Limit theory for moderate deviations from a unit root

An asymptotic theory is given for autoregressive time series with a root of the form p(n) = 1 + c/k(n), which represents moderate deviations from unity when (k(n))(n is an element of N) is a deterministic sequence increasing to infinity at a rate slower than n, so that k(n) = o(n) as n -> infinity. For c < 0, the results provide a root nk(n) rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the root n and n convergence rates for the stationary (k(n) = 1) and conventional local to unity (k(n) = n) cases. For c > 0, the serial correlation coefficient is shown to have a k(n)rho(n)(n) convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when p(n) > 1. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for k(n) = 1, where the convergence rate of the serial correlation coefficient is (1 + c)(n) and no invariance principle applies. (c) 2005 Elsevier B.V. All rights reserved.