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# Introduction to Econometrics by G. S. Maddala and Kajal Lahiri

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Maintaining G.S. Maddala’s brilliant expository style of cutting through the technical superstructure to reveal only essential details, while retaining the nerve centre of the subject matter, Professor Kajal Lahiri has brought forward this new edition of one of the most important textbooks in its field.

G.S.Maddala was one of the leading figures in field of econometrics for more than 30 years until he passed away in 1999. At the time of his death, he held the University Eminent Scholar Professorship in the Department of Economics at Ohio State University. His previous affiliations include Stanford University, University of Rochester and University of Florida.

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.

Part I Introduction and the Linear Regression Model CHAPTER 1 What is Econometrics?

1.1 What Is Econometrics?

1.2 Economic and Econometric Models

1.3 The Aims and Methodology of Econometrics

1.4 What Constitutes a Test of an Economic Theory?

Summary and an Outline of the Book

2 Statistical Background and Matrix Algebra

2.1 Introduction

2.2 Probability

2.3 Random Variables and Probability Distributions

2.4 The Normal Probability Distribution and Related Distributions

2.5 Classical Statistical Inference 21

2.6 Properties of Estimators 23

2.7 Sampling Distributions for Samples from a Normal Population 26

2.8 Interval Estimation 27

2.9 Testing of Hypotheses 28

2.10 Relationship Between Confidence Interval Procedures and Tests of Hypotheses 32

Summary 33

Exercises 34

Appendix to Chapter 2 41

3 Simple Regression 59

3.1 Introduction

3.2 Specification of the Relationships

3.3 The Method of Moments

3.4 Method of Least Squares

3.5 Statistical Inference in the Linear Regression Model

3.6 Analysis of Variance for the Simple Regression Model

3.7 Prediction with the Simple Regression Model

3.8 Outliers

3.9 Alternative Functional Forms for Regression Equations

*3.10 Inverse Prediction in the Least Squares Regression Model

*3.11 Stochastic Regressors

*3.12 The Regression Fallacy

Summary

Exercises

Appendix to Chapter 3

4 Multiple Regression

4.1 Introduction

4.2 A Model with Two Explanatory Variables

4.3 Statistical Inference in the Multiple Regression Model

4.4 Interpretation of the Regression Coefficients

4.5 Partial Correlations and Multiple Correlation

4.6 Relationships Among Simple, Partial, and Multiple Correlation Coefficients

4.7 Prediction in the Multiple Regression Model

4.8 Analysis of Variance and Tests of Hypotheses

4.9 Omission of Relevant Variables and Inclusion of Irrelevant Variables

4.10 Degrees of Freedom and R2

4.11 Tests for Stability

*4.12 The LR, W, and LM Tests

Summary

Exercises

Appendix to Chapter 4

Data Sets

5 Heteroskedasticity 201

5.1 Introduction

5.2 Detection of Heteroskedasticity

5.3 Consequences of Heteroskedasticity

5.4 Solutions to the Heteroskedasticity Problem

5.5 Heteroskedasticity and the Use of Deflators

*5.6 Testing the Linear Versus Log-Linear Functional

Form

Summary

Exercises

Appendix to Chapter 5

6 Autocorrelation

6.1 Introduction

6.2 Durbin-Watson Test

6.3 Estimation in Levels Versus First Differences

6.4 Estimation Procedures with Autocorrelated Errors

6.5 Effect of AR(1) Errors on OLS Estimates

6.6 Some Further Comments on the DW Test

6.7 Tests for Serial Correlation in Models with Lagged Dependent Variables 248X CONTENTS

6.8 A General Test for Higher-Order Serial Correlation: The LM Test

6.9 Strategies When the DW Test Statistic Is Significant

6.10 Trends and Random Walks

*6.11 ARCH Models and Serial Correlation

Summary 265

Exercises 267

7 Multicollinearity 269

7.1 Introduction

