Limit theory for moderate deviations from a unit root

Abstract

An asymptotic theory is given for autoregressive time series with a root of the form ρn=1+c/kn, which represents moderate deviations from unity when (kn)n∈N is a deterministic sequence increasing to infinity at a rate slower than n, so that kn=o(n) as n→∞. For c<0, the results provide a nkn rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the n and n convergence rates for the stationary (kn=1) and conventional local to unity (kn=n) cases. For c>0, the serial correlation coefficient is shown to have a knρnn convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when ρn>1. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for kn=1, where the convergence rate of the serial correlation coefficient is (1+c)n and no invariance principle applies.