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Linear Regression
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Linear Regression
A Mathematical Introduction



July 2018 | 272 pages | SAGE Publications, Inc

Damodar N. Gujarati’s Linear Regression: A Mathematical Introduction presents linear regression theory in a rigorous, but approachable manner that is accessible to students in all social sciences. This concise title goes step-by-step through the intricacies, and theory and practice of regression analysis. The technical discussion is provided in a clear style that doesn’t overwhelm the reader with abstract mathematics. End-of-chapter exercises test mastery of the content and advanced discussion of some of the topics is offered in the appendices.

Data sets accompanying this book are available for download:
Chapter 4 Data: Wages for Workers
Chapter 6 Data: Earnings and Educational Attainment
Definitions of Variables: Chapter 4 and Chapter 6 Data
 
List of Figures
 
Series Editor’s Introduction
 
Preface
 
About the Author
 
About the Author
 
Chapter 1: The Linear Regression Model (LRM)
1.1 Introduction  
1.2 Meaning of “Linear” in Linear Regression  
1.3 Estimation of the LRM: An Algebraic Approach  
1.4 Goodness of Fit of a Regression Model: The Coefficient of Determination (R2)  
1.5 R2 for Regression Through the Origin  
1.6 An Example: The Determination of the Hourly Wages in the United States  
1.7 Summary  
Exercises  
Appendix 1A: Derivation of the Normal Equations  
 
Chapter 2: The Classical Linear Regression Model (CLRM)
2.1 Assumptions of the CLRM  
2.2 The Sampling or Probability Distributions of the OLS Estimators  
2.3 Properties of OLS Estimators: The Gauss–Markov Theorem  
2.4 Estimating Linear Functions of the OLS Parameters  
2.5 Large-Sample Properties of OLS Estimators  
2.6 Summary  
Exercises  
 
Chapter 3: The Classical Normal Linear Regression Model: The Method of Maximum Likelihood (ML)
3.1 Introduction  
3.2 The Mechanics of ML  
3.3 The Likelihood Function of the k-Variable Regression Model  
3.4 Properties of the ML Method  
3.5 Summary  
Exercises  
Appendix 3A: Asymptotic Efficiency of the ML Estimators of the LRM  
 
Chapter 4: Linear Regression Model: Distribution Theory and Hypothesis Testing
4.1 Introduction  
4.2 Types of Hypotheses  
4.3 Procedure for Hypothesis Testing  
4.4 The Determination of Hourly Wages in the United States  
4.5 Testing Hypotheses About an Individual Regression Coefficient  
4.6 Testing the Hypothesis That All the Regressors Collectively Have No Influence on the Regressand  
4.7 Testing the Incremental Contribution of a Regressor  
4.8 Confidence Interval for the Error Variance s 2  
4.9 Large-Sample Tests of Hypotheses  
4.10 Summary  
Exercises  
Appendix 4A: Constrained Least Squares: OLS Estimation Under Linear Restrictions  
 
Chapter 5: Generalized Least Squares (GLS): Extensions of the Classical Linear Regression Model
5.1 Introduction  
5.2 Estimation of B With a Nonscalar Covariance Matrix  
5.3 Estimated Generalized Least Squares  
5.4 Heteroscedasticity and Weighted Least Squares  
5.5 White’s Heteroscedasticity-Consistent Standard Errors  
5.6 Autocorrelation  
5.7 Summary  
Exercises  
Appendix 5A: ML Estimation of GLS  
 
Chapter 6: Extensions of the Classical Linear Regression Model: The Case of Stochastic or Endogenous Regressors
6.1 Introduction  
6.2 X and u Are Distributed Independently  
6.3 X and u Are Contemporaneously Uncorrelated  
6.4 X and u Are Neither Independently Distributed Nor Contemporaneously Uncorrelated  
6.5 The Case of k Regressors  
6.6 What Is the Solution? The Method of Instrumental Variables (IVs)  
6.7 Hypothesis Testing Under IV Estimation  
6.8 Practical Problems in the Application of the IV Method  
6.9 Regression Involving More Than One Endogenous Regressor  
6.10 An Illustrative Example: Earnings and Educational Attainment of Youth in the United States  
6.11 Regression Involving More Than One Endogenous Regressor  
6.12 Summary  
Appendix 6A: Properties of OLS When Random X and u Are Independently Distributed  
Appendix 6B: Properties of OLS Estimators When Random X and u Are Contemporaneously Uncorrelated  
 
Chapter 7: Selected Topics in Linear Regression
7.1 Introduction  
7.2 The Nature of Multicollinearity  
7.3 Model Specification Errors  
7.4 Qualitative or Dummy Regressors  
7.5 Nonnormal Error Term  
7.6 Summary  
Exercises  
Appendix 7A: Ridge Regression: A Solution to Perfect Collinearity  
Appendix 7B: Specification Errors  
 
Appendix A: Basics of Matrix Algebra
A.1 Definitions  
A.2 Types of Matrices  
A.3 Matrix Operations  
A.4 Matrix Transposition  
A.5 Matrix Inversion  
A.6 Determinants  
A.7 Rank of a Matrix  
A.8 Finding the Inverse of a Square Matrix  
A.9 Trace of a Square Matrix  
A.10 Quadratic Forms and Definite Matrices  
A.11 Eigenvalues and Eigenvectors  
A.12 Vector and Matrix Differentiation  
 
Appendix B: Essentials of Large-Sample Theory
B.1 Some Inequalities  
B.2 Types of Convergence  
B.3 The Order of Magnitude of a Sequence  
B.4 The Order of Magnitude of a Stochastic Sequence  
 
Appendix C: Small- and Large-Sample Properties of Estimators
C.1 Small-Sample Properties of Estimators  
C.2 Large-Sample Properties of Estimators  
 
Appendix D: Some Important Probability Distributions
D.1 The Normal Distribution and the Z Test  
D.2 The Gamma Distribution  
D.3 The Chi-Square (? 2) Distribution and the ? 2 Test  
D.4 Student’s t Distribution  
D.5 Fisher’s F Distribution  
D.6 Relationships Among Probability Distributions  
D.7 Uniform Distributions  
D.8 Some Special Features of the Normal Distribution  
 
Index

“This is a nifty volume that complements the series of ‘Little Green Books’ nicely. It offers a blend of the abstract and the concrete, presenting both ‘the math’ and the ‘how-to’ that will be of use to both experienced and novice users.”

 
Wendy L. Martinek
Binghamton University

“Damodar Gujariti brings his world-class expertise as an econometrician to bear on explicating the fundamentals of the math behind regression analysis, the most widely-used social science research tool around. His presentation shows clarity, understanding and range, always with good applied illustrations.”

Michael S. Lewis-Beck
University of Iowa

“This text is a useful monograph on linear models theory. The writing is clear and derivations insightful.”

Jay Verkuilen
CUNY Graduate Center
Key features

KEY FEATURES: 

  • This book offers with unique brevity a discussion of the basics of regression analysis. 
  • Discussion of the method of Maximum Likelihood (ML), a topic which does not receive much attention in other texts, is covered.
  • Succinct discussion of distribution theory offers discussion of estimation and hypothesis—topics that are the foundation of statistical inference.
  • Two large data based examples of regression analysis, allowing with practical problems that are illustrated with the data, show how regression theory is applied in practice. 
 

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