Limit Theory for Moderate Deviations from Unity - PHILLIPS, Peter C. B. (Institutional Knowledge at Singapore Management University, 2007)
An asymptotic theory is given for autoregressive time series with a root of the form [rho]n=1+c/kn, which represents moderate deviations from unity when is a deterministic sequence increasing to infinity at a rate slower than n, so that kn=o(n) as n-->[infinity]. For c<0, the results provide a rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the 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 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 kn=1, where the convergence rate of the serial correlation coefficient is (1+c)n and no invariance principle applies.