Parameter estimation for a misspecified arma model with infinite variance innovations

Abstract

We consider a general linear model\(X_t = \sum\nolimits_{j = - \infty }^\infty {\psi _j Z_{t - j} } \), where the innovations Zt belong to the domain of attraction of an α-stable law for α<2, so that neither Zt nor Xt have a finite variance. We do not assume that (Xt) is a standardARMA process of the form φ(B)Xt=ϕ(B)Zt, but we fit anARMA process of a given order to the data X1,...,Xn by estimating the coefficients of φ and ϕ. Given that (Xt) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)1/α, which compares favorably with the L2 situation with rate\(\sqrt n \). In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)1/α. We also give an analytic expression for the limit distribution.