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Linear Regression
A Mathematical Introduction

- Damodar N. Gujarati - West Point, New York, USA

**Volume:**177

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.

Chapter 4 Data: Wages for Workers

Chapter 6 Data: Earnings and Educational Attainment

Definitions of Variables: Chapter 4 and Chapter 6 Data

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

C.1 Small-Sample Properties of Estimators |

C.2 Large-Sample Properties of Estimators |

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 |

“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.”

**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.”

**University of Iowa**

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

**CUNY Graduate Center**