An Approximate Wavelet MLE of Short- and Long-Memory Parameters

By design, a wavelet's strength rests in its ability simultaneously to localize a process 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 an N x N matrix operator by allowing the N largest elements of the wavelet-represented operator adequately to represent the matrix operator. This property allows many dense matrices to have a sparse representation when transformed by wavelets. In this paper, we generalize the long-memory parameter estimator of McCoy and Walden (1996) simultaneously to estimate short- and long-memory parameters. Using the sparse wavelet representation of a matrix operator, we are able adequately 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's coefficient 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.