Basic Econometrics

Contents
Preface vii
Acknowledgments xi
Introduction 1
I.1 What is Econometrics? 1
I.2 Why a Separate Discipline? 2
I.3 Methodology of Econometrics 2
I.4 Types of Econometrics 9
I.5 Mathematical and Statistical Prerequisites 10
I.6 The Role of the Computer 10
I.7 Suggestions for Further Reading 10
PART 1 Single-Equation Regression Models
1. The Nature of Regression Analysis 15
1.1 Historical Origin of the Term Regression 15
1.2 The Modern Interpretation of Regression 15
1.3 Statistical versus Deterministic Relationships 19
1.4 Regression versus Causation 19
1.5 Regression versus Correlation 20
1.6 Terminology and Notation 20
1.7 The Nature and Sources of Data for Economic Analysis 21
Summary and Conclusions 28
Multiple Choice Questions 29
Exercises 32
Key to Multiple Choice Questions 37
2. Two-Variable Regression Analysis: Some Basic Ideas 38
2.1 A Hypothetical Example 38
2.2 The Concept of Population Regression Function (PRF) 41
2.3 The Meaning of the Term Linear 42
2.4 Stochastic Specifi cation of PRF 43
xvi Contents
2.5 The Signifi cance of the Stochastic Disturbance Term 45
2.6 The Sample Regression Function (SRF) 46
2.7 Illustrative Examples 49
Summary and Conclusions 51
Multiple Choice Questions 51
Exercises 54
Key to Multiple Choice Questions 60
3. Two-Variable Regression Model: The Problem of Estimation 61
3.1 The Method of Ordinary Least Squares 61
3.2 The Classical Linear Regression Model: The Assumptions Underlying the
Method of Least Squares 67
3.3 Precision or Standard Errors of Least-Squares Estimates 74
3.4 Properties of Least-Squares Estimators: The Gauss–Markov Theorem 76
3.5 The Coeffi cient of Determination r2
: A Measure of “Goodness of Fit” 78
3.6 A Numerical Example 83
3.7 Illustrative Examples 86
3.8 A Note on Monte Carlo Experiments 88
Summary and Conclusions 89
Multiple Choice Questions 90
Exercises 93
Key to Multiple Choice Questions 99
Appendix 3A 100
4. Classical Normal Linear Regression Model (CNLRM) 105
4.1 The Probability Distribution of Disturbances ui
105
4.2 The Normality Assumption for ui
106
4.3 Properties of OLS Estimators under the Normality Assumption 107
4.4 The Method of Maximum Likelihood (ML) 109
Summary and Conclusions 110
Appendix 4A 113
5. Two-Variable Regression: Interval Estimation and Hypothesis Testing 115
5.1 Statistical Prerequisites 115
5.2 Interval Estimation: Some Basic Ideas 115
5.3 Confi dence Intervals for Regression Coeffi cients b1 and b2 117
5.4 Confi dence Interval for s2 119
5.5 Hypothesis Testing: General Comments 120
5.6 Hypothesis Testing: The Confi dence-Interval Approach 121
5.7 Hypothesis Testing: The Test-of-Signifi cance Approach 122
5.8 Hypothesis Testing: Some Practical Aspects 127
5.9 Regression Analysis and Analysis of Variance 131
5.10 Application of Regression Analysis: The Problem of Prediction 133
5.11 Reporting the Results of Regression Analysis 136
5.12 Evaluating the Results of Regression Analysis 137
Summary and Conclusions 140
Multiple Choice Questions 141
Exercises 146
Contents xvii
Key to Multiple Choice Questions 154
Appendix 5A 155
6. Extensions of the Two-Variable Linear Regression Model 159
6.1 Regression through the Origin 159
6.2 Scaling and Units of Measurement 166
6.3 Regression on Standardized Variables 170
6.4 Functional Forms of Regression Models 171
6.5 How to Measure Elasticity: The Log-Linear Model 172
6.6 Semilog Models: Log–Lin and Lin–Log Models 175
6.7 Reciprocal Models 179
6.8 Choice of Functional Form 184
6.9 A Note on the Nature of the Stochastic Error Term: Additive versus
Multiplicative Stochastic Error Term 186
Summary and Conclusions 187
Multiple Choice Questions 188
Exercises 190
Key to Multiple Choice Questions 196
Appendix 6A 197
7. Multiple Regression Analysis: The Problem of Estimation 203