7.2 Some Illustrative Examples

7.3 Some Measures of Multicollinearity

7.4 Problems with Measuring Multicollinearity

7.5 Solutions to the Multicollinearity Problem:

Ridge Regression

7.6 Principal Component Regression

7.7 Dropping Variables

7.8 Miscellaneous Other Solutions

Summary

Exercises

Appendix to Chapter 7

8 Dummy Variables and Truncated Variables

8.1 Introduction

8.2 Dummy Variables for Changes in the Intercept Term

8.3 Dummy Variables for Changes in Slope Coefficients

8.4 Dummy Variables for Cross-Equation Constraints

8.5 Dummy Variables for Testing Stability of

Regression Coefficients

8.6 Dummy Variables Under Heteroskedasticity and Autocorrelation

8.7 Dummy Dependent Variables

8.8 The Linear Probability Model and the Linear Discriminant Function

8.9 The Probit and Logit Models

8.10 Illustrative Example

8.11 Truncated Variables: The Tobit Model

Summary

Exercises

9 Simultaneous Equations Models

9.1 Introduction

9.2 Endogenous and Exogenous Variables

9.3 The Identification Problem: Identification Through Reduced Form

9.4 Necessary and Sufficient Conditions for Identification

9.5 Methods of Estimation: The Instrumental Variable Method

9.6 Methods of Estimation: The Two-Stage Least Squares Method

9.7 The Question of Normalization

*9.8 The Limited-Information Maximum Likelihood Method

*9.9 On the Use of OLS in the Estimation of Simultaneous-Equations Models

*9.10 Exogeneity and Causality

Summary 395

Exercises 397

Appendix to Chapter 9 400

10 Models of Expectations 405

10.1 Introduction

10.2 Naive Models of Expectations

10.4 Estimation with the Adaptive Expectations Model

10.5 Two Illustrative Examples

10.6 Expectational Variables and Adjustment Lags

10.8 Alternative Distributed Lag Models: Polynomial Lags

10.9 Rational Lags

10.10 Rational Expectations

10.11 Tests for Rationality

10.12 Estimation of a Demand and Supply Model Under Rational Expectations

10.13 The Serial Correlation Problem in Rational Expectations Models

Summary

Exercises

11 Errors in Variables

11.1 Introduction

11.2 The Classical Solution for a Single-Equation Model with One Explanatory Variable

11.3 The Single-Equation Model with Two Explanatory Variables

11.4 Reverse Regression

11.5 Instrumental Variable Methods

11.6 Proxy Variables

11.7 Some Other Problems

Summary

Exercises

12 Diagnostic Checking, Model Selection, and Specification Testing

12.1 Introduction

12.2 Diagnostic Tests Based on Least Squares Residuals

12.3 Problems with Least Squares Residuals

12.4 Some Other Types of Residuals 481

12.5 DFFITS and Bounded Influence Estimation

12.6 Model Selection

12.7 Selection of Regressors

12.8 Implied F-Ratios for the Various Criteria

12.9 Cross-Validation

12.10 Hausman’s Specification Error Test

12.11 The Plosser-Schwert-White Differencing Test

12.12 Tests for Nonnested Hypotheses

Summary

Exercises

Appendix to Chapter 12

13 Introduction to Time-Series Analysis 525

13.1 Introduction

13.2 Two Methods of Time-Series Analysis: Frequency

Domain and Time Domain

13.3 Stationary and Nonstationary Time Series

13.4 Some Useful Models for Time Series

13.5 Estimation of AR, MA, and ARMA Models

13.6 The Box-Jenkins Approach

13.7 R2 Measures in Time-Series Models

Summary

Exercises

Data Sets

14 Vector Autoregressions, Unit Roots, and Cointegration

14.1 Introduction

14.2 Vector Autoregressions

14.3 Problems with VAR Models in Practice

14.4 Unit Roots

14.5 Unit Root Tests

14.6 Cointegration

14.7 The Cointegrating Regression

14.8 Vector Autoregressions and Cointegration

14.9 Cointegration and Error Correction Models

14.10 Tests for Cointegration

14.11 Cointegration and Testing of the REH and MEH

14.12 A Summary Assessment of Cointegration

Summary 602

Exercises 603

APPENDIX: Tables 609

Author Index 623

Subject Index 627