Principles of Econometrics

Preface v

List of Examples xxi

1 An Introduction to Econometrics 1

1.1 Why Study Econometrics? 1

1.2 What Is Econometrics About? 2

1.2.1 Some Examples 3

1.3 The Econometric Model 4

1.3.1 Causality and Prediction 5

1.4 How Are Data Generated? 5

1.4.1 Experimental Data 6

1.4.2 Quasi-Experimental Data 6

1.4.3 Nonexperimental Data 7

1.5 Economic Data Types 7

1.5.1 Time-Series Data 7

1.5.2 Cross-Section Data 8

1.5.3 Panel or Longitudinal Data 9

1.6 The Research Process 9

1.7 Writing an Empirical Research Paper 11

1.7.1 Writing a Research Proposal 11

1.7.2 A Format for Writing a Research Report 11

1.8 Sources of Economic Data 13

1.8.1 Links to Economic Data on the Internet 13

1.8.2 Interpreting Economic Data 14

1.8.3 Obtaining the Data 14

Probability Primer 15

P.1 Random Variables 16

P.2 Probability Distributions 17

P.3 Joint, Marginal, and Conditional Probabilities 20

P.3.1 Marginal Distributions 20

P.3.2 Conditional Probability 21

P.3.3 Statistical Independence 21

P.4 A Digression: Summation Notation 22

P.5 Properties of Probability Distributions 23

P.5.1 Expected Value of a Random Variable 24

P.5.2 Conditional Expectation 25

P.5.3 Rules for Expected Values 25

P.5.4 Variance of a Random Variable 26

P.5.5 Expected Values of Several Random Variables 27

P.5.6 Covariance Between Two Random Variables 27

P.6 Conditioning 29

P.6.1 Conditional Expectation 30

P.6.2 Conditional Variance 31

P.6.3 Iterated Expectations 32

P.6.4 Variance Decomposition 33

P.6.5 Covariance Decomposition 34

P.7 The Normal Distribution 34

P.7.1 The Bivariate Normal Distribution 37

P.8 Exercises 39

2 The Simple Linear Regression Model 46

2.1 An Economic Model 47

2.2 An Econometric Model 49

2.2.1 Data Generating Process 51

2.2.2 The Random Error and Strict Exogeneity 52

2.2.3 The Regression Function 53

2.2.4 Random Error Variation 54

2.2.5 Variation in x 56

2.2.6 Error Normality 56

2.2.7 Generalizing the Exogeneity Assumption 56

2.2.8 Error Correlation 57

2.2.9 Summarizing the Assumptions 58

2.3 Estimating the Regression Parameters 59

2.3.1 The Least Squares Principle 61

2.3.2 Other Economic Models 65

2.4 Assessing the Least Squares Estimators 66

2.4.1 The Estimator b2 67

2.4.2 The Expected Values of b1 and b2 68

2.4.3 Sampling Variation 69

2.4.4 The Variances and Covariance of b1 and b2 69

2.5 The Gauss–Markov Theorem 72

2.6 The Probability Distributions of the Least Squares Estimators 73

2.7 Estimating the Variance of the Error Term 74

2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators 74

2.7.2 Interpreting the Standard Errors 76

2.8 Estimating Nonlinear Relationships 77

2.8.1 Quadratic Functions 77

2.8.2 Using a Quadratic Model 77

2.8.3 A Log-Linear Function 79

2.8.4 Using a Log-Linear Model 80

2.8.5 Choosing a Functional Form 82

2.9 Regression with Indicator Variables 82

2.10 The Independent Variable 84

2.10.1 Random and Independent x 84

2.10.2 Random and Strictly Exogenous x 86

2.10.3 Random Sampling 87

2.11 Exercises 89

2.11.1 Problems 89

2.11.2 Computer Exercises 93

Appendix 2A Derivation of the Least Squares Estimates 98

Appendix 2B Deviation from the Mean Form of b2 99

Appendix 2C b2 Is a Linear Estimator 100

Appendix 2D Derivation of Theoretical Expression for b2 100

Appendix 2E Deriving the Conditional Variance of b2 100

Appendix 2F Proof of the Gauss–Markov Theorem 102