7.1 The Three-Variable Model: Notation and Assumptions 203
7.2 Interpretation of Multiple Regression Equation 205
7.3 The Meaning of Partial Regression Coeffi cients 205
7.4 OLS and ML Estimation of the Partial Regression Coeffi cients 207
7.5 The Multiple Coeffi cient of Determination R2
and the
Multiple Coeffi cient of Correlation R 210
7.6 An Illustrative Example 212
7.7 Simple Regression in the Context of Multiple Regression:
Introduction to Specifi cation Bias 214
7.8 R2
and the Adjusted R2 215
7.9 The Cobb–Douglas Production Function: More on Functional Form 220
7.10 Polynomial Regression Models 223
*7.11 Partial Correlation Coeffi cients 226
Summary and Conclusions 228
Multiple Choice Questions 228
Exercises 231
Key to Multiple Choice Questions 243
Appendix 7A 243
8. Multiple Regression Analysis: The Problem of Inference 249
8.1 The Normality Assumption Once Again 249
8.2 Hypothesis Testing in Multiple Regression: General Comments 250
8.3 Hypothesis Testing about Individual Regression Coeffi cients 251
8.4 Testing the Overall Signifi cance of the Sample Regression 253
8.5 Testing the Equality of Two Regression Coeffi cients 262
8.6 Restricted Least Squares: Testing Linear Equality Restrictions 264
8.7 Testing for Structural or Parameter Stability of Regression Models: The Chow Test 270
xviii Contents
8.8 Prediction with Multiple Regression 275
8.9 The Troika of Hypothesis Tests: The Likelihood Ratio (LR), Wald (W), and
Lagrange Multiplier (LM) Tests 275
8.10 Testing the Functional Form of Regression: Choosing between
Linear and Log–Linear Regression Models 276
Summary and Conclusions 278
Multiple Choice Questions 278
Exercises 281
Key to Multiple Choice Questions 292
Appendix 8A 292
9. Dummy Variable Regression Models 295
9.1 The Nature of Dummy Variables 295
9.2 ANOVA Models 296
9.3 ANOVA Models with Two Qualitative Variables 300
9.4 Regression with a Mixture of Quantitative and Qualitative Regressors:
The ANCOVA Models 302
9.5 The Dummy Variable Alternative to the Chow Test 303
9.6 Interaction Effects Using Dummy Variables 306
9.7 The Use of Dummy Variables in Seasonal Analysis 307
9.8 Piecewise Linear Regression 311
9.9 Panel Data Regression Models 314
9.10 Some Technical Aspects of the Dummy Variable Technique 314
9.11 Topics for Further Study 316
9.12 A Concluding Example 316
Summary and Conclusions 320
Multiple Choice Questions 320
Exercises 324
Key to Multiple Choice Questions 332
Appendix 9A 332
PART 2 Relaxing the Assumptions of the Classical Model
10. Multicollinearity: What Happens If the Regressors are Correlated? 339
10.1 The Nature of Multicollinearity 340
10.2 Estimation in the Presence of Perfect Multicollinearity 342
10.3 Estimation in the Presence of “High” but “Imperfect” Multicollinearity 344
10.4 Multicollinearity: Much Ado about Nothing? Theoretical Consequences of
Multicollinearity 344
10.5 Practical Consequences of Multicollinearity 346
10.6 An Illustrative Example 351
10.7 Detection of Multicollinearity 356
10.8 Remedial Measures 360
10.9 Is Multicollinearity Necessarily Bad? Maybe Not, If the Objective
Is Prediction Only 365
10.10 An Extended Example: The Longley Data 365
Summary and Conclusions 368
Contents xix
Multiple Choice Questions 369
Exercises 372
Key to Multiple Choice Questions 385
11. Heteroscedasticity: What Happens if the Error Variance is Nonconstant? 386
11.1 The Nature of Heteroscedasticity 386
11.2 OLS Estimation in the Presence of Heteroscedasticity 391
11.3 The Method of Generalized Least Squares (GLS) 392
11.4 Consequences of Using OLS in the Presence of Heteroscedasticity 395
11.5 Detection of Heteroscedasticity 397
11.6 Remedial Measures 410
11.7 Concluding Examples 416
11.8 A Caution about Overreacting to Heteroscedasticity 420
Summary and Conclusions 421
