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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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向量，",[1802,2015,2017,2030],{"className":2016},[1809],[1802,2018,2020],{"className":2019},[1813],[1815,2021,2022],{"xmlns":1817},[1820,2023,2024,2028],{},[1823,2025,2026],{},[1826,2027,1835],{},[1850,2029,1835],{"encoding":1852},[1802,2031,2033],{"className":2032,"ariaHidden":1847},[1857],[1802,2034,2036,2039],{"className":2035},[1861],[1802,2037],{"className":2038,"style":1866},[1865],[1802,2040,1835],{"className":2041,"style":1895},[1870,1871]," 是包含常数项的 ",[1802,2044,2046,2065],{"className":2045},[1809],[1802,2047,2049],{"className":2048},[1813],[1815,2050,2051],{"xmlns":1817},[1820,2052,2053,2062],{},[1823,2054,2055,2057,2059],{},[1826,2056,1970],{},[1830,2058,1973],{},[1826,2060,2061],{},"K",[1850,2063,2064],{"encoding":1852},"n\\times K",[1802,2066,2068,2086],{"className":2067,"ariaHidden":1847},[1857],[1802,2069,2071,2074,2077,2080,2083],{"className":2070},[1861],[1802,2072],{"className":2073,"style":1990},[1865],[1802,2075,1970],{"className":2076},[1870,1871],[1802,2078],{"className":2079,"style":1872},[1876],[1802,2081,1973],{"className":2082},[1906],[1802,2084],{"className":2085,"style":1872},[1876],[1802,2087,2089,2092],{"className":2088},[1861],[1802,2090],{"className":2091,"style":1866},[1865],[1802,2093,2061],{"className":2094,"style":2095},[1870,1871],"margin-right:0.0715em;"," 设计矩阵，",[1802,2098,2100,2114],{"className":2099},[1809],[1802,2101,2103],{"className":2102},[1813],[1815,2104,2105],{"xmlns":1817},[1820,2106,2107,2111],{},[1823,2108,2109],{},[1826,2110,1838],{},[1850,2112,2113],{"encoding":1852},"\\beta",[1802,2115,2117],{"className":2116,"ariaHidden":1847},[1857],[1802,2118,2120,2123],{"className":2119},[1861],[1802,2121],{"className":2122,"style":1891},[1865],[1802,2124,1838],{"className":2125,"style":1899},[1870,1871]," 是参数，",[1802,2128,2130,2143],{"className":2129},[1809],[1802,2131,2133],{"className":2132},[1813],[1815,2134,2135],{"xmlns":1817},[1820,2136,2137,2141],{},[1823,2138,2139],{},[1826,2140,1844],{},[1850,2142,1844],{"encoding":1852},[1802,2144,2146],{"className":2145,"ariaHidden":1847},[1857],[1802,2147,2149,2153],{"className":2148},[1861],[1802,2150],{"className":2151,"style":2152},[1865],"height:0.4306em;",[1802,2154,1844],{"className":2155},[1870,1871]," 是误差。OLS 选择 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使残差平方和最小：",[1802,2189,2191],{"className":2190},[1805],[1802,2192,2194,2273],{"className":2193},[1809],[1802,2195,2197],{"className":2196},[1813],[1815,2198,2199],{"xmlns":1817,"display":1818},[1820,2200,2201,2270],{},[1823,2202,2203,2211,2213,2216,2219,2231,2235,2237,2240,2242,2244,2255,2257,2259,2261,2263,2265,2267],{},[2204,2205,2206,2208],"mover",{"accent":1847},[1826,2207,1838],{},[1830,2209,2210],{},"^",[1830,2212,1832],{},[1826,2214,2215],{},"arg",[1830,2217,2218],{},"⁡",[2220,2221,2222,2229],"munder",{},[1823,2223,2224,2227],{},[1826,2225,2226],{},"min",[1830,2228,2218],{},[1826,2230,2171],{},[1830,2232,2234],{"stretchy":2233},"false","(",[1826,2236,1828],{},[1830,2238,2239],{},"−",[1826,2241,1835],{},[1826,2243,2171],{},[2245,2246,2247,2250],"msup",{},[1830,2248,2249],{"stretchy":2233},")",[1830,2251,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},"normal","0em","′",[1830,2256,2234],{"stretchy":2233},[1826,2258,1828],{},[1830,2260,2239],{},[1826,2262,1835],{},[1826,2264,2171],{},[1830,2266,2249],{"stretchy":2233},[1826,2268,2269],{"mathvariant":2252},".",[1850,2271,2272],{"encoding":1852},"\\hat\\beta=\\arg\\min_b(Y-Xb)'(Y-Xb).",[1802,2274,2276,2349,2437,2502],{"className":2275,"ariaHidden":1847},[1857],[1802,2277,2279,2283,2340,2343,2346],{"className":2278},[1861],[1802,2280],{"className":2281,"style":2282},[1865],"height:1.1523em;vertical-align:-0.1944em;",[1802,2284,2287],{"className":2285},[1870,2286],"accent",[1802,2288,2292,2331],{"className":2289},[2290,2291],"vlist-t","vlist-t2",[1802,2293,2296,2326],{"className":2294},[2295],"vlist-r",[1802,2297,2301,2312],{"className":2298,"style":2300},[2299],"vlist","height:0.9579em;",[1802,2302,2304,2309],{"style":2303},"top:-3em;",[1802,2305],{"className":2306,"style":2308},[2307],"pstrut","height:3em;",[1802,2310,1838],{"className":2311,"style":1899},[1870,1871],[1802,2313,2315,2318],{"style":2314},"top:-3.2634em;",[1802,2316],{"className":2317,"style":2308},[2307],[1802,2319,2323],{"className":2320,"style":2322},[2321],"accent-body","left:-0.1667em;",[1802,2324,2210],{"className":2325},[1870],[1802,2327,2330],{"className":2328},[2329],"vlist-s","​",[1802,2332,2334],{"className":2333},[2295],[1802,2335,2338],{"className":2336,"style":2337},[2299],"height:0.1944em;",[1802,2339],{},[1802,2341],{"className":2342,"style":1877},[1876],[1802,2344,1832],{"className":2345},[1881],[1802,2347],{"className":2348,"style":1877},[1876],[1802,2350,2352,2356,2365,2369,2421,2425,2428,2431,2434],{"className":2351},[1861],[1802,2353],{"className":2354,"style":2355},[1865],"height:1.5021em;vertical-align:-0.7521em;",[1802,2357,2360,2361],{"className":2358},[2359],"mop","ar",[1802,2362,2364],{"style":2363},"margin-right:0.0139em;","g",[1802,2366],{"className":2367,"style":2368},[1876],"margin-right:0.1667em;",[1802,2370,2373],{"className":2371},[2359,2372],"op-limits",[1802,2374,2376,2412],{"className":2375},[2290,2291],[1802,2377,2379,2409],{"className":2378},[2295],[1802,2380,2383,2399],{"className":2381,"style":2382},[2299],"height:0.6679em;",[1802,2384,2386,2389],{"style":2385},"top:-2.3479em;margin-left:0em;",[1802,2387],{"className":2388,"style":2308},[2307],[1802,2390,2396],{"className":2391},[2392,2393,2394,2395],"sizing","reset-size6","size3","mtight",[1802,2397,2171],{"className":2398},[1870,1871,2395],[1802,2400,2401,2404],{"style":2303},[1802,2402],{"className":2403,"style":2308},[2307],[1802,2405,2406],{},[1802,2407,2226],{"className":2408},[2359],[1802,2410,2330],{"className":2411},[2329],[1802,2413,2415],{"className":2414},[2295],[1802,2416,2419],{"className":2417,"style":2418},[2299],"height:0.7521em;",[1802,2420],{},[1802,2422,2234],{"className":2423},[2424],"mopen",[1802,2426,1828],{"className":2427,"style":1872},[1870,1871],[1802,2429],{"className":2430,"style":1872},[1876],[1802,2432,2239],{"className":2433},[1906],[1802,2435],{"className":2436,"style":1872},[1876],[1802,2438,2440,2444,2447,2450,2487,2490,2493,2496,2499],{"className":2439},[1861],[1802,2441],{"className":2442,"style":2443},[1865],"height:1.0519em;vertical-align:-0.25em;",[1802,2445,1835],{"className":2446,"style":1895},[1870,1871],[1802,2448,2171],{"className":2449},[1870,1871],[1802,2451,2454,2457],{"className":2452},[2453],"mclose",[1802,2455,2249],{"className":2456},[2453],[1802,2458,2461],{"className":2459},[2460],"msupsub",[1802,2462,2464],{"className":2463},[2290],[1802,2465,2467],{"className":2466},[2295],[1802,2468,2471],{"className":2469,"style":2470},[2299],"height:0.8019em;",[1802,2472,2474,2478],{"style":2473},"top:-3.113em;margin-right:0.05em;",[1802,2475],{"className":2476,"style":2477},[2307],"height:2.7em;",[1802,2479,2481],{"className":2480},[2392,2393,2394,2395],[1802,2482,2484],{"className":2483},[1870,2395],[1802,2485,2254],{"className":2486},[1870,2395],[1802,2488,2234],{"className":2489},[2424],[1802,2491,1828],{"className":2492,"style":1872},[1870,1871],[1802,2494],{"className":2495,"style":1872},[1876],[1802,2497,2239],{"className":2498},[1906],[1802,2500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",[1802,2525,2527,2566],{"className":2526},[1809],[1802,2528,2530],{"className":2529},[1813],[1815,2531,2532],{"xmlns":1817},[1820,2533,2534,2563],{},[1823,2535,2536,2542,2544,2546,2548,2550,2556,2558,2560],{},[2245,2537,2538,2540],{},[1826,2539,1835],{},[1830,2541,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1830,2543,2234],{"stretchy":2233},[1826,2545,1828],{},[1830,2547,2239],{},[1826,2549,1835],{},[2204,2551,2552,2554],{"accent":1847},[1826,2553,1838],{},[1830,2555,2210],{},[1830,2557,2249],{"stretchy":2233},[1830,2559,1832],{},[1975,2561,2562],{},"0",[1850,2564,2565],{"encoding":1852},"X'(Y-X\\hat\\beta)=0",[1802,2567,2569,2625,2689],{"className":2568,"ariaHidden":1847},[1857],[1802,2570,2572,2576,2610,2613,2616,2619,2622],{"className":2571},[1861],[1802,2573],{"className":2574,"style":2575},[1865],"height:1.0019em;vertical-align:-0.25em;",[1802,2577,2579,2582],{"className":2578},[1870],[1802,2580,1835],{"className":2581,"style":1895},[1870,1871],[1802,2583,2585],{"className":2584},[2460],[1802,2586,2588],{"className":2587},[2290],[1802,2589,2591],{"className":2590},[2295],[1802,2592,2595],{"className":2593,"style":2594},[2299],"height:0.7519em;",[1802,2596,2598,2601],{"style":2597},"top:-3.063em;margin-right:0.05em;",[1802,2599],{"className":2600,"style":2477},[2307],[1802,2602,2604],{"className":2603},[2392,2393,2394,2395],[1802,2605,2607],{"className":2606},[1870,2395],[1802,2608,2254],{"className":2609},[1870,2395],[1802,2611,2234],{"className":2612},[2424],[1802,2614,1828],{"className":2615,"style":1872},[1870,1871],[1802,2617],{"className":2618,"style":1872},[1876],[1802,2620,2239],{"className":2621},[1906],[1802,2623],{"className":2624,"style":1872},[1876],[1802,2626,2628,2632,2635,2677,2680,2683,2686],{"className":2627},[1861],[1802,2629],{"className":2630,"style":2631},[1865],"height:1.2079em;vertical-align:-0.25em;",[1802,2633,1835],{"className":2634,"style":1895},[1870,1871],[1802,2636,2638],{"className":2637},[1870,2286],[1802,2639,2641,2669],{"className":2640},[2290,2291],[1802,2642,2644,2666],{"className":2643},[2295],[1802,2645,2647,2655],{"className":2646,"style":2300},[2299],[1802,2648,2649,2652],{"style":2303},[1802,2650],{"className":2651,"style":2308},[2307],[1802,2653,1838],{"className":2654,"style":1899},[1870,1871],[1802,2656,2657,2660],{"style":2314},[1802,2658],{"className":2659,"style":2308},[2307],[1802,2661,2663],{"className":2662,"style":2322},[2321],[1802,2664,2210],{"className":2665},[1870],[1802,2667,2330],{"className":2668},[2329],[1802,2670,2672],{"className":2671},[2295],[1802,2673,2675],{"className":2674,"style":2337},[2299],[1802,2676],{},[1802,2678,2249],{"className":2679},[2453],[1802,2681],{"className":2682,"style":1877},[1876],[1802,2684,1832],{"className":2685},[1881],[1802,2687],{"className":2688,"style":1877},[1876],[1802,2690,2692,2695],{"className":2691},[1861],[1802,2693],{"className":2694,"style":2009},[1865],[1802,2696,2562],{"className":2697},[1870],"，如果 ",[1802,2700,2702,2722],{"className":2701},[1809],[1802,2703,2705],{"className":2704},[1813],[1815,2706,2707],{"xmlns":1817},[1820,2708,2709,2719],{},[1823,2710,2711,2717],{},[2245,2712,2713,2715],{},[1826,2714,1835],{},[1830,2716,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1826,2718,1835],{},[1850,2720,2721],{"encoding":1852},"X'X",[1802,2723,2725],{"className":2724,"ariaHidden":1847},[1857],[1802,2726,2728,2731,2763],{"className":2727},[1861],[1802,2729],{"className":2730,"style":2594},[1865],[1802,2732,2734,2737],{"className":2733},[1870],[1802,2735,1835],{"className":2736,"style":1895},[1870,1871],[1802,2738,2740],{"className":2739},[2460],[1802,2741,2743],{"className":2742},[2290],[1802,2744,2746],{"className":2745},[2295],[1802,2747,2749],{"className":2748,"style":2594},[2299],[1802,2750,2751,2754],{"style":2597},[1802,2752],{"className":2753,"style":2477},[2307],[1802,2755,2757],{"className":2756},[2392,2393,2394,2395],[1802,2758,2760],{"className":2759},[1870,2395],[1802,2761,2254],{"className":2762},[1870,2395],[1802,2764,1835],{"className":2765,"style":1895},[1870,1871]," 可逆，则：",[1802,2768,2770],{"className":2769},[1805],[1802,2771,2773,2823],{"className":2772},[1809],[1802,2774,2776],{"className":2775},[1813],[1815,2777,2778],{"xmlns":1817,"display":1818},[1820,2779,2780,2820],{},[1823,2781,2782,2788,2790,2792,2798,2800,2810,2816,2818],{},[2204,2783,2784,2786],{"accent":1847},[1826,2785,1838],{},[1830,2787,2210],{},[1830,2789,1832],{},[1830,2791,2234],{"stretchy":2233},[2245,2793,2794,2796],{},[1826,2795,1835],{},[1830,2797,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1826,2799,1835],{},[2245,2801,2802,2804],{},[1830,2803,2249],{"stretchy":2233},[1823,2805,2806,2808],{},[1830,2807,2239],{},[1975,2809,1977],{},[2245,2811,2812,2814],{},[1826,2813,1835],{},[1830,2815,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1826,2817,1828],{},[1826,2819,2269],{"mathvariant":2252},[1850,2821,2822],{"encoding":1852},"\\hat\\beta=(X'X)^{-1}X'Y.",[1802,2824,2826,2883],{"className":2825,"ariaHidden":1847},[1857],[1802,2827,2829,2832,2874,2877,2880],{"className":2828},[1861],[1802,2830],{"className":2831,"style":2282},[1865],[1802,2833,2835],{"className":2834},[1870,2286],[1802,2836,2838,2866],{"className":2837},[2290,2291],[1802,2839,2841,2863],{"className":2840},[2295],[1802,2842,2844,2852],{"className":2843,"style":2300},[2299],[1802,2845,2846,2849],{"style":2303},[1802,2847],{"className":2848,"style":2308},[2307],[1802,2850,1838],{"className":2851,"style":1899},[1870,1871],[1802,2853,2854,2857],{"style":2314},[1802,2855],{"className":2856,"style":2308},[2307],[1802,2858,2860],{"className":2859,"style":2322},[2321],[1802,2861,2210],{"className":2862},[1870],[1802,2864,2330],{"className":2865},[2329],[1802,2867,2869],{"className":2868},[2295],[1802,2870,2872],{"className":2871,"style":2337},[2299],[1802,2873],{},[1802,2875],{"className":2876,"style":1877},[1876],[1802,2878,1832],{"className":2879},[1881],[1802,2881],{"className":2882,"style":1877},[1876],[1802,2884,2886,2890,2893,2925,2928,2964,2996,2999],{"className":2885},[1861],[1802,2887],{"className":2888,"style":2889},[1865],"height:1.1141em;vertical-align:-0.25em;",[1802,2891,2234],{"className":2892},[2424],[1802,2894,2896,2899],{"className":2895},[1870],[1802,2897,1835],{"className":2898,"style":1895},[1870,1871],[1802,2900,2902],{"className":2901},[2460],[1802,2903,2905],{"className":2904},[2290],[1802,2906,2908],{"className":2907},[2295],[1802,2909,2911],{"className":2910,"style":2470},[2299],[1802,2912,2913,2916],{"style":2473},[1802,2914],{"className":2915,"style":2477},[2307],[1802,2917,2919],{"className":2918},[2392,2393,2394,2395],[1802,2920,2922],{"className":2921},[1870,2395],[1802,2923,2254],{"className":2924},[1870,2395],[1802,2926,1835],{"className":2927,"style":1895},[1870,1871],[1802,2929,2931,2934],{"className":2930},[2453],[1802,2932,2249],{"className":2933},[2453],[1802,2935,2937],{"className":2936},[2460],[1802,2938,2940],{"className":2939},[2290],[1802,2941,2943],{"className":2942},[2295],[1802,2944,2947],{"className":2945,"style":2946},[2299],"height:0.8641em;",[1802,2948,2949,2952],{"style":2473},[1802,2950],{"className":2951,"style":2477},[2307],[1802,2953,2955],{"className":2954},[2392,2393,2394,2395],[1802,2956,2958,2961],{"className":2957},[1870,2395],[1802,2959,2239],{"className":2960},[1870,2395],[1802,2962,1977],{"className":2963},[1870,2395],[1802,2965,2967,2970],{"className":2966},[1870],[1802,2968,1835],{"className":2969,"style":1895},[1870,1871],[1802,2971,2973],{"className":2972},[2460],[1802,2974,2976],{"className":2975},[2290],[1802,2977,2979],{"className":2978},[2295],[1802,2980,2982],{"className":2981,"style":2470},[2299],[1802,2983,2984,2987],{"style":2473},[1802,2985],{"className":2986,"style":2477},[2307],[1802,2988,2990],{"className":2989},[2392,2393,2394,2395],[1802,2991,2993],{"className":2992},[1870,2395],[1802,2994,2254],{"className":2995},[1870,2395],[1802,2997,1828],{"className":2998,"style":1872},[1870,1871],[1802,3000,2269],{"className":3001},[1870],[1793,3003,3004],{"id":3004},"学习目标",[3006,3007,3008,3012,3015,3018,3021],"ul",{},[3009,3010,3011],"li",{},"从最小化问题推导正规方程和 OLS 估计量；",[3009,3013,3014],{},"用投影矩阵解释拟合值与残差；",[3009,3016,3017],{},"使用 FWL 定理理解多元回归系数；",[3009,3019,3020],{},"推导遗漏变量偏误的方向和条件；",[3009,3022,3023],{},"理解函数形式、交互项和对数模型的解释。",[1793,3025,3027],{"id":3026},"_2-几何解释","2. 几何解释",[1798,3029,3030,3031,1955,3165,3193,3194,3222,3223,3362,3363,3391],{},"拟合值 ",[1802,3032,3034,3062],{"className":3033},[1809],[1802,3035,3037],{"className":3036},[1813],[1815,3038,3039],{"xmlns":1817},[1820,3040,3041,3059],{},[1823,3042,3043,3049,3051,3053],{},[2204,3044,3045,3047],{"accent":1847},[1826,3046,1828],{},[1830,3048,2210],{},[1830,3050,1832],{},[1826,3052,1835],{},[2204,3054,3055,3057],{"accent":1847},[1826,3056,1838],{},[1830,3058,2210],{},[1850,3060,3061],{"encoding":1852},"\\hat Y=X\\hat\\beta",[1802,3063,3065,3114],{"className":3064,"ariaHidden":1847},[1857],[1802,3066,3068,3072,3105,3108,3111],{"className":3067},[1861],[1802,3069],{"className":3070,"style":3071},[1865],"height:0.9468em;",[1802,3073,3075],{"className":3074},[1870,2286],[1802,3076,3078],{"className":3077},[2290],[1802,3079,3081],{"className":3080},[2295],[1802,3082,3084,3092],{"className":3083,"style":3071},[2299],[1802,3085,3086,3089],{"style":2303},[1802,3087],{"className":3088,"style":2308},[2307],[1802,3090,1828],{"className":3091,"style":1872},[1870,1871],[1802,3093,3095,3098],{"style":3094},"top:-3.2523em;",[1802,3096],{"className":3097,"style":2308},[2307],[1802,3099,3102],{"className":3100,"style":3101},[2321],"left:-0.25em;",[1802,3103,2210],{"className":3104},[1870],[1802,3106],{"className":3107,"style":1877},[1876],[1802,3109,1832],{"className":3110},[1881],[1802,3112],{"className":3113,"style":1877},[1876],[1802,3115,3117,3120,3123],{"className":3116},[1861],[1802,3118],{"className":3119,"style":2282},[1865],[1802,3121,1835],{"className":3122,"style":1895},[1870,1871],[1802,3124,3126],{"className":3125},[1870,2286],[1802,3127,3129,3157],{"className":3128},[2290,2291],[1802,3130,3132,3154],{"className":3131},[2295],[1802,3133,3135,3143],{"className":3134,"style":2300},[2299],[1802,3136,3137,3140],{"style":2303},[1802,3138],{"className":3139,"style":2308},[2307],[1802,3141,1838],{"className":3142,"style":1899},[1870,1871],[1802,3144,3145,3148],{"style":2314},[1802,3146],{"className":3147,"style":2308},[2307],[1802,3149,3151],{"className":3150,"style":2322},[2321],[1802,3152,2210],{"className":3153},[1870],[1802,3155,2330],{"className":3156},[2329],[1802,3158,3160],{"className":3159},[2295],[1802,3161,3163],{"className":3162,"style":2337},[2299],[1802,3164],{},[1802,3166,3168,3181],{"className":3167},[1809],[1802,3169,3171],{"className":3170},[1813],[1815,3172,3173],{"xmlns":1817},[1820,3174,3175,3179],{},[1823,3176,3177],{},[1826,3178,1828],{},[1850,3180,1828],{"encoding":1852},[1802,3182,3184],{"className":3183,"ariaHidden":1847},[1857],[1802,3185,3187,3190],{"className":3186},[1861],[1802,3188],{"className":3189,"style":1866},[1865],[1802,3191,1828],{"className":3192,"style":1872},[1870,1871]," 在 ",[1802,3195,3197,3210],{"className":3196},[1809],[1802,3198,3200],{"className":3199},[1813],[1815,3201,3202],{"xmlns":1817},[1820,3203,3204,3208],{},[1823,3205,3206],{},[1826,3207,1835],{},[1850,3209,1835],{"encoding":1852},[1802,3211,3213],{"className":3212,"ariaHidden":1847},[1857],[1802,3214,3216,3219],{"className":3215},[1861],[1802,3217],{"className":3218,"style":1866},[1865],[1802,3220,1835],{"className":3221,"style":1895},[1870,1871]," 的列空间上的正交投影，残差 ",[1802,3224,3226,3256],{"className":3225},[1809],[1802,3227,3229],{"className":3228},[1813],[1815,3230,3231],{"xmlns":1817},[1820,3232,3233,3253],{},[1823,3234,3235,3241,3243,3245,3247],{},[2204,3236,3237,3239],{"accent":1847},[1826,3238,1844],{},[1830,3240,2210],{},[1830,3242,1832],{},[1826,3244,1828],{},[1830,3246,2239],{},[2204,3248,3249,3251],{"accent":1847},[1826,3250,1828],{},[1830,3252,2210],{},[1850,3254,3255],{"encoding":1852},"\\hat u=Y-\\hat