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Wavelet-Based Estimation for Seasonal Long-Memory Processes

We introduce the multiscale analysis of seasonal persistent processes, that is, time series models with a singularity in their spectral density function at one or more frequencies in [0,1/2]. The discrete wavelet packet transform (DWPT) and a nondecimated version of it known as the maximal overlap DWPT (MODWPT) are introduced as alternative methods to Fourier-based techniques for analyzing time series that exhibit seasonal long memory. The approximate log-linear relationship between the wavelet packet variance and frequency is used to produce a least squares estimator of the fractional difference parameter. Approximate maximum likelihood estimation is performed by replacing the variance/covariance matrix with a diagonalized matrix based on the DWPT. Simulations are performed to compare the wavelet-based techniques with the spectral estimate-based techniques for both least squares and maximum likelihood procedures. An application of this methodology to atmospheric and economic time series is used for demonstration purposes.