Multiple Choice Questions 421
Exercises 424
Key to Multiple Choice Questions 432
Appendix 11A 432
12. Autocorrelation: What Happens if the Error Terms are Correlated? 436
12.1 The Nature of the Problem 437
12.2 OLS Estimation in the Presence of Autocorrelation 443
12.3 The BLUE Estimator in the Presence of Autocorrelation 445
12.4 Consequences of Using OLS in the Presence of Autocorrelation 446
12.5 Relationship between Wages and Productivity in the Business Sector of the
United States, 1960–2005 451
12.6 Detecting Autocorrelation 453
12.7 What to do when you fi nd Autocorrelation: Remedial Measures 463
12.8 Model Mis-Specifi cation versus Pure Autocorrelation 463
12.9 Correcting for (Pure) Autocorrelation: The Method of Generalized
Least Squares (GLS) 464
12.10 The Newey–West Method of Correcting the OLS Standard Errors 470
12.11 OLS versus FGLS and HAC 470
12.12 Additional Aspects of Autocorrelation 471
12.13 A Concluding Example 472
Summary and Conclusions 474
Multiple Choice Questions 475
Exercises 478
Key to Multiple Choice Questions 490
Appendix 12A 491
13. Econometric Modeling: Model Specifi cation and Diagnostic Testing 492
13.1 Model Selection Criteria 493
13.2 Types of Specifi cation Errors 493
13.3 Consequences of Model Specifi cation Errors 495
13.4 Tests of Specifi cation Errors 499
13.5 Errors of Measurement 506
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13.6 Incorrect Specifi cation of the Stochastic Error Term 510
13.7 Nested versus Non-Nested Models 510
13.8 Tests of Non-Nested Hypotheses 511
13.9 Model Selection Criteria 516
13.10 Additional Topics in Econometric Modeling 520
13.11 Concluding Examples 524
13.12 Non-Normal Errors and Stochastic Regressors 533
13.13 A Word to the Practitioner 535
Summary and Conclusions 536
Multiple Choice Questions 537
Exercises 540
Key to Multiple Choice Questions 546
Appendix 13A 546
PART 3 Topics in Econometrics
14. Nonlinear Regression Models 553
14.1 Intrinsically Linear and Intrinsically Nonlinear Regression Models 553
14.2 Estimation of Linear and Nonlinear Regression Models 555
14.3 Estimating Nonlinear Regression Models: The Trial-and-Error Method 555
14.4 Approaches to Estimating Nonlinear Regression Models 557
14.5 Illustrative Examples 558
Summary and Conclusions 562
Multiple Choice Questions 563
Exercises 565
Key to Multiple Choice Questions 566
Appendix 14A 567
15. Qualitative Response Regression Models 570
15.1 The Nature of Qualitative Response Models 570
15.2 The Linear Probability Model (LPM) 572
15.3 Applications of LPM 578
15.4 Alternatives to LPM 581
15.5 The Logit Model 582
15.6 Estimation of the Logit Model 584
15.7 The Grouped Logit (Glogit) Model: A Numerical Example 587
15.8 The Logit Model for Ungrouped or Individual Data 590
15.9 The Probit Model 594
15.10 Logit and Probit Models 599
15.11 The Tobit Model 602
15.12 Modeling Count Data: The Poisson Regression Model 604
15.13 Further Topics in Qualitative Response Regression Models 607
Summary and Conclusions 609
Multiple Choice Questions 610
Exercises 613
Key to Multiple Choice Questions 620
Appendix 15A 620
Contents xxi
16. Panel Data Regression Models 622
16.1 Why Panel Data? 623
16.2 Panel Data: An Illustrative Example 624
16.3 Pooled OLS Regression or Constant Coeffi cients Model 625
16.4 The Fixed Effect Least-Squares Dummy Variable (LSDV) Model 627
16.5 The Fixed-Effect Within-Group (WG) Estimator 630
16.6 The Random Effects Model (REM) 633
16.7 Properties of Various Estimators 637
16.8 Fixed Effects versus Random Effects Model: Some Guidelines 637
16.9 Panel Data Regressions: Some Concluding Comments 638
16.10 Some Illustrative Examples 639
Summary and Conclusions 644
Multiple Choice Questions 645
Exercises 648
Key to Multiple Choice Questions 651