Appendix 2G Proofs of Results Introduced in Section 2.10 103

2G.1 The Implications of Strict Exogeneity 103

2G.2 The Random and Independent x Case 103

2G.3 The Random and Strictly Exogenous x Case 105

2G.4 Random Sampling 106

Appendix 2H Monte Carlo Simulation 106

2H.1 The Regression Function 106

2H.2 The Random Error 107

2H.3 Theoretically True Values 107

2H.4 Creating a Sample of Data 108

2H.5 Monte Carlo Objectives 109

2H.6 Monte Carlo Results 109

2H.7 Random-x Monte Carlo Results 110

3 Interval Estimation and Hypothesis Testing 112

3.1 Interval Estimation 113

3.1.1 The t-Distribution 113

3.1.2 Obtaining Interval Estimates 115

3.1.3 The Sampling Context 116

3.2 Hypothesis Tests 118

3.2.1 The Null Hypothesis 118

3.2.2 The Alternative Hypothesis 118

3.2.3 The Test Statistic 119

3.2.4 The Rejection Region 119

3.2.5 A Conclusion 120

3.3 Rejection Regions for Specific Alternatives 120

3.3.1 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 120

3.3.2 One-Tail Tests with Alternative ‘‘Less Than’’ (<) 121

3.3.3 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (≠) 122

3.4 Examples of Hypothesis Tests 123

3.5 The p-Value 126

3.6 Linear Combinations of Parameters 129

3.6.1 Testing a Linear Combination of Parameters 131

3.7 Exercises 133

3.7.1 Problems 133

3.7.2 Computer Exercises 139

Appendix 3A Derivation of the t-Distribution 144

Appendix 3B Distribution of the t-Statistic under H1 145

Appendix 3C Monte Carlo Simulation 147

3C.1 Sampling Properties of Interval Estimators 148

3C.2 Sampling Properties of Hypothesis Tests 149

3C.3 Choosing the Number of Monte Carlo Samples 149

3C.4 Random-x Monte Carlo Results 150

4 Prediction, Goodness-of-Fit, and Modeling Issues 152

4.1 Least Squares Prediction 153

4.2 Measuring Goodness-of-Fit 156

4.2.1 Correlation Analysis 158

4.2.2 Correlation Analysis and R2 158

4.3 Modeling Issues 160

4.3.1 The Effects of Scaling the Data 160

4.3.2 Choosing a Functional Form 161

4.3.3 A Linear-Log Food Expenditure Model 163

4.3.4 Using Diagnostic Residual Plots 165

4.3.5 Are the Regression Errors Normally Distributed? 167

4.3.6 Identifying Influential Observations 169

4.4 Polynomial Models 171

4.4.1 Quadratic and Cubic Equations 171

4.5 Log-Linear Models 173

4.5.1 Prediction in the Log-Linear Model 175

4.5.2 A Generalized R2 Measure 176

4.5.3 Prediction Intervals in the Log-Linear Model 177

4.6 Log-Log Models 177

4.7 Exercises 179

4.7.1 Problems 179

4.7.2 Computer Exercises 185

Appendix 4A Development of a Prediction Interval 192

Appendix 4B The Sum of Squares Decomposition 193

Appendix 4C Mean Squared Error: Estimation and Prediction 193

5 The Multiple Regression Model 196

5.1 Introduction 197

5.1.1 The Economic Model 197

5.1.2 The Econometric Model 198

5.1.3 The General Model 202

5.1.4 Assumptions of the Multiple Regression Model 203

5.2 Estimating the Parameters of the Multiple Regression Model 205

5.2.1 Least Squares Estimation Procedure 205

5.2.2 Estimating the Error Variance σ2 207

5.2.3 Measuring Goodness-of-Fit 208

5.2.4 Frisch–Waugh–Lovell (FWL) Theorem 209

5.3 Finite Sample Properties of the Least Squares Estimator 211

5.3.1 The Variances and Covariances of the Least Squares Estimators 212

5.3.2 The Distribution of the Least Squares Estimators 214

5.4 Interval Estimation 216

5.4.1 Interval Estimation for a Single Coefficient 216

