Principles of EconometricsT h i r d E d i t i o n

R.Carter HillLouisiana State University

William E. GriffithsUniversity of Melbourne

Guay C. Lim |University of Melbourne

B I C E N T E N N I A L

John Wiley & Sons, Inc.

Contents

Preface ix

Chapter 1 An Introduction to Econometrics 1

1.1 Why Study Econometrics? 11.2 What is Econometrics About? 2

1.2.1 Some Examples 31.3 The Econometric Model 41.4 How Do We Obtain Data? 5

1.4.1 Experimental Data 51.4.2 Nonexperirnental Data 5

1.5 Statistical Inference "• 61.6 A Research Format 7

Chapter 2 The Simple Linear Regression Model 8

Learning Objectives 8Keywords 92.1 An Economic Model 92.2 An Econometric Model 12

2.2.1 Introducing the Error Term 152.3 Estimating the Regression Parameters 18

2.3.1 The Least Squares Principle 202.3.2 Estimates for the Food Expenditure Function 222.3.3 Interpreting the Estimates 22

2.3.3a Elasticities 232.3.3b Prediction 242.3.3c Computer Output 24

2.3.4 Other Economic Models 242.4 Assessing the Least Squares Estimators 26

2.4.1 The Estimator b2 272.4.2 The Expected Values of b\ and b2 272.4.3 Repeated Sampling 282.4.4 The Variances and Covariance of b\ and b2 29

2.5 The Gauss-Markov Theorem 312.6 The Probability Distributions of the Least Squares Estimators 322.7 Estimating the Variance of the Error Term 33

2.7.1 Estimating the Variances and Covariances of the Least SquaresEstimators 34

Kvi CONTENTS

2.7.2 Calculations^for the Food Expenditure Data 352.8 Exercises 36

2.8.1 Problems 362.8.2 Computer Exercises 39

Appendix 2A Derivation of the Least Squares Estimates 42Appendix 2B Deviation from the Mean Form of b2 43Appendix 2C b2 is a Linear Estimator 44Appendix 2D Derivation of Theoretical Expression for b2 44Appendix 2E Deriving the Variance of b2 45Appendix 2F Proof of the Gauss-Markov Theorem 46

Chapter 3 Interval Estimation and Hypothesis Testing 48

Learning Objectives 48Keywords 483.1 Interval Estimation 49

3.1.1 The /-distribution 493.1.2 Obtaining Interval Estimates 513.1.3 An Illustration 523.1.4 The Repeated Sampling Context 53

3.2 Hypothesis Tests 543.2.1 -1 The Null Hypothesis 553.2.2 The Alternative Hypothesis 553.2.3 The Test Statistic 553.2.4 The Rejection Region 553.2.5 A Conclusion 56

3.3 Rejection Regions for Specific Alternatives 563.3.1 One-Tail Tests with Alternative "Greater Than" (>) 563.3.2 One-Tail Tests with Alternative "Less Than" « ) 573.3.3 Two-Tail Tests with Alternative "Not Equal To" (^) 58

3.4 Examples of Hypothesis Tests 593.4.1 Right-Tail Tests 59

3.4.1a One-Tail Test of Significance 593.4.1b One-Tail Test of an Economic Hypothesis 60

3.4.2 Left-Tail Tests 613.4.3 Two-Tail Tests 62

3.4.3a Two-Tail Test of an Economic Hypothesis 623.4.3b Two-Tail Test of Significance 63

3.5 The p-value 643.5.1 p-value for a Right-Tail Test 653.5.2 p-value for a Left-Tail Test 663.5.3 p-value for a Two-Tail Test 663.5.4 p-value for a Two-Tail Test of Significance 67

3.6 Exercises 683.6.1 Problems 683.6.2 Computer Exercises 69

Appendix 3A Derivation of the /-Distribution 72Appendix 3B Distribution of the f-Statistic Under Hi 73

CONTENTS xvii

Chapter 4 Prediction, Goodness^jf-Fit, and Modeling Issues 75

Learning Objectives 75Keywords 764.1 Least Squares Prediction 76

4.1.1 Prediction in the Food Expenditure Model 794.2 Measuring Goodness-of-Fit 80

4.2.1 Correlation Analysis 814.2.2 Correlation Analysis and R2 824.2.3 The Food Expenditure Example 824.2.4 Reporting the Results 83

