An Introduction to Econometric Theory

A GUIDE TO ECONOMICS, STATISTICS AND FINANCE THAT EXPLORES THE MATHEMATICAL FOUNDATIONS UNDERLING ECONOMETRIC METHODS An Introduction to Econometric Theory offers a text to help in the mastery of the mathematics that underlie econometric methods and includes a detailed study of matrix algebra and distribution theory. Designed to be an accessible resource, the text explains in clear language why things are being done, and how previous material informs a current argument. The style is deliberately informal with numbered theorems and lemmas avoided. However, very few technical results are quoted without some form of explanation, demonstration or proof. The author—a noted expert in the field—covers a wealth of topics including: simple regression, basic matrix algebra, the general linear model, distribution theory, the normal distribution, properties of least squares, unbiasedness and efficiency, eigenvalues, statistical inference in regression, t and F tests, the partitioned regression, specification analysis, random regressor theory, introduction to asymptotics and maximum likelihood. Each of the chapters is supplied with a collection of exercises, some of which are straightforward and others more challenging. This important text: Presents a guide for teaching econometric methods to undergraduate and graduate students of economics, statistics or finance Offers proven classroom-tested material Contains sets of exercises that accompany each chapter Includes a companion website that hosts additional materials, a solution manual and lecture slides Written for undergraduates and graduate students of economics, statistics or finance, An Introduction to Econometric Theory is an essential beginner's guide to the underpinnings of econometrics.

Table of Contents:
List of Figures ix Preface xi About the CompanionWebsite xv Part I Fitting 1 1 Elementary Data Analysis 3 1.1 Variables and Observations 3 1.2 Summary Statistics 4 1.3 Correlation 6 1.4 Regression 10 1.5 Computing the Regression Line 12 1.6 Multiple Regression 16 1.7 Exercises 18 2 Matrix Representation 21 2.1 Systems of Equations 21 2.2 Matrix Algebra Basics 23 2.3 Rules of Matrix Algebra 26 2.4 Partitioned Matrices 27 2.5 Exercises 28 3 Solving the Matrix Equation 31 3.1 Matrix Inversion 31 3.2 Determinant and Adjoint 34 3.3 Transposes and Products 37 3.4 Cramer’s Rule 38 3.5 Partitioning and Inversion 39 3.6 A Note on Computation 41 3.7 Exercises 43 4 The Least Squares Solution 47 4.1 Linear Dependence and Rank 47 4.2 The General Linear Regression 50 4.3 Definite Matrices 52 4.4 Matrix Calculus 56 4.5 Goodness of Fit 57 4.6 Exercises 59 Part II Modelling 63 5 Probability Distributions 65 5.1 A Random Experiment 65 5.2 Properties of the Normal Distribution 68 5.3 Expected Values 72 5.4 Discrete Random Variables 75 5.5 Exercises 80 6 More on Distributions 83 6.1 Random Vectors 83 6.2 The Multivariate Normal Distribution 84 6.3 Other Continuous Distributions 87 6.4 Moments 90 6.5 Conditional Distributions 92 6.6 Exercises 94 7 The Classical RegressionModel 97 7.1 The Classical Assumptions 97 7.2 The Model 99 7.3 Properties of Least Squares 101 7.4 The Projection Matrices 103 7.5 The Trace 104 7.6 Exercises 106 8 The Gauss-Markov Theorem 109 8.1 A Simple Example 109 8.2 Efficiency in the General Model 111 8.3 Failure of the Assumptions 113 8.4 Generalized Least Squares 114 8.5 Weighted Least Squares 116 8.6 Exercises 118 Part III Testing 121 9 Eigenvalues and Eigenvectors 123 9.1 The Characteristic Equation 123 9.2 Complex Roots 124 9.3 Eigenvectors 126 9.4 Diagonalization 128 9.5 Other Properties 130 9.6 An Interesting Result 131 9.7 Exercises 133 10 The Gaussian RegressionModel 135 10.1 Testing Hypotheses 135 10.2 Idempotent Quadratic Forms 137 10.3 Confidence Regions 140 10.4 t Statistics 141 10.5 Tests of Linear Restrictions 144 10.6 Constrained Least Squares 146 10.7 Exercises 149 11 Partitioning and Specification 153 11.1 The Partitioned Regression 153 11.2 Frisch-Waugh-Lovell Theorem 155 11.3 Misspecification Analysis 156 11.4 Specification Testing 159 11.5 Stability Analysis 160 11.6 Prediction Tests 162 11.7 Exercises 163 Part IV Extensions 167 12 Random Regressors 169 12.1 Conditional Probability 169 12.2 Conditional Expectations 170 12.3 StatisticalModels Contrasted 174 12.4 The Statistical Assumptions 176 12.5 Properties of OLS 178 12.6 The Gaussian Model 182 12.7 Exercises 183 13 Introduction to Asymptotics 187 13.1 The Lawof Large Numbers 187 13.2 Consistent Estimation 192 13.3 The Central LimitTheorem 195 13.4 Asymptotic Normality 198 13.5 Multiple Regression 201 13.6 Exercises 203 14 Asymptotic Estimation Theory 207 14.1 Large Sample Efficiency 207 14.2 Instrumental Variables 208 14.3 Maximum Likelihood 210 14.4 Gaussian ML 213 14.5 Properties of ML Estimators 214 14.6 Likelihood Inference 216 14.7 Exercises 218 Part V Appendices 221 A The Binomial Coefficients 223 B The Exponential Function 225 C Essential Calculus 227 D The Generalized Inverse 229 Recommended Reading 233 Index 235

About the Author :
JAMES DAVIDSON is Professor of Econometrics at the University of Exeter. He has also held teaching posts at the University of Warwick, the London School of Economics, the University of Wales Aberystwyth and Cardiff University, as well as visiting positions at the University of California, Berkeley, the University of California, San Diego, and Central European University, Budapest.