5.4.2 Interval Estimation for a Linear Combination of Coefficients 217

5.5 Hypothesis Testing 218

5.5.1 Testing the Significance of a Single Coefficient 219

5.5.2 One-Tail Hypothesis Testing for a Single Coefficient 220

5.5.3 Hypothesis Testing for a Linear Combination of Coefficients 221

5.6 Nonlinear Relationships 222

5.7 Large Sample Properties of the Least Squares Estimator 227

5.7.1 Consistency 227

5.7.2 Asymptotic Normality 229

5.7.3 Relaxing Assumptions 230

5.7.4 Inference for a Nonlinear Function of Coefficients 232

5.8 Exercises 234

5.8.1 Problems 234

5.8.2 Computer Exercises 240

Appendix 5A Derivation of Least Squares Estimators 247

Appendix 5B The Delta Method 248

5B.1 Nonlinear Function of a Single Parameter 248

5B.2 Nonlinear Function of Two Parameters 249

Appendix 5C Monte Carlo Simulation 250

5C.1 Least Squares Estimation with Chi-Square Errors 250

5C.2 Monte Carlo Simulation of the Delta Method 252

Appendix 5D Bootstrapping 254

5D.1 Resampling 255

5D.2 Bootstrap Bias Estimate 256

5D.3 Bootstrap Standard Error 256

5D.4 Bootstrap Percentile Interval Estimate 257

5D.5 Asymptotic Refinement 258

6 Further Inference in the Multiple Regression Model 260

6.1 Testing Joint Hypotheses: The F-test 261

6.1.1 Testing the Significance of the Model 264

6.1.2 The Relationship Between t- and F-Tests 265

6.1.3 More General F-Tests 267

6.1.4 Using Computer Software 268

6.1.5 Large Sample Tests 269

6.2 The Use of Nonsample Information 271

6.3 Model Specification 273

6.3.1 Causality versus Prediction 273

6.3.2 Omitted Variables 275

6.3.3 Irrelevant Variables 277

6.3.4 Control Variables 278

6.3.5 Choosing a Model 280

6.3.6 RESET 281

6.4 Prediction 282

6.4.1 Predictive Model Selection Criteria 285

6.5 Poor Data, Collinearity, and Insignificance 288

6.5.1 The Consequences of Collinearity 289

6.5.2 Identifying and Mitigating Collinearity 290

6.5.3 Investigating Influential Observations 293

6.6 Nonlinear Least Squares 294

6.7 Exercises 297

6.7.1 Problems 297

6.7.2 Computer Exercises 303

Appendix 6A The Statistical Power of F-Tests 311

Appendix 6B Further Results from the FWL Theorem 315

7 Using Indicator Variables 317

7.1 Indicator Variables 318

7.1.1 Intercept Indicator Variables 318

7.1.2 Slope-Indicator Variables 320

7.2 Applying Indicator Variables 323

7.2.1 Interactions Between Qualitative Factors 323

7.2.2 Qualitative Factors with Several Categories 324

7.2.3 Testing the Equivalence of Two Regressions 326

7.2.4 Controlling for Time 328

7.3 Log-Linear Models 329

7.3.1 A Rough Calculation 330

7.3.2 An Exact Calculation 330

7.4 The Linear Probability Model 331

7.5 Treatment Effects 332

7.5.1 The Difference Estimator 334

7.5.2 Analysis of the Difference Estimator 334

7.5.3 The Differences-in-Differences Estimator 338

7.6 Treatment Effects and Causal Modeling 342

7.6.1 The Nature of Causal Effects 342

7.6.2 Treatment Effect Models 343

7.6.3 Decomposing the Treatment Effect 344

7.6.4 Introducing Control Variables 345

7.6.5 The Overlap Assumption 347

7.6.6 Regression Discontinuity Designs 347

7.7 Exercises 351

7.7.1 Problems 351

7.7.2 Computer Exercises 358

Appendix 7A Details of Log-Linear Model Interpretation 366

Appendix 7B Derivation of the Differences-in-Differences Estimator 366

Appendix 7C The Overlap Assumption: Details 367

8 Heteroskedasticity 368

8.1 The Nature of Heteroskedasticity 369