4.3 Modeling Issues 844.3.1 The Effects of Scaling the Data 844.3.2 Choosing a Functional Form 864.3.3 The Food Expenditure Model 874.3.4 Are the Regression Errors Normally Distributed? 894.3.5 Another Empirical Example 90

4.4 Log-Linear Models 934.4.1 A Growth Model 944.4.2 A Wage Equation 944.4.3 Prediction in the Log-Linear Model 954.4.4 A Generalized R2 Measure 964.4.5 Prediction Intervals in the Log-Linear Model 96

4.5 Exercises 974.5.1 Problems 974.5.2 Computer Exercises 98

Appendix 4A Development of a Prediction Interval 101Appendix 4B The Sum of Squares Decomposition 103Appendix 4C The Log-Normal Distribution 103

Chapter 5 The Multiple Regression Model 105

Learning Objectives 105Keywords 1055.1 Introduction 106

5.1.1 The Economic Model 1065.1.2 The Econometric Model 108

5.1.2a The General Model 1095.1.2b The Assumptions of the Model 110

5.2 Estimating the Parameters of the Multiple Regression Model 1115.2.1 Least Squares Estimation Procedure 1115.2.2 Least Squares Estimates Using Hamburger Chain Data 1125.2.3 Estimation of the Error Variance cr2 114

5.3 Sampling Properties of the Least Squares Estimator 1155.3.1 The Variances and Covanances of the Least Squares Estimators 1155.3.2 The Properties of the Least Squares Estimators Assuming

Normally Distributed Errors 1175.4 Interval Estimation 1185.5 Hypothesis Testing for a Single Coefficient 120

5.5.1 Testing the Significance of a Single Coefficient 120

xviii CONTENTS

5.5.2 One-Tail Hypothesis Testing for a Single Coefficient 1225.5.2a Testing For Elastic Demand 1225.5.2b Testing Advertising Effectiveness 123

5.6 Measuring Goodness-of-Fit 1245.6.1 Reporting the Regression Results 126

5.7 Exercises 1275.7.1 Problems 1275.7.2 Computer Exercises 129

Appendix 5A Derivation of Least Squares Estimators 133

Chapter 6 Further Inference in the Multiple Regression Model 134

Learning Objectives 134Keywords 1356.1 The F-Test 135

6.1.1 The Relationship Between t- and F-Tests 1386.2 Testing the Significance of a Model 1386.3 An Extended Model 1406.4 Testing Some Economic Hypotheses 142

6.4.1 The Significance of Advertising 1426.4.2 The Optimal Level of Advertising 142

f 6.4.2a A One-Tail Test with More than One Parameter 1446.4.3 Using Computer Software 145

6.5 The Use of Nonsample Information 1466.6 Model Specification 148

6.6.1 Omitted Variables 1496.6.2 Irrelevant Variables 1506.6.3 Choosing the Model 151

6.6.3a The RESET Test 1516.7 Poor Data, Collinearity, and Insignificance 153

6.7.1 The Consequences of Collinearity 1536.7.2 An Example 1546.7.3 Identifying and Mitigating Collinearity 155

6.8 Prediction 1566.9 Exercises 157

6.9.1 Problems 1576.9.2 Computer Exercises 160

Appendix 6A Chi-Square and F-Tests: More Details 163Appendix 6B Omitted-Variable Bias: A Proof 165

Chapter 7 Nonlinear Relationships 166

Learning Objectives 166Keywords 1667.1 Polynomials 167

7.1.1 Cost and "Product Curves 1677.1.2 A Wage Equation 169

7.2 Dummy Variables 1707.2.1 Intercept Dummy Variables 171

7.2.1a Choosing the Reference Group 172

CONTENTS xix

7.2.2 Slope Dummy Variables^ 1727.2.3 An Example: The University Effect on House Prices 174

7.3 Applying Dummy Variables 1757.3.1 Interactions between Qualitative Factors 1757.3.2 Qualitative Factors with Several Categories 1777.3.3 Testing the Equivalence of Two Regressions 1797.3.4 Controlling for Time 181