Y",[1802,3257,3259,3306,3325],{"className":3258,"ariaHidden":1847},[1857],[1802,3260,3262,3265,3297,3300,3303],{"className":3261},[1861],[1802,3263],{"className":3264,"style":2183},[1865],[1802,3266,3268],{"className":3267},[1870,2286],[1802,3269,3271],{"className":3270},[2290],[1802,3272,3274],{"className":3273},[2295],[1802,3275,3277,3285],{"className":3276,"style":2183},[2299],[1802,3278,3279,3282],{"style":2303},[1802,3280],{"className":3281,"style":2308},[2307],[1802,3283,1844],{"className":3284},[1870,1871],[1802,3286,3287,3290],{"style":2303},[1802,3288],{"className":3289,"style":2308},[2307],[1802,3291,3294],{"className":3292,"style":3293},[2321],"left:-0.2222em;",[1802,3295,2210],{"className":3296},[1870],[1802,3298],{"className":3299,"style":1877},[1876],[1802,3301,1832],{"className":3302},[1881],[1802,3304],{"className":3305,"style":1877},[1876],[1802,3307,3309,3313,3316,3319,3322],{"className":3308},[1861],[1802,3310],{"className":3311,"style":3312},[1865],"height:0.7667em;vertical-align:-0.0833em;",[1802,3314,1828],{"className":3315,"style":1872},[1870,1871],[1802,3317],{"className":3318,"style":1872},[1876],[1802,3320,2239],{"className":3321},[1906],[1802,3323],{"className":3324,"style":1872},[1876],[1802,3326,3328,3331],{"className":3327},[1861],[1802,3329],{"className":3330,"style":3071},[1865],[1802,3332,3334],{"className":3333},[1870,2286],[1802,3335,3337],{"className":3336},[2290],[1802,3338,3340],{"className":3339},[2295],[1802,3341,3343,3351],{"className":3342,"style":3071},[2299],[1802,3344,3345,3348],{"style":2303},[1802,3346],{"className":3347,"style":2308},[2307],[1802,3349,1828],{"className":3350,"style":1872},[1870,1871],[1802,3352,3353,3356],{"style":3094},[1802,3354],{"className":3355,"style":2308},[2307],[1802,3357,3359],{"className":3358,"style":3101},[2321],[1802,3360,2210],{"className":3361},[1870]," 与 ",[1802,3364,3366,3379],{"className":3365},[1809],[1802,3367,3369],{"className":3368},[1813],[1815,3370,3371],{"xmlns":1817},[1820,3372,3373,3377],{},[1823,3374,3375],{},[1826,3376,1835],{},[1850,3378,1835],{"encoding":1852},[1802,3380,3382],{"className":3381,"ariaHidden":1847},[1857],[1802,3383,3385,3388],{"className":3384},[1861],[1802,3386],{"className":3387,"style":1866},[1865],[1802,3389,1835],{"className":3390,"style":1895},[1870,1871]," 的每一列正交：",[1802,3393,3395],{"className":3394},[1805],[1802,3396,3398,3427],{"className":3397},[1809],[1802,3399,3401],{"className":3400},[1813],[1815,3402,3403],{"xmlns":1817,"display":1818},[1820,3404,3405,3424],{},[1823,3406,3407,3413,3419,3421],{},[2245,3408,3409,3411],{},[1826,3410,1835],{},[1830,3412,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[2204,3414,3415,3417],{"accent":1847},[1826,3416,1844],{},[1830,3418,2210],{},[1830,3420,1832],{},[1975,3422,3423],{},"0.",[1850,3425,3426],{"encoding":1852},"X'\\hat u=0.",[1802,3428,3430,3508],{"className":3429,"ariaHidden":1847},[1857],[1802,3431,3433,3436,3468,3499,3502,3505],{"className":3432},[1861],[1802,3434],{"className":3435,"style":2470},[1865],[1802,3437,3439,3442],{"className":3438},[1870],[1802,3440,1835],{"className":3441,"style":1895},[1870,1871],[1802,3443,3445],{"className":3444},[2460],[1802,3446,3448],{"className":3447},[2290],[1802,3449,3451],{"className":3450},[2295],[1802,3452,3454],{"className":3453,"style":2470},[2299],[1802,3455,3456,3459],{"style":2473},[1802,3457],{"className":3458,"style":2477},[2307],[1802,3460,3462],{"className":3461},[2392,2393,2394,2395],[1802,3463,3465],{"className":3464},[1870,2395],[1802,3466,2254],{"className":3467},[1870,2395],[1802,3469,3471],{"className":3470},[1870,2286],[1802,3472,3474],{"className":3473},[2290],[1802,3475,3477],{"className":3476},[2295],[1802,3478,3480,3488],{"className":3479,"style":2183},[2299],[1802,3481,3482,3485],{"style":2303},[1802,3483],{"className":3484,"style":2308},[2307],[1802,3486,1844],{"className":3487},[1870,1871],[1802,3489,3490,3493],{"style":2303},[1802,3491],{"className":3492,"style":2308},[2307],[1802,3494,3496],{"className":3495,"style":3293},[2321],[1802,3497,2210],{"className":3498},[1870],[1802,3500],{"className":3501,"style":1877},[1876],[1802,3503,1832],{"className":3504},[1881],[1802,3506],{"className":3507,"style":1877},[1876],[1802,3509,3511,3514],{"className":3510},[1861],[1802,3512],{"className":3513,"style":2009},[1865],[1802,3515,3423],{"className":3516},[1870],[1798,3518,3519],{},"投影矩阵为：",[1802,3521,3523],{"className":3522},[1805],[1802,3524,3526,3603],{"className":3525},[1809],[1802,3527,3529],{"className":3528},[1813],[1815,3530,3531],{"xmlns":1817,"display":1818},[1820,3532,3533,3600],{},[1823,3534,3535,3543,3545,3547,3549,3555,3557,3567,3573,3575,3578,3585,3587,3590,3592,3598],{},[3536,3537,3538,3541],"msub",{},[1826,3539,3540],{},"P",[1826,3542,1835],{},[1830,3544,1832],{},[1826,3546,1835],{},[1830,3548,2234],{"stretchy":2233},[2245,3550,3551,3553],{},[1826,3552,1835],{},[1830,3554,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1826,3556,1835],{},[2245,3558,3559,3561],{},[1830,3560,2249],{"stretchy":2233},[1823,3562,3563,3565],{},[1830,3564,2239],{},[1975,3566,1977],{},[2245,3568,3569,3571],{},[1826,3570,1835],{},[1830,3572,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[1830,3574,1848],{"separator":1847},[1876,3576],{"width":3577},"2em",[3536,3579,3580,3583],{},[1826,3581,3582],{},"M",[1826,3584,1835],{},[1830,3586,1832],{},[1826,3588,3589],{},"I",[1830,3591,2239],{},[3536,3593,3594,3596],{},[1826,3595,3540],{},[1826,3597,1835],{},[1830,3599,1848],{"separator":1847},[1850,3601,3602],{"encoding":1852},"P_X=X(X'X)^{-1}X',\n\\qquad\nM_X=I-P_X,",[1802,3604,3606,3666,3841,3859],{"className":3605,"ariaHidden":1847},[1857],[1802,3607,3609,3613,3657,3660,3663],{"className":3608},[1861],[1802,3610],{"className":3611,"style":3612},[1865],"height:0.8333em;vertical-align:-0.15em;",[1802,3614,3616,3620],{"className":3615},[1870],[1802,3617,3540],{"className":3618,"style":3619},[1870,1871],"margin-right:0.1389em;",[1802,3621,3623],{"className":3622},[2460],[1802,3624,3626,3648],{"className":3625},[2290,2291],[1802,3627,3629,3645],{"className":3628},[2295],[1802,3630,3633],{"className":3631,"style":3632},[2299],"height:0.3283em;",[1802,3634,3636,3639],{"style":3635},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1802,3637],{"className":3638,"style":2477},[2307],[1802,3640,3642],{"className":3641},[2392,2393,2394,2395],[1802,3643,1835],{"className":3644,"style":1895},[1870,1871,2395],[1802,3646,2330],{"className":3647},[2329],[1802,3649,3651],{"className":3650},[2295],[1802,3652,3655],{"className":3653,"style":3654},[2299],"height:0.15em;",[1802,3656],{},[1802,3658],{"className":3659,"style":1877},[1876],[1802,3661,1832],{"className":3662},[1881],[1802,3664],{"className":3665,"style":1877},[1876],[1802,3667,3669,3672,3675,3678,3710,3713,3748,3780,3783,3787,3790,3832,3835,3838],{"className":3668},[1861],[1802,3670],{"className":3671,"style":2889},[1865],[1802,3673,1835],{"className":3674,"style":1895},[1870,1871],[1802,3676,2234],{"className":3677},[2424],[1802,3679,3681,3684],{"className":3680},[1870],[1802,3682,1835],{"className":3683,"style":1895},[1870,1871],[1802,3685,3687],{"className":3686},[2460],[1802,3688,3690],{"className":3689},[2290],[1802,3691,3693],{"className":3692},[2295],[1802,3694,3696],{"className":3695,"style":2470},[2299],[1802,3697,3698,3701],{"style":2473},[1802,3699],{"className":3700,"style":2477},[2307],[1802,3702,3704],{"className":3703},[2392,2393,2394,2395],[1802,3705,3707],{"className":3706},[1870,2395],[1802,3708,2254],{"className":3709},[1870,2395],[1802,3711,1835],{"className":3712,"style":1895},[1870,1871],[1802,3714,3716,3719],{"className":3715},[2453],[1802,3717,2249],{"className":3718},[2453],[1802,3720,3722],{"className":3721},[2460],[1802,3723,3725],{"className":3724},[2290],[1802,3726,3728],{"className":3727},[2295],[1802,3729,3731],{"className":3730,"style":2946},[2299],[1802,3732,3733,3736],{"style":2473},[1802,3734],{"className":3735,"style":2477},[2307],[1802,3737,3739],{"className":3738},[2392,2393,2394,2395],[1802,3740,3742,3745],{"className":3741},[1870,2395],[1802,3743,2239],{"className":3744},[1870,2395],[1802,3746,1977],{"className":3747},[1870,2395],[1802,3749,3751,3754],{"className":3750},[1870],[1802,3752,1835],{"className":3753,"style":1895},[1870,1871],[1802,3755,3757],{"className":3756},[2460],[1802,3758,3760],{"className":3759},[2290],[1802,3761,3763],{"className":3762},[2295],[1802,3764,3766],{"className":3765,"style":2470},[2299],[1802,3767,3768,3771],{"style":2473},[1802,3769],{"className":3770,"style":2477},[2307],[1802,3772,3774],{"className":3773},[2392,2393,2394,2395],[1802,3775,3777],{"className":3776},[1870,2395],[1802,3778,2254],{"className":3779},[1870,2395],[1802,3781,1848],{"className":3782},[1923],[1802,3784],{"className":3785,"style":3786},[1876],"margin-right:2em;",[1802,3788],{"className":3789,"style":2368},[1876],[1802,3791,3793,3797],{"className":3792},[1870],[1802,3794,3582],{"className":3795,"style":3796},[1870,1871],"margin-right:0.109em;",[1802,3798,3800],{"className":3799},[2460],[1802,3801,3803,3824],{"className":3802},[2290,2291],[1802,3804,3806,3821],{"className":3805},[2295],[1802,3807,3809],{"className":3808,"style":3632},[2299],[1802,3810,3812,3815],{"style":3811},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[1802,3813],{"className":3814,"style":2477},[2307],[1802,3816,3818],{"className":3817},[2392,2393,2394,2395],[1802,3819,1835],{"className":3820,"style":1895},[1870,1871,2395],[1802,3822,2330],{"className":3823},[2329],[1802,3825,3827],{"className":3826},[2295],[1802,3828,3830],{"className":3829,"style":3654},[2299],[1802,3831],{},[1802,3833],{"className":3834,"style":1877},[1876],[1802,3836,1832],{"className":3837},[1881],[1802,3839],{"className":3840,"style":1877},[1876],[1802,3842,3844,3847,3850,3853,3856],{"className":3843},[1861],[1802,3845],{"className":3846,"style":3312},[1865],[1802,3848,3589],{"className":3849,"style":1895},[1870,1871],[1802,3851],{"className":3852,"style":1872},[1876],[1802,3854,2239],{"className":3855},[1906],[1802,3857],{"className":3858,"style":1872},[1876],[1802,3860,3862,3866,3906],{"className":3861},[1861],[1802,3863],{"className":3864,"style":3865},[1865],"height:0.8778em;vertical-align:-0.1944em;",[1802,3867,3869,3872],{"className":3868},[1870],[1802,3870,3540],{"className":3871,"style":3619},[1870,1871],[1802,3873,3875],{"className":3874},[2460],[1802,3876,3878,3898],{"className":3877},[2290,2291],[1802,3879,3881,3895],{"className":3880},[2295],[1802,3882,3884],{"className":3883,"style":3632},[2299],[1802,3885,3886,3889],{"style":3635},[1802,3887],{"className":3888,"style":2477},[2307],[1802,3890,3892],{"className":3891},[2392,2393,2394,2395],[1802,3893,1835],{"className":3894,"style":1895},[1870,1871,2395],[1802,3896,2330],{"className":3897},[2329],[1802,3899,3901],{"className":3900},[2295],[1802,3902,3904],{"className":3903,"style":3654},[2299],[1802,3905],{},[1802,3907,1848],{"className":3908},[1923],[1798,3910,3911,3912,4041,4042,4171],{},"因此 ",[1802,3913,3915,3943],{"className":3914},[1809],[1802,3916,3918],{"className":3917},[1813],[1815,3919,3920],{"xmlns":1817},[1820,3921,3922,3940],{},[1823,3923,3924,3930,3932,3938],{},[2204,3925,3926,3928],{"accent":1847},[1826,3927,1828],{},[1830,3929,2210],{},[1830,3931,1832],{},[3536,3933,3934,3936],{},[1826,3935,3540],{},[1826,3937,1835],{},[1826,3939,1828],{},[1850,3941,3942],{"encoding":1852},"\\hat Y=P_XY",[1802,3944,3946,3992],{"className":3945,"ariaHidden":1847},[1857],[1802,3947,3949,3952,3983,3986,3989],{"className":3948},[1861],[1802,3950],{"className":3951,"style":3071},[1865],[1802,3953,3955],{"className":3954},[1870,2286],[1802,3956,3958],{"className":3957},[2290],[1802,3959,3961],{"className":3960},[2295],[1802,3962,3964,3972],{"className":3963,"style":3071},[2299],[1802,3965,3966,3969],{"style":2303},[1802,3967],{"className":3968,"style":2308},[2307],[1802,3970,1828],{"className":3971,"style":1872},[1870,1871],[1802,3973,3974,3977],{"style":3094},[1802,3975],{"className":3976,"style":2308},[2307],[1802,3978,3980],{"className":3979,"style":3101},[2321],[1802,3981,2210],{"className":3982},[1870],[1802,3984],{"className":3985,"style":1877},[1876],[1802,3987,1832],{"className":3988},[1881],[1802,3990],{"className":3991,"style":1877},[1876],[1802,3993,3995,3998,4038],{"className":3994},[1861],[1802,3996],{"className":3997,"style":3612},[1865],[1802,3999,4001,4004],{"className":4000},[1870],[1802,4002,3540],{"className":4003,"style":3619},[1870,1871],[1802,4005,4007],{"className":4006},[2460],[1802,4008,4010,4030],{"className":4009},[2290,2291],[1802,4011,4013,4027],{"className":4012},[2295],[1802,4014,4016],{"className":4015,"style":3632},[2299],[1802,4017,4018,4021],{"style":3635},[1802,4019],{"className":4020,"style":2477},[2307],[1802,4022,4024],{"className":4023},[2392,2393,2394,2395],[1802,4025,1835],{"className":4026,"style":1895},[1870,1871,2395],[1802,4028,2330],{"className":4029},[2329],[1802,4031,4033],{"className":4032},[2295],[1802,4034,4036],{"className":4035,"style":3654},[2299],[1802,4037],{},[1802,4039,1828],{"className":4040,"style":1872},[1870,1871],"、",[1802,4043,4045,4073],{"className":4044},[1809],[1802,4046,4048],{"className":4047},[1813],[1815,4049,4050],{"xmlns":1817},[1820,4051,4052,4070],{},[1823,4053,4054,4060,4062,4068],{},[2204,4055,4056,4058],{"accent":1847},[1826,4057,1844],{},[1830,4059,2210],{},[1830,4061,1832],{},[3536,4063,4064,4066],{},[1826,4065,3582],{},[1826,4067,1835],{},[1826,4069,1828],{},[1850,4071,4072],{"encoding":1852},"\\hat u=M_XY",[1802,4074,4076,4122],{"className":4075,"ariaHidden":1847},[1857],[1802,4077,4079,4082,4113,4116,4119],{"className":4078},[1861],[1802,4080],{"className":4081,"style":2183},[1865],[1802,4083,4085],{"className":4084},[1870,2286],[1802,4086,4088],{"className":4087},[2290],[1802,4089,4091],{"className":4090},[2295],[1802,4092,4094,4102],{"className":4093,"style":2183},[2299],[1802,4095,4096,4099],{"style":2303},[1802,4097],{"className":4098,"style":2308},[2307],[1802,4100,1844],{"className":4101},[1870,1871],[1802,4103,4104,4107],{"style":2303},[1802,4105],{"className":4106,"style":2308},[2307],[1802,4108,4110],{"className":4109,"style":3293},[2321],[1802,4111,2210],{"className":4112},[1870],[1802,4114],{"className":4115,"style":1877},[1876],[1802,4117,1832],{"className":4118},[1881],[1802,4120],{"className":4121,"style":1877},[1876],[1802,4123,4125,4128,4168],{"className":4124},[1861],[1802,4126],{"className":4127,"style":3612},[1865],[1802,4129,4131,4134],{"className":4130},[1870],[1802,4132,3582],{"className":4133,"style":3796},[1870,1871],[1802,4135,4137],{"className":4136},[2460],[1802,4138,4140,4160],{"className":4139},[2290,2291],[1802,4141,4143,4157],{"className":4142},[2295],[1802,4144,4146],{"className":4145,"style":3632},[2299],[1802,4147,4148,4151],{"style":3811},[1802,4149],{"className":4150,"style":2477},[2307],[1802,4152,4154],{"className":4153},[2392,2393,2394,2395],[1802,4155,1835],{"className":4156,"style":1895},[1870,1871,2395],[1802,4158,2330],{"className":4159},[2329],[1802,4161,4163],{"className":4162},[2295],[1802,4164,4166],{"className":4165,"style":3654},[2299],[1802,4167],{},[1802,4169,1828],{"className":4170,"style":1872},[1870,1871],"。矩阵语言揭示了 OLS 的两个重要事实：它是一个线性投影，且残差与所有回归解释变量正交。但正交不等于因果独立。",[1793,4173,4175],{"id":4174},"_3-fwl-定理与控制变量","3. FWL 定理与控制变量",[1798,4177,4178],{},"考虑：",[1802,4180,4182],{"className":4181},[1805],[1802,4183,4185,4220],{"className":4184},[1809],[1802,4186,4188],{"className":4187},[1813],[1815,4189,4190],{"xmlns":1817,"display":1818},[1820,4191,4192,4217],{},[1823,4193,4194,4196,4198,4201,4204,4206,4208,4211,4213,4215],{},[1826,4195,1828],{},[1830,4197,1832],{},[1826,4199,4200],{},"τ",[1826,4202,4203],{},"D",[1830,4205,1841],{},[1826,4207,1835],{},[1826,4209,4210],{},"γ",[1830,4212,1841],{},[1826,4214,1844],{},[1826,4216,2269],{"mathvariant":2252},[1850,4218,4219],{"encoding":1852},"Y=\\tau D+X\\gamma+u.",[1802,4221,4223,4241,4264,4286],{"className":4222,"ariaHidden":1847},[1857],[1802,4224,4226,4229,4232,4235,4238],{"className":4225},[1861],[1802,4227],{"className":4228,"style":1866},[1865],[1802,4230,1828],{"className":4231,"style":1872},[1870,1871],[1802,4233],{"className":4234,"style":1877},[1876],[1802,4236,1832],{"className":4237},[1881],[1802,4239],{"className":4240,"style":1877},[1876],[1802,4242,4244,4247,4251,4255,4258,4261],{"className":4243},[1861],[1802,4245],{"className":4246,"style":3312},[1865],[1802,4248,4200],{"className":4249,"style":4250},[1870,1871],"margin-right:0.1132em;",[1802,4252,4203],{"className":4253,"style":4254},[1870,1871],"margin-right:0.0278em;",[1802,4256],{"className":4257,"style":1872},[1876],[1802,4259,1841],{"className":4260},[1906],[1802,4262],{"className":4263,"style":1872},[1876],[1802,4265,4267,4270,4273,4277,4280,4283],{"className":4266},[1861],[1802,4268],{"className":4269,"style":3865},[1865],[1802,4271,1835],{"className":4272,"style":1895},[1870,1871],[1802,4274,4210],{"className":4275,"style":4276},[1870,1871],"margin-right:0.0556em;",[1802,4278],{"className":4279,"style":1872},[1876],[1802,4281,1841],{"className":4282},[1906],[1802,4284],{"className":4285,"style":1872},[1876],[1802,4287,4289,4292,4295],{"className":4288},[1861],[1802,4290],{"className":4291,"style":2152},[1865],[1802,4293,1844],{"className":4294},[1870,1871],[1802,4296,2269],{"className":4297},[1870],[1798,4299,4300,4301,4329,4330,4358,4359,4492,4493,4329,4521,4549,4550,4679],{},"先把 ",[1802,4302,4304,4317],{"className":4303},[1809],[1802,4305,4307],{"className":4306},[1813],[1815,4308,4309],{"xmlns":1817},[1820,4310,4311,4315],{},[1823,4312,4313],{},[1826,4314,4203],{},[1850,4316,4203],{"encoding":1852},[1802,4318,4320],{"className":4319,"ariaHidden":1847},[1857],[1802,4321,4323,4326],{"className":4322},[1861],[1802,4324],{"className":4325,"style":1866},[1865],[1802,4327,4203],{"className":4328,"style":4254},[1870,1871]," 对 ",[1802,4331,4333,4346],{"className":4332},[1809],[1802,4334,4336],{"className":4335},[1813],[1815,4337,4338],{"xmlns":1817},[1820,4339,4340,4344],{},[1823,4341,4342],{},[1826,4343,1835],{},[1850,4345,1835],{"encoding":1852},[1802,4347,4349],{"className":4348,"ariaHidden":1847},[1857],[1802,4350,4352,4355],{"className":4351},[1861],[1802,4353],{"className":4354,"style":1866},[1865],[1802,4356,1835],{"className":4357,"style":1895},[1870,1871]," 回归，得到残差 ",[1802,4360,4362,4391],{"className":4361},[1809],[1802,4363,4365],{"className":4364},[1813],[1815,4366,4367],{"xmlns":1817},[1820,4368,4369,4388],{},[1823,4370,4371,4378,4380,4386],{},[2204,4372,4373,4375],{"accent":1847},[1826,4374,4203],{},[1830,4376,4377],{},"~",[1830,4379,1832],{},[3536,4381,4382,4384],{},[1826,4383,3582],{},[1826,4385,1835],{},[1826,4387,4203],{},[1850,4389,4390],{"encoding":1852},"\\tilde D=M_XD",[1802,4392,4394,4443],{"className":4393,"ariaHidden":1847},[1857],[1802,4395,4397,4401,4434,4437,4440],{"className":4396},[1861],[1802,4398],{"className":4399,"style":4400},[1865],"height:0.9202em;",[1802,4402,4404],{"className":4403},[1870,2286],[1802,4405,4407],{"className":4406},[2290],[1802,4408,4410],{"className":4409},[2295],[1802,4411,4413,4421],{"className":4412,"style":4400},[2299],[1802,4414,4415,4418],{"style":2303},[1802,4416],{"className":4417,"style":2308},[2307],[1802,4419,4203],{"className":4420,"style":4254},[1870,1871],[1802,4422,4424,4427],{"style":4423},"top:-3.6023em;",[1802,4425],{"className":4426,"style":2308},[2307],[1802,4428,4431],{"className":4429,"style":4430},[2321],"left:-0.1944em;",[1802,4432,4377],{"className":4433},[1870],[1802,4435],{"className":4436,"style":1877},[1876],[1802,4438,1832],{"className":4439},[1881],[1802,4441],{"className":4442,"style":1877},[1876],[1802,4444,4446,4449,4489],{"className":4445},[1861],[1802,4447],{"className":4448,"style":3612},[1865],[1802,4450,4452,4455],{"className":4451},[1870],[1802,4453,3582],{"className":4454,"style":3796},[1870,1871],[1802,4456,4458],{"className":4457},[2460],[1802,4459,4461,4481],{"className":4460},[2290,2291],[1802,4462,4464,4478],{"className":4463},[2295],[1802,4465,4467],{"className":4466,"style":3632},[2299],[1802,4468,4469,4472],{"style":3811},[1802,4470],{"className":4471,"style":2477},[2307],[1802,4473,4475],{"className":4474},[2392,2393,2394,2395],[1802,4476,1835],{"className":4477,"style":1895},[1870,1871,2395],[1802,4479,2330],{"className":4480},[2329],[1802,4482,4484],{"className":4483},[2295],[1802,4485,4487],{"className":4486,"style":3654},[2299],[1802,4488],{},[1802,4490,4203],{"className":4491,"style":4254},[1870,1871],"；再把 ",[1802,4494,4496,4509],{"className":4495},[1809],[1802,4497,4499],{"className":4498},[1813],[1815,4500,4501],{"xmlns":1817},[1820,4502,4503,4507],{},[1823,4504,4505],{},[1826,4506,1828],{},[1850,4508,1828],{"encoding":1852},[1802,4510,4512],{"className":4511,"ariaHidden":1847},[1857],[1802,4513,4515,4518],{"className":4514},[1861],[1802,4516],{"className":4517,"style":1866},[1865],[1802,4519,1828],{"className":4520,"style":1872},[1870,1871],[1802,4522,4524,4537],{"className":4523},[1809],[1802,4525,4527],{"className":4526},[1813],[1815,4528,4529],{"xmlns":1817},[1820,4530,4531,4535],{},[1823,4532,4533],{},[1826,4534,1835],{},[1850,4536,1835],{"encoding":1852},[1802,4538,4540],{"className":4539,"ariaHidden":1847},[1857],[1802,4541,4543,4546],{"className":4542},[1861],[1802,4544],{"className":4545,"style":1866},[1865],[1802,4547,1835],{"className":4548,"style":1895},[1870,1871]," 残差化，得到 ",[1802,4551,4553,4581],{"className":4552},[1809],[1802,4554,4556],{"className":4555},[1813],[1815,4557,4558],{"xmlns":1817},[1820,4559,4560,4578],{},[1823,4561,4562,4568,4570,4576],{},[2204,4563,4564,4566],{"accent":1847},[1826,4565,1828],{},[1830,4567,4377],{},[1830,4569,1832],{},[3536,4571,4572,4574],{},[1826,4573,3582],{},[1826,4575,1835],{},[1826,4577,1828],{},[1850,4579,4580],{"encoding":1852},"\\tilde Y=M_XY",[1802,4582,4584,4630],{"className":4583,"ariaHidden":1847},[1857],[1802,4585,4587,4590,4621,4624,4627],{"className":4586},[1861],[1802,4588],{"className":4589,"style":4400},[1865],[1802,4591,4593],{"className":4592},[1870,2286],[1802,4594,4596],{"className":4595},[2290],[1802,4597,4599],{"className":4598},[2295],[1802,4600,4602,4610],{"className":4601,"style":4400},[2299],[1802,4603,4604,4607],{"style":2303},[1802,4605],{"className":4606,"style":2308},[2307],[1802,4608,1828],{"className":4609,"style":1872},[1870,1871],[1802,4611,4612,4615],{"style":4423},[1802,4613],{"className":4614,"style":2308},[2307],[1802,4616,4618],{"className":4617,"style":3101},[2321],[1802,4619,4377],{"className":4620},[1870],[1802,4622],{"className":4623,"style":1877},[1876],[1802,4625,1832],{"className":4626},[1881],[1802,4628],{"className":4629,"style":1877},[1876],[1802,4631,4633,4636,4676],{"className":4632},[1861],[1802,4634],{"className":4635,"style":3612},[1865],[1802,4637,4639,4642],{"className":4638},[1870],[1802,4640,3582],{"className":4641,"style":3796},[1870,1871],[1802,4643,4645],{"className":4644},[2460],[1802,4646,4648,4668],{"className":4647},[2290,2291],[1802,4649,4651,4665],{"className":4650},[2295],[1802,4652,4654],{"className":4653,"style":3632},[2299],[1802,4655,4656,4659],{"style":3811},[1802,4657],{"className":4658,"style":2477},[2307],[1802,4660,4662],{"className":4661},[2392,2393,2394,2395],[1802,4663,1835],{"className":4664,"style":1895},[1870,1871,2395],[1802,4666,2330],{"className":4667},[2329],[1802,4669,4671],{"className":4670},[2295],[1802,4672,4674],{"className":4673,"style":3654},[2299],[1802,4675],{},[1802,4677,1828],{"className":4678,"style":1872},[1870,1871],"。FWL 定理说明：",[1802,4681,4683],{"className":4682},[1805],[1802,4684,4686,4747],{"className":4685},[1809],[1802,4687,4689],{"className":4688},[1813],[1815,4690,4691],{"xmlns":1817,"display":1818},[1820,4692,4693,4744],{},[1823,4694,4695,4701,4703,4742],{},[2204,4696,4697,4699],{"accent":1847},[1826,4698,4200],{},[1830,4700,2210],{},[1830,4702,1832],{},[4704,4705,4706,4724],"mfrac",{},[1823,4707,4708,4718],{},[2245,4709,4710,4716],{},[2204,4711,4712,4714],{"accent":1847},[1826,4713,4203],{},[1830,4715,4377],{},[1830,4717,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[2204,4719,4720,4722],{"accent":1847},[1826,4721,1828],{},[1830,4723,4377],{},[1823,4725,4726,4736],{},[2245,4727,4728,4734],{},[2204,4729,4730,4732],{"accent":1847},[1826,4731,4203],{},[1830,4733,4377],{},[1830,4735,2254],{"mathvariant":2252,"lspace":2253,"rspace":2253},[2204,4737,4738,4740],{"accent":1847},[1826,4739,4203],{},[1830,4741,4377],{},[1826,4743,2269],{"mathvariant":2252},[1850,4745,4746],{"encoding":1852},"\\hat\\tau=\\frac{\\tilde D'\\tilde Y}{\\tilde D'\\tilde D}.",[1802,4748,4750,4796],{"className":4749,"ariaHidden":1847},[1857],[1802,4751,4753,4756,4787,4790,4793],{"className":4752},[1861],[1802,4754],{"className":4755,"style":2183},[1865],[1802,4757,4759],{"className":4758},[1870,2286],[1802,4760,4762],{"className":4761},[2290],[1802,4763,4765],{"className":4764},[2295],[1802,4766,4768,4776],{"className":4767,"style":2183},[2299],[1802,4769,4770,4773],{"style":2303},[1802,4771],{"className":4772,"style":2308},[2307],[1802,4774,4200],{"className":4775,"style":4250},[1870,1871],[1802,4777,4778,4781],{"style":2303},[1802,4779],{"className":4780,"style":2308},[2307],[1802,4782,4784],{"className":4783,"style":3293},[2321],[1802,4785,2210],{"className":4786},[1870],[1802,4788],{"className":4789,"style":1877},[1876],[1802,4791,1832],{"className":4792},[1881],[1802,4794],{"className":4795,"style":1877},[1876],[1802,4797,4799,4803,5051],{"className":4798},[1861],[1802,4800],{"className":4801,"style":4802},[1865],"height:2.4074em;vertical-align:-0.8102em;",[1802,4804,4806,4810,5048],{"className":4805},[1870],[1802,4807],{"className":4808},[2424,4809],"nulldelimiter",[1802,4811,4813],{"className":4812},[4704],[1802,4814,4816,5039],{"className":4815},[2290,2291],[1802,4817,4819,5036],{"className":4818},[2295],[1802,4820,4823,4925,4936],{"className":4821,"style":4822},[2299],"height:1.5972em;",[1802,4824,4826,4829],{"style":4825},"top:-2.1898em;",[1802,4827],{"className":4828,"style":2308},[2307],[1802,4830,4832,4894],{"className":4831},[1870],[1802,4833,4835,4866],{"className":4834},[1870],[1802,4836,4838],{"className":4837},[1870,2286],[1802,4839,4841],{"className":4840},[2290],[1802,4842,4844],{"className":4843},[2295],[1802,4845,4847,4855],{"className":4846,"style":4400},[2299],[1802,4848,4849,4852],{"style":2303},[1802,4850],{"className":4851,"style":2308},[2307],[1802,4853,4203],{"className":4854,"style":4254},[1870,1871],[1802,4856,4857,4860],{"style":4423},[1802,4858],{"className":4859,"style":2308},[2307],[1802,4861,4863],{"className":4862,"style":4430},[2321],[1802,4864,4377],{"className":4865},[1870],[1802,4867,4869],{"className":4868},[2460],[1802,4870,4872],{"className":4871},[2290],[1802,4873,4875],{"className":4874},[2295],[1802,4876,4879],{"className":4877,"style":4878},[2299],"height:0.6779em;",[1802,4880,4882,4885],{"style":4881},"top:-2.989em;margin-right:0.05em;",[1802,4883],{"className":4884,"style":2477},[2307],[1802,4886,4888],{"className":4887},[2392,2393,2394,2395],[1802,4889,4891],{"className":4890},[1870,2395],[1802,4892,2254],{"className":4893},[1870,2395],[1802,4895,4897],{"className":4896},[1870,2286],[1802,4898,4900],{"className":4899},[2290],[1802,4901,4903],{"className":4902},[2295],[1802,4904,4906,4914],{"className":4905,"style":4400},[2299],[1802,4907,4908,4911],{"style":2303},[1802,4909],{"className":4910,"style":2308},[2307],[1802,4912,4203],{"className":4913,"style":4254},[1870,1871],[1802,4915,4916,4919],{"style":4423},[1802,4917],{"className":4918,"style":2308},[2307],[1802,4920,4922],{"className":4921,"style":4430},[2321],[1802,4923,4377],{"className":4924},[1870],[1802,4926,4928,4931],{"style":4927},"top:-3.23em;",[1802,4929],{"className":4930,"style":2308},[2307],[1802,4932],{"className":4933,"style":4935},[4934],"frac-line","border-bottom-width:0.04em;",[1802,4937,4939,4942],{"style":4938},"top:-3.677em;",[1802,4940],{"className":4941,"style":2308},[2307],[1802,4943,4945,5005],{"className":4944},[1870],[1802,4946,4948,4979],{"className":4947},[1870],[1802,4949,4951],{"className":4950},[1870,2286],[1802,4952,4954],{"className":4953},[2290],[1802,4955,4957],{"className":4956},[2295],[1802,4958,4960,4968],{"className":4959,"style":4400},[2299],[1802,4961,4962,4965],{"style":2303},[1802,4963],{"className":4964,"style":2308},[2307],[1802,4966,4203],{"className":4967,"style":4254},[1870,1871],[1802,4969,4970,4973],{"style":4423},[1802,4971],{"className":4972,"style":2308},[2307],[1802,4974,4976],{"className":4975,"style":4430},[2321],[1802,4977,4377],{"className":4978},[1870],[1802,4980,4982],{"className":4981},[2460],[1802,4983,4985],{"className":4984},[2290],[1802,4986,4988],{"className":4987},[2295],[1802,4989,4991],{"className":4990,"style":2594},[2299],[1802,4992,4993,4996],{"style":2597},[1802,4994],{"className":4995,"style":2477},[2307],[1802,4997,4999],{"className":4998},[2392,2393,2394,2395],[1802,5000,5002],{"className":5001},[1870,2395],[1802,5003,2254],{"className":5004},[1870,2395],[1802,5006,5008],{"className":5007},[1870,2286],[1802,5009,5011],{"className":5010},[2290],[1802,5012,5014],{"className":5013},[2295],[1802,5015,5017,5025],{"className":5016,"style":4400},[2299],[1802,5018,5019,5022],{"style":2303},[1802,5020],{"className":5021,"style":2308},[2307],[1802,5023,1828],{"className":5024,"style":1872},[1870,1871],[1802,5026,5027,5030],{"style":4423},[1802,5028],{"className":5029,"style":2308},[2307],[1802,5031,5033],{"className":5032,"style":3101},[2321],[1802,5034,4377],{"className":5035},[1870],[1802,5037,2330],{"className":5038},[2329],[1802,5040,5042],{"className":5041},[2295],[1802,5043,5046],{"className":5044,"style":5045},[2299],"height:0.8102em;",[1802,5047],{},[1802,5049],{"className":5050},[2453,4809],[1802,5052,2269],{"className":5053},[1870],[1798,5055,5056,5057,5085,5086,5114,5115,5143,5144,5172],{},"所以多元回归中的 ",[1802,5058,5060,5073],{"className":5059},[1809],[1802,5061,5063],{"className":5062},[1813],[1815,5064,5065],{"xmlns":1817},[1820,5066,5067,5071],{},[1823,5068,5069],{},[1826,5070,4203],{},[1850,5072,4203],{"encoding":1852},[1802,5074,5076],{"className":5075,"ariaHidden":1847},[1857],[1802,5077,5079,5082],{"className":5078},[1861],[1802,5080],{"className":5081,"style":1866},[1865],[1802,5083,4203],{"className":5084,"style":4254},[1870,1871]," 系数来自：在剔除 ",[1802,5087,5089,5102],{"className":5088},[1809],[1802,5090,5092],{"className":5091},[1813],[1815,5093,5094],{"xmlns":1817},[1820,5095,5096,5100],{},[1823,5097,5098],{},[1826,5099,1835],{},[1850,5101,1835],{"encoding":1852},[1802,5103,5105],{"className":5104,"ariaHidden":1847},[1857],[1802,5106,5108,5111],{"className":5107},[1861],[1802,5109],{"className":5110,"style":1866},[1865],[1802,5112,1835],{"className":5113,"style":1895},[1870,1871]," 可解释部分之后，",[1802,5116,5118,5131],{"className":5117},[1809],[1802,5119,5121],{"className":5120},[1813],[1815,5122,5123],{"xmlns":1817},[1820,5124,5125,5129],{},[1823,5126,5127],{},[1826,5128,4203],{},[1850,5130,4203],{"encoding":1852},[1802,5132,5134],{"className":5133,"ariaHidden":1847},[1857],[1802,5135,5137,5140],{"className":5136},[1861],[1802,5138],{"className":5139,"style":1866},[1865],[1802,5141,4203],{"className":5142,"style":4254},[1870,1871]," 的剩余变异与 ",[1802,5145,5147,5160],{"className":5146},[1809],[1802,5148,5150],{"className":5149},[1813],[1815,5151,5152],{"xmlns":1817},[1820,5153,5154,5158],{},[1823,5155,5156],{},[1826,5157,1828],{},[1850,5159,1828],{"encoding":1852},[1802,5161,5163],{"className":5162,"ariaHidden":1847},[1857],[1802,5164,5166,5169],{"className":5165},[1861],[1802,5167],{"className":5168,"style":1866},[1865],[1802,5170,1828],{"className":5171,"style":1872},[1870,1871]," 的剩余变异之间的线性关系。",[1798,5174,5175],{},"这解释了为什么控制变量选择重要：控制变量改变了比较所使用的变异，也可能打开新的偏误路径。