Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete‐time linear process {xt}, a one‐sided moving average of independent identically distributed random variables {εt}, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving‐average coefficients b(j) such that εt is invertible in terms of the present and possibly infinite past values of {xt}. By treating {xt} as if it is second‐order stationary, a normalized spectral density function f(μ) is defined in terms of the b(j) and, having observed x1, ..., xT, an autoregression of order k is fitted by the well‐known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k←∞ as T←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f(μ), the convergence rate being T−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering ‘non‐invertible’ models.