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

Let {xt} (t = 0, ±1, ±2, ...) be a linear process, xt = [epsilon]t + b(l) [epsilon]t - 1 + · · ·, where {[epsilon]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 [delta] [set membership, variant] (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 + · · · = [epsilon]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 [infinity] simultaneously but sufficiently slowly with T, shown to be consistent, the order of consistency being T-1/[phi], [phi] > [delta]. The finite sample behaviour is examined by a simulation study.