7.3.4a Seasonal Dummies 1817.3.4b Annual Dummies 1827.3.4c Regime Effects 182

7.4 Interactions Between Continuous Variables 1827.5 Log-Linear Models 184

7.5.1 Dummy Variables 1857.5.1a A Rough Calculation 1857.5.1b An Exact Calculation 185

7.5.2 Interaction and Quadratic Terms 1867.6 Exercises 186

7.6.1 Problems 1867.6.2 Computer Exercises 190

Appendix 7A Details of Log-Linear Model Interpretation 195

IChapter 8 Heterosltedastidty 197Learning Objectives 197Keywords 1978.1 The Nature of Heteroskedasticity 1978.2 Using the Least Squares Estimator 2018.3 The Generalized Least Squares Estimator 202

8.3.1 Transforming The Model 2038.3.2 Estimating the Variance Function 2058.3.3 A Heteroskedastic Partition 208

8.4 Detecting Heteroskedasticity 2118.4.1 Residual Plots 2118.4.2 The Goldfeld-Quandt Test 2118.4.3 Testing the Variance Function 212

8.4.3a The White Test 2158.4.3b Testing the Food Expenditure Example 215

8.5 Exercises 2168.5.1 Problems 2168.5.2 Computer Exercises 219

Appendix 8A Properties of the Least Squares Estimator 222Appendix 8B Variance Function Tests for Heteroskedasticity 224

Chapter 9 Dynamic Models, Autocorrelation and Forecasting 226

Learning Objectives 226Keywords 2269.1 Introduction 2279.2 Lags in the Error Term: Autocorrelation 230

9.2.1 Area Response Model for Sugar Cane 230

xx CONTENTS

9.2.2 First Order Autoregressive Errors 2319.3 Estimating an AR(1) Error Model 235

9.3.1 Least Squares Estimation 2359.3.2 Nonlinear Least Squares Estimation 236

9.3.2a Generalized Least Squares Estimation 2379.3.3 Estimating a More General Model 237

9.4 Testing for Autocorrelation 2399.4.1 Residual Correlogram 2399.4.2 A Lagrange Multiplier Test 2429.4.3 Recapping and Looking Forward 243

9.5 An Introduction to Forecasting: Autoregressive Models 2449.6 Finite Distributed Lags 2489.7 Autoregressive Distributed Lag Models 2509.8 Exercises 253

9.8.1 Problems 2539.8.2 Computer Exercises 255

Appendix 9A Generalized Least Squares Estimation 259Appendix 9B The Durbin-Watson Test 261

9B.1 The Durbin-Watson Bounds Test 263Appendix 9C Deriving ARDL Lag Weights 264

9C.1 The Geometric Lag 264f 9C.2 Lag Weights for More General ARDL Models 265

Appendix 9D Forecasting:Exponential Smoothing 266

Chapter 10 Random Regressors and Moment Based Estimation 268

Learning Objectives 268Keywords 26910.1 Linear Regression with Random x's 270

10.1.1 The Small Sample Properties of the Least Squares Estimator 27010.1.2 Asymptotic Properties of the Least Squares Estimator: x Not

Random 27110.1.3 Asymptotic Properties of the Least Squares Estimator:

x Random 27210.1.4 Why Least Squares Fails 273

10.2 Cases in Which x and e are Correlated 27410.2.1 Measurement Error 27410.2.2 Omitted Variables 27510.2.3 Simultaneous Equations Bias 27610.2.4 Lagged Dependent Variable Models with Serial Correlation 276

10.3 Estimators Based on the Method of Moments 27610.3.1 Method of Moments Estimation of a Population Mean and

Variance 27710.3.2 Method of Moments Estimation in the Simple Linear

Regression Model 27810.3.3 Instrumental Variables Estimation in the Simple Linear

Regression Model 27810.3.3a The Importance of Using Strong Instruments 27910.3.3b An Illustration Using Simulated Data 28010.3.3c An Illustration Using a Wage Equation 281

CONTENTS xxi

10.3.4 Instrumental Variables-*Estimation with Surplus Instruments 28210.3.4a An Illustration Using Simulated Data 28410.3.4b An Illustration Using a Wage Equation 284

