},[2116],[1841,7152],{"className":7153,"style":2112},[2090],[1841,7155,7157,7161,7193,7224,7264,7267,7270],{"className":7156},[1875],[1841,7158],{"className":7159,"style":7160},[1879],"height:1.1413em;vertical-align:-0.25em;",[1841,7162,7164,7167],{"className":7163},[1884],[1841,7165,3109],{"className":7166,"style":3211},[1884,1885],[1841,7168,7170],{"className":7169},[2040],[1841,7171,7173],{"className":7172},[2044],[1841,7174,7176],{"className":7175},[2049],[1841,7177,7179],{"className":7178,"style":5382},[2053],[1841,7180,7181,7184],{"style":4867},[1841,7182],{"className":7183,"style":2062},[2061],[1841,7185,7187],{"className":7186},[2066,2067,2068,2069],[1841,7188,7190],{"className":7189},[1884,2069],[1841,7191,6670],{"className":7192},[1884,2069],[1841,7194,7196,7199],{"className":7195},[2600],[1841,7197,2552],{"className":7198},[2600],[1841,7200,7202],{"className":7201},[2040],[1841,7203,7205],{"className":7204},[2044],[1841,7206,7208],{"className":7207},[2049],[1841,7209,7212],{"className":7210,"style":7211},[2053],"height:0.8913em;",[1841,7213,7214,7217],{"style":4867},[1841,7215],{"className":7216,"style":2062},[2061],[1841,7218,7220],{"className":7219},[2066,2067,2068,2069],[1841,7221,6980],{"className":7222,"style":7223},[1884,1885,2069],"margin-right:0.0278em;",[1841,7225,7227,7230],{"className":7226},[1884],[1841,7228,3067],{"className":7229,"style":3282},[1884,1885],[1841,7231,7233],{"className":7232},[2040],[1841,7234,7236,7256],{"className":7235},[2044,2045],[1841,7237,7239,7253],{"className":7238},[2049],[1841,7240,7242],{"className":7241,"style":2054},[2053],[1841,7243,7244,7247],{"style":3298},[1841,7245],{"className":7246,"style":2062},[2061],[1841,7248,7250],{"className":7249},[2066,2067,2068,2069],[1841,7251,1944],{"className":7252},[1884,1885,2069],[1841,7254,2077],{"className":7255},[2076],[1841,7257,7259],{"className":7258},[2049],[1841,7260,7262],{"className":7261,"style":2084},[2053],[1841,7263],{},[1841,7265],{"className":7266,"style":2091},[2090],[1841,7268,1948],{"className":7269},[2095],[1841,7271],{"className":7272,"style":2091},[2090],[1841,7274,7276,7279,7282,7285,7288,7291,7294,7297,7329,7332,7372],{"className":7275},[1875],[1841,7277],{"className":7278,"style":7031},[1879],[1841,7280,6991],{"className":7281,"style":7223},[1884,1885],[1841,7283,2541],{"className":7284},[2572],[1841,7286,3109],{"className":7287,"style":3211},[1884,1885],[1841,7289,2552],{"className":7290},[2600],[1841,7292,7000],{"className":7293},[1884],[1841,7295,2541],{"className":7296},[2572],[1841,7298,7300,7303],{"className":7299},[1884],[1841,7301,3109],{"className":7302,"style":3211},[1884,1885],[1841,7304,7306],{"className":7305},[2040],[1841,7307,7309],{"className":7308},[2044],[1841,7310,7312],{"className":7311},[2049],[1841,7313,7315],{"className":7314,"style":5382},[2053],[1841,7316,7317,7320],{"style":4867},[1841,7318],{"className":7319,"style":2062},[2061],[1841,7321,7323],{"className":7322},[2066,2067,2068,2069],[1841,7324,7326],{"className":7325},[1884,2069],[1841,7327,6670],{"className":7328},[1884,2069],[1841,7330,2552],{"className":7331},[2600],[1841,7333,7335,7338],{"className":7334},[1884],[1841,7336,2009],{"className":7337},[1884,1885],[1841,7339,7341],{"className":7340},[2040],[1841,7342,7344,7364],{"className":7343},[2044,2045],[1841,7345,7347,7361],{"className":7346},[2049],[1841,7348,7350],{"className":7349,"style":2054},[2053],[1841,7351,7352,7355],{"style":2148},[1841,7353],{"className":7354,"style":2062},[2061],[1841,7356,7358],{"className":7357},[2066,2067,2068,2069],[1841,7359,1944],{"className":7360},[1884,1885,2069],[1841,7362,2077],{"className":7363},[2076],[1841,7365,7367],{"className":7366},[2049],[1841,7368,7370],{"className":7369,"style":2084},[2053],[1841,7371],{},[1841,7373,2015],{"className":7374},[1884],[2517,7376,7378],{"id":7377},"reading-the-notation","Reading the notation",[1798,7380,7381,7382,7532],{},"SARIMA",[1841,7383,7385,7429],{"className":7384},[1844],[1841,7386,7388],{"className":7387},[1848],[1850,7389,7390],{"xmlns":1852},[1854,7391,7392,7426],{},[1857,7393,7394,7396,7398,7400,7402,7404,7406,7408,7410,7412,7414,7416,7418,7420],{},[1946,7395,2541],{"stretchy":2540},[1961,7397,1963],{},[1946,7399,2741],{"separator":1871},[1961,7401,1963],{},[1946,7403,2741],{"separator":1871},[1961,7405,1963],{},[1946,7407,2552],{"stretchy":2540},[1946,7409,2541],{"stretchy":2540},[1961,7411,3444],{},[1946,7413,2741],{"separator":1871},[1961,7415,1963],{},[1946,7417,2741],{"separator":1871},[1961,7419,1963],{},[1936,7421,7422,7424],{},[1946,7423,2552],{"stretchy":2540},[1961,7425,6670],{},[1864,7427,7428],{"encoding":1866},"(1,1,1)(0,1,1)_{12}",[1841,7430,7432],{"className":7431,"ariaHidden":1871},[1870],[1841,7433,7435,7438,7441,7444,7447,7450,7453,7456,7459,7462,7465,7468,7471,7474,7477,7480,7483,7486,7489],{"className":7434},[1875],[1841,7436],{"className":7437,"style":2565},[1879],[1841,7439,2541],{"className":7440},[2572],[1841,7442,1963],{"className":7443},[1884],[1841,7445,2741],{"className":7446},[2827],[1841,7448],{"className":7449,"style":2831},[2090],[1841,7451,1963],{"className":7452},[1884],[1841,7454,2741],{"className":7455},[2827],[1841,7457],{"className":7458,"style":2831},[2090],[1841,7460,1963],{"className":7461},[1884],[1841,7463,2552],{"className":7464},[2600],[1841,7466,2541],{"className":7467},[2572],[1841,7469,3444],{"className":7470},[1884],[1841,7472,2741],{"className":7473},[2827],[1841,7475],{"className":7476,"style":2831},[2090],[1841,7478,1963],{"className":7479},[1884],[1841,7481,2741],{"className":7482},[2827],[1841,7484],{"className":7485,"style":2831},[2090],[1841,7487,1963],{"className":7488},[1884],[1841,7490,7492,7495],{"className":7491},[2600],[1841,7493,2552],{"className":7494},[2600],[1841,7496,7498],{"className":7497},[2040],[1841,7499,7501,7524],{"className":7500},[2044,2045],[1841,7502,7504,7521],{"className":7503},[2049],[1841,7505,7507],{"className":7506,"style":2145},[2053],[1841,7508,7509,7512],{"style":2148},[1841,7510],{"className":7511,"style":2062},[2061],[1841,7513,7515],{"className":7514},[2066,2067,2068,2069],[1841,7516,7518],{"className":7517},[1884,2069],[1841,7519,6670],{"className":7520},[1884,2069],[1841,7522,2077],{"className":7523},[2076],[1841,7525,7527],{"className":7526},[2049],[1841,7528,7530],{"className":7529,"style":2084},[2053],[1841,7531],{}," means:",[1809,7534,7535,7538,7541],{},[1812,7536,7537],{},"one