17. Dynamic Econometric Models: Autoregressive and Distributed-Lag Models 652
17.1 The Role of “Time,” or “Lag,” in Economics 653
17.2 The Reasons for Lags 657
17.3 Estimation of Distributed-Lag Models 658
17.4 The Koyck Approach to Distributed-Lag Models 659
17.5 Rationalization of the Koyck Model: The Adaptive Expectations Model 664
17.6 Another Rationalization of the Koyck Model: The Stock
Adjustment, or Partial Adjustment, Model 666
17.7 Combination of Adaptive Expectations and Partial Adjustment Models 668
17.8 Estimation of Autoregressive Models 669
17.9 The Method of Instrumental Variables (IV) 670
17.10 Detecting Autocorrelation in Autoregressive Models: Durbin h Test 671
17.11 A Numerical Example: The Demand for Money in Canada,
1979–I to 1988–IV 673
17.12 Illustrative Examples 676
17.13 The Almon Approach to Distributed-Lag Models:
The Almon or Polynomial Distributed Lag (PDL) 679
17.14 Causality in Economics: The Granger Causality Test 686
Summary and Conclusions 692
Multiple Choice Questions 693
Exercises 696
Key to Multiple Choice Questions 705
Appendix 17A 705
PART 4 Simultaneous-Equation Models and Time Series Econometrics
18. Simultaneous-Equation Models 709
18.1 The Nature of Simultaneous-Equation Models 709
18.2 Examples of Simultaneous-Equation Models 710
18.3 The Simultaneous-Equation Bias: Inconsistency of OLS Estimators 715
18.4 The Simultaneous-Equation Bias: A Numerical Example 718
Summary and Conclusions 720
xxii Contents
Multiple Choice Questions 720
Exercises 721
Key to Multiple Choice Questions 725
19. The Identifi cation Problem 726
19.1 Notations and Defi nitions 726
19.2 The Identifi cation Problem 729
19.3 Rules for Identifi cation 736
19.4 A Test of Simultaneity 740
19.5 Tests for Exogeneity 743
Summary and Conclusions 743
Multiple Choice Questions 744
Exercises 746
Key to Multiple Choice Questions 750
20. Simultaneous-Equation Methods 751
20.1 Approaches to Estimation 751
20.2 Recursive Models and Ordinary Least Squares 753
20.3 Estimation of a Just Identifi ed Equation: The Method of Indirect Least Squares (ILS) 755
20.4 Estimation of an Overidentifi ed Equation: The Method of
Two-Stage Least Squares (2SLS) 758
20.5 2SLS: A Numerical Example 761
20.6 Illustrative Examples 764
Summary and Conclusions 770
Multiple Choice Questions 771
Exercises 773
Key to Multiple Choice Questions 777
Appendix 20A 777
21. Time Series Econometrics: Some Basic Concepts 780
21.1 A Look at Selected U.S. Economic Time Series 781
21.2 Key Concepts 782
21.3 Stochastic Processes 783
21.4 Unit Root Stochastic Process 787
21.5 Trend Stationary (TS) and Difference Stationary (DS)
Stochastic Processes 788
21.6 Integrated Stochastic Processes 789
21.7 The Phenomenon of Spurious Regression 790
21.8 Tests of Stationarity 791
21.9 The Unit Root Test 797
21.10 Transforming Nonstationary Time Series 802
21.11 Cointegration: Regression of a Unit Root Time Series on
Another Unit Root Time Series 805
21.12 Some Economic Applications 808
Summary and Conclusions 811
Multiple Choice Questions 812
Exercises 815
Key to Multiple Choice Questions 819
Contents xxiii
22. Time Series Econometrics: Forecasting 820
22.1 Approaches to Economic Forecasting 820
22.2 AR, MA, and ARIMA Modeling of Time Series Data 822
22.3 The Box–Jenkins (BJ) Methodology 824
22.4 Identifi cation 825
22.5 Estimation of the ARIMA Model 829
22.6 Diagnostic Checking 829
22.7 Forecasting 830
22.8 Further Aspects of the BJ Methodology 831
22.9 Vector Autoregression (VAR) 831
22.10 Measuring Volatility in Financial Time Series: The ARCH and GARCH Models 838
22.11 Concluding Examples 843
Summary and Conclusions 845
Multiple Choice Questions 846
Exercises 848
Key to Multiple Choice Questions 850
Appendix D* Statistical Tables 851
Selected Bibliography 868
Index 873