Wavelet-Based Estimation Procedures For Seasonal Long-Memory Models

The appearance of long-range dependence has been observed in a wide variety of real-word time series. So called long-memory models, which exhibit a slowly decaying autocovariance sequence and a pole at frequency zero in their spectral density function, have been used to characterize long-range dependence parsimoniously. A generalization of such models allows the pole in the spectral density function to be placed anywhere in the frequency interval causing a slowly decaying oscillating autocovariance sequence. This is known as the so called seasonal long-memory model. While an exact method for maximizing the likelihood exists and a semiparametric Whittle approximation has been proposed, we investigate two estimating procedures using the discrete wavelet packet transform: an approximate maximum likelihood method and an ordinary least squares method. We utilize the known decorrelating properties of the wavelet transform to allow us to assume a simplified variance-covariance structure for the seasonal long-memory model. We describe our computational procedures and explore the versatility gained by using the wavelet transform. As an example, we fit a seasonal long-memory model to an observed time series. The proposed wavelet-based techniques offer useful and computationally efficient alternatives to previous time and frequency domain methods.