Econometric Methods

vi ECONOMETRICMETHODS 1.3 To derive cov(a, b) 1.4 Gauss-Markov theorem 1.5 To derive var(e0) Problem 2 Further Aspects of Two-Variable Relationships , 2.1 Time as a Regressor 2.1.1 Constant Growth Curves 2.1.2 Numerical Example 22 Transformations of Variables 2.2.1 Log-Log Transformations 2.2.2 Semilog Transformations 2.2.3 Reciprocal Transformations 2.3 An Empirical Example of a Nonl~nearRelation: U.S. Inflation and Unemployment 2.4 Lagged Dependent Variable as Regressor 2.4.1 An Introduction to Asymptotics 2.4.2 Convergence in Probability 2.4.3 Convergence in Distribution 2.4.4 The Autoregressive Equation 2.5 Stationary and Nonstationary Series 2.5.1 Unit Root 2.5.2 Numerical Illustration 2.6 Maximum Likelihood Estimation of the Autoregressive Equation 2.6.1 Maximum Likelihood Estimators 2.6.2 Properties of Maximum Likelihood Estimators Appendix 2.1 Change of variables in density functions 2.2 Maximum likelihood estimators for the AR(1) model Problems 3 The k-Variable Linear Equation 3.1 Matrix Formulation of the k-Variable Model 3.1.1 The Algebra of Least Squares 3.1.2 Decomposition of the Sum of Squares 3.1.3 Equation in Deviation Form 3.2 Partial Correlation Coefficients 3.2.1 Sequential Buildup of the Explained Sum of Squares 3.2.2 Partial Correlation Coefficients and Multiple Regression Coefficients 3.2.3 General Treatment of Partial Correlation and Multiple Regression Coefficients 3.3 The Geometry of Least Squares 3.4 Inference in the k-Variable Equation 3.4.1 Assumptions 3.4.2 Mean and Variance of b 36 36 37 37 41 42 43 43 44 45 46 47 49 52 53 54 55 56 57 59 59 61 61 63 64 65 66 69 70 70 72 73 76 78 81 82 83 86 86 87 Contents vii 3.4.3 Estimation of u2 3.4.4 Gauss-Markov Theorem 3.4.5 Testing Linear Hypotheses about p 3.4.6 Restricted and Unrestricted Regressions 3.4.7 Fitting the Restricted Regression 3.5 Prediction Appendix 3.1 To prove r12.3 = (112 - r13r23)/ 3.2 Solving for a single regression coefficient in a multiple regression 3.3 To show that minimizing a'u subject to X'a = c gives J'm Jm u = x(x'x)-lc 3.4 Derivation of the restricted estimator b, Problems 4 Some Tests of the k-Variable Linear Equation for Specification Error 4.1 Specification Error 4.1.1 Possible Problems with u 4.1.2 Possible Problems with X 4.1.3 Possible Problems with p 4.2 Model Evaluation and Diagnostic Tests 4.3 Tests of Parameter Constancy 4.3.1 The Chow Forecast Test 4.3.2 The Hansen Test 4.3.3 Tests Based on Recursive Estimation 4.3.4 One-Step Ahead Prediction Errors 4.3.5 CUSUM and CUSUMSQ Tests 4.3.6 A More General Test of Specification Error: The Ramsey RESET Test 4.4 A Numerical Illustration 45 Tests of Structural Change 4.5.1 Test of One Structural Change 4.5.2 Tests of Slope Coefficients 4.5.3 Tests of Intercepts 4.5.4 Summary 4.5.5 A Numerical Example 4.5.6 Extensions @ Dummy Variables 4.6.1 Introduction 4.6.2 Seasonal Dummies 4.6.3 Qualitative Variables 4.6.4 Two or More Sets of Dummy Variables 4.6.5 A Numerical Example Appendix 4.1 TO show var(d) = u2[In2+ X2(X;Xl)-'X;] Problems 89 89 90 95 96 99 100 101 103 103 104 109 109 110 110 111 112 113 113 116 117 118 119 121 121 126 127 128 128 129 130 132 133 133 134 135 137 137 138 139 viii ECONOMETRIC METHODS I 5 Maximum Likelihood (ML), Generalized Least Squares (GLS), and Instrumental Variable (IV) Estimators 5.1 Maximum Likelihood Estimators 5.1.1 Properties of Maximum Likelihood Estimators 5.2 ML Estimation of the Linear Model 5.3 Likelihood Ratio, Wald, and Lagrange Multiplier Tests 5.3.1 Likelihood Ratio (LR) Tests 5.3.2 The Wald (W) Test 5.3.3 Lagrange Multiplier (LM) Test 5.4 ML Estimation of the Linear Model with Nonspherical Disturbances 5.4.1 Generalized Least Squares 5.5 Instrumental