Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers

Donald Andrews ()

No 790, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: 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: "The 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.

Keywords: Uniform law of large Numbers; consistency; nonlinear econometric models (search for similar items in EconPapers)
Pages: 26 pages
Date: 1986-04
Note: CFP 693.
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (11)

Published in Econometrica (November 1987), 55(6): 1465-1471

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