控制变量不是“越多越科学”。",[1793,5177,5179],{"id":5178},"_4-遗漏变量偏误","4. 遗漏变量偏误",[1798,5181,5182],{},"真实模型为：",[1802,5184,5186],{"className":5185},[1805],[1802,5187,5189,5231],{"className":5188},[1809],[1802,5190,5192],{"className":5191},[1813],[1815,5193,5194],{"xmlns":1817,"display":1818},[1820,5195,5196,5228],{},[1823,5197,5198,5200,5202,5208,5210,5212,5219,5222,5224,5226],{},[1826,5199,1828],{},[1830,5201,1832],{},[3536,5203,5204,5206],{},[1826,5205,1838],{},[1975,5207,1977],{},[1826,5209,4203],{},[1830,5211,1841],{},[3536,5213,5214,5216],{},[1826,5215,1838],{},[1975,5217,5218],{},"2",[1826,5220,5221],{},"Z",[1830,5223,1841],{},[1826,5225,1844],{},[1830,5227,1848],{"separator":1847},[1850,5229,5230],{"encoding":1852},"Y=\\beta_1D+\\beta_2Z+u,",[1802,5232,5234,5252,5312,5370],{"className":5233,"ariaHidden":1847},[1857],[1802,5235,5237,5240,5243,5246,5249],{"className":5236},[1861],[1802,5238],{"className":5239,"style":1866},[1865],[1802,5241,1828],{"className":5242,"style":1872},[1870,1871],[1802,5244],{"className":5245,"style":1877},[1876],[1802,5247,1832],{"className":5248},[1881],[1802,5250],{"className":5251,"style":1877},[1876],[1802,5253,5255,5258,5300,5303,5306,5309],{"className":5254},[1861],[1802,5256],{"className":5257,"style":1891},[1865],[1802,5259,5261,5264],{"className":5260},[1870],[1802,5262,1838],{"className":5263,"style":1899},[1870,1871],[1802,5265,5267],{"className":5266},[2460],[1802,5268,5270,5292],{"className":5269},[2290,2291],[1802,5271,5273,5289],{"className":5272},[2295],[1802,5274,5277],{"className":5275,"style":5276},[2299],"height:0.3011em;",[1802,5278,5280,5283],{"style":5279},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[1802,5281],{"className":5282,"style":2477},[2307],[1802,5284,5286],{"className":5285},[2392,2393,2394,2395],[1802,5287,1977],{"className":5288},[1870,2395],[1802,5290,2330],{"className":5291},[2329],[1802,5293,5295],{"className":5294},[2295],[1802,5296,5298],{"className":5297,"style":3654},[2299],[1802,5299],{},[1802,5301,4203],{"className":5302,"style":4254},[1870,1871],[1802,5304],{"className":5305,"style":1872},[1876],[1802,5307,1841],{"className":5308},[1906],[1802,5310],{"className":5311,"style":1872},[1876],[1802,5313,5315,5318,5358,5361,5364,5367],{"className":5314},[1861],[1802,5316],{"className":5317,"style":1891},[1865],[1802,5319,5321,5324],{"className":5320},[1870],[1802,5322,1838],{"className":5323,"style":1899},[1870,1871],[1802,5325,5327],{"className":5326},[2460],[1802,5328,5330,5350],{"className":5329},[2290,2291],[1802,5331,5333,5347],{"className":5332},[2295],[1802,5334,5336],{"className":5335,"style":5276},[2299],[1802,5337,5338,5341],{"style":5279},[1802,5339],{"className":5340,"style":2477},[2307],[1802,5342,5344],{"className":5343},[2392,2393,2394,2395],[1802,5345,5218],{"className":5346},[1870,2395],[1802,5348,2330],{"className":5349},[2329],[1802,5351,5353],{"className":5352},[2295],[1802,5354,5356],{"className":5355,"style":3654},[2299],[1802,5357],{},[1802,5359,5221],{"className":5360,"style":2095},[1870,1871],[1802,5362],{"className":5363,"style":1872},[1876],[1802,5365,1841],{"className":5366},[1906],[1802,5368],{"className":5369,"style":1872},[1876],[1802,5371,5373,5376,5379],{"className":5372},[1861],[1802,5374],{"className":5375,"style":1916},[1865],[1802,5377,1844],{"className":5378},[1870,1871],[1802,5380,1848],{"className":5381},[1923],[1798,5383,5384,5385,4329,5413,5441],{},"但研究者只回归 ",[1802,5386,5388,5401],{"className":5387},[1809],[1802,5389,5391],{"className":5390},[1813],[1815,5392,5393],{"xmlns":1817},[1820,5394,5395,5399],{},[1823,5396,5397],{},[1826,5398,1828],{},[1850,5400,1828],{"encoding":1852},[1802,5402,5404],{"className":5403,"ariaHidden":1847},[1857],[1802,5405,5407,5410],{"className":5406},[1861],[1802,5408],{"className":5409,"style":1866},[1865],[1802,5411,1828],{"className":5412,"style":1872},[1870,1871],[1802,5414,5416,5429],{"className":5415},[1809],[1802,5417,5419],{"className":5418},[1813],[1815,5420,5421],{"xmlns":1817},[1820,5422,5423,5427],{},[1823,5424,5425],{},[1826,5426,4203],{},[1850,5428,4203],{"encoding":1852},[1802,5430,5432],{"className":5431,"ariaHidden":1847},[1857],[1802,5433,5435,5438],{"className":5434},[1861],[1802,5436],{"className":5437,"style":1866},[1865],[1802,5439,4203],{"className":5440,"style":4254},[1870,1871],"。在简单线性情形下，短回归系数近似为：",[1802,5443,5445],{"className":5444},[1805],[1802,5446,5448,5520],{"className":5447},[1809],[1802,5449,5451],{"className":5450},[1813],[1815,5452,5453],{"xmlns":1817,"display":1818},[1820,5454,5455,5517],{},[1823,5456,5457,5467,5469,5475,5477,5483,5515],{},[3536,5458,5459,5465],{},[2204,5460,5461,5463],{"accent":1847},[1826,5462,1838],{},[1830,5464,4377],{},[1975,5466,1977],{},[1830,5468,1832],{},[3536,5470,5471,5473],{},[1826,5472,1838],{},[1975,5474,1977],{},[1830,5476,1841],{},[3536,5478,5479,5481],{},[1826,5480,1838],{},[1975,5482,5218],{},[4704,5484,5485,5502],{},[1823,5486,5487,5490,5492,5494,5496,5498,5500],{},[1826,5488,5489],{"mathvariant":2252},"Cov",[1830,5491,2218],{},[1830,5493,2234],{"stretchy":2233},[1826,5495,4203],{},[1830,5497,1848],{"separator":1847},[1826,5499,5221],{},[1830,5501,2249],{"stretchy":2233},[1823,5503,5504,5507,5509,5511,5513],{},[1826,5505,5506],{"mathvariant":2252},"Var",[1830,5508,2218],{},[1830,5510,2234],{"stretchy":2233},[1826,5512,4203],{},[1830,5514,2249],{"stretchy":2233},[1826,5516,2269],{"mathvariant":2252},[1850,5518,5519],{"encoding":1852},"\\tilde\\beta_1=\\beta_1+\\beta_2\\frac{\\operatorname{Cov}(D,Z)}{\\operatorname{Var}(D)}.",[1802,5521,5523,5620,5675],{"className":5522,"ariaHidden":1847},[1857],[1802,5524,5526,5530,5611,5614,5617],{"className":5525},[1861],[1802,5527],{"className":5528,"style":5529},[1865],"height:1.1257em;vertical-align:-0.1944em;",[1802,5531,5533,5577],{"className":5532},[1870],[1802,5534,5536],{"className":5535},[1870,2286],[1802,5537,5539,5569],{"className":5538},[2290,2291],[1802,5540,5542,5566],{"className":5541},[2295],[1802,5543,5546,5554],{"className":5544,"style":5545},[2299],"height:0.9313em;",[1802,5547,5548,5551],{"style":2303},[1802,5549],{"className":5550,"style":2308},[2307],[1802,5552,1838],{"className":5553,"style":1899},[1870,1871],[1802,5555,5557,5560],{"style":5556},"top:-3.6134em;",[1802,5558],{"className":5559,"style":2308},[2307],[1802,5561,5563],{"className":5562,"style":2322},[2321],[1802,5564,4377],{"className":5565},[1870],[1802,5567,2330],{"className":5568},[2329],[1802,5570,5572],{"className":5571},[2295],[1802,5573,5575],{"className":5574,"style":2337},[2299],[1802,5576],{},[1802,5578,5580],{"className":5579},[2460],[1802,5581,5583,5603],{"className":5582},[2290,2291],[1802,5584,5586,5600],{"className":5585},[2295],[1802,5587,5589],{"className":5588,"style":5276},[2299],[1802,5590,5591,5594],{"style":5279},[1802,5592],{"className":5593,"style":2477},[2307],[1802,5595,5597],{"className":5596},[2392,2393,2394,2395],[1802,5598,1977],{"className":5599},[1870,2395],[1802,5601,2330],{"className":5602},[2329],[1802,5604,5606],{"className":5605},[2295],[1802,5607,5609],{"className":5608,"style":3654},[2299],[1802,5610],{},[1802,5612],{"className":5613,"style":1877},[1876],[1802,5615,1832],{"className":5616},[1881],[1802,5618],{"className":5619,"style":1877},[1876],[1802,5621,5623,5626,5666,5669,5672],{"className":5622},[1861],[1802,5624],{"className":5625,"style":1891},[1865],[1802,5627,5629,5632],{"className":5628},[1870],[1802,5630,1838],{"className":5631,"style":1899},[1870,1871],[1802,5633,5635],{"className":5634},[2460],[1802,5636,5638,5658],{"className":5637},[2290,2291],[1802,5639,5641,5655],{"className":5640},[2295],[1802,5642,5644],{"className":5643,"style":5276},[2299],[1802,5645,5646,5649],{"style":5279},[1802,5647],{"className":5648,"style":2477},[2307],[1802,5650,5652],{"className":5651},[2392,2393,2394,2395],[1802,5653,1977],{"className":5654},[1870,2395],[1802,5656,2330],{"className":5657},[2329],[1802,5659,5661],{"className":5660},[2295],[1802,5662,5664],{"className":5663,"style":3654},[2299],[1802,5665],{},[1802,5667],{"className":5668,"style":1872},[1876],[1802,5670,1841],{"className":5671},[1906],[1802,5673],{"className":5674,"style":1872},[1876],[1802,5676,5678,5682,5722,5821],{"className":5677},[1861],[1802,5679],{"className":5680,"style":5681},[1865],"height:2.363em;vertical-align:-0.936em;",[1802,5683,5685,5688],{"className":5684},[1870],[1802,5686,1838],{"className":5687,"style":1899},[1870,1871],[1802,5689,5691],{"className":5690},[2460],[1802,5692,5694,5714],{"className":5693},[2290,2291],[1802,5695,5697,5711],{"className":5696},[2295],[1802,5698,5700],{"className":