10.3.5 Instrumental Variables Estimation in a General Model 28510.3.5a Hypothesis Testing with Instrumental Variables

Estimates 28610.3.5b Goodness-of-Fit with Instrumental Variables

Estimates 28610.4 Specification Tests 286

10.4.1 The Hausman Test for Endogeneity 28710.4.2 Testing for Weak Instruments 28810.4.3 Testing Instrument Validity 28910.4.4 Numerical Examples Using Simulated Data 290

10.4.4a The Hausman Test 29010.4.4b Test for Weak Instruments 29010.4.4c Testing Surplus Moment Conditions 291

10.4.5 Specification Tests for the Wage Equation 29110.5 Exercises 292

10.5.1 Problems 29210.5.2 Computer Exercises 293

Appendix 10A Conditional and Iterated Expectations 29710A.1 Conditional Expectations 29710A.2 Iterated Expectations 29810A.3 Regression Model Applications 298

Appendix 10B The Inconsistency of Least Squares 299Appendix 10C The Consistency of the IV Estimator 300Appendix 10D The Logic of the Hausman Test 301

Chapter 11 Simultaneous Equations Models 303

Learning Objectives 303Keywords 30311.1 A Supply and Demand Model 30411.2 The Reduced Form Equations 30611.3 The Failure of Least Squares 30711.4 The Identification Problem 30711.5 Two-Stage Least Squares Estimation 309

11.5.1 The General Two-Stage Least Squares EstimationProcedure 310

11.5.2 The Properties of the Two-Stage Least Squares Estimator 31111.6 An Example of Two-Stage Least Squares Estimation 311

11.6.1 Identification 31211.6.2 The Reduced Form Equations 31211.6.3 The Structural Equations 313

11.7 Supply and Demand at the Fulton Fish Market 31411.7.1 Identification 31511.7.2 The Reduced Form Equations 31511.7.3 Two-Stage Least Squares Estimation of Fish Demand 317

11.8 Exercises 31811.8.1 Problems 318

CONTENTS

11.8.2 Computer Exercises 319Appendix 11A An Algebraic Explanation of the Failure of

Least Squares 323

Chapter 12 Nonstationary Time-Series Data and Cointegration 325

Learning Objectives 325Keywords 32512.1 Stationary and Nonstationary Variables 326

12.1.1 The First-Order Autoregressive Model 32812.1.2 Random Walk Models 331

12.2 Spurious Regressions 33312.3 Unit Root Tests for Stationarity 335

12.3.1 Dickey-Fuller Test 1 (No Constant and No Trend) 33512.3.2 Dickey-Fuller Test 2 (With Constant But No Trend) 33512.3.3 Dickey-Fuller Test 3 (With Constant and With Trend) 33612.3.4 The Dickey-Fuller Testing Procedure 33612.3.5 The Dickey-Fuller Tests: An Example 33712.3.6 Order of Integration 338

12.4 Cointegration 33912.4.1 An Example of a Cointegration Test 340

12.5 Regression When There is No Cointegration 34012.5.1 First Difference Stationary 34112.5.2 Trend Stationary 342

12.6 Exercises 34212.6.1 Problems 34212.6.2 Computer Exercises 344

Chapter 13 VEC and VAR Models:An Introduction to Macroeconometrics 346

Learning Objectives 346Keywords 34613.1 VEC and VAR Models 34713.2 Estimating a Vector Error Correction Model 349

13.2.1 Example 34913.3 Estimating a VAR Model 35113.4 Impulse Responses and Variance Decompositions 352

13.4.1 Impulse Response Functions ^ 35213.4.1a The Univariate Case 35213.4.1b The Bivariate Case 353

13.4.2 Forecast Error Variance Decompositions 35513.4.2a Univariate Analysis 35513.4.2b Bivariate Analysis 35613.4.2c The General Case 357

13.5 Exercises 35713.5.1 Problems 35713.5.2 Computer Exercises 358

Appendix 13A The Identification Problem 361

CONTENTS xxiii

Chapter 14 Time-Varying Volatility and ARCH Models:An Introduction to Financial Econometrics 363

Learning Objectives 363Keywords 36314.1 The ARCH Model 364

14.1.1 Conditional and Unconditional Forecasts 36514.2 Time-Varying Volatility 36514.3 Testing, Estimating and Forecasting 369

