EconometricsMod

¡' U

Jeffrey M.Wooldridge

rief Contents

Chapter 1 The Nature of Econometrics and Economic Data

PART 1: REGRESSION ANALYSIS WITH CROSS-SECTIONAL DATA

Chapter 2 The Simple Regression ModelChapter 3 Múltiple Regression Analysis: EstimationChapter 4 Múltiple Regression Analysis: InferenceChapter 5 Múltiple Regression Analysis: OLS AsymptoÜcsChapter 6 Múltiple Regression Analysis: Further IssuesChapter 7 Múltiple Regression Analysis with Qualitative Information:

Binary (or Dummy) VariablesChapter 8 HeteroskedasticityChapter 9 More on Specification and Data Issues

PART 2: REGRESSION ANALYSIS WITH TIME SERIES DATA

Chapter 10 Basic Regression Analysis with Time Series DataChapter 1 1 Further Issues in Using OLS with Time Series DataChapter 12 Serial Correlation and Heteroskedasticity ín Time

Series Regressions

PART 3: ADVANCED TOPICS

Chapter 13 Pooling Cross Sections across Time: Simple PanelData Methods

Chapter 14 Advanced Panel Data MethodsChapter 15 Instrumental Variables Estimation and Two Stage Least SquaresChapter 16 Simultaneous Equations ModelsChapter 1 7 Limited Dependen! Variable Models and Sample

Selection CorrectionsChapter 18 Advanced Time Series TopicsChapter 19 Carrying Out an Empirical Project

APPENDICES

Appendix A Basic Mathematical ToolsAppendix B Fundamentáis of ProbabilityAppendix C Fundamentáis of Mathematical StatisticsAppendix D Summary of Matrix AlgebraAppendix E The Linear Regression Model in Matrix FormAppendix F Answers to Chapter QuestionsAppendix G Statistical Tables

ReferencesGlossaryíndex

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Contents

C H A P T E R I

The Nature of Econometricsand Economic Data 1

1.1 What Is Econometrics? 11.2 Steps in Empirical Economic Analysis 21.3 The Structure of Economic Data 5

Cross-Sectional Data 5Time Series Data 8Pooled Cross Sections 9Panel or Longitudinal Dala 10A Commení on Data Structures 12

1.4 Causality and the Notion of Ceteris Paribus inEconometric Analysis 12

Summary 17Key Terms 17Problems 17Computer Exercises 18

P A R T i

Regression Analysis withCross-Sectional Data 21

C H A P T E R Z

The Simple Regression Model 22

2.1 Definition of the Simple RegressionModel 22

2.2 Deriving the Ordinary Least SquaresEsti mates 27A Nole on Terminology 35

2.3 Properties of OLS on Any Sample of Data 36Fitted Valúes and Residuals 36Algébrate Properties of OLS Statistics 37Goodness-of-Fit 40

2.4 Units of Measurement and FunctionalForm 41The Effects ofChanging Units of Measuremen!on OLS Statistics 41Incorporating Nonlinearities in SimpleRegression 43The Meaning of "Linear" Regression 46

2.5 Expected Valúes and Variances of the OLSEstimators 46Vnbiasedness of OLS 47Variances ofthe OLS Estimators 52Estimating the Error Variance 56

2.6 Regression through the Origin 58Summary 59Key Terms 60Problems 61Computer Exercises 64Appendix 2A 66

C H A P T E R 3

Múltiple Regression Analysis:Estimation 68

3.1 Motivation for Múltiple Regression 68The Model with Two Independen!Variables 68The Model with k Independen! Variables 71

3.2 Mechanics and Interpretaron of Ordinary LeastSquares 73Obíaining the OLS Estímales 73¡nterpreting ¡he OLS Regression Equation 74On the Meaning of "Holding Other FactorsFixed" in Múltiple Regression 77Changing More Than One Independen! VariableSimultaneously 77OLS Fitted Valúes and Residuals 77A "Partialling Out" ¡nterpretation ofMúltiple Regression 78Comparison of Simple and Múltiple RegressionEstimates 79Goodness-of-Fit 80Regression through the Origin 83

3.3 The Expected Valué of the OLSEstimators 84Including Irrelevant Variables in a RegressionModel 89Omitted Variable Bias: The Simple Case 89Omitted Variable Bias: More General Cases 93

3.4 The Variance of the OLS Estimators 94The Components ofthe OLS Variances:Multicollinearity 95

Contents

Variances m Misspecified Models 99Estimating a2: Standard Errors oftheOLS Estimators 101

