Limit theory for moderate deviations from a unit root

An asymptotic theory is given for autoregressive time series with a root of the form ρ n = 1 + c / k n , which represents moderate deviations from unity when ( k n ) n ∈ N is a deterministic sequence increasing to infinity at a rate slower than n, so that k n = o ( n ) as n → ∞ . For c < 0 , the results provide a nk n rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the 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 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 k n = 1 , where the convergence rate of the serial correlation coefficient is ( 1 + c ) n and no invariance principle applies.