An asymptotic theory is given for autoregressive time series with a root of the form ρ n = 1 + c/ n α , 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 n α ρ n n convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when ρ n > 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.
Phillips, Peter C.B. and Magdalinos, Tassos, "Limit Theory for Moderate Deviations from a Unit Root" (2004). Cowles Foundation Discussion Papers. 1750.