Wavelet-based parameter estimation for polynomial contaminated fractionally differenced processes

We consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component. Trend, i.e., large scale variations in the series that are best modeled outside of a stochastic framework, is often confounded with low-frequency stochastic fluctuations. This problem is particularly evident in models such as fractionally differenced (FD) processes, which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components. We use the discrete wavelet transform (DWT) to estimate parameters for stationary and nonstationary FD processes in a model of polynomial trend plus FD noise. Using Daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations. Parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the DWT decorrelates FD processes approximately. We consider this decorrelation in detail, examining the between- and within-scale wavelet correlations separately. Better between-scale decorrelation can be achieved by increasing the length of the wavelet filter, whereas the within-scale correlations can be handled via explicit modeling by a low-order autoregressive process. We demonstrate our methodology by applying it to a popular climate dataset.