Estimation of the Impulse-Response Coefficients of a Linear Process with Infinite Variance

Let {xt} (t = 0, ±1, ±2, ...) be a linear process, xt = ϵt + b(l) ϵt − 1 + · · ·, where {ϵt} 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 + · · · = ϵt, 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.