ordinary difference and one annual seasonal difference;",[1812,7539,7540],{},"one non-seasonal AR and MA term;",[1812,7542,7543],{},"one seasonal MA term linking shocks 12 months apart.",[1798,7545,7546],{},"It does not mean the order is appropriate. Compare it with seasonal naive and inspect whether double differencing created strong negative autocorrelation.",[1793,7548,7550],{"id":7549},"_6-cointegration-preserves-a-long-run-relation","6. Cointegration preserves a long-run relation",[1798,7552,7553,7554,7624],{},"Suppose ",[1841,7555,7557,7575],{"className":7556},[1844],[1841,7558,7560],{"className":7559},[1848],[1850,7561,7562],{"xmlns":1852},[1854,7563,7564,7572],{},[1857,7565,7566],{},[1936,7567,7568,7570],{},[1860,7569,5474],{},[1860,7571,1944],{},[1864,7573,7574],{"encoding":1866},"x_t",[1841,7576,7578],{"className":7577,"ariaHidden":1871},[1870],[1841,7579,7581,7584],{"className":7580},[1875],[1841,7582],{"className":7583,"style":5839},[1879],[1841,7585,7587,7590],{"className":7586},[1884],[1841,7588,5474],{"className":7589},[1884,1885],[1841,7591,7593],{"className":7592},[2040],[1841,7594,7596,7616],{"className":7595},[2044,2045],[1841,7597,7599,7613],{"className":7598},[2049],[1841,7600,7602],{"className":7601,"style":2054},[2053],[1841,7603,7604,7607],{"style":2148},[1841,7605],{"className":7606,"style":2062},[2061],[1841,7608,7610],{"className":7609},[2066,2067,2068,2069],[1841,7611,1944],{"className":7612},[1884,1885,2069],[1841,7614,2077],{"className":7615},[2076],[1841,7617,7619],{"className":7618},[2049],[1841,7620,7622],{"className":7621,"style":2084},[2053],[1841,7623],{}," is a random walk and",[1841,7626,7628],{"className":7627},[1921],[1841,7629,7631,7669],{"className":7630},[1844],[1841,7632,7634],{"className":7633},[1848],[1850,7635,7636],{"xmlns":1852,"display":1930},[1854,7637,7638,7666],{},[1857,7639,7640,7646,7648,7650,7656,7658,7664],{},[1936,7641,7642,7644],{},[1860,7643,1941],{},[1860,7645,1944],{},[1946,7647,1948],{},[1961,7649,2735],{},[1936,7651,7652,7654],{},[1860,7653,5474],{},[1860,7655,1944],{},[1946,7657,1954],{},[1936,7659,7660,7662],{},[1860,7661,5635],{},[1860,7663,1944],{},[1946,7665,2741],{"separator":1871},[1864,7667,7668],{"encoding":1866},"y_t=2x_t+u_t,",[1841,7670,7672,7727,7786],{"className":7671,"ariaHidden":1871},[1870],[1841,7673,7675,7678,7718,7721,7724],{"className":7674},[1875],[1841,7676],{"className":7677,"style":1913},[1879],[1841,7679,7681,7684],{"className":7680},[1884],[1841,7682,1941],{"className":7683,"style":5519},[1884,1885],[1841,7685,7687],{"className":7686},[2040],[1841,7688,7690,7710],{"className":7689},[2044,2045],[1841,7691,7693,7707],{"className":7692},[2049],[1841,7694,7696],{"className":7695,"style":2054},[2053],[1841,7697,7698,7701],{"style":5670},[1841,7699],{"className":7700,"style":2062},[2061],[1841,7702,7704],{"className":7703},[2066,2067,2068,2069],[1841,7705,1944],{"className":7706},[1884,1885,2069],[1841,7708,2077],{"className":7709},[2076],[1841,7711,7713],{"className":7712},[2049],[1841,7714,7716],{"className":7715,"style":2084},[2053],[1841,7717],{},[1841,7719],{"className":7720,"style":2091},[2090],[1841,7722,1948],{"className":7723},[2095],[1841,7725],{"className":7726,"style":2091},[2090],[1841,7728,7730,7734,7737,7777,7780,7783],{"className":7729},[1875],[1841,7731],{"className":7732,"style":7733},[1879],"height:0.7944em;vertical-align:-0.15em;",[1841,7735,2735],{"className":7736},[1884],[1841,7738,7740,7743],{"className":7739},[1884],[1841,7741,5474],{"className":7742},[1884,1885],[1841,7744,7746],{"className":7745},[2040],[1841,7747,7749,7769],{"className":7748},[2044,2045],[1841,7750,7752,7766],{"className":7751},[2049],[1841,7753,7755],{"className":7754,"style":2054},[2053],[1841,7756,7757,7760],{"style":2148},[1841,7758],{"className":7759,"style":2062},[2061],[1841,7761,7763],{"className":7762},[2066,2067,2068,2069],[1841,7764,1944],{"className":7765},[1884,1885,2069],[1841,7767,2077],{"className":7768},[2076],[1841,7770,7772],{"className":7771},[2049],[1841,7773,7775],{"className":7774,"style":2084},[2053],[1841,7776],{},[1841,7778],{"className":7779,"style":2112},[2090],[1841,7781,1954],{"className":7782},[2116],[1841,7784],{"className":7785,"style":2112},[2090],[1841,7787,7789,7792,7832],{"className":7788},[1875],[1841,7790],{"className":7791,"style":1913},[1879],[1841,7793,7795,7798],{"className":7794},[1884],[1841,7796,5635],{"className":7797},[1884,1885],[1841,7799,7801],{"className":7800},[2040],[1841,7802,7804,7824],{"className":7803},[2044,2045],[1841,7805,7807,7821],{"className":7806},[2049],[1841,7808,7810],{"className":7809,"style":2054},[2053],[1841,7811,7812,7815],{"style":2148},[1841,7813],{"className":7814,"style":2062},[2061],[1841,7816,7818],{"className":7817},[2066,2067,2068,2069],[1841,7819,1944],{"className":7820},[1884,1885,2069],[1841,7822,2077],{"className":7823},[2076],[1841,7825,7827],{"className":7826},[2049],[1841,7828,7830],{"className":7829,"style":2084},[2053],[1841,7831],{},[1841,7833,2741],{"className":7834},[2827],[1798,7836,7837,7838,7908],{},"where ",[1841,7839,7841,7859],{"className":7840},[1844],[1841,7842,7844],{"className":7843},[1848],[1850,7845,7846],{"xmlns":1852},[1854,7847,7848,7856],{},[1857,7849,7850],{},[1936,7851,7852,7854],{},[1860,7853,5635],{},[1860,7855,1944],{},[1864,7857,7858],{"encoding":1866},"u_t",[1841,7860,7862],{"className":7861,"ariaHidden":1871},[1870],[1841,7863,7865,7868],{"className":7864},[1875],[1841,7866],{"className":7867,"style":5839},[1879],[1841,7869,7871,7874],{"className":7870},[1884],[1841,7872,5635],{"className":7873},[1884,1885],[1841,7875,7877],{"className":7876},[2040],[1841,7878,7880,7900],{"className":7879},[2044,2045],[1841,7881,7883,7897],{"className":7882},[2049],[1841,7884,7886],{"className":7885,"style":2054},[2053],[1841,7887,7888,7891],{"style":2148},[1841,7889],{"className":7890,"style":2062},[2061],[1841,7892,7894],{"className":7893},[2066,2067,2068,2069],[1841,7895,1944],{"className":7896},[1884,1885,2069],[1841,7898,2077],{"className":7899},[2076],[1841,7901,7903],{"className":7902},[2049],[1841,7904,7906],{"className":7905,"style":2084},[2053],[1841,7907],{}," is stationary. Both levels are non-stationary, but",[1841,7910,7912],{"className":7911},[1921],[1841,7913,7915,7951],{"className":7914},[1844],[1841,7916,7918],{"className":7917},[1848],[1850,7919,7920],{"xmlns":1852,"display":1930},[1854,7921,7922,7948],{},[1857,7923,7924,7930,7932,7934,7940,7942],{},[1936,7925,7926,7928],{},[1860,7927,1941],{},[1860,7929,1944],{},[1946,7931,1974],{},[1961,7933,2735],{},[1936,7935,7936,7938],{},[1860,7937,5474],{},[1860,7939,1944],{},[1946,7941,1948],{},[1936,7943,7944,7946],{},[1860,7945,5635],{},[1860,7947,1944],{},[1864,7949,7950],{"encoding":1866},"y_t-2x_t=u_t",[1841,7952,7954,8010,8068],{"className":7953,"ariaHidden":1871},[1870],[1841,7955,7957,7961,8001,8004,8007],{"className":7956},[1875],[1841,7958],{"className":7959,"style":7960},[1879],"height:0.7778em;vertical-align:-0.1944em;",[1841,7962,7964,7967],{"className":7963},[1884],[1841,7965,1941],{"className":7966,"style":5519},[1884,1885],[1841,7968,7970],{"className":7969},[2040],[1841,7971,7973,7993],{"className":7972},[2044,2045],[1841,7974,7976,7990],{"className":7975},[2049],[1841,7977,7979],{"className":7978,"style":2054},[2053],[1841,7980,7981,7984],{"style":5670},[1841,7982],{"className":7983,"style":2062},[2061],[1841,7985,7987],{"className":7986},[2066,2067,2068,2069],[1841,7988,1944],{"className":7989},[1884,1885,2069],[1841,7991,2077],{"className":7992},[2076],[1841,7994,7996],{"className":7995},[2049],[1841,7997,7999],{"className":7998,"style":2084},[2053],[1841,8000],{},[1841,8002],{"className":8003,"style":2112},[2090],[1841,8005,1974],{"className":8006},[2116],[1841,8008],{"className":8009,"style":2112},[2090],[1841,8011,8013,8016,8019,8059,8062,8065],{"className":8012},[1875],[1841,8014],{"className":8015,"style":7733},[1879],[1841,8017,2735],{"className":8018},[1884],[1841,8020,8022,8025],{"className":8021},[1884],[1841,8023,5474],{"className":8024},[1884,1885],[1841,8026,8028],{"className":8027},[2040],[1841,8029,8031,8051],{"className":8030},[2044,2045],[1841,8032,8034,8048],{"className":8033},[2049],[1841,8035,8037],{"className":8036,"style":2054},[2053],[1841,8038,8039,8042],{"style":2148},[1841,8040],{"className":8041,"style":2062},[2061],[1841,8043,8045],{"className":8044},[2066,2067,2068,2069],[1841,8046,1944],{"className":8047},[1884,1885,2069],[1841,8049,2077],{"className":8050},[2076],[1841,8052,8054],{"className":8053},[2049],[1841,8055,8057],{"className":8056,"style":2084},[2053],[1841,8058],{},[1841,8060],{"className":8061,"style":2091},[2090],[1841,8063,1948],{"className":8064},[2095],[1841,8066],{"className":8067,"style":2091},[2090],[1841,8069,8071,8074],{"className":8070},[1875],[1841,8072],{"className":8073,"style":5839},[1879],[1841,8075,8077,8080],{"className":8076},[1884],[1841,8078,5635],{"className":8079},[1884,1885],[1841,8081,8083],{"className":8082},[2040],[1841,8084,8086,8106],{"className":8085},[2044,2045],[1841,8087,8089,8103],{"className":8088},[2049],[1841,8090,8092],{"className":8091,"style":2054},[2053],[1841,8093,8094,8097],{"style":2148},[1841,8095],{"className":8096,"style":2062},[2061],[1841,8098,8100],{"className":8099},[2066,2067,2068,2069],[1841,8101,1944],{"className":8102},[1884,1885,2069],[1841,8104,2077],{"className":8105},[2076],[1841,8107,8109],{"className":8108},[2049],[1841,8110,8112],{"className":8111,"style":2084},[2053],[1841,8113],{},[1798,8115,8116,8117,2015],{},"is stationary. The pair is cointegrated. Differencing both series independently would hide the equilibrium error ",[1841,8118,8120,8160],{"className":8119},[1844],[1841,8121,8123],{"className":8122},[1848],[1850,8124,8125],{"xmlns":1852},[1854,8126,8127,8157],{},[1857,8128,8129,8141,8143,8145],{},[1936,8130,8131,8133],{},[1860,8132,1941],{},[1857,8134,8135,8137,8139],{},[1860,8136,1944],{},[1946,8138,1974],{},[1961,8140,1963],{},[1946,8142,1974],{},[1961,8144,2735],{},[1936,8146,8147,8149],{},[1860,8148,5474],{},[1857,8150,8151,8153,8155],{},[1860,8152,1944],{},[1946,8154,1974],{},[1961,8156,1963],{},[1864,8158,8159],{"encoding":1866},"y_{t-1}-2x_{t-1}",[1841,8161,8163,8228],{"className":8162,"ariaHidden":1871},[1870],[1841,8164,8166,8170,8219,8222,8225],{"className":8165},[1875],[1841,8167],{"className":8168,"style":8169},[1879],"height:0.7917em;vertical-align:-0.2083em;",[1841,8171,8173,8176],{"className":8172},[1884],[1841,8174,1941],{"className":8175,"style":5519},[1884,1885],[1841,8177,8179],{"className":8178},[