3.5 Efficiency of OLS: The Gauss-MarkovTheorem 102

Summary 104Key Térras 105Problems 105Computer Exercises 110Appendix 3A 113

C H A P T E R 4

Múltiple Regression Analysis:Inference 117

4.1 Sampling Distributions of the OLSEstimators 117

4.2 Testing Hypotheses about a Single PopulationParameter: The i Test 120Testing against One-Sided Alternatives ¡23Two-Sided Ahernatives 128Testing Other Hypotheses about )3. 130Cotnputing p-Valuesfor t Tests 133A Reminder on the Language of ClassicalHypothesis Testing 135Economic, or Practical, versus StatisticalSignificance 135

4.3 Confidence Intervals 1384.4 Testing Hypotheses about a Single Linear

Combination ofthe Parameters 1404.5 Testing Múltiple Linear Restrictions:

The F Test 143Testing Exclusión Restrictions 143Relationship between F and t Staüstics 149The R-Squared Form of the F Statistic 150Computing p-Valuesfor F Tests 151The F Statistic for Overall Significance of aRegression 152Testing General Linear Restrictions 153

4.6 Reporting Regression Results 154Summary 156KeyTerms 158Problems 159Computer Exercises 163

C H A P T E R 5

Múltiple Regression Analysis:OLS Asymptotics 167

5.1 Consistency 167Deriving the Inconsistency in OLS 170

5.2 Asymptotic Normafity and Large SampleInference 172Other Large Sample Tests: The LagrangeMultiplier Statistic 176

5.3 Asymptotic Efficiency of OLS 179Summary 180KeyTerms 181Problems 181Computer Exercises 181Appendix 5A 182

C H A P T E R 6

Múltiple Regression Analysis:Further Issues 184

6.1 Effects of Data Scaling on OLSStaüstics 184Beta Coefficients 187

6.2 More on Funcíional Form 189More on Using Logarithmic FunctionalForms 189Models with Quadratics 192Models with Interaction Terms 197

6.3 More on Goodness-of-Fit and Selectionol'Regressors 199Adjusted R-Squared 200Using Adjusted R-Squared lo Choose betweenNonnested Models 201Controlling for Too Many Factors inRegression Analysis 203Adding Regressors to Reduce the ErrorVariance 205

6.4 Prediction and Residual Analysis 206Confidence Intervals forPredictions 206Residual Analysis 209Predicting y When log(y) ís the Dependen!Variable 210

Summary 215Key Terms 215Problems 216Computer Exercises 218Appendix 6A 223

C H A P T E R 7

Múltiple Regression Analysis withQualitative Information: Binary(or Dummy) Variables 225

7.1 Describing Qualitative Information 225

Conten ts

7.2 A Single Dummy Independeré Variable 226Iníerpreñng Coefficienls on Dummy ExplanatoryVariables When ¡he Dependen! Variable Islog(y) 231

7.3 Using Dummy Variables for MúltipleCategories 233Incorporaüng Ordinal Informationby UsingDummy Variables 235

7.4 Interactions Involving Dummy Variables 238Interactions among Dummy Variables 238Allowing for Different Slopes 239Testingfor Differences in Regression Functionsacross Groups 243

1.5 A Binary Dependen! Variable: The LinearProbability Model 246

7.6 More on Policy Analysis and ProgramEvaluation 251

Summary 254Key Terms 255Problems 255Computer Exercises 258

C H A P T E R 8

Heteroskedasticity 264

8.1 Consequences of Heteroskedasticity for OLS 2648.2 Heteroskedasticity-Robust Inference after

OLS Estimation 265Computing Heteroskedasticity-RobustLM Tests 269

8.3 Testing for Heteroskedasticity 271The WhiTe Test for Heteroskedasticity 274

8.4 Weighted Least Squares Estimation 276The Heteroskedasticity Is Known up to aMulíiplicative Constant 277The Heteroskedasticity Funclion Must BeEstimated: Feasible GLS 282What Ifthe Assumed HeteroskedasticityFunction Is Wrong? 287Prediction and Prediction Intervals withHeteroskedasticity 289

8.5 The Linear Probability Model Revisited 290Surnmary 293Key Terms 294Problems 294Computer Exercises 296

C H A P T E R 9

More on Specification andData Issues 300

9.1 Functional Form Misspecification 300

RESET as a General Test for Functional FormMisspecification 303Tests againsí Nonnested Alternatives 305