8.2 Heteroskedasticity in the Multiple Regression Model 370

8.2.1 The Heteroskedastic Regression Model 371

8.2.2 Heteroskedasticity Consequences for the OLS Estimator 373

8.3 Heteroskedasticity Robust Variance Estimator 374

8.4 Generalized Least Squares: Known Form of Variance 375

8.4.1 Transforming the Model: Proportional Heteroskedasticity 375

8.4.2 Weighted Least Squares: Proportional Heteroskedasticity 377

8.5 Generalized Least Squares: Unknown Form of Variance 379

8.5.1 Estimating the Multiplicative Model 381

8.6 Detecting Heteroskedasticity 383

8.6.1 Residual Plots 384

8.6.2 The Goldfeld–Quandt Test 384

8.6.3 A General Test for Conditional Heteroskedasticity 385

8.6.4 The White Test 387

8.6.5 Model Specification and Heteroskedasticity 388

8.7 Heteroskedasticity in the Linear Probability Model 390

8.8 Exercises 391

8.8.1 Problems 391

8.8.2 Computer Exercises 401

Appendix 8A Properties of the Least Squares Estimator 407

Appendix 8B Lagrange Multiplier Tests for Heteroskedasticity 408

Appendix 8C Properties of the Least Squares Residuals 410

8C.1 Details of Multiplicative Heteroskedasticity Model 411

Appendix 8D Alternative Robust Sandwich Estimators 411

Appendix 8E Monte Carlo Evidence: OLS, GLS, and FGLS 414

9 Regression with Time-Series Data: Stationary Variables 417

9.1 Introduction 418

9.1.1 Modeling Dynamic Relationships 420

9.1.2 Autocorrelations 424

9.2 Stationarity and Weak Dependence 427

9.3 Forecasting 430

9.3.1 Forecast Intervals and Standard Errors 433

9.3.2 Assumptions for Forecasting 435

9.3.3 Selecting Lag Lengths 436

9.3.4 Testing for Granger Causality 437

9.4 Testing for Serially Correlated Errors 438

9.4.1 Checking the Correlogram of the Least Squares Residuals 439

9.4.2 Lagrange Multiplier Test 440

9.4.3 Durbin–Watson Test 443

9.5 Time-Series Regressions for Policy Analysis 443

9.5.1 Finite Distributed Lags 445

9.5.2 HAC Standard Errors 448

9.5.3 Estimation with AR(1) Errors 452

9.5.4 Infinite Distributed Lags 456

9.6 Exercises 463

9.6.1 Problems 463

9.6.2 Computer Exercises 468

Appendix 9A The Durbin–Watson Test 476

9A.1 The Durbin–Watson Bounds Test 478

Appendix 9B Properties of an AR(1) Error 479

10 Endogenous Regressors and Moment-Based Estimation 481

10.1 Least Squares Estimation with Endogenous Regressors 482

10.1.1 Large Sample Properties of the OLS Estimator 483

10.1.2 Why Least Squares Estimation Fails 484

10.1.3 Proving the Inconsistency of OLS 486

10.2 Cases inWhich x and e are Contemporaneously Correlated 487

10.2.1 Measurement Error 487

10.2.2 Simultaneous Equations Bias 488

10.2.3 Lagged-Dependent Variable Models with Serial Correlation 489

10.2.4 Omitted Variables 489

10.3 Estimators Based on the Method of Moments 490

10.3.1 Method of Moments Estimation of a Population Mean and Variance 490

10.3.2 Method of Moments Estimation in the Simple Regression Model 491

10.3.3 Instrumental Variables Estimation in the Simple Regression Model 492

10.3.4 The Importance of Using Strong Instruments 493

10.3.5 Proving the Consistency of the IV Estimator 494

10.3.6 IV Estimation Using Two-Stage Least Squares (2SLS) 495

10.3.7 Using Surplus Moment Conditions 496

10.3.8 Instrumental Variables Estimation in the Multiple Regression Model 498

10.3.9 Assessing Instrument Strength Using the First-Stage Model 500

10.3.10 Instrumental Variables Estimation in a General Model 502

