Parameter Estimation for Infinite Variance Fractional ARIMA

Consider the fractional ARIMA time series with innovations that
have infinite variance. This is a finite parameter model which exhibits
both long-range dependence (long memory) and high variability. We prove
the consistency of an estimator of the unknown parameters which is based
on the periodogram and derive its asymptotic distribution. This shows
that the results of Mikosch, Gadrich, Kliippelberg and Adler for ARMA
time series remain valid for fractional ARIMA with long-range dependence.
We also extend the limit theorem for sample autocovariances of
infinite variance moving averages developed in Davis and Resnick to
moving averages whose coefficients are not absolutely summable.
1. Introduction and main results. This paper is concerned with the
estimation of the parameters of the fractional ARIMA time series {Xn)
defined by the equations
where the innovations 2, have infinite variance and where d is a positive
fractional number. B and A denote the backward and differencing operator,
respectively. Because of the presence of the fractional d, the time series (1.1)
has not only infinite variance, but also exhibits long-range dependence (long
memory). For more details, see [24], [I91 and [20].

Our goal is to estimate both d and the coefficients of the polynomials
and 0,by using a variant of Whittle's method. For a stationary Gaussian
time series with spectral density g(h, PI, -T < h < T, Whittle's method,
which provides an estimate of P, requires replacing the inverse covariance
matrix that appears in the Gaussian likelihood by a Toeplitz (covariance)
matrix with spectral density l/g and then maximizing the quadratic form.
Hannan [I51 applied Whittle's method to finite variance ARMA time series,
that is, to (1.1) with d = 0. He proved that the estimator is consistent and
asymptotically normal. An ARMA time series, however, has short range
dependence because the correlations decrease exponentially fast. Fox and
Gaqqu [12] extended this result to Gaussian time series with long-range
dependence such as fractional Gaussian noise or fractional ARIMA by appealing
to a central limit theorem for weighted quadratic forms whose weights are chosen in such a way as to compensate for the long-range dependence.
Fox and Taqqu's result, which was later generalized to the full maximum
likelihood by Dahlhaus [8], is the basis of one of the most commonly used
techniques for estimating the intensity of long-range dependence in Gaussian
time series (see [21). Giraitis and Surgailis [I31 extended Fox and Taqqu's
result to finite variance innovations without Gaussian assumptions and
Heyde and Gay [I71 to random fields.