9.2 Using Proxy Variables for UnobservedExplanatory Variables 306Using Lagged Dependent Variables as ProxyVariables 310A Different Slant on MúltipleRegression 312

9.3 Models with Random Slopes 3139.4 Properties of OLS under Measurement

Error 315Measurement Error in the DependentVariable 316Measurement Error in an ExplanatoryVariable 318

9.5 Missing Data, Nonrandom Samples, andOutlying Observations 322Missing Data 322Nonrandom Samples 323Outliers and ¡nfluential Observations 325

9.6 Least Absolute Deviations Estimation 330Summary 331Key Terms 332Problems 332Computer Exercises 334

P A R T 2

Regression Analysis withTime Series Data 339C H A P T E R 1 0

Basic Regression Analysis with TimeSeries Data 340

10.1 The Nature of Time Series Data 34010.2 Examples of Time Series Regression

Models 342Static Models 342Finite Distributed Lag Models 342A Convention about the Time Index 345

] 0.3 Finite Sample Properties of OLS under ClassicalAssumptíons 345Unbiasedness of OLS 345The Variances oflhe OLS Estimators and theGauss-Markov Theorem 349Inference under the Classical Linear ModelAssumptions 351

10.4 Functional Form, Dummy Variables, andIndex Numbers 353

Conté nts

10.5 Trends and Seasonality 360Chámete riz,ing Trending Time Seríes 360Using Trending Variables in RegressionAnalysis 363A Detrending Interpretarían of Kegressions wilha Time Trena 365Computing R-Squared when the Dependen!Variable Is Trending 366Seasonality 368

Summary 370Key Terms 371Problems 371Computer Exercises 373

C H A P T E R 1 1

Further Issues in Using OLS withTime Series Data 377

11.1 Stationary and Weakly DependentTime Series 377Stationary and Nonslationary TimeSeríes 378Weakly Dependent Time Seríes 379

11.2 Asymptotic Properties of OLS 38111.3 Using Highly Persisten! Time Series in

Regression Analysis 388Highly Persisten! Time Seríes 388Transformations on Highly PersisíentTime Series 393Deciding Whether a Time SeríesIs 1(1) 394

11.4 Dynamically Complete Models and the Absenceof Serial Correlation 396

11.5 The Homoskedasticity Assumption for TimeSeries Models 399

Summary 400Key Terms 401Problems 401Computer Exercises 404

CHAPTER 12

Serial Correlation andHeteroskedasticity in TimeSeries Regressions 408

12.1 Properties of OLS with Serially CorrelatedErrors 408Unbiasedness and Consistency 408Efficiency and Inference 409Goodness-of-Fit 410

Serial Correlation in ¡he Presence ofLaggedDependent Variables 411

12.2 Testing for Serial Correlation 412A l TeslforAR(I) Serial Correlaíion withStrictly Exogenous Regressors 412The Durbin-Watson Test under ClassicalAssumplions 415Testing for AR(1) Sería! Correlation wiíhou!Slrícíly Exogenous Regressors 416Testing for Higher Order SerialCorrelation 417

12.3 Correcting for Serial Correlation with StrictlyExogenous Regressors 419Obtaining the Best Linear Unbiased EstimatorintheAR(l)Model 419Feasible GLS Eslimation with ARfl)Errors 421Comparing OLS and FGLS 423Correcting for Higher Order SerialCorrelation 425

12.4 Differencing and Serial Correlation 42612.5 Serial Correlation-Robust Inference

after OLS 42812.6 Heteroskedasticity in Time Series

Regressions 432Heteroskedasticity-Robust Statisücs 432Testing for Heteroskedasticity 432Autoregressive ConditionalHeteroskedasticity 433Heteroskedasticity and Seria! Correlation inRegression Models 435

Summary 437Key Terms 437Problems 438Computer Exercises 438

P A R T 3

Advanced Topics 443

CHAPTER 1 3

Pooling Cross Sections acrossTime:Simple Panel Data Methods 444

13.1 Pooling Independen! Cross Sectionsacross Time 445The Chow Test for Structural Changeacross Time 449

13.2 Policy Analysis with Pooled Cross Sections 45013.3 Two-Period Panel Data Analysis 455

Organizing Pane! Data 461

Contents

13.4 Policy Analysis with Two-PeriodPanel Data 462

13.5 Differencing with More Than TwoTime Periods 465Potentiat Pitfalls in Firsl DifferencingPanel Data 470

Summary 471Key Terms 471Problems 471Computer Exercises 473Appendix 13A 478

CHAPTER 14

Advanced Panel Data Methods 481

14.1 Fixed Effects Estimation 481The Dummy Variable Regression 485Fixed Effects or First Differencing? 487Fixed Effects with ünbalanced Panels 488

