Regression Asymptotics Using Martingale Convergence Methods

Abstract

Weak convergence of partial sums and multilinear forms in independent random variables and linear processes to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and conceptually simple method for obtaining such forms of convergence. The method relies on the fact that the econometric quantities of interest involve discrete time martingales or semimartingales and shows how in the limit these quantities become continuous martingales and semimartingales. The limit theory itself uses very general convergence results for semimartingales that were obtained in work by Jacod and Shiryaev (2003). The theory that is developed here is applicable in a wide range of econometric models and many examples are given.

One notable outcome of the new approach is that it provides a unified treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity, as both these cases are subsumed within the martingale convergence approach and different rates of convergence are accommodated in a natural way. The approach is also useful in developing asymptotics for certain nonlinear functions of integrated processes, which are now receiving attention in econometric applications, and some new results in this area are presented. The paper is partly of pedagogical interest and the conceptual simplicity of the methods is appealing. Since this is the first time the methods have been used in econometrics, the exposition is presented in some detail with illustrations of new derivations of some well-known existing results, as well as some new asymptotic results and the unification of the limit theory for autoregression.

Keywords: Semimartingale, martingale, convergence, stochastic integrals, bilinear forms, multilinear forms, U-statistics, unit root, stationarity, Brownian motion, invariance principle, unification

JEL Classification: C13, C14, C32