2040],[1841,8180,8182,8211],{"className":8181},[2044,2045],[1841,8183,8185,8208],{"className":8184},[2049],[1841,8186,8188],{"className":8187,"style":2145},[2053],[1841,8189,8190,8193],{"style":5670},[1841,8191],{"className":8192,"style":2062},[2061],[1841,8194,8196],{"className":8195},[2066,2067,2068,2069],[1841,8197,8199,8202,8205],{"className":8198},[1884,2069],[1841,8200,1944],{"className":8201},[1884,1885,2069],[1841,8203,1974],{"className":8204},[2116,2069],[1841,8206,1963],{"className":8207},[1884,2069],[1841,8209,2077],{"className":8210},[2076],[1841,8212,8214],{"className":8213},[2049],[1841,8215,8217],{"className":8216,"style":2216},[2053],[1841,8218],{},[1841,8220],{"className":8221,"style":2112},[2090],[1841,8223,1974],{"className":8224},[2116],[1841,8226],{"className":8227,"style":2112},[2090],[1841,8229,8231,8235,8238],{"className":8230},[1875],[1841,8232],{"className":8233,"style":8234},[1879],"height:0.8528em;vertical-align:-0.2083em;",[1841,8236,2735],{"className":8237},[1884],[1841,8239,8241,8244],{"className":8240},[1884],[1841,8242,5474],{"className":8243},[1884,1885],[1841,8245,8247],{"className":8246},[2040],[1841,8248,8250,8279],{"className":8249},[2044,2045],[1841,8251,8253,8276],{"className":8252},[2049],[1841,8254,8256],{"className":8255,"style":2145},[2053],[1841,8257,8258,8261],{"style":2148},[1841,8259],{"className":8260,"style":2062},[2061],[1841,8262,8264],{"className":8263},[2066,2067,2068,2069],[1841,8265,8267,8270,8273],{"className":8266},[1884,2069],[1841,8268,1944],{"className":8269},[1884,1885,2069],[1841,8271,1974],{"className":8272},[2116,2069],[1841,8274,1963],{"className":8275},[1884,2069],[1841,8277,2077],{"className":8278},[2076],[1841,8280,8282],{"className":8281},[2049],[1841,8283,8285],{"className":8284,"style":2216},[2053],[1841,8286],{},[1798,8288,8289],{},"A two-variable error-correction equation may be",[1841,8291,8293],{"className":8292},[1921],[1841,8294,8296,8382],{"className":8295},[1844],[1841,8297,8299],{"className":8298},[1848],[1850,8300,8301],{"xmlns":1852,"display":1930},[1854,8302,8303,8379],{},[1857,8304,8305,8308,8314,8316,8319,8321,8333,8335,8337,8349,8351,8353,8355,8357,8369,8371,8377],{},[1860,8306,8307],{"mathvariant":2014},"Δ",[1936,8309,8310,8312],{},[1860,8311,1941],{},[1860,8313,1944],{},[1946,8315,1948],{},[1860,8317,8318],{},"α",[1946,8320,2541],{"stretchy":2540},[1936,8322,8323,8325],{},[1860,8324,1941],{},[1857,8326,8327,8329,8331],{},[1860,8328,1944],{},[1946,8330,1974],{},[1961,8332,1963],{},[1946,8334,1974],{},[1961,8336,2735],{},[1936,8338,8339,8341],{},[1860,8340,5474],{},[1857,8342,8343,8345,8347],{},[1860,8344,1944],{},[1946,8346,1974],{},[1961,8348,1963],{},[1946,8350,2552],{"stretchy":2540},[1946,8352,1954],{},[1860,8354,4636],{"mathvariant":2014},[1860,8356,8307],{"mathvariant":2014},[1936,8358,8359,8361],{},[1860,8360,1941],{"mathvariant":1940},[1857,8362,8363,8365,8367],{},[1860,8364,1944],{},[1946,8366,1974],{},[1961,8368,1963],{},[1946,8370,1954],{},[1936,8372,8373,8375],{},[1860,8374,2009],{},[1860,8376,1944],{},[1860,8378,2015],{"mathvariant":2014},[1864,8380,8381],{"encoding":1866},"\\Delta y_t=\\alpha(y_{t-1}-2x_{t-1})\n+\\Gamma\\Delta\\mathbf 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Recent work on ",[5613,9244,9248],{"href":9245,"rel":9246},"https:\u002F\u002Fproceedings.mlr.press\u002Fv244\u002Fzambon24a.html",[9247],"nofollow","mixed-type probabilistic reconciliation"," extends this idea when a hierarchy combines counts and continuous quantities. Coherence is a logical requirement; accuracy gains remain empirical and must be backtested at every level.",[1793,9251,9253],{"id":9252},"_8-model-choice-map","8. Model-choice map",[4162,9255,9256,9269],{},[4165,9257,9258],{},[4168,9259,9260,9263,9266],{},[4171,9261,9262],{},"Data question",[4171,9264,9265],{},"First model",[4171,9267,9268],{},"Add only if needed",[4209,9270,9271,9282,9293,9304,9315],{},[4168,9272,9273,9276,9279],{},[4214,9274,9275],{},"one stationary series",[4214,9277,9278],{},"ARMA",[4214,9280,9281],{},"nonlinear\u002Fvolatility structure",[4168,9283,9284,9287,9290],{},[4214,9285,9286],{},"one seasonal series",[4214,9288,9289],{},"seasonal naive + regression\u002FSARIMA",[4214,9291,9292],{},"multiple seasonalities or state space",[4168,9294,9295,9298,9301],{},[4214,9296,9297],{},"several stationary series",[4214,9299,9300],{},"VAR",[4214,9302,9303],{},"structural identification or shrinkage",[4168,9305,9306,9309,9312],{},[4214,9307,9308],{},"non-stationary levels with stable spread",[4214,9310,9311],{},"VECM",[4214,9313,9314],{},"breaks or time-varying relation",[4168,9316,9317,9320,9323],{},[4214,9318,9319],{},"many aggregation levels",[4214,9321,9322],{},"local models + reconciliation",[4214,9324,9325],{},"joint probabilistic hierarchy",[1793,9327,9329],{"id":9328},"practice","Practice",[9331,9332,9333,9424,9516,9519,9522,9852],"ol",{},[1812,9334,9335,9336,9423],{},"Multiply the worked VAR matrix by ",[1841,9337,9339,9367],{"className":9338},[1844],[1841,9340,9342],{"className":9341},[1848],[1850,9343,9344],{"xmlns":1852},[1854,9345,9346,9364],{},[1857,9347,9348,9350,9353,9355,9358],{},[1946,9349,2541],{"stretchy":2540},[1961,9351,9352],{},"0.38",[1946,9354,2741],{"separator":1871},[1961,9356,9357],{},"0.11",[3841,9359,9360,9362],{},[1946,9361,2552],{"stretchy":2540},[1946,9363,3848],{"mathvariant":2014,"lspace":3847,"rspace":3847},[1864,9365,9366],{"encoding":1866},"(0.38,0.11)'",[1841,9368,9370],{"className":9369,"ariaHidden":1871},[1870],[1841,9371,9373,9376,9379,9382,9385,9388,9391],{"className":9372},[1875],[1841,9374],{"className":9375,"style":3927},[1879],[1841,9377,2541],{"className":9378},[2572],[1841,9380,9352],{"className":9381},[1884],[1841,9383,2741],{"className":9384},[2827],[1841,9386],{"className":9387,"style":2831},[2090],[1841,9389,9357],{"className":9390},[1884],[1841,9392,9394,9397],{"className":9393},[2600],[1841,9395,2552],{"className":9396},[2600],[1841,9398,9400],{"className":9399},[2040],[1841,9401,9403],{"className":9402},[2044],[1841,9404,9406],{"className":9405},[2049],[1841,9407,9409],{"className":9408,"style":3961},[2053],[1841,9410,9411,9414],{"style":2802},[1841,9412],{"className":9413,"style":2062},[2061],[1841,9415,9417],{"className":9416},[2066,2067,2068,2069],[1841,9418,9420],{"className":9419},[1884,2069],[1841,9421,3848],{"className":9422},[1884,2069]," to verify the horizon-3 response.",[1812,9425,9426,9427,9515],{},"For ",[1841,9428,9430,9457],{"className":9429},[1844],[1841,9431,9433],{"className":9432},[1848],[1850,9434,9435],{"xmlns":1852},[1854,9436,9437,9454],{},[1857,9438,9439,9441,9443,9446,9448,9450,9452],{},[1860,9440,1862],{},[1946,9442,1948],{},[1961,9444,9445],{},"8",[1946,9447,2741],{"separator":1871},[1860,9449,1798],{},[1946,9451,1948],{},[1961,9453,4246],{},[1864,9455,9456],{"encoding":1866},"k=8,p=3",[1841,9458,9460,9478,9506],{"className":9459,"ariaHidden":1871},[1870],[1841,9461,9463,9466,9469,9472,9475],{"className":9462},[1875],[1841,9464],{"className":9465,"style":1880},[1879],[1841,9467,1862],{"className":9468,"style":1886},[1884,1885],[1841,9470],{"className":9471,"style":2091},[2090],[1841,9473,1948],{"className":9474},[2095],[1841,9476],{"className":9477,"style":2091},[2090],[1841,9479,9481,9485,9488,9491,9494,9497,9500,9503],{"className":9480},[1875],[1841,9482],{"className":9483,"style":9484},[1879],"height:0.8389em;vertical-align:-0.1944em;",[1841,9486,9445],{"className":9487},[1884],[1841,9489,2741],{"className":9490},[2827],[1841,9492],{"className":9493,"style":2831},[2090],[1841,9495,1798],{"className":9496},[1884,1885],[1841,9498],{"className":9499,"style":2091},[2090],[1841,9501,1948],{"className":9502},[2095],[1841,9504],{"className":9505,"style":2091},[2090],[1841,9507,9509,9512],{"className":9508},[1875],[1841,9510],{"className":9511,"style":2651},[1879],[1841,9513,4246],{"className":9514},[1884],", count VAR regression coefficients with intercepts.",[1812,9517,9518],{},"Give a predictive but non-causal explanation for why temperature lags improve electricity-demand forecasts.",[1812,9520,9521],{},"Explain why seasonal indicators and seasonal differencing encode different counterfactuals.",[1812,9523,8703,9524,2655,9719,9775,9776,9851],{},[1841,9525,9527,9573],{"className":9526},[1844],[1841,9528,9530],{"className":9529},[1848],[1850,9531,9532],{"xmlns":1852},[1854,9533,9534,9570],{},[1857,9535,9536,9548,9550,9552,9564,9566,9568],{},[1936,9537,9538,9540],{},[1860,9539,1941],{},[1857,9541,9542,9544,9546],{},[1860,9543,1944],{},[1946,9545,1974],{},[1961,9547,1963],{},[1946,9549,1974],{},[1961,9551,2735],{},[1936,9553,9554,9556],{},[1860,9555,5474],{},[1857,9557,9558,9560,9562],{},[1860,9559,1944],{},[1946,9561,1974],{},[1961,9563,1963],{},[1946,9565,1948],{},[1946,9567,1974],{},[1961,9569,2620],{},[1864,9571,9572],{"encoding":1866},"y_{t-1}-2x_{t-1}=-5",[1841,9574,9576,9640,9707],{"className":9575,"ariaHidden":1871},[1870],[1841,9577,9579,9582,9631,9634,9637],{"className":9578},[1875],[1841,9580],{"className":9581,"style":8169},[1879],[1841,9583,9585,9588],{"className":9584},[1884],[1841,9586,1941],{"className":9587,"style":5519},[1884,1885],[1841,9589,9591],{"className":9590},[2040],[1841,9592,9594,9623],{"className":9593},[2044,2045],[1841,9595,9597,9620],{"className":9596},[2049],[1841,9598,9600],{"className":9599,"style":2145},[2053],[1841,9601,9602,9605],{"style":5670},[1841,9603],{"className":9604,"style":2062},[2061],[1841,9606,9608],{"className":9607},[2066,2067,2068,2069],[1841,9609,9611,9614,9617],{"className":9610},[1884,2069],[1841,9612,1944],{"className":9613},[1884,1885,2069],[1841,9615,1974],{"className":9616},[2116,2069],[1841,9618,1963],{"className":9619},[1884,2069],[1841,9621,2077],{"className":9622},[2076],[1841,9624,9626],{"className":9625},[2049],[1841,9627,9629],{"className":9628,"style":2216},[2053],[1841,9630],{},[1841,9632],{"className":9633,"style":2112},[2090],[1841,9635,1974],{"className":9636},[2116],[1841,9638],{"className":9639,"style":2112},[2090],[1841,9641,9643,9646,9649,9698,9701,9704],{"className":9642},[1875],[1841,9644],{"className":9645,"style":8234},[1879],[1841,9647,2735],{"className":9648},[1884],[1841,9650,9652,9655],{"className":9651},[1884],[1841,9653,5474],{"className":9654},[1884,1885],[1841,9656,9658],{"className":9657},[2040],[1841,9659,9661,9690],{"className":9660},[2044,2045],[1841,9662,9664,9687],{"className":9663},[2049],[1841,9665,9667],{"className":9666,"style":2145},[2053],[1841,9668,9669,9672],{"style":2148},[1841,9670],{"className":9671,"style":2062},[2061],[1841,9673,9675],{"className":9674},[2066,2067,2068,2069],[1841,9676,9678,9681,9684],{"className":9677},[1884,2069],[1841,9679,1944],{"className":9680},[1884,1885,2069],[1841,9682,1974],{"className":9683},[2116,2069],[1841,9685,1963],{"className":9686},[1884,2069],[1841,9688,2077],{"className":9689},[2076],[1841,9691,9693],{"className":9692},[2049],[1841,9694,9696],{"className":9695,"style":2216},[2053],[1841,9697],{},[1841,9699],{"className":9700,"style":2091},[2090],[1841,9702,1948],{"className":9703},[2095],[1841,9705],{"className":9706,"style":2091},[2090],[1841,9708,9710,9713,9716],{"className":9709},[1875],[1841,9711],{"className":9712,"style":8755},[1879],[1841,9714,1974],{"className":9715},[1884],[1841,9717,2620],{"className":9718},[1884],[1841,9720,9722,9742],{"