Wavelet-Based Parameter Estimation for Trend Contaminated Fractionally Differenced Processes

A common problem in the analysis of time series is how to deal with a possible trend component, which is usually thought of as large scale (or low frequency) variations or patterns in the series that might be best modeled separately from the rest of the series. Trend is often confounded with low frequency stochastic fluctuations, particularly in the case of models such as fractionally differenced (FD) processes, that can account for long memory dependence (slowly decaying autocorrelation) and can be extended to encompass nonstationary processes exhibiting quite significant low frequency components. In this paper we assume a model of polynomial trend plus FD noise and apply the discrete wavelet transform (DWT) to separate a time series into pieces that can be used to estimate both the FD parameters and the trend. The estimation of the FD parameters is based on an approximate maximum likelihood approach that is made possible by the fact that the DWT decorrelates FD processes approximately. We demonstrate our methodology by applying it to a popular climate dataset.