An alternative maximum likelihood estimator of long-memory processes using compactly supported wavelets

In this paper we apply compactly supported wavelets to the ARFIMA(p,d,q) long-memory process to develop an alternative maximum likelihood estimator of the differencing parameter, d, that is invariant to unknown means, model specification, and contamination. We show that this class of time series have wavelet transforms whose covariance matrix is sparse when the wavelet is compactly supported. It is shown that the sparse covariance matrix can be approximated to a high level of precision by a matrix equal to the covariance matrix except with the off-diagonal elements set equal to zero. This diagonal matrix is shown to reduce the order of calculating the likelihood function to an order smaller than those associated with the exact MLE method. We test the robustness of the wavelet MLE of the fractional differencing parameter to a variety of compactly supported wavelets, series length, and contamination levels by generating ARFIMA(p, d, q) processes for different values of p, d, and q, and calculating the wavelet MLE using only the main diagonal elements of its covariance matrix. In our simulations we find the wavelet MLE to be superior to the approximate frequency MLE when estimating contaminated ARFIMA(0,d,0), and uncontaminated ARFIMA(1,d,0) and ARFIMA(0, d, 1) processes except when the MA parameter is close to one. We also find the wavelet MLE to be robust to model specification and as such is an attractive alternative semiparameter estimator to the Geweke, Porter-Hudak estimator. (C) 2000 Elsevier Science B.V. All rights reserved.