The new edition continues to provide a large number of worked examples, and some shorter data sets. Further data sets and additional supplementary material to assist both the student and lecturer are available on the companion website www.wileyeurope.com/college/maddala

New features for the fourth edition:

- Chapters 5 and 6, on Heteroscedasticity and Autocorrelation, now reflect the latest professional practice in dealing with these common variations of the basic regression model.
- Chapter 10 includes extensive discussion on diagnostic checking in linear models, various nested and non-nested model selection procedures, specification testing, data transformations, and tests for non-normality.
- The first three chapters of Part III cover an introduction to time-series analysis, including the Box–Jenkins approach, forecasting and seasonality, models of expectations and distributed lag models, and vector auto-regressions, unit roots, and cointegration.
- Chapters 15 and 16 cover, respectively, the latest developments in panel data analysis and various re-sampling methods for use in small sample inference.

**Kajal Lahiri** is Distinguished Professor of Economics, and Health Policy, and Management and Behaviour at the State University of New York, Albany where he is also Director of the Econometric Research Institute. Professor Lahiri is an Honorary Fellow of the International Institute of Forecasters.

Foreword xvii

Preface to the Fourth Edition xix

**Part I Introduction and the Linear Regression Model 1**

**CHAPTER 1 What is Econometrics? 3**

1.1 What is econometrics? 3

1.2 Economic and econometric models 4

1.3 The aims and methodology of econometrics 6

1.4 What constitutes a test of an economic theory? 8

**CHAPTER 2 Statistical Background and Matrix Algebra 11**

2.1 Introduction 11

2.2 Probability 12

2.3 Random variables and probability distributions 17

2.4 The normal probability distribution and related distributions 18

2.5 Classical statistical inference 21

2.6 Properties of estimators 22

2.7 Sampling distributions for samples from a normal population 26

2.8 Interval estimation 26

2.9 Testing of hypotheses 28

2.10 Relationship between confidence interval procedures and tests of hypotheses 31

2.11 Combining independent tests 32

**CHAPTER 3 Simple Regression 59**

3.1 Introduction 59

3.2 Specification of the relationships 61

3.3 The method of moments 65

3.4 The method of least squares 68

3.5 Statistical inference in the linear regression model 76

3.6 Analysis of variance for the simple regression model 83

3.7 Prediction with the simple regression model 85

3.8 Outliers 88

3.9 Alternative functional forms for regression equations 95

*3.10 Inverse prediction in the least squares regression model1 99

*3.11 Stochastic regressors 102

*3.12 The regression fallacy 102

**CHAPTER 4 Multiple Regression 127**

4.1 Introduction 127

4.2 A model with two explanatory variables 129

4.3 Statistical inference in the multiple regression model 134

4.4 Interpretation of the regression coefficients 143

4.5 Partial correlations and multiple correlation 146

4.6 Relationships among simple, partial, and multiple correlation coefficients 147

4.7 Prediction in the multiple regression model 153

4.8 Analysis of variance and tests of hypotheses 155

4.9 Omission of relevant variables and inclusion of irrelevant variables 160

4.10 Degrees of freedom and *R*2 165

4.11 Tests for stability 169

4.12 The LR, W, and LM tests 176

**Part II Violation of the Assumptions of the Basic Regression Model 209**

**CHAPTER 5 Heteroskedasticity 211**

5.1 Introduction 211

5.2 Detection of heteroskedasticity 214

5.3 Consequences of heteroskedasticity 219

5.4 Solutions to the heteroskedasticity problem 221

5.5 Heteroskedasticity and the use of deflators 224

5.6 Testing the linear versus log-linear functional form 228

**CHAPTER 6 Autocorrelation 239**

6.1 Introduction 239

6.2 The Durbin–Watson test 240

6.3 Estimation in levels versus first differences 242

6.4 Estimation procedures with autocorrelated errors 246

6.5 Effect of AR(1) errors on OLS estimates 250

6.6 Some further comments on the DW test 254

6.7 Tests for serial correlation in models with lagged dependent variables 257

6.8 A general test for higher-order serial correlation: The LM test 259

6.9 Strategies when the DW test statistic is significant 261

*6.10 Trends and random walks 266

*6.11 ARCH models and serial correlation 271

6.12 Some comments on the DW test and Durbin’s *h*-test and *t*-test 272

**CHAPTER 7 Multicollinearity 279**

7.1 Introduction 279

7.2 Some illustrative examples 280

7.3 Some measures of multicollinearity 283

7.4 Problems with measuring multicollinearity 286

7.5 Solutions to the multicollinearity problem: Ridge regression 290

