ON PERIODOGRAM BASED LEAST-SQUARES ESTIMATION OF THE LONG MEMORY PARAMETER OF FARMA PROCESSES

In [1] a periodogram based least squares regression method was proposed to get an estimation delta of the long memory parameter delta is-an-element-of (- 1/2, 1/2) of the FARMA process (3). It was suspected for the case delta > 0 and was proved for the case delta < 0 that there exists a sequence delta(n) of estimators under consideration for which delta(n) is asymptotically normal, namely delta(n) -> N(delta, a(n)) holds, where lim a(n) = 0. However we shall state that the large-sample behaviour of the FARMA's periodogram for very low frequencies is unusual, so we shall have to improve the considerations of [1]. In this paper a sufficient condition for the asymptotical ''goodness'' of the periodogram, will be given. For Gaussian FARMA processes by each delta is-an-element-of (- 1/2, 1/2) \ {0} but one at most, it happens to be necessary condition too. With the aid of this condition we shall prove (both for delta < 0 and for delta > 0) the asymptotical normality of delta(n) arising from some modification of the estimation procedure of [1].