An approximate wavelet MLE of short- and long-memory parameters

By design a wavelet's strength rests in its ability to localize a process simultaneously in time-scale space. The wavelet's ability to localize a time series in time-scale space directly leads to the computational efficiency of the wavelet representation of a N x N matrix operator by allowing the N largest elements of the wavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. This property allows many dense matrices to have sparse representation when transformed by wavelets. In this paper we generalize the long-memory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrix operator, we are able to approximate an ARFIMA model's likelihood function with the series' wavelet coefficients and their variances. Maximization of this approximate likelihood function over the short and long-memory parameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model. By simultaneously maximizing the likelihood function over both the short and long-memory parameters and using only the wavelet coefficient's variances, the approximate wavelet MLE provides a fast alternative to the frequency-domain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLE to be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas the frequency-domain MLE dramatically deteriorates as the moving average parameter approaches the boundaries of invertibility.