14.3.1 Testing for ARCH Effects 36914.3.2 Estimating ARCH Models 36914.3.3 Forecasting Volatility 370

14.4 Extensions 37114.4.1 The GARCH Model—Generalized ARCH 37114.4.2 Allowing for an Asymmetric Effect 37214.4.3 GARCH-in-Mean and Time-Varying Risk Premium 374

14.5 Exercises 37514.5.1 Problems 37514.5.2 Computer Exercises 376

Chapter 15 Panel Data Models 382

Learning Objectives " 382Keywords 38215.1 Grunfeld's Investment Data 38415.2 Sets of Regression Equations 38515.3 Seemingly Unrelated Regressions 387

15.3.1 Separate or Joint Estimation? 38915.3.2 Testing Cross-Equation Hypotheses 390

15.4 The Fixed Effects Model 39115.4.1 A Dummy Variable Model 39115.4.2 The Fixed Effects Estimator 39315.4.3 Fixed Effects Estimation Using a Microeconomic Panel 396

15.5 The Random Effects Model 39815.5.1 Error Term Assumptions 39915.5.2 Testing for Random Effects 40015.5.3 Estimation of the Random Effects Model 40115.5.4 An Example Using the NLS Data 40215.5.5 Comparing Fixed and Random Effects Estimators 403

15.5.5a Endogeneity in the Random Effects Model 40315.5.5b The Fixed Effects Estimator in a Random

Effects Model 40415.5.5c A Hausman Test 404

15.6 Exercises 40615.6.1 Problems 40615.6.2 Computer Exercises 408

Appendix 15A Estimation of Error Components 415

xxiv CONTENTS

Chapter 16 Qualitative^and Limited Dependent Variable Models 417

Learning Objectives 417Keywords 41716.1 Models with Binary Dependent Variables 418

16.1.1 The Linear Probability Model 41916.1.2 The Probit Model 42116.1.3 Interpretation of the Probit Model 42216.1.4 Maximum Likelihood Estimation of the Probit Model 42316.1.5 An Example 424

16.2 The Logit Model for Binary Choice 42516.3 Multinomial Logit 426

16.3.1 Multinomial Logit Choice Probabilities 42716.3.2 Maximum Likelihood Estimation 42716.3.3 Post-Estimation Analysis 42816.3.4 An Example 429

16.4 Conditional Logit 43116.4.1 Conditional Logit Choice Probabilities 43116.4.2 Post-Estimation Analysis 43216.4.3 An Example 433

16.5 Ordered Choice Models 43316.5.1. Ordinal Probit Choice Probabilities 43416.5.2' -Estimation and Interpretation 435

437437438439440441441442444445446447448449450

and Sources of Economic Data 45717.1 Selecting a Topic for an Economics Project 457

17.1.1 Choosing a Topic 45717.1.2 Writing an Abstract 458

17.2 A Format for Writing a Research Report 45817.3 Sources of Economic Data 460

17.3.1 Links to Economic Data on the Internet 46017.3.2 Traditional Sources of Economic Data 46117.3.3 Interpreting Economic Data 461

17.4 Exercises 462

16.5.316.6 Models

16.6.116.6.216.6.3

16.7 Limited16.7.116.7.216.7.316.7.416.7.516.7.6

An Examplefor Count Data

Maximum Likelihood EstimationInterpretation in the Poisson Regression ModelAn Example

Dependent VariablesCensored DataA Monte Carlo ExperimentMaximum Likelihood EstimationTobit Model InterpretationAn ExampleSample Selection16.7.6a The Econometric Model16.7.6b Heckit Example: Wages of Married Women

16.8 Exercises

Chapter 17 Writing an Empirical Research Report,

CONTENTS xxv

Appendix A Review of Math Essentials 463

Learning Objectives 463Keywords 463A.I Summation 464A.2 Some Basics 465

A.2.1 Numbers 465A.2.2 Exponents 466A.2.3 Scientific Notation 466A.2.4 Logarithms and the Number e 466

A.3 Linear Relationships 468A.3.1 Elasticity 469

A.4 Nonlinear Relationships 470A.4.1 Quadratic Function 471A.4.2 Cubic Function 471A.4.3 Reciprocal Function 472A.4.4 Log-Log Function 473A.4.5 Log-Linear Function 473A.4.6 Approximating Logarithms 473A.4.7 Approximating Logarithms in the Log-Linear Model 474A.4.8 Linear-Log Function 475