className":9721},[1844],[1841,9723,9725],{"className":9724},[1848],[1850,9726,9727],{"xmlns":1852},[1854,9728,9729,9739],{},[1857,9730,9731,9733,9735,9737],{},[1860,9732,8318],{},[1946,9734,1948],{},[1946,9736,1974],{},[1961,9738,3454],{},[1864,9740,9741],{"encoding":1866},"\\alpha=-0.2",[1841,9743,9745,9763],{"className":9744,"ariaHidden":1871},[1870],[1841,9746,9748,9751,9754,9757,9760],{"className":9747},[1875],[1841,9749],{"className":9750,"style":5486},[1879],[1841,9752,8318],{"className":9753,"style":8452},[1884,1885],[1841,9755],{"className":9756,"style":2091},[2090],[1841,9758,1948],{"className":9759},[2095],[1841,9761],{"className":9762,"style":2091},[2090],[1841,9764,9766,9769,9772],{"className":9765},[1875],[1841,9767],{"className":9768,"style":8755},[1879],[1841,9770,1974],{"className":9771},[1884],[1841,9773,3454],{"className":9774},[1884],", what is the error-correction contribution to ",[1841,9777,9779,9799],{"className":9778},[1844],[1841,9780,9782],{"className":9781},[1848],[1850,9783,9784],{"xmlns":1852},[1854,9785,9786,9796],{},[1857,9787,9788,9790],{},[1860,9789,8307],{"mathvariant":2014},[1936,9791,9792,9794],{},[1860,9793,1941],{},[1860,9795,1944],{},[1864,9797,9798],{"encoding":1866},"\\Delta y_t",[1841,9800,9802],{"className":9801,"ariaHidden":1871},[1870],[1841,9803,9805,9808,9811],{"className":9804},[1875],[1841,9806],{"className":9807,"style":4610},[1879],[1841,9809,8307],{"className":9810},[1884],[1841,9812,9814,9817],{"className":9813},[1884],[1841,9815,1941],{"className":9816,"style":5519},[1884,1885],[1841,9818,9820],{"className":9819},[2040],[1841,9821,9823,9843],{"className":9822},[2044,2045],[1841,9824,9826,9840],{"className":9825},[2049],[1841,9827,9829],{"className":9828,"style":2054},[2053],[1841,9830,9831,9834],{"style":5670},[1841,9832],{"className":9833,"style":2062},[2061],[1841,9835,9837],{"className":9836},[2066,2067,2068,2069],[1841,9838,1944],{"className":9839},[1884,1885,2069],[1841,9841,2077],{"className":9842},[2076],[1841,9844,9846],{"className":9845},[2049],[1841,9847,9849],{"className":9848,"style":2084},[2053],[1841,9850],{},"?",[1812,9853,9854,9855,9924,9925,10058],{},"For the R example, substitute the printed ",[1841,9856,9858,9875],{"className":9857},[1844],[1841,9859,9861],{"className":9860},[1848],[1850,9862,9863],{"xmlns":1852},[1854,9864,9865,9873],{},[1857,9866,9867],{},[1936,9868,9869,9871],{},[1860,9870,4636],{"mathvariant":2014},[1961,9872,3444],{},[1864,9874,4641],{"encoding":1866},[1841,9876,9878],{"className":9877,"ariaHidden":1871},[1870],[1841,9879,9881,9884],{"className":9880},[1875],[1841,9882],{"className":9883,"style":2445},[1879],[1841,9885,9887,9890],{"className":9886},[1884],[1841,9888,4636],{"className":9889},[1884],[1841,9891,9893],{"className":9892},[2040],[1841,9894,9896,9916],{"className":9895},[2044,2045],[1841,9897,9899,9913],{"className":9898},[2049],[1841,9900,9902],{"className":9901,"style":2145},[2053],[1841,9903,9904,9907],{"style":2148},[1841,9905],{"className":9906,"style":2062},[2061],[1841,9908,9910],{"className":9909},[2066,2067,2068,2069],[1841,9911,3444],{"className":9912},[1884,2069],[1841,9914,2077],{"className":9915},[2076],[1841,9917,9919],{"className":9918},[2049],[1841,9920,9922],{"className":9921,"style":2084},[2053],[1841,9923],{}," into ",[1841,9926,9928,9958],{"className":9927},[1844],[1841,9929,9931],{"className":9930},[1848],[1850,9932,9933],{"xmlns":1852},[1854,9934,9935,9955],{},[1857,9936,9937,9939,9945,9951,9953],{},[1860,9938,1959],{},[1936,9940,9941,9943],{},[1860,9942,4636],{"mathvariant":2014},[1961,9944,3444],{},[3841,9946,9947,9949],{},[1860,9948,1959],{},[1860,9950,2738],{"mathvariant":2014},[1946,9952,1954],{},[1860,9954,4345],{"mathvariant":2014},[1864,9956,9957],{"encoding":1866},"A\\Gamma_0A^\\top+\\Sigma",[1841,9959,9961,10049],{"className":9960,"ariaHidden":1871},[1870],[1841,9962,9964,9968,9971,10011,10040,10043,10046],{"className":9963},[1875],[1841,9965],{"className":9966,"style":9967},[1879],"height:0.9991em;vertical-align:-0.15em;",[1841,9969,1959],{"className":9970},[1884,1885],[1841,9972,9974,9977],{"className":9973},[1884],[1841,9975,4636],{"className":9976},[1884],[1841,9978,9980],{"className":9979},[2040],[1841,9981,9983,10003],{"className":9982},[2044,2045],[1841,9984,9986,10000],{"className":9985},[2049],[1841,9987,9989],{"className":9988,"style":2145},[2053],[1841,9990,9991,9994],{"style":2148},[1841,9992],{"className":9993,"style":2062},[2061],[1841,9995,9997],{"className":9996},[2066,2067,2068,2069],[1841,9998,3444],{"className":9999},[1884,2069],[1841,10001,2077],{"className":10002},[2076],[1841,10004,10006],{"className":10005},[2049],[1841,10007,10009],{"className":10008,"style":2084},[2053],[1841,10010],{},[1841,10012,10014,10017],{"className":10013},[1884],[1841,10015,1959],{"className":10016},[1884,1885],[1841,10018,10020],{"className":10019},[2040],[1841,10021,10023],{"className":10022},[2044],[1841,10024,10026],{"className":10025},[2049],[1841,10027,10029],{"className":10028,"style":2787},[2053],[1841,10030,10031,10034],{"style":2802},[1841,10032],{"className":10033,"style":2062},[2061],[1841,10035,10037],{"className":10036},[2066,2067,2068,2069],[1841,10038,2738],{"className":10039},[1884,2069],[1841,10041],{"className":10042,"style":2112},[2090],[1841,10044,1954],{"className":10045},[2116],[1841,10047],{"className":10048,"style":2112},[2090],[1841,10050,10052,10055],{"className":10051},[1875],[1841,10053],{"className":10054,"style":2924},[1879],[1841,10056,4345],{"className":10057},[1884]," and verify one matrix entry by