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-scalespace. The wavelet's ability to localize a time series in time-scale space directly leads to the computationalefficiency of the wavelet representation of a N £ N matrix operator by allowing the N largest elements of thewavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. Thisproperty 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 estimatesimultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrixoperator, we are able to approximate an ARFIMA model's likelihood function with the series' wavelet coefficientsand their variances. Maximization of this approximate likelihood function over the short and long-memoryparameter 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 andusing only the wavelet coefficient's variances, the approximate wavelet MLE provides a fast alternative to thefrequency-domain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLEto be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas thefrequency-domain MLE dramatically deteriorates as the moving average parameter approaches the boundariesof invertibility.