内容

目次

Preface viii I Fitting 1 1 Elementary Data Analysis 2 1.1 Variables and Observations 2 1.2 Summary Statistics 4 1.3 Correlation 6 1.4 Regression 9 1.5 Computing the Regression Line 12 1.6 Multiple Regression 16 1.7 Exercises 19 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 29 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 44 4 The Least Squares Solution 47 4.1 Linear Dependence and Rank 47 4.2 The General Linear Regression 51 4.3 De.nite Matrices 53 4.4 Matrix Calculus 56 4.5 Goodness of Fit 58 4.6 Exercises 61 II Modelling 64 5 Probability Distributions 65 5.1 A Random Experiment 65 5.2 Properties of the Normal Distribution 69 5.3 Expected Values 72 5.4 Discrete Random Variables 75 5.5 Exercises 82 6 More on Distributions 84 6.1 Random Vectors 84 6.2 The Multivariate Normal Distribution 85 6.3 Other Continuous Distributions 88 6.4 Moments 91 6.5 Conditional Distributions 93 6.6 Exercises 96 7 The Classical Regression Model 98 7.1 The Classical Assumptions 98 7.2 The Model 100 7.3 Properties of Least Squares 102 7.4 The Projection Matrices 105 7.5 The Trace 106 7.6 Exercises 109 8 The Gauss-Markov Theorem 111 8.1 A Simple Example 111 8.2 E￠ ciency in the General Model 113 8.3 Failure of the Assumptions 115 8.4 Generalized Least Squares 116 8.5 Weighted Least Squares 118 8.6 Exercises 121 III Testing 123 9 Eigenvalues and Eigenvectors 124 9.1 The Characteristic Equation 124 9.2 Complex Roots 126 9.3 Eigenvectors 127 9.4 Diagonalization 129 9.5 Other Properties 131 9.6 An Interesting Result 133 9.7 Exercises 135 10 The Gaussian Regression Model 137 10.1 Testing Hypotheses 137 10.2 Idempotent Quadratic Forms 139 10.3 Con.dence Regions 142 10.4 t Statistics 144 10.5 Tests of Linear Restrictions 147 10.6 Constrained Least Squares 149 10.7 Exercises 152 11 Partitioning and Speci.cation 155 11.1 The Partitioned Regression 155 11.2 Frisch-Waugh-Lovell Theorem 157 11.3 Misspeci.cation Analysis 158 11.4 Speci.cation Testing 162 11.5 Stability Analysis 163 11.6 Prediction Tests 165 11.7 Exercises 167 IV Extensions 170 12 Random Regressors 171 12.1 Conditional Probability 171 12.2 Conditional Expectations 173 12.3 Statistical Models Contrasted 176 12.4 The Statistical Assumptions 179 12.5 Properties of OLS 181 12.6 The Gaussian Model 184 12.7 Exercises 187 13 Introduction to Asymptotics 189 13.1 The Law of Large Numbers 190 13.2 Consistent Estimation 194 13.3 The Central Limit Theorem 197 13.4 Asymptotic Normality 201 13.5 Multiple Regression 204 13.6 Exercises 207 14 Asymptotic Estimation Theory 209 14.1 Large Sample E￠ ciency 209 14.2 Instrumental Variables 210 14.3 Maximum Likelihood 213 14.4 Gaussian ML 216 14.5 Properties of ML Estimators 217 14.6 Likelihood Inference 219 14.7 Exercises 221 V Appendices 224 A The Binomial Coe￠ cients 225 B The Exponential Function 227 C Essential Calculus 229 D The Generalized Inverse 231 Recommended Reading 234 Acknowledgements 236 Index 236 Figures 1.1 Long and short UK interest rates 6 1.2 Scatter plot of the interest rate series. 7 1.3 The regression line 10 1.4 The regression line and the data 10 1.5 The regression residual 11 1.6 Plot of y = 2 + 2x1 + x2 17 5.1 Archery target scatter 65 5.2 Frequency contours 66 5.3 Bivariate normal density 67 5.4 Normal p.d.f. 71 5.5 Binomial probabilities and the normal density function. 78 5.6 Prussian cavalry data and predictions 80 6.1 The standard Cauchy density function 90 10.1 Con.dence regions for k = 2 143 13.1 P.d.f of the sum of three uniform r.v.s, with normal curve for comparison. 199