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

Consider a discrete‐time linear process { x t }, a one‐sided moving average of independent identically distributed random variables {ε t }, 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 ε t is invertible in terms of the present and possibly infinite past values of { x t }. By treating { x t } 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 x 1 , ..., x T , 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.