ESTIMATION OF THE IMPULSE-RESPONSE COEFFICIENTS OF A LINEAR PROCESS WITH INFINITE VARIANCE

Let (xt) (t = 0, ±1, ±2, …) be a linear process, xt = ε(lunate)t + b(l) ε(lunate)t - 1 + · · ·, where (ε(lunate)t) is a sequence of independent identically distributed random variables with the common distribution in the domain of attraction of a symmetric stable law of index δ ∈ (0, 2), and the b(j) are real coefficients. Under the additional assumption that xt also has an autoregressive representation, xt + a(1) xt - 1 + · · · = ε(lunate)t, the question of estimating the b(j) from a realization of T consecutive observations of (xt) is considered. Two different "autoregressive" estimators of the b(j) are examined, and by requiring that the order, k, of the fitted autoregression approaches ∞ simultaneously but sufficiently slowly with T, shown to be consistent, the order of consistency being T-1/φ, φ > δ. The finite sample behaviour is examined by a simulation study. © 1993 Academic Press Inc.