The convergence of multivariate ‘unit root’ distributions to their asymptotic limits: The case of money-income causality

Abstract

We examine the quality of recently developed asymptotic approximations to the sampling distributions of various statistics in levels regressions when the regressors have unit roots. The calculations were performed using a bivariate probability model typical of some considered in applied macroeconomic research: the parameters of the model were obtained by estimating a VAR using postwar U.S. money and industrial production growth rates, resulting in pseudo-data that are I(1) with drifts. With 100 observations the asymptotic approximations are often found to be adequate; with 400 observations they are generally good. In addition, when the statistics have nonstandard distributions, both the asymptotic and exact distributions differ substantially from the usual normal or χ 2 distributions that would apply were the regressors stationary.