5699,"style":5276},[2299],[1802,5701,5702,5705],{"style":5279},[1802,5703],{"className":5704,"style":2477},[2307],[1802,5706,5708],{"className":5707},[2392,2393,2394,2395],[1802,5709,5218],{"className":5710},[1870,2395],[1802,5712,2330],{"className":5713},[2329],[1802,5715,5717],{"className":5716},[2295],[1802,5718,5720],{"className":5719,"style":3654},[2299],[1802,5721],{},[1802,5723,5725,5728,5818],{"className":5724},[1870],[1802,5726],{"className":5727},[2424,4809],[1802,5729,5731],{"className":5730},[4704],[1802,5732,5734,5809],{"className":5733},[2290,2291],[1802,5735,5737,5806],{"className":5736},[2295],[1802,5738,5741,5766,5774],{"className":5739,"style":5740},[2299],"height:1.427em;",[1802,5742,5744,5747],{"style":5743},"top:-2.314em;",[1802,5745],{"className":5746,"style":2308},[2307],[1802,5748,5750,5757,5760,5763],{"className":5749},[1870],[1802,5751,5753],{"className":5752},[2359],[1802,5754,5506],{"className":5755},[1870,5756],"mathrm",[1802,5758,2234],{"className":5759},[2424],[1802,5761,4203],{"className":5762,"style":4254},[1870,1871],[1802,5764,2249],{"className":5765},[2453],[1802,5767,5768,5771],{"style":4927},[1802,5769],{"className":5770,"style":2308},[2307],[1802,5772],{"className":5773,"style":4935},[4934],[1802,5775,5776,5779],{"style":4938},[1802,5777],{"className":5778,"style":2308},[2307],[1802,5780,5782,5788,5791,5794,5797,5800,5803],{"className":5781},[1870],[1802,5783,5785],{"className":5784},[2359],[1802,5786,5489],{"className":5787,"style":2363},[1870,5756],[1802,5789,2234],{"className":5790},[2424],[1802,5792,4203],{"className":5793,"style":4254},[1870,1871],[1802,5795,1848],{"className":5796},[1923],[1802,5798],{"className":5799,"style":2368},[1876],[1802,5801,5221],{"className":5802,"style":2095},[1870,1871],[1802,5804,2249],{"className":5805},[2453],[1802,5807,2330],{"className":5808},[2329],[1802,5810,5812],{"className":5811},[2295],[1802,5813,5816],{"className":5814,"style":5815},[2299],"height:0.936em;",[1802,5817],{},[1802,5819],{"className":5820},[2453,4809],[1802,5822,2269],{"className":5823},[1870],[1798,5825,5826,5827,4329,5855,5883,5884,3362,5912,5940],{},"偏误方向由两条路径共同决定：",[1802,5828,5830,5843],{"className":5829},[1809],[1802,5831,5833],{"className":5832},[1813],[1815,5834,5835],{"xmlns":1817},[1820,5836,5837,5841],{},[1823,5838,5839],{},[1826,5840,5221],{},[1850,5842,5221],{"encoding":1852},[1802,5844,5846],{"className":5845,"ariaHidden":1847},[1857],[1802,5847,5849,5852],{"className":5848},[1861],[1802,5850],{"className":5851,"style":1866},[1865],[1802,5853,5221],{"className":5854,"style":2095},[1870,1871],[1802,5856,5858,5871],{"className":5857},[1809],[1802,5859,5861],{"className":5860},[1813],[1815,5862,5863],{"xmlns":1817},[1820,5864,5865,5869],{},[1823,5866,5867],{},[1826,5868,1828],{},[1850,5870,1828],{"encoding":1852},[1802,5872,5874],{"className":5873,"ariaHidden":1847},[1857],[1802,5875,5877,5880],{"className":5876},[1861],[1802,5878],{"className":5879,"style":1866},[1865],[1802,5881,1828],{"className":5882,"style":1872},[1870,1871]," 的作用，以及 ",[1802,5885,5887,5900],{"className":5886},[1809],[1802,5888,5890],{"className":5889},[1813],[1815,5891,5892],{"xmlns":1817},[1820,5893,5894,5898],{},[1823,5895,5896],{},[1826,5897,4203],{},[1850,5899,4203],{"encoding":1852},[1802,5901,5903],{"className":5902,"ariaHidden":1847},[1857],[1802,5904,5906,5909],{"className":5905},[1861],[1802,5907],{"className":5908,"style":1866},[1865],[1802,5910,4203],{"className":5911,"style":4254},[1870,1871],[1802,5913,5915,5928],{"className":5914},[1809],[1802,5916,5918],{"className":5917},[1813],[1815,5919,5920],{"xmlns":1817},[1820,5921,5922,5926],{},[1823,5923,5924],{},[1826,5925,5221],{},[1850,5927,5221],{"encoding":1852},[1802,5929,5931],{"className":5930,"ariaHidden":1847},[1857],[1802,5932,5934,5937],{"className":5933},[1861],[1802,5935],{"className":5936,"style":1866},[1865],[1802,5938,5221],{"className":5939,"style":2095},[1870,1871]," 的相关。只有知道两者方向，才可以判断高估还是低估。",[5942,5943,5945],"h3",{"id":5944},"教育工资例子","教育—工资例子",[1798,5947,5948],{},"若能力提高工资且能力更高者教育更多，则省略能力通常使教育回报被高估；若教育更多的人来自不利背景而背景降低工资，偏误方向可能相反。实际研究不能只凭直觉说“能力偏误一定向上”，要画出因果图或写出协方差路径。",[1793,5950,5952],{"id":5951},"_5-函数形式","5. 函数形式",[5954,5955,5956,5972],"table",{},[5957,5958,5959],"thead",{},[5960,5961,5962,5966,5969],"tr",{},[5963,5964,5965],"th",{},"形式",[5963,5967,5968],{},"模型",[5963,5970,5971],{},"主要解释",[5973,5974,5975,6296,6641,6983],"tbody",{},[5960,5976,5977,5981,6165],{},[5978,5979,5980],"td",{},"水平—水平",[5978,5982,5983],{},[1802,5984,5986,6022],{"className":5985},[1809],[1802,5987,5989],{"className":5988},[1813],[1815,5990,5991],{"xmlns":1817},[1820,5992,5993,6019],{},[1823,5994,5995,5997,5999,6005,6007,6013,6015,6017],{},[1826,5996,1828],{},[1830,5998,1832],{},[3536,6000,6001,6003],{},[1826,6002,1838],{},[1975,6004,2562],{},[1830,6006,1841],{},[3536,6008,6009,6011],{},[1826,6010,1838],{},[1975,6012,1977],{},[1826,6014,1835],{},[1830,6016,1841],{},[1826,6018,1844],{},[1850,6020,6021],{"encoding":1852},"Y=\\beta_0+\\beta_1X+u",[1802,6023,6025,6043,6098,6156],{"className":6024,"ariaHidden":1847},[1857],[1802,6026,6028,6031,6034,6037,6040],{"className":6027},[1861],[1802,6029],{"className":6030,"style":1866},[1865],[1802,6032,1828],{"className":6033,"style":1872},[1870,1871],[1802,6035],{"className":6036,"style":1877},[1876],[1802,6038,1832],{"className":6039},[1881],[1802,6041],{"className":6042,"style":1877},[1876],[1802,6044,6046,6049,6089,6092,6095],{"className":6045},[1861],[1802,6047],{"className":6048,"style":1891},[1865],[1802,6050,6052,6055],{"className":6051},[1870],[1802,6053,1838],{"className":6054,"style":1899},[1870,1871],[1802,6056,6058],{"className":6057},[2460],[1802,6059,6061,6081],{"className":6060},[2290,2291],[1802,6062,6064,6078],{"className":6063},[2295],[1802,6065,6067],{"className":6066,"style":5276},[2299],[1802,6068,6069,6072],{"style":5279},[1802,6070],{"className":6071,"style":2477},[2307],[1802,6073,6075],{"className":6074},[2392,2393,2394,2395],[1802,6076,2562],{"className":6077},[1870,2395],[1802,6079,2330],{"className":6080},[2329],[1802,6082,6084],{"className":6083},[2295],[1802,6085,6087],{"className":6086,"style":3654},[2299],[1802,6088],{},[1802,6090],{"className":6091,"style":1872},[1876],[1802,6093,1841],{"className":6094},[1906],[1802,6096],{"className":6097,"style":1872},[1876],[1802,6099,6101,6104,6144,6147,6150,6153],{"className":6100},[1861],[1802,6102],{"className":6103,"style":1891},[1865],[1802,6105,6107,6110],{"className":6106},[1870],[1802,6108,1838],{"className":6109,"style":1899},[1870,1871],[1802,6111,6113],{"className":6112},[2460],[1802,6114,6116,6136],{"className":6115},[2290,2291],[1802,6117,6119,6133],{"className":6118},[2295],[1802,6120,6122],{"className":6121,"style":5276},[2299],[1802,6123,6124,6127],{"style":5279},[1802,6125],{"className":6126,"style":2477},[2307],[1802,6128,6130],{"className":6129},[2392,2393,2394,2395],[1802,6131,1977],{"className":6132},[1870,2395],[1802,6134,2330],{"className":6135},[2329],[1802,6137,6139],{"className":6138},[2295],[1802,6140,6142],{"className":6141,"style":3654},[2299],[1802,6143],{},[1802,6145,1835],{"className":6146,"style":1895},[1870,1871],[1802,6148],{"className":6149,"style":1872},[1876],[1802,6151,1841],{"className":6152},[1906],[1802,6154],{"className":6155,"style":1872},[1876],[1802,6157,6159,6162],{"className":6158},[1861],[1802,6160],{"className":6161,"style":2152},[1865],[1802,6163,1844],{"className":6164},[1870,1871],[5978,6166,6167,6195,6196,6224,6225,6295],{},[1802,6168,6170,6183],{"className":6169},[1809],[1802,6171,6173],{"className":6172},[1813],[1815,6174,6175],{"xmlns":1817},[1820,6176,6177,6181],{},[1823,6178,6179],{},[1826,6180,1835],{},[1850,6182,1835],{"encoding":1852},[1802,6184,6186],{"className":6185,"ariaHidden":1847},[1857],[1802,6187,6189,6192],{"className":6188},[1861],[1802,6190],{"className":6191,"style":1866},[1865],[1802,6193,1835],{"className":6194,"style":1895},[1870,1871]," 增加一单位，",[1802,6197,6199,6212],{"className":6198},[1809],[1802,6200,6202],{"className":6201},[1813],[1815,6203,6204],{"xmlns":1817},[1820,6205,6206,6210],{},[1823,6207,6208],{},[1826,6209,1828],{},[1850,6211,1828],{"encoding":1852},[1802,6213,6215],{"className":6214,"ariaHidden":1847},[1857],[1802,6216,6218,6221],{"className":6217},[1861],[1802,6219],{"className":6220,"style":1866},[1865],[1802,6222,1828],{"className":6223,"style":1872},[1870,1871]," 变化 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X+u",[1802,7038,7040,7066,7121,7190],{"className":7039,"ariaHidden":1847},[1857],[1802,7041,7043,7046,7051,7054,7057,7060,7063],{"className":7042},[1861],[1802,7044],{"className":7045,"style":1891},[1865],[1802,7047,6359,7049],{"className":7048},[2359],[1802,7050,2364],{"style":2363},[1802,7052],{"className":7053,"style":2368},[1876],[1802,7055,1828],{"className":7056,"style":1872},[1870,1871],[1802,7058],{"className":7059,"style":1877},[1876],[1802,7061,1832],{"className":7062},[1881],[1802,7064],{"className":7065,"style":1877},[1876],[1802,7067,7069,7072,7112,7115,7118],{"className":7068},[1861],[1802,7070],{"className":7071,"style":1891},[1865],[1802,7073,7075,7078],{"className":7074},[1870],[1802,7076,1838],{"className":7077,"style":1899},[1870,1871],[1802,7079,7081],{"className":7080},[2460],[1802,7082,7084,7104],{"className":7083},[2290,2291],[1802,7085,7087,7101],{"className":7086},[2295],[1802,7088,7090],{"className":7089,"style":5276},[2299],[1802,7091,7092,7095],{"style":5279},[1802,7093],{"className":7094,"style":2477},[2307],[1802,7096,7098],{"className":7097},[2392,2393,2394,2395],[1802,7099,2562],{"className":7100},[1870,2395],[1802,7102,2330],{"className":7103},[2329],[1802,7105,7107],{"className":7106},[2295],[1802,7108,7110],{"className":7109,"style":3654},[2299],[1802,7111],{},[1802,7113],{"className":7114,"style":1872},[1876],[1802,7116,1841],{"className":7117},[1906],[1802,7119],{"className":7120,"style":1872},[1876],[1802,7122,7124,7127,7167,7170,7175,7178,7181,7184,7187],{"className":7123},