7.6 Principal component regression 292

7.7 Dropping variables 297

7.8 Miscellaneous other solutions 300

**CHAPTER 8 Dummy Variables and Truncated Variables 313**

8.1 Introduction 313

8.2 Dummy variables for changes in the intercept term 314

8.3 Dummy variables for changes in slope coefficients 319

8.4 Dummy variables for cross-equation constraints 322

8.5 Dummy variables for testing stability of regression coefficients 324

8.6 Dummy variables under heteroskedasticity and autocorrelation 327

8.7 Dummy dependent variables 329

8.8 The linear probability model and the linear discriminant function 329

8.9 The probit and logit models 333

8.10 Truncated variables: The tobit model 343

**CHAPTER 9 Simultaneous Equation Models 355**

9.1 Introduction 355

9.2 Endogenous and exogenous variables 357

9.3 The identification problem: Identification through reduced form 357

9.4 Necessary and sufficient conditions for identification 362

9.5 Methods of estimation: The instrumental variable method 365

9.6 Methods of estimation: The two-stage least squares method 371

9.7 The question of normalization 378

*9.8 The limited-information maximum likelihood method 379

*9.9 On the use of OLS in the estimation of simultaneous equation models 380

*9.10 Exogeneity and causality 386

9.11 Some problems with instrumental variable methods 392

**CHAPTER 10 Diagnostic Checking, Model Selection, and Specification Testing 401**

10.1 Introduction 401

10.2 Diagnostic tests based on least squares residuals 402

10.3 Problems with least squares residuals 404

10.4 Some other types of residual 405

10.5 DFFITS and bounded influence estimation 411

10.6 Model selection 414

10.7 Selection of regressors 419

10.8 Implied *F*-ratios for the various criteria 423

10.9 Cross-validation 427

10.10 Hausman’s specification error test 428

10.11 The Plosser–Schwert–White differencing test 435

10.12 Tests for nonnested hypotheses 436

10.13 Nonnormality of errors 440

10.14 Data transformations 441

**CHAPTER 11 Errors in Variables 451**

11.1 Introduction 451

11.2 The classical solution for a single-equation model with one explanatory variable 452

11.3 The single-equation model with two explanatory variables 455

11.4 Reverse regression 463

11.5 Instrumental variable methods 465

11.6 Proxy variables 468

11.7 Some other problems 471

**Part III Special Topics 479**

**CHAPTER 12 Introduction to Time-Series Analysis 481**

12.1 Introduction 481

12.2 Two methods of time-series analysis: Frequency domain and time domain 482

12.3 Stationary and nonstationary time series 482

12.4 Some useful models for time series 485

12.5 Estimation of AR, MA, and ARMA models 492

12.6 The Box–Jenkins approach 496

12.7 *R*2 measures in time-series models 503

**CHAPTER 13 Models of Expectations and Distributed Lags 509**

13.1 Models of expectations 509

13.2 Naive models of expectations 510

13.3 The adaptive expectations model 512

13.4 Estimation with the adaptive expectations model 514

13.5 Two illustrative examples 516

13.6 Expectational variables and adjustment lags 520

13.7 Partial adjustment with adaptive expectations 524

13.8 Alternative distributed lag models: Polynomial lags 526

13.9 Rational lags 533

13.10 Rational expectations 534

13.11 Tests for rationality 536

13.12 Estimation of a demand and supply model under rational expectations 538

13.13 The serial correlation problem in rational expectations models 544

**CHAPTER 14 Vector Autoregressions, Unit Roots, and Cointegration 551**

14.1 Introduction 551

14.2 Vector autoregressions 551

14.3 Problems with VAR models in practice 553

14.4 Unit roots 554

14.5 Unit root tests 555

14.6 Cointegration 563

14.7 The cointegrating regression 564

14.8 Vector autoregressions and cointegration 567

14.9 Cointegration and error correction models 571

14.10 Tests for cointegration 571

14.11 Cointegration and testing of the REH and MEH 572

14.12 A summary assessment of cointegration 574

**CHAPTER 15 Panel Data Analysis 583**

15.1 Introduction 583

15.2 The LSDV or fixed effects model 584

15.3 The random effects model 586

15.4 Fixed effects versus random effects 589

15.5 Dynamic panel data models 591

15.6 Panel data models with correlated effects and simultaneity 593

15.7 Errors in variables in panel data 595

15.8 The SUR model 597

15.9 The random coefficient model 597

**CHAPTER 16 Small-Sample Inference: Resampling Methods 601**

16.1 Introduction 601

16.2 Monte Carlo methods 602

16.3 Resampling methods: Jackknife and bootstrap 603

16.4 Bootstrap confidence intervals 605

16.5 Hypothesis testing with the bootstrap 606

16.6 Bootstrapping residuals versus bootstrapping the data 607

16.7 Non-IID errors and nonstationary models 607

**Appendix 611**

**Index 621**