Discrete time parametric models with long memory and infinite variance

Abstract

We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator β n for β is consistent ( β n converges to the true value β 0 in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences ( n logn ) 1/ga(β n − β 0) .