[1861],[1802,7125],{"className":7126,"style":1891},[1865],[1802,7128,7130,7133],{"className":7129},[1870],[1802,7131,1838],{"className":7132,"style":1899},[1870,1871],[1802,7134,7136],{"className":7135},[2460],[1802,7137,7139,7159],{"className":7138},[2290,2291],[1802,7140,7142,7156],{"className":7141},[2295],[1802,7143,7145],{"className":7144,"style":5276},[2299],[1802,7146,7147,7150],{"style":5279},[1802,7148],{"className":7149,"style":2477},[2307],[1802,7151,7153],{"className":7152},[2392,2393,2394,2395],[1802,7154,1977],{"className":7155},[1870,2395],[1802,7157,2330],{"className":7158},[2329],[1802,7160,7162],{"className":7161},[2295],[1802,7163,7165],{"className":7164,"style":3654},[2299],[1802,7166],{},[1802,7168],{"className":7169,"style":2368},[1876],[1802,7171,6359,7173],{"className":7172},[2359],[1802,7174,2364],{"style":2363},[1802,7176],{"className":7177,"style":2368},[1876],[1802,7179,1835],{"className":7180,"style":1895},[1870,1871],[1802,7182],{"className":7183,"style":1872},[1876],[1802,7185,1841],{"className":7186},[1906],[1802,7188],{"className":7189,"style":1872},[1876],[1802,7191,7193,7196],{"className":7192},[1861],[1802,7194],{"className":7195,"style":2152},[1865],[1802,7197,1844],{"className":7198},[1870,1871],[5978,7200,7201,7270],{},[1802,7202,7204,7221],{"className":7203},[1809],[1802,7205,7207],{"className":7206},[1813],[1815,7208,7209],{"xmlns":1817},[1820,7210,7211,7219],{},[1823,7212,7213],{},[3536,7214,7215,7217],{},[1826,7216,1838],{},[1975,7218,1977],{},[1850,7220,6245],{"encoding":1852},[1802,7222,7224],{"className":7223,"ariaHidden":1847},[1857],[1802,7225,7227,7230],{"className":7226},[1861],[1802,7228],{"className":7229,"style":1891},[1865],[1802,7231,7233,7236],{"className":7232},[1870],[1802,7234,1838],{"className":7235,"style":1899},[1870,1871],[1802,7237,7239],{"className":7238},[2460],[1802,7240,7242,7262],{"className":7241},[2290,2291],[1802,7243,7245,7259],{"className":7244},[2295],[1802,7246,7248],{"className":7247,"style":5276},[2299],[1802,7249,7250,7253],{"style":5279},[1802,7251],{"className":7252,"style":2477},[2307],[1802,7254,7256],{"className":7255},[2392,2393,2394,2395],[1802,7257,1977],{"className":7258},[1870,2395],[1802,7260,2330],{"className":7261},[2329],[1802,7263,7265],{"className":7264},[2295],[1802,7266,7268],{"className":7267,"style":3654},[2299],[1802,7269],{}," 是弹性",[1798,7272,7273,7274,7336],{},"对数形式改变了参数含义，也会改变误差结构和样本处理。不能为了让 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更高而机械取对数。",[1793,7338,7340],{"id":7339},"_6-交互项与异质性","6. 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education+\\beta_2 female\n+\\beta_3(education\\times 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},[1870,1871],[1802,7769,7387],{"className":7770},[1870,1871],[1802,7772,7390],{"className":7773},[1870,1871],[1802,7775,7393],{"className":7776},[1870,1871],[1802,7778,7396],{"className":7779},[1870,1871],[1802,7781,7399],{"className":7782},[1870,1871],[1802,7784,1970],{"className":7785},[1870,1871],[1802,7787],{"className":7788,"style":1872},[1876],[1802,7790,1973],{"className":7791},[1906],[1802,7793],{"className":7794,"style":1872},[1876],[1802,7796,7798,7801,7804,7807,7810,7813,7816,7819,7822,7825],{"className":7797},[1861],[1802,7799],{"className":7800,"style":2508},[1865],[1802,7802,7412],{"className":7803,"style":7686},[1870,1871],[1802,7805,7379],{"className":7806},[1870,1871],[1802,7808,7693],{"className":7809},[1870,1871],[1802,7811,7422],{"className":7812,"style":7697},[1870,1871],[1802,7814,7379],{"className":7815},[1870,1871],[1802,7817,2249],{"className":7818},[2453],[1802,7820],{"className":7821,"style":1872},[1876],[1802,7823,1841],{"className":7824},[1906],[1802,7826],{"className":7827,"style":1872},[1876],[1802,7829,7831,7834,7837],{"className":7830},[1861],[1802,7832],{"className":7833,"style":2152},[1865],[1802,7835,1844],{"className":7836},[1870,1871],[1802,7838,2269],{"className":7839},[1870],[1798,7841,7842,7843,7912,7913,8046,8047,8117],{},"男性的教育斜率是 ",[1802,7844,7846,7863],{"className":7845},[1809],[1802,7847,7849],{"className":7848},[1813],[1815,7850,7851],{"xmlns":1817},[1820,7852,7853,7861],{},[1823,7854,7855],{},[3536,7856,7857,7859],{},[1826,7858,1838],{},[1975,7860,1977],{},[1850,7862,6245],{"encoding":1852},[1802,7864,7866],{"className":7865,"ariaHidden":1847},[1857],[1802,7867,7869,7872],{"className":7868},[1861],[1802,7870],{"className":7871,"style":1891},[1865],[1802,7873,7875,7878],{"className":7874},[1870],[1802,7876,1838],{"className":7877,"style":1899},[1870,1871],[1802,7879,7881],{"className":7880},[2460],[1802,7882,7884,7904],{"className":7883},[2290,2291],[1802,7885,7887,7901],{"className":7886},[2295],[1802,7888,7890],{"className":7889,"style":5276},[2299],[1802,7891,7892,7895],{"style":5279},[1802,7893],{"className":7894,"style":2477},[2307],[1802,7896,7898],{"className":7897},[2392,2393,2394,2395],[1802,7899,1977],{"className":7900},[1870,2395],[1802,7902,2330],{"className":7903},[2329],[1802,7905,7907],{"className":7906},[2295],[1802,7908,7910],{"className":7909,"style":3654},[2299],[1802,7911],{},"，女性的教育斜率是 ",[1802,7914,7916,7942],{"className":7915},[1809],[1802,7917,7919],{"className":7918},[1813],[1815,7920,7921],{"xmlns":1817},[1820,7922,7923,7939],{},[1823,7924,7925,7931,7933],{},[3536,7926,7927,7929],{},[1826,7928,1838],{},[1975,7930,1977],{},[1830,7932,1841],{},[3536,7934,7935,7937],{},[1826,7936,1838],{},[1975,7938,7433],{},[1850,7940,7941],{"encoding":1852},"\\beta_1+\\beta_3",[1802,7943,7945,8000],{"className":7944,"ariaHidden":1847},[1857],[1802,7946,7948,7951,7991,7994,7997],{"className":7947},[1861],[1802,7949],{"className":7950,"style":1891},[1865],[1802,7952,7954,7957],{"className":7953},[1870],[1802,7955,1838],{"className":7956,"style":1899},[1870,1871],[1802,7958,7960],{"className":7959},[2460],[1802,7961,7963,7983],{"className":7962},[2290,2291],[1802,7964,7966,7980],{"className":7965},[2295],[1802,7967,7969],{"className":7968,"style":5276},[2299],[1802,7970,7971,7974],{"style":5279},[1802,7972],{"className":7973,"style":2477},[2307],[1802,7975,7977],{"className":7976},[2392,2393,2394,2395],[1802,7978,1977],{"className":7979},[1870,2395],[1802,7981,2330],{"className":7982},[2329],[1802,7984,7986],{"className":7985},[2295],[1802,7987,7989],{"className":7988,"style":3654},[2299],[1802,7990],{},[1802,7992],{"className":7993,"style":1872},[1876],[1802,7995,1841],{"className":7996},[1906],[1802,7998],{"className":7999,"style":1872},[1876],[1802,8001,8003,8006],{"className":8002},[1861],[1802,8004],{"className":8005,"style":1891},[1865],[1802,8007,8009,8012],{"className":8008},[1870],[1802,8010,1838],{"className":8011,"style":1899},[1870,1871],[1802,8013,8015],{"className":8014},[2460],[1802,8016,8018,8038],{"className":8017},[2290,2291],[1802,8019,8021,8035],{"className":8020},[2295],[1802,8022,8024],{"className":8023,"style":5276},[2299],[1802,8025,8026,8029],{"style":5279},[1802,8027],{"className":8028,"style":2477},[2307],[1802,8030,8032],{"className":8031},[2392,2393,2394,2395],[1802,8033,7433],{"className":8034},[1870,2395],[1802,8036,2330],{"className":8037},[2329],[1802,8039,8041],{"className":8040},[2295],[1802,8042,8044],{"className":8043,"style":3654},[2299],[1802,8045],{},"。含交互项时，不能把 ",[1802,8048,8050,8068],{"className":8049},[1809],[1802,8051,8053],{"className":8052},[1813],[1815,8054,8055],{"xmlns":1817},[1820,8056,8057,8065],{},[1823,8058,8059],{},[3536,8060,8061,8063],{},[1826,8062,1838],{},[1975,8064,7433],{},[1850,8066,8067],{"encoding":1852},"\\beta_3",[1802,8069,8071],{"className":8070,"ariaHidden":1847},[1857],[1802,8072,8074,8077],{"className":8073},[1861],[1802,8075],{"className":8076,"style":1891},[1865],[1802,8078,8080,8083],{"className":8079},[1870],[1802,8081,1838],{"className":8082,"style":1899},[1870,1871],[1802,8084,8086],{"className":8085},[2460],[1802,8087,8089,8109],{"className":8088},[2290,2291],[1802,8090,8092,8106],{"className":8091},[2295],[1802,8093,8095],{"className":8094,"style":5276},[2299],[1802,8096,8097,8100],{"style":5279},[1802,8098],{"className":8099,"style":2477},[2307],[1802,8101,8103],{"className":8102},[2392,2393,2394,2395],[1802,8104,7433],{"className":8105},[1870,2395],[1802,8107,2330],{"className":8108},[2329],[1802,8110,8112],{"className":8111},[2295],[1802,8113,8115],{"className":8114,"style":3654},[2299],[1802,8116],{}," 单独称为“女性效应”；要在有意义的教育水平上计算边际效应。",[1793,8119,8120],{"id":8120},"研究生扩展",[3006,8122,8123,8126,8129,8132,8135],{},[3009,8124,8125],{},"投影定理与最佳线性预测；",[3009,8127,8128],{},"Frisch–Waugh–Lovell 在高维控制和固定效应中的应用；",[3009,8130,8131],{},"partialling-out 与双重机器学习；",[3009,8133,8134],{},"高杠杆点、影响函数和稳健回归；",[3009,8136,8137],{},"规格搜索、多重检验和预先注册。",[1793,8139,8140],{"id":8140},"自测题",[8142,8143,8144,8147,8150,8153,8156],"ol",{},[3009,8145,8146],{},"为什么 OLS 残差与解释变量正交不能证明外生性？",[3009,8148,8149],{},"用 FWL 定理解释控制家庭背景后的教育系数变化。",[3009,8151,8152],{},"遗漏变量偏误何时为零？",[3009,8154,8155],{},"对数—对数模型中系数如何解释？",[3009,8157,8158],{},"交互项存在时，如何计算某一组的边际效应？",[1793,8160,8161],{"id":8161},"下一章",[1798,8163,8164],{},"OLS 给出点估计，但研究者还必须说明估计的不确定性。下一章讨论 t\u002FF 检验、异方差、聚类和 bootstrap。",{"title":10,"searchDepth":8166,"depth":8166,"links":8167},2,[8168,8169,8170,8171,8172,8176,8177,8178,8179,8180],{"id":1795,"depth":8166,"text":1796},{"id":3004,"depth":8166,"text":3004},{"id":3026,"depth":8166,"text":3027},{"id":4174,"depth":8166,"text":4175},{"id":5178,"depth":8166,"text":5179,"children":8173},[8174],{"id":5944,"depth":8175,"text":5945},3,{"id":5951,"depth":8166,"text":5952},{"id":7339,"depth":8166,"text":7340},{"id":8120,"depth":8166,"text":8120},{"id":8140,"depth":8166,"text":8140},{"id":8161,"depth":8166,"text":8161},"推导普通最小二乘、解释 FWL 定理并分析函数形式与遗漏变量偏误。","md",{},true,{"title":1334,"description":8181},"5xV9tR45u3rX1WCTxwZcTl2LUYnbXIm9QEPNzF6YXWo",[8188,8190],{"title":1330,"path":1331,"stem":1332,"description":8189,"children":-1},"从样本、总体、条件期望和线性投影开始建立计量经济学语言。",{"title":1338,"path":1339,"stem":1340,"description":8191,"children":-1},"从抽样分布、t\u002FF 检验到异方差、聚类和 bootstrap 推断。",1785754747403]