Wavelet estimation of a local long memