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This paper presents several stochastic equicontinuity results that are useful for establishing the asymptotic properties of estimators and tests in parametric, semiparametric, and nonparametric econometric models. In particular, they can be applied straightforwardly in the estimation and testing results of Andrews (1989b). The paper takes various stochastic equicontinuity results from the probability literature, which rely on entropy conditions of one sort or another, and provides primitive conditions under which the entropy conditions hold. This yields stochastic equicontinuity results that are readily applicable in a variety of contexts. This paper also presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1989b). The results allow for near epoch dependent non-identically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., regression based on residuals), and nonparametric regression on index functions. Some of the results make use of the stochastic equicontinuity results of the paper.
Andrews, Donald W.K., "Asymptotics for Semiparametric Econometric Models: II. Stochastic Equicontinuity and Nonparametric Kernel Estimation" (1989). Cowles Foundation Discussion Papers. 1153.