10.3.11 Additional Issues When Using IV Estimation 504

10.4 Specification Tests 505

10.4.1 The Hausman Test for Endogeneity 505

10.4.2 The Logic of the Hausman Test 507

10.4.3 Testing Instrument Validity 508

10.5 Exercises 510

10.5.1 Problems 510

10.5.2 Computer Exercises 516

Appendix 10A Testing for Weak Instruments 520

10A.1 A Test for Weak Identification 521

10A.2 Testing for Weak Identification: Conclusions 525

Appendix 10B Monte Carlo Simulation 525

10B.1 Illustrations Using Simulated Data 526

10B.2 The Sampling Properties of IV/2SLS 528

11 Simultaneous Equations Models 531

11.1 A Supply and Demand Model 532

11.2 The Reduced-Form Equations 534

11.3 The Failure of Least Squares Estimation 535

11.3.1 Proving the Failure of OLS 535

11.4 The Identification Problem 536

11.5 Two-Stage Least Squares Estimation 538

11.5.1 The General Two-Stage Least Squares Estimation Procedure 539

11.5.2 The Properties of the Two-Stage Least Squares Estimator 540

11.6 Exercises 545

11.6.1 Problems 545

11.6.2 Computer Exercises 551

Appendix 11A 2SLS Alternatives 557

11A.1 The k-Class of Estimators 557

11A.2 The LIML Estimator 558

11A.3 Monte Carlo Simulation Results 562

12 Regression with Time-Series Data: Nonstationary Variables 563

12.1 Stationary and Nonstationary Variables 564

12.1.1 Trend Stationary Variables 567

12.1.2 The First-Order Autoregressive Model 570

12.1.3 Random Walk Models 572

12.2 Consequences of Stochastic Trends 574

12.3 Unit Root Tests for Stationarity 576

12.3.1 Unit Roots 576

12.3.2 Dickey–Fuller Tests 577

12.3.3 Dickey–Fuller Test with Intercept and No Trend 577

12.3.4 Dickey–Fuller Test with Intercept and Trend 579

12.3.5 Dickey–Fuller Test with No Intercept and No Trend 580

12.3.6 Order of Integration 581

12.3.7 Other Unit Root Tests 582

12.4 Cointegration 582

12.4.1 The Error Correction Model 584

12.5 Regression When There Is No Cointegration 585

12.6 Summary 587

12.7 Exercises 588

12.7.1 Problems 588

12.7.2 Computer Exercises 592

13 Vector Error Correction and Vector Autoregressive Models 597

13.1 VEC and VAR Models 598

13.2 Estimating a Vector Error Correction Model 600

13.3 Estimating a VAR Model 601

13.4 Impulse Responses and Variance Decompositions 603

13.4.1 Impulse Response Functions 603

13.4.2 Forecast Error Variance Decompositions 605

13.5 Exercises 607

13.5.1 Problems 607

13.5.2 Computer Exercises 608

Appendix 13A The Identification Problem 612

14 Time-Varying Volatility and ARCH Models 614

14.1 The ARCH Model 615

14.2 Time-Varying Volatility 616

14.3 Testing, Estimating, and Forecasting 620

14.4 Extensions 622

14.4.1 The GARCH Model—Generalized ARCH 622

14.4.2 Allowing for an Asymmetric Effect 623

14.4.3 GARCH-in-Mean and Time-Varying Risk Premium 624

14.4.4 Other Developments 625

14.5 Exercises 626

14.5.1 Problems 626

14.5.2 Computer Exercises 627

15 Panel Data Models 634

15.1 The Panel Data Regression Function 636

15.1.1 Further Discussion of Unobserved Heterogeneity 638

15.1.2 The Panel Data Regression Exogeneity Assumption 639

15.1.3 Using OLS to Estimate the Panel Data Regression 639

15.2 The Fixed Effects Estimator 640

15.2.1 The Difference Estimator: T = 2 640

15.2.2 The Within Estimator: T = 2 642

15.2.3 The Within Estimator: T > 2 643

15.2.4 The Least Squares Dummy Variable Model 644

15.3 Panel Data Regression Error Assumptions 646

15.3.1 OLS Estimation with Cluster-Robust Standard Errors 648