14.2 Random Effects Models 489Random Effects or Fixed Effects? 493

14.3 Applying Panel Data Methods to OtherData Structures 494

Summary 496Key Terms 496Problems 497Computer Exercises 498Appendix 14A 503

C H A P T E R 1 5

Instrumental Variables Estimationand Two Stage Least Squares 506

15.1 Motivation: Omitted Variables in a SimpleRegression Model 507Statistical Inference with theIV Estímalo r 510Properlies of IV with a Poor Instrumenta!Variable 514Computing R-Squared afterIV Estimation 516

15.2 IV Estimation of the Múltiple RegressionModel 517

15.3 Two Stage Least Squares 521A Single Endogenous ExplanatoryVariable 521MulticoUinearity and 2SLS 523Múltiple Endogenous ExplanatoryVariables 524Testing Múltiple Hypotheses after2SLS Estimation 525

15.4 IV Solutions ío Errors-in-VariablesProblems 525

15.5 Testing for Endogeneity and TestingOveridentifying Restrictions 527Testing for Endogeneity 527Testing Overidentification Restrictions 529

15.6 2SLS with Heteroskedasticity 53 i15.7 Applying 2SLS to Time Seríes Equations 53115.8 Applying 2SLS to Pooled Cross Sections and

Panel Data 534Summary 536Key Terms 536Problems 536Computer Exercises 539Appendix 15A 543

CHAPTER 16

Simultaneous Equations Models 546

16.1 The Nature of Simultaneous EquationsModels 546

16.2 Simultaneity Bias in OLS 55016.3 Identifying and Estimating a Structurai

Equation 552Identification in a Two-Equation System 552Estimation by 2SLS 557

16.4 Systems with More Than Two Equations 559Identification in Systems with Three orMore Equations 559Estimation 560

16.5 Simultaneous Equations Models withTime Series 560

16.6 Simultaneous Equations Models withPanel Data 564

Summary 566Key Terms 567Problems 567Computer Exercises 570

C H A P T E R 1 7

Limited Dependent VariableModels and Sample SelectionCorrections 574

17.1 Logit and Probit Models for BinaryResponse 575Specifying Logit and Probit Models 575Máximum Likelihood Estimation of Logit andProbit Models 578Testing Múltiple Hypotheses 579

Contents

Interpreting the Logií and ProbilEstimares 580

17.2 The Tobit Model for Córner SolutíonResponses 587Inlerpreting the Tobil Estimates 589Specification Issues in TobiT Models 594

17.3 The Poisson Regression Model 59517.4 Censored and Truncated Regression

Models 600Censored Regression Models 601Truncated Regression Models 604

17.5 Sample Selection Corrections 606When Is OLS on the Selecled SampleConsistent? 607Incidental Truncaíion 608

Summary 612Key Terms 613Problerns 614Computer Exercises 615Appendix 17A 620Appendix 17B 621

C H A P T L R 18

Advanced Time Series Topics 623

18.1 Infinite Distributed Lag Models 624The Geometric (or Koyck) Distributed Lag 626Rationa! Distributed Lag Models 628

18.2 Testing for Unit Roots 63018.3 Spurious Regression 63618.4 Cointegration and Error Correction Models 637

Cointegration 637Error Correction Models 643

18.5 Forecasting 645Types of Regression Models UsedforForecasting 646One-Step-Ahead Forecasñng 647Comparing One-Step-Ahead Forecasts 651Multipie-Step-Ahead Forecasts 652Forecasting Trending, Seasonal, and IntegratedProcesses 655

Summary 660Key Terms 661Problerns 661Computer Exercises 663

CHATTER 19

Carrying Out an Empírica!Project 668

19.1 Posing a Question 668

19.2 Literature Review 67019.3DataCollection 671

Deciding on the Appropriate Data Sel 671Entering and Storing Your Data 672Inspecting, Cleaning, and SummarizingYour Data 673

19.4 Econometric Analysis 67519.5 Writing an Empirical Paper 678

Introduction 678Conceptual (or Theoretical) Framework 679Econometric Models and EstimationMethods 679The Data 682Results 682Conclusions 683Style Hints 684

