Limit Theory for Moderate Deviations from a Unit Root

An asymptotic theory is given for autoregressive time series with a root of the form rho_{n} = 1+c/n^{alpha}, which represents moderate deviations from unity when alpha in (0,1). The limit theory is obtained using a combination of a functional law to a diffusion on D[0,infinity) and a central limit law to a scalar normal variate. For c < 0, the results provide a n^{(1+alpha)/2} rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the square root of n and n convergence rates for the stationary (alpha = 0) and conventional (alpha = 1) local to unity cases. For c > 0, the serial correlation coefficient is shown to have a n^{alpha}rho_{n}^{n} convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when rho_{n} > 1. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for alpha = 0, where the convergence rate of the serial correlation coefficient is (1 + c)^{n} and no invariance principle applies.