Local Asymptotic Specification Error Analysis

Abstract

An approximation to the inconsistency introduced by imposing an incorrect restriction on a parametric model is given. The approximation can be applied to estimators generated by optimizing any objective function satisfying certain regularity conditions. Examples given include analysis of misspecification in discrete choice and time-series models estimated by maximum likelihood, and in a nonlinear regression model. SPECIFICATION ERROR ANALYSIS in the linear regression model has been studied by Theil [1], who gives formulas for, e.g., the effect of leaving out relevant variables on the expected values of the estimators of the coefficients of the included variables. In this paper we suggest analogous formulas for estimators obtained by optimizing an objective function subject to restrictions. We have in mind maximizing (1/n) x loglikelihood and will usually use this terminology. We consider the effect on the limit of the restricted estimator of a small violation of the restrictions. In the linear regression case our formula coincides with that given by Theil. In order to keep our results widely applicable and to avoid a mass of unnecessary detail we make assumptions on the asymptotic behavior of the loglikelihood function itself, rather than on the data-generating process per se. Many alternative sets of assumptions on the data densities can lead to the behavior we require of the loglikelihood functions. These will not be pursued here. The interested reader is referred to, e.g., White [12] for the case of independent observations and Kohn [8] for the time-series case. 1. GENERAL FORMULAS The general approach we take is based on a linear approximation to the likelihood function at the maximum likelihood estimator. It is in this sense that our analysis is local. For some models the local and global specification error results coincide; a well known case is the effect of omitted regressors in the linear regression model. Essentially the only cases involve linearity, although often there is agreement regarding the signs of the inconsistency. We show below that the local and global results even fail to coincide in the case of misspecified AR processes. Generally however, the global results are unknown.2 Taylor expansions are typically used together with assumptions on the data generating process to obtain the asymptotic distribution of the maximum likelihood estimator (Cramer [3]). In this paper we will not concern ourselves with asymptotic distributions of Vn -normed MLE's since these have been worked