hand.",[10060,10061,10063],"legacy-details",{"title":10062},"Answers",[9331,10064,10065,10314,10449,10452,10455,10578],{},[1812,10066,10067,2655,10191,2015],{},[1841,10068,10070,10113],{"className":10069},[1844],[1841,10071,10073],{"className":10072},[1848],[1850,10074,10075],{"xmlns":1852},[1854,10076,10077,10110],{},[1857,10078,10079,10081,10083,10085,10087,10089,10091,10093,10095,10097,10099,10101,10103,10105,10107],{},[1946,10080,2541],{"stretchy":2540},[1961,10082,3447],{},[1946,10084,2552],{"stretchy":2540},[1946,10086,2541],{"stretchy":2540},[1961,10088,9352],{},[1946,10090,2552],{"stretchy":2540},[1946,10092,1954],{},[1946,10094,2541],{"stretchy":2540},[1961,10096,3454],{},[1946,10098,2552],{"stretchy":2540},[1946,10100,2541],{"stretchy":2540},[1961,10102,9357],{},[1946,10104,2552],{"stretchy":2540},[1946,10106,1948],{},[1961,10108,10109],{},"0.25",[1864,10111,10112],{"encoding":1866},"(0.6)(0.38)+(0.2)(0.11)=0.25",[1841,10114,10116,10149,10182],{"className":10115,"ariaHidden":1871},[1870],[1841,10117,10119,10122,10125,10128,10131,10134,10137,10140,10143,10146],{"className":10118},[1875],[1841,10120],{"className":10121,"style":2565},[1879],[1841,10123,2541],{"className":10124},[2572],[1841,10126,3447],{"className":10127},[1884],[1841,10129,2552],{"className":10130},[2600],[1841,10132,2541],{"className":10133},[2572],[1841,10135,9352],{"className":10136},[1884],[1841,10138,2552],{"className":10139},[2600],[1841,10141],{"className":10142,"style":2112},[2090],[1841,10144,1954],{"className":10145},[2116],[1841,10147],{"className":10148,"style":2112},[2090],[1841,10150,10152,10155,10158,10161,10164,10167,10170,10173,10176,10179],{"className":10151},[1875],[1841,10153],{"className":10154,"style":2565},[1879],[1841,10156,2541],{"className":10157},[2572],[1841,10159,3454],{"className":10160},[1884],[1841,10162,2552],{"className":10163},[2600],[1841,10165,2541],{"className":10166},[2572],[1841,10168,9357],{"className":10169},[1884],[1841,10171,2552],{"className":10172},[2600],[1841,10174],{"className":10175,"style":2091},[2090],[1841,10177,1948],{"className":10178},[2095],[1841,10180],{"className":10181,"style":2091},[2090],[1841,10183,10185,10188],{"className":10184},[1875],[1841,10186],{"className":10187,"style":2651},[1879],[1841,10189,10109],{"className":10190},[1884],[1841,10192,10194,10236],{"className":10193},[1844],[1841,10195,10197],{"className":10196},[1848],[1850,10198,10199],{"xmlns":1852},[1854,10200,10201,10233],{},[1857,10202,10203,10205,10207,10209,10211,10213,10215,10217,10219,10221,10223,10225,10227,10229,10231],{},[1946,10204,2541],{"stretchy":2540},[1961,10206,3463],{},[1946,10208,2552],{"stretchy":2540},[1946,10210,2541],{"stretchy":2540},[1961,10212,9352],{},[1946,10214,2552],{"stretchy":2540},[1946,10216,1954],{},[1946,10218,2541],{"stretchy":2540},[1961,10220,3470],{},[1946,10222,2552],{"stretchy":2540},[1946,10224,2541],{"stretchy":2540},[1961,10226,9357],{},[1946,10228,2552],{"stretchy":2540},[1946,10230,1948],{},[1961,10232,4252],{},[1864,10234,10235],{"encoding":1866},"(0.1)(0.38)+(0.5)(0.11)=0.093",[1841,10237,10239,10272,10305],{"className":10238,"ariaHidden":1871},[1870],[1841,10240,10242,10245,10248,10251,10254,10257,10260,10263,10266,10269],{"className":10241},[1875],[1841,10243],{"className":10244,"style":2565},[1879],[1841,10246,2541],{"className":10247},[2572],[1841,10249,3463],{"className":10250},[1884],[1841,10252,2552],{"className":10253},[2600],[1841,10255,2541],{"className":10256},[2572],[1841,10258,9352],{"className":10259},[1884],[1841,10261,2552],{"className":10262},[2600],[1841,10264],{"className":10265,"style":2112},[2090],[1841,10267,1954],{"className":10268},[2116],[1841,10270],{"className":10271,"style":2112},[2090],[1841,10273,10275,10278,10281,10284,10287,10290,10293,10296,10299,10302],{"className":10274},[1875],[1841,10276],{"className":10277,"style":2565},[1879],[1841,10279,2541],{"className":10280},[2572],[1841,10282,3470],{"className":10283},[1884],[1841,10285,2552],{"className":10286},[2600],[1841,10288,2541],{"className":10289},[2572],[1841,10291,9357],{"className":10292},[1884],[1841,10294,2552],{"className":10295},[2600],[1841,10297],{"className":10298,"style":2091},[2090],[1841,10300,1948],{"className":10301},[2095],[1841,10303],{"className":10304,"style":2091},[2090],[1841,10306,10308,10311],{"className":10307},[1875],[1841,10309],{"className":10310,"style":2651},[1879],[1841,10312,4252],{"className":10313},[1884],[1812,10315,10316,10448],{},[1841,10317,10319,10361],{"className":10318},[1844],[1841,10320,10322],{"className":10321},[1848],[1850,10323,10324],{"xmlns":1852},[1854,10325,10326,10358],{},[1857,10327,10328,10330,10332,10334,10336,10338,10340,10342,10344,10346,10348,10351,10353,10355],{},[1860,10329,1862],{},[1946,10331,2541],{"stretchy":2540},[1860,10333,1862],{},[1860,10335,1798],{},[1946,10337,1954],{},[1961,10339,1963],{},[1946,10341,2552],{"stretchy":2540},[1946,10343,1948],{},[1961,10345,9445],{},[1946,10347,2541],{"stretchy":2540},[1961,10349,10350],{},"25",[1946,10352,2552],{"stretchy":2540},[1946,10354,1948],{},[1961,10356,10357],{},"200",[1864,10359,10360],{"encoding":1866},"k(kp+1)=8(25)=200",[1841,10362,10364,10391,10412,10439],{"className":10363,"ariaHidden":1871},[1870],[1841,10365,10367,10370,10373,10376,10379,10382,10385,10388],{"className":10366},[1875],[1841,10368],{"className":10369,"style":2565},[1879],[1841,10371,1862],{"className":10372,"style":1886},[1884,1885],[1841,10374,2541],{"className":10375},[2572],[1841,10377,1862],{"className":10378,"style":1886},[1884,1885],[1841,10380,1798],{"className":10381},[1884,1885],[1841,10383],{"className":10384,"style":2112},[2090],[1841,10386,1954],{"className":10387},[2116],[1841,10389],{"className":10390,"style":2112},[2090],[1841,10392