This paper establishes the asymptotic normality of series estimators for nonparametric regression models. Gallant's Fourier flexible form estimators, trigonometric series estimators, and polynomial series estimators are prime examples of the estimators covered by the results. The results apply to a wide variety of estimates in the regression model under consideration, including derivatives and integrals of the regression function. The errors in the model may be homoskedastic or heteroskedastic. The paper also considers series estimators for additive interactive regression, semiparametric regression, and semiparametric index regression models, and shows them to be consistent and asymptotically normal.

This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. Results in the literature provide a condition on the growth rate of the lag truncation parameter as T → ∞ that is sufficient for consistency. No results are available, however, regarding the choice of lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of weighting scheme. In consequence, available estimators are not entirely operational and the relative merits of the estimators are unknown. This paper addresses these problems. The asymptotic truncated mean squared errors of estimators in a given class are determined and compared. Asymptotically optimal kernel/weighting scheme and bandwidth/lag truncation parameters are obtained using an asymptotic truncated mean squared error criterion. Using these results, data-dependent automatic bandwidth/lag truncation parameters are introduced. The finite sample properties of the estimators are analyzed via Monte Carlo simulation.

This paper is concerned with the use of power properties of tests in econometric applications. Inverse power functions are defined. These functions are designed to yield summary measures of power that facilitate the interpretation of test results in practice. Simple approximations are introduced for the inverse power functions of Wald, likelihood ratio, Lagrange multiplier, and Hausman tests. These approximations readily convey the general qualitative features of the power of a test. Examples are provided to illustrate their usefulness in interpreting test results. A COMMON PROBLEM faced in applied econometrics is that of interpreting the results of a hypothesis test when the test fails to reject the null hypothesis. Most practitioners realize that just because a test fails to reject a hypothesis one cannot claim to accept it. Nevertheless, it is common for this to be ignored, since the practitioner is often in a position where he would like the outcome of the test to provide useful inferences whether or not the test rejects. The purpose of this paper is to introduce inverse power (IP) summary measures that enable the practitioner to avoid such errors and make valid inferences when a test fails to reject the null hypothesis. These summary measures are widely applicable, easy to use (especially in the common case of a test concerning a single restriction), and simple to compute. When a test rejects the null hypothesis, the implication is that the data are inconsistent with each parameter point in the null in the sense that the probabil- ity of type I error for each point is small, viz., a or less. Correspondingly, when a test fails to reject the null hypothesis an analogous statement is needed regarding the error probabilities for points in the alternative hypothesis. It is not the case that all points in the alternative are inconsistent with the data in the sense that their probability of type II error is small (a or less). It is possible, however, to determine the region S in the alternative parameter space that is inconsistent with the data in this sense. The IP function introduced below evaluated at

This paper extends the Pearson chi-square testing method to nondynam ic parametric econometric models, in particular, to models with covar iates. The paper establishes the asymptotic distribution of the test statistic when the test statistic is based on data-dependent random cells of a general form and on an arbitrary asymptotically normal estimator. These results a re attained by extending recent probabilistic results for the weak convergence of empirical processes indexed by sets. The chi-square test that is introduced can be used to test goodness-of-fit of a parametric model, as well as to test particular aspects of the parametric model that are of interest.

A basic tool of modern econometrics is a uniform law of large numbers (LLN). It is a primary ingredient used in proving consistency and asymptotic normality of parametric and nonparametric estimators in nonlinear econometric models. Thus, in a well-known review article, Burguete, Gallant, and Sousa [8, p. 162] introduce a uniform LLN with the statement: following theorem is the result upon which the asymptotic theory of nonlinear econometrics rests. So pervasive is the use of uniform LLNs, that numerous authors appeal to an unspecified generic uniform LLN. Others appeal to some specific result. The purpose of this paper is to provide a generic uniform LLN that is sufficiently general to incorporate most applications of uniform LLNs in the nonlinear econometrics literature. In summary, the paper presents a result that can be used to turn state of the art pointwise LLNs into uniform LLNs over compact sets, with the addition of a single smoothness condition -- either a Lipschitz condition or a derivative condition. The latter is particularly easy to verify, and is implied by common assumptions used to prove asymptotic normality of estimators. Thus, the additional condition is not particularly restrictive. In contrast to other uniform LLNs that appear in the literature, the one given here allows the full range of heterogeneity of summands (i.e., non-identical distributions), and temporal dependence, that is available with pointwise LLNs.

THIS PAPER INVESTIGATES a property of estimators called stability. The stability exponent of an estimator is a measure of the magnitude of the effect of any single observation in the sample on the realized value of the estimator. A number of reasons related to robustness suggest that often it is desirable for an estimator to be relatively insensitive to any particular observation in the sample, i.e., to have high stability. In addition, it is useful for diagnostic purposes to have knowledge of the stability exponents of different estimators, in order to know which estimators are likely to rely more heavily on some single observation. The paper is organized as follows: Section 1 introduces the basic idea contained in the paper, motivates it, and summarizes the results in an informal manner. Section 2 presents definitions, assumptions, and the general results. For purposes of illustration, the linear regression model with the least squares estimator is used as a running example throughout this section. Section 3 discusses numerous additional applications of the general results. An Appendix contains proofs of

Donald W. K. Andrews, A Note on the Unbiasedness of Feasible GLS, Quasi-Maximum Likelihood, Robust, Adaptive, and Spectral Estimators of the Linear Model, Econometrica, Vol. 54, No. 3 (May, 1986), pp. 687-698