15.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors 650

15.4 The Random Effects Estimator 651

15.4.1 Testing for Random Effects 653

15.4.2 A Hausman Test for Endogeneity in the Random Effects Model 654

15.4.3 A Regression-Based Hausman Test 656

15.4.4 The Hausman–Taylor Estimator 658

15.4.5 Summarizing Panel Data Assumptions 660

15.4.6 Summarizing and Extending Panel Data Model Estimation 661

15.5 Exercises 663

15.5.1 Problems 663

15.5.2 Computer Exercises 670

Appendix 15A Cluster-Robust Standard Errors: Some Details 677

Appendix 15B Estimation of Error Components 679

16 Qualitative and Limited Dependent Variable Models 681

16.1 Introducing Models with Binary Dependent Variables 682

16.1.1 The Linear Probability Model 683

16.2 Modeling Binary Choices 685

16.2.1 The Probit Model for Binary Choice 686

16.2.2 Interpreting the Probit Model 687

16.2.3 Maximum Likelihood Estimation of the Probit Model 690

16.2.4 The Logit Model for Binary Choices 693

16.2.5 Wald Hypothesis Tests 695

16.2.6 Likelihood Ratio Hypothesis Tests 696

16.2.7 Robust Inference in Probit and Logit Models 698

16.2.8 Binary Choice Models with a Continuous Endogenous Variable 698

16.2.9 Binary Choice Models with a Binary Endogenous Variable 699

16.2.10 Binary Endogenous Explanatory Variables 700

16.2.11 Binary Choice Models and Panel Data 701

16.3 Multinomial Logit 702

16.3.1 Multinomial Logit Choice Probabilities 703

16.3.2 Maximum Likelihood Estimation 703

16.3.3 Multinomial Logit Postestimation Analysis 704

16.4 Conditional Logit 707

16.4.1 Conditional Logit Choice Probabilities 707

16.4.2 Conditional Logit Postestimation Analysis 708

16.5 Ordered Choice Models 709

16.5.1 Ordinal Probit Choice Probabilities 710

16.5.2 Ordered Probit Estimation and Interpretation 711

16.6 Models for Count Data 713

16.6.1 Maximum Likelihood Estimation of the Poisson Regression Model 713

16.6.2 Interpreting the Poisson Regression Model 714

16.7 Limited Dependent Variables 717

16.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model 717

16.7.2 Truncated Regression 718

16.7.3 Censored Samples and Regression 718

16.7.4 Tobit Model Interpretation 720

16.7.5 Sample Selection 723

16.8 Exercises 725

16.8.1 Problems 725

16.8.2 Computer Exercises 733

Appendix 16A Probit Marginal Effects: Details 739

16A.1 Standard Error of Marginal Effect at a Given Point 739

16A.2 Standard Error of Average Marginal Effect 740

Appendix 16B Random Utility Models 741

16B.1 Binary Choice Model 741

16B.2 Probit or Logit? 742

Appendix 16C Using Latent Variables 743

16C.1 Tobit (Tobit Type I) 743

16C.2 Heckit (Tobit Type II) 744

Appendix 16D A Tobit Monte Carlo Experiment 745

A Mathematical Tools 748

A.1 Some Basics 749

A.1.1 Numbers 749

A.1.2 Exponents 749

A.1.3 Scientific Notation 749

A.1.4 Logarithms and the Number e 750

A.1.5 Decimals and Percentages 751

A.1.6 Logarithms and Percentages 751

A.2 Linear Relationships 752

A.2.1 Slopes and Derivatives 753

A.2.2 Elasticity 753

A.3 Nonlinear Relationships 753

A.3.1 Rules for Derivatives 754

A.3.2 Elasticity of a Nonlinear Relationship 757

A.3.3 Second Derivatives 757

A.3.4 Maxima and Minima 758

A.3.5 Partial Derivatives 759

A.3.6 Maxima and Minima of Bivariate Functions 760

A.4 Integrals 762

A.4.1 Computing the Area Under a Curve 762

A.5 Exercises 764

B Probability Concepts 768

