COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 790 "Consistency in Nonlinear Econometric Models: Donald W.K. Andrews April 1986 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. |
