COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1471 Limit Theory for Moderate Deviations from a Unit Root Peter C. B. Phillips and Tassos Magdalinos July 2004 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 squareroot 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 rhon > 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. Keywords: Central limit theory; Diffusion, Explosive autoregression, Local to unity, Moderate deviations, Unit root distribution AMS 1991 subject classification: 62M10 JEL classification: C22 |