B.1 Discrete Random Variables 769

B.1.1 Expected Value of a Discrete Random Variable 769

B.1.2 Variance of a Discrete Random Variable 770

B.1.3 Joint, Marginal, and Conditional Distributions 771

B.1.4 Expectations Involving Several Random Variables 772

B.1.5 Covariance and Correlation 773

B.1.6 Conditional Expectations 774

B.1.7 Iterated Expectations 774

B.1.8 Variance Decomposition 774

B.1.9 Covariance Decomposition 777

B.2 Working with Continuous Random Variables 778

B.2.1 Probability Calculations 779

B.2.2 Properties of Continuous Random Variables 780

B.2.3 Joint, Marginal, and Conditional Probability Distributions 781

B.2.4 Using Iterated Expectations with Continuous Random Variables 785

B.2.5 Distributions of Functions of Random Variables 787

B.2.6 Truncated Random Variables 789

B.3 Some Important Probability Distributions 789

B.3.1 The Bernoulli Distribution 790

B.3.2 The Binomial Distribution 790

B.3.3 The Poisson Distribution 791

B.3.4 The Uniform Distribution 792

B.3.5 The Normal Distribution 793

B.3.6 The Chi-Square Distribution 794

B.3.7 The t-Distribution 796

B.3.8 The F-Distribution 797

B.3.9 The Log-Normal Distribution 799

B.4 Random Numbers 800

B.4.1 Uniform Random Numbers 805

B.5 Exercises 806

C Review of Statistical Inference 812

C.1 A Sample of Data 813

C.2 An Econometric Model 814

C.3 Estimating the Mean of a Population 815

C.3.1 The Expected Value of Y 816

C.3.2 The Variance of Y 817

C.3.3 The Sampling Distribution of Y 817

C.3.4 The Central Limit Theorem 818

C.3.5 Best Linear Unbiased Estimation 820

C.4 Estimating the Population Variance and Other Moments 820

C.4.1 Estimating the Population Variance 821

C.4.2 Estimating Higher Moments 821

C.5 Interval Estimation 822

C.5.1 Interval Estimation: σ2 Known 822

C.5.2 Interval Estimation: σ2 Unknown 825

C.6 Hypothesis Tests About a Population Mean 826

C.6.1 Components of Hypothesis Tests 826

C.6.2 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 828

C.6.3 One-Tail Tests with Alternative ‘‘Less Than’’ (<) 829

C.6.4 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (≠) 829

C.6.5 The p-Value 831

C.6.6 A Comment on Stating Null and Alternative Hypotheses 832

C.6.7 Type I and Type II Errors 833

C.6.8 A Relationship Between Hypothesis Testing and Confidence Intervals 833

C.7 Some Other Useful Tests 834

C.7.1 Testing the Population Variance 834

C.7.2 Testing the Equality of Two Population Means 834

C.7.3 Testing the Ratio of Two Population Variances 835

C.7.4 Testing the Normality of a Population 836

C.8 Introduction to Maximum Likelihood Estimation 837

C.8.1 Inference with Maximum Likelihood Estimators 840

C.8.2 The Variance of the Maximum Likelihood Estimator 841

C.8.3 The Distribution of the Sample Proportion 842

C.8.4 Asymptotic Test Procedures 843

C.9 Algebraic Supplements 848

C.9.1 Derivation of Least Squares Estimator 848

C.9.2 Best Linear Unbiased Estimation 849

C.10 Kernel Density Estimator 851

C.11 Exercises 854

C.11.1 Problems 854

C.11.2 Computer Exercises 857

D Statistical Tables 862

TableD.1 Cumulative Probabilities for the Standard Normal Distribution 𝚽(z) = P(Z ≤ z) 862

TableD.2 Percentiles of the t-distribution 863

TableD.3 Percentiles of the Chi-square Distribution 864

TableD.4 95th Percentile for the F-distribution 865

TableD.5 99th Percentile for the F-distribution 866

TableD.6 Standard Normal pdf Values 𝛟(z) 867

Index 869