Limit Theory for Moderate Deviations from a Unit Root

An asymptotic theory is given for autoregressive time series with a root of the form ρn = 1 + c/nα, which represents moderate deviations from unity when α in (0,1). The limit theory is obtained using a combination of a functional law to a diffusion on D[0,∞) and a central limit law to a scalar normal variate. For c > 0, the results provide a n(1+α)/2 rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the squareroot of n and n convergence rates for the stationary (α = 0) and conventional (α = 1) local to unity cases. For c > 0, the serial correlation coefficient is shown to have a nαρnn 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 α = 0, where the convergence rate of the serial correlation coefficient is (1 + c)n and no invariance principle applies.