,10394,10397,10400,10403,10406,10409],{"className":10393},[1875],[1841,10395],{"className":10396,"style":2565},[1879],[1841,10398,1963],{"className":10399},[1884],[1841,10401,2552],{"className":10402},[2600],[1841,10404],{"className":10405,"style":2091},[2090],[1841,10407,1948],{"className":10408},[2095],[1841,10410],{"className":10411,"style":2091},[2090],[1841,10413,10415,10418,10421,10424,10427,10430,10433,10436],{"className":10414},[1875],[1841,10416],{"className":10417,"style":2565},[1879],[1841,10419,9445],{"className":10420},[1884],[1841,10422,2541],{"className":10423},[2572],[1841,10425,10350],{"className":10426},[1884],[1841,10428,2552],{"className":10429},[2600],[1841,10431],{"className":10432,"style":2091},[2090],[1841,10434,1948],{"className":10435},[2095],[1841,10437],{"className":10438,"style":2091},[2090],[1841,10440,10442,10445],{"className":10441},[1875],[1841,10443],{"className":10444,"style":2651},[1879],[1841,10446,10357],{"className":10447},[1884],", before 36 innovation-covariance parameters.",[1812,10450,10451],{},"Temperature may proxy a common seasonal\u002Fcalendar process; predictive timing alone does not identify an intervention effect.",[1812,10453,10454],{},"Indicators assume repeating mean levels; seasonal differencing assumes shocks can accumulate across the same season.",[1812,10456,10457,10548,10549,10577],{},[1841,10458,10460,10494],{"className":10459},[1844],[1841,10461,10463],{"className":10462},[1848],[1850,10464,10465],{"xmlns":1852},[1854,10466,10467,10491],{},[1857,10468,10469,10471,10473,10475,10477,10479,10481,10483,10485,10487,10489],{},[1946,10470,2541],{"stretchy":2540},[1946,10472,1974],{},[1961,10474,3454],{},[1946,10476,2552],{"stretchy":2540},[1946,10478,2541],{"stretchy":2540},[1946,10480,1974],{},[1961,10482,2620],{},[1946,10484,2552],{"stretchy":2540},[1946,10486,1948],{},[1946,10488,1954],{},[1961,10490,1963],{},[1864,10492,10493],{"encoding":1866},"(-0.2)(-5)=+1",[1841,10495,10497,10536],{"className":10496,"ariaHidden":1871},[1870],[1841,10498,10500,10503,10506,10509,10512,10515,10518,10521,10524,10527,10530,10533],{"className":10499},[1875],[1841,10501],{"className":10502,"style":2565},[1879],[1841,10504,2541],{"className":10505},[2572],[1841,10507,1974],{"className":10508},[1884],[1841,10510,3454],{"className":10511},[1884],[1841,10513,2552],{"className":10514},[2600],[1841,10516,2541],{"className":10517},[2572],[1841,10519,1974],{"className":10520},[1884],[1841,10522,2620],{"className":10523},[1884],[1841,10525,2552],{"className":10526},[2600],[1841,10528],{"className":10529,"style":2091},[2090],[1841,10531,1948],{"className":10532},[2095],[1841,10534],{"className":10535,"style":2091},[2090],[1841,10537,10539,10542,10545],{"className":10538},[1875],[1841,10540],{"className":10541,"style":8755},[1879],[1841,10543,1954],{"className":10544},[1884],[1841,10546,1963],{"className":10547},[1884],", moving ",[1841,10550,10552,10565],{"className":10551},[1844],[1841,10553,10555],{"className":10554},[1848],[1850,10556,10557],{"xmlns":1852},[1854,10558,10559,10563],{},[1857,10560,10561],{},[1860,10562,1941],{},[1864,10564,1941],{"encoding":1866},[1841,10566,10568],{"className":10567,"ariaHidden":1871},[1870],[1841,10569,10571,10574],{"className":10570},[1875],[1841,10572],{"className":10573,"style":1913},[1879],[1841,10575,1941],{"className":10576,"style":5519},[1884,1885]," upward toward the relation.",[1812,10579,10580],{},"Any entry should agree up to printed rounding; the executable residual uses full precision.",[1793,10582,10584],{"id":10583},"takeaway","Takeaway",[1798,10586,10587,10588,10593,10594,10598],{},"Multivariate and seasonal models are block-matrix extensions of the univariate theory, but identification becomes harder as dimension grows. The ",[5613,10589,10592],{"href":10590,"rel":10591},"https:\u002F\u002Fpmc.ncbi.nlm.nih.gov\u002Farticles\u002FPMC11729849\u002F",[9247],"2024\u002F25 VARMA review"," is a graduate bridge from these classical equations to current identification and estimation research. Continue to ",[5613,10595,10597],{"href":10596},"..\u002F06-spectral-analysis\u002F","Spectral Analysis"," to study covariance by frequency rather than lag.",{"title":10,"searchDepth":10600,"depth":10600,"links":10601},2,[10602,10603,10604,10609,10610,10611,10612,10615,10616,10617,10618,10619],{"id":1795,"depth":10600,"text":1796},{"id":1803,"depth":10600,"text":1804},{"id":1835,"depth":10600,"text":1836,"children":10605},[10606,10608],{"id":2519,"depth":10607,"text":2520},3,{"id":2711,"depth":10607,"text":2712},{"id":3396,"depth":10600,"text":3397},{"id":4272,"depth":10600,"text":4273},{"id":5456,"depth":10600,"text":5457},{"id":6646,"depth":10600,"text":6647,"children":10613},[10614],{"id":7377,"depth":10607,"text":7378},{"id":7549,"depth":10600,"text":7550},{"id":8829,"depth":10600,"text":8830},{"id":9252,"depth":10600,"text":9253},{"id":9328,"depth":10600,"text":9329},{"id":10583,"depth":10600,"text":10584},"Express vector autoregressions, seasonal operators, cointegration, and forecast constraints with block matrices and stable linear systems.","md",{"sidebar":10623},{"order":10624},5,true,{"title":983,"description":10620},"LS-Q7XA1oT2Oiyp_0Iaj1gwBsgYPuymkhvIC2JXZHGA",[10629,10631],{"title":977,"path":978,"stem":979,"description":10630,"children":-1},"Derive AR estimators from regression and moment equations, evaluate exact Gaussian likelihoods with covariance matrices, and interpret uncertainty.",{"title":989,"path":990,"stem":991,"description":10632,"children":-1},"Translate autocovariance into frequency, estimate spectra, diagnose cycles, and avoid aliasing and filtering leakage.",1785754738338]