tailieunhanh - Handbook of Econometrics Vols1-5 _ Chapter 15

Chapter 15 APPROXIMATING THE DISTRIBUTIONS OF ECONOMETRIC ESTIMATORS AND TEST STATISTICS THOMAS J. ROTHENBERG The comparison of different hypotheses, . of competing models, is the basis of model specification. It may be performed along two main lines. | Chapter 15 APPROXIMATING THE DISTRIBUTIONS OF ECONOMETRIC ESTIMATORS AND TEST STATISTICS THOMAS J. ROTHENBERG University of California Berkeley Contents 1. Introduction 882 2. Alternative approximation methods 884 . Preliminaries 884 . Curve-fitting 886 . Transformations 887 . Asymptotic expansions 889 . Ad hoc methods 891 3. Edgeworth approximations 892 . Sums of independent random variables 893 . A general expansion 896 . Non-normal expansions 900 4. Second-order comparisons of estimators 902 . General approach 903 . Optimality criteria 904 . Second-order efficient estimators 905 . Deficiency 907 . Generalizations 908 5. Second-order comparisons of tests 909 . General approach 909 . Some results when q 1 912 . Results for the multiparameter case 915 . Confidence intervals 917 6. Some examples 918 . Simultaneous equations 918 . Autoregressive models 925 . Generalized least squares 929 . Departures from normality 930 7. Conclusions 931 References 932 1 am indebted to Christopher Cavanagh for his help in preparing this survey and to Donald Andrews and James Reeds for comments on an earlier draft. Research support from the National Science Foundation is gratefully acknowledged. Handbook of Econometrics Volume II Edited by 7. Griliches and . Intriligator Elsevier Science Publishers BV 1984 882 T. J. Rothenberg 1. Introduction Exact finite-sample probability distributions of estimators and test statistics are available in convenient form only for simple functions of the data and when the likelihood function is completely specified. Often in econometrics these conditions are not satisfied and inference is based on approximations to the sampling distributions. Typically large sample methods of approximation based on the central limit theorem are employed. For example if 0n is an estimator of a parameter 0 based on a sample of size n it is sometimes possible to find a function o 0 such that the distribution