Summary 687Key Terms 687Sample Empirical Projects 687List of Journals 692Data Sources 693

A P P E N D I X A

Basic Mathematical Tools 695

A. 1 The Summation Operator and DescriptiveStatistics 695

A.2 Properties of Linear Functions 697A.3 Proportions and Percentages 699A.4 Some Special Functions and

Their Properties 702Quadratic Functions 702The Natural Logarithm 704The Exponential Function 708

A.5 Differential Calculus 709Summary 711Key Terms 711Problems 711

APPENDIX B

Fundamentáis of Probability 714

B.l Random Variables and Their ProbabilityDistributions 714Discreíe Random Variables 715Conünuous Random Variables 717

B.2 Joint Distributions, Condítional Distributions,and Independence 719Joint Distributions and Independence 719Condiüonal Distributions 721

B.3 Features of Probability Distributions 722

Contents

A Mensure of Central Tendency: TheExpected Valué 722Properlies of Expected Valúes 724Anorher Measure of Central Tendency:The Median 725Measures of Variability: Variance and StandardDeviation 726Variance 726Standard Deviation 728Standardizing a Random Variable 728Skewness and Kurtosis 729

B.4 Features of Joint and ConditionalDistributions 729Measures of Association: Covañance andCorreíation 729Covañance 729Correíation Coefficiem 731Variance ofSums of Random Variables 732Conditional Expecíation 733Properties of Conditional Expecíation 734Conditional Variance 736

B.5 The Normal and Related Distributions 737The Normal Distributíon 737The Standard Normal Distributíon 738Additional Properlies ofíhe NormalDistribution 740The Chi-Square Distribution 741The i Distribution 741The F Distribution 743

Summary 744Key Terms 744Problems 745

A P P E N D I X C

Fundamentáis of MathematicalStatistics 747

C.l Populations, Parameters, and RandomSampling 747Sampling 748

C.2 Finite Sample Properties of Estimators 748Estimators and Estímales 749Unbiasedness 750The Sampling Variance of Estimaíors 752Efficiency 754

C.3 Asymptotic or Larger Sample Propertiesof Estimators 755Consistency 755Asymptotic Normality 758

C.4 General Approaches to ParameterEstimation 760

Method of Moments 760Máximum Likelihood 761LeastSquares 762

C,5 Interval Estimation and ConfídenceIntervals 762The Nature of Interval Estimation 762Confídence Inten'als for ¡he Mean/rom aNormaüy Distñbuted Population 764A Simple Rule of Thumb for a 95%Confídence Inter\>al 768Asymptotic Confidence Intervals forNonnormal Populaíions 768

C.6 Hypothesis Testing 770Fundamentáis of Hypothesis Testing 770Testing Hypotheses ahout the Mean in aNormal Population 772Asymptotic Tesis for NonnormalPopulations 774Compuíing and Using p-Values 776The Relationship between Confídence Intervalsand Hypothesis Testing 779Practica! versus Síatisticai Significance 780

C.l Remarks on Notation 781Summary 782Key Terms 782Problems 783

APPENDIX D

Summary of Matrix Algebra 788

D.l Basic Definitions 788D.2 Matrix Operations 789

Matrix Addition 789Scalar Mulüplication 790Matrix Multiplication 790Transpose 791Partitioned Matrix Multiplication 792Trace 792Inverse 792

D.3 Linear Independence and Rankof a Matrix 793

D.4 Quadratic Fomis and Positive DefiniteMatrices 793

D.5 Idempotent Matrices 794D.6 Differentiation of Linear and Quadratic

Forms 795D.7 Moments and Distributions of

Random Vectors 795Expecled Valué 795Variance-Covañance Matrix 795Multivariate Normal Distribution 796

Contents xi

Chi-Square Distributiont Distribution 797F Distribution 797

Summary 797Key Terms 797Problems 798

796APPENDIX F

APPENDIX E

The Linear Regression Model inMatrix Form 799

E.l The Model and Ordinary Least SquaresEstimation 799

E.2 Finite Sample Properties of OLS 801E.3 Statistical Inference 805E.4 Some Asymptotic Analysis 807

Watd Statistics for Testing MúltipleHypotheses 809

Summary 810Key Terms 811Problems 811

Answers to Chapter Questions 813

APPENDIX cStatistical Tables 823

References 830Glossary 835Index 849