Inference of a generalized long memory process in the wavelet domain

Consider the discrete wavelet transform (DWT) of a time series $X=\{X{t},t\in\BBZ\}$ with weakly stationary $K$ th differences. Such time series are encountered in many situations, including unit root or long memory processes. If the wavelet has at least $K$ vanishing moments, the DWT is jointly stationary across scales provided that small scales coefficients are reshaped in appropriate blocks to cope with the downsampling embedded in the DWT. Our goal is to compute the covariance matrix or the joint spectral density of the DWT, given the autocovariance of the $K$th differences. We assume that the DWT corresponds to a multiresolution analysis (MRA), which allows us to compute wavelet coefficients iteratively through a succession of finite impulse response (FIR) filters and downsampling. This iterative scheme, however, is not suitable for our purpose in the case where the process $X$ itself is not stationary. Hence, we first derive an iterative algorithm with the same DWT output but with input the $K$th differences of the time series. An iterative low complexity scheme is then deduced to compute the exact covariance matrix and spectral density of the DWT. This new algorithm is an opportunity to investigate how using exact DWT covariance computations improves previously proposed statistical methods that rely on approximated computations. Numerical experiments are used for comparisons. Two cases are examined: 1) a local semi-parametric likelihood estimation of long memory processes in the wavelet domain and 2) the computation of a test statistic for detecting change points in the wavelet domain for long memory processes. A real data set analysis is also presented. Namely, economic structural changes are investigated by looking for change points in the daily S&P 500 absolute log returns.