A Simple Asymptotic Analysis of Residual-Based Statistics

Abstract

What's the asymptotic null distribution of a rank-based serial autocorrelation test applied to residuals of an estimated GARCH model? What's the limiting distribution of estimated ACD parameters applied to the residuals of some first-stage modelling procedure? This paper addresses the often occuring situation in econometrics of applying standard statistics to residuals instead of innovations. The paper provides a simple and unified way of calculating the necessary adjustment in the limiting distribution, be it of tests or estimators. On the technical side, we also provide a novel approach to this problem using Le Cam's theory of convergence of experiments (in this paper restricted to Gaussian shift experiments). The resulting formula is simple and the regularity conditions required fairly minimal. Numerous examples show the strength and wide applicability of our approach.

Keywords: Asymptotic size, discretized estimators, goodness-of-fit tests, local asymptotic normality, rank statistics, structural break tests, temporal dependence tests, two-stage inference

JEL Classification: C32, C51, C52