A.5 Exercises J 476

Appendix B Review of Probability Concepts 478

Learning Objectives 478Keywords 479B.I Random Variables 479B.2 Probability Distributions 480B.3 Joint, Marginal and Conditional Probability Distributions 483

B.3.1 Marginal Distributions 484B.3.2 Conditional Probability 484B.3.3 A Simple Experiment 486

B.4 Properties of Probability Distributions 487B.4.1 Mean, Median and Mode 487B.4.2 Expected Values of Functions of a Random Variable 488B.4.3 Expected Values of Several Random Variables 490B.4.4 The Simple Experiment Again 492

B.5 Some Important Probability Distributions , 493B.5.1 The Normal Distribution N 493B.5.2 The Chi-Square Distribution 495B.5.3 The f-Distribution 495B.5.4 The F-Distribution 496

B.6 Exercises 497

Appendix C Review of Statistical Inference 501

Learning Objectives 501

Keywords 502

xxvi CONTENTS

C.I A Sample of Data '"'* 502C.2 An Econometric Model 504C.3 Estimating the Mean of a Population 504

C.3.1 The Expected Value of Y 506C.3.2 The Variance of Y 506C.3.3 The Sampling Distribution of Y 507C.3.4 The Central Limit Theorem 508C.3.5 Best Linear Unbiased Estimation 509

C.4 Estimating the Population Variance and Other Moments 509C.4.1 Estimating the Population Variance 510C.4.2 Estimating Higher Moments 511C.4.3 The Hip Data 511C.4.4 Using the Estimates 512

C.5 Interval Estimation 512C.5.1 Interval Estimation: a2 Known 513C.5.2 A Simulation 514C.5.3 Interval Estimation: a2 Unknown 515C.5.4 A Simulation (Continued) 517C.5.5 Interval Estimation Using the Hip Data 517

C.6 Hypothesis Tests About a Population Mean 517C.6.1 Components of Hypothesis Tests 517

fc.6.1a The Null Hypothesis 518C6.1b The Alternative Hypothesis 518C.6.1c The Test Statistic 518C.6.Id The Rejection Region 519C.6.1e A Conclusion 519

C.6.2 One-Tail Tests with Alternative "Greater Than" (>) 519C.6.3 One-Tail Tests with Alternative "Less Than" (<) 519C.6.4 Two-Tail Tests with Alternative "Not Equal To" (^) 520C.6.5 Example of a One-Tail Test Using the Hip Data 520C.6.6 Example of a Two-Tail Test Using Hip Data 521C.6.7 The p- Value 522C.6.8 A Comment on Stating Null and Alternative Hypotheses 523C.6.9 Type I and Type II Errors 524C.6.10 A Relationship Between Hypothesis Testing and Confidence

Intervals 525C.7 Some Other Useful Tests 525

C.7.1 Testing the Population Variance 525C.7.2 Testing the Equality of Two Population Means 526C.7.3 Testing the Ratio of Two Population Variances 527C.7.4 Testing the Normality of a Population 527

C.8 Introduction to Maximum Likelihood Estimation 528C.8.1 Inference with Maximum Likelihood Estimators 532C.8.2 The Variance of the Maximum Likelihood Estimator 533C.8.3 The Distribution of the Sample Proportion 534C.8.4 Asymptotic Test Procedures 536

C.8.4a The Likelihood Ratio (LR) Test 536C.8.4b The Wald Test 538C.8.4c The Lagrange Multiplier (LM) Test 539

CONTENTS xxvii

C.9 Algebraic Supplements (Optional) 541C.9.1 Derivation of Least Squares Estimator 541C.9.2 Best Linear Unbiased Estimation 543

CIO Exercises 544

Appendix D Answers to Selected Exercises 548

Appendix E Tables 572Table 1 Cumulative Probabilities for the Standard Normal Distribution 572Table 2 Percentiles for the f-Distribution 573Table 3 Percentiles for the Chi-square Distribution 574Table 4 95th Percentile for the F-Distribution 575Table 5 99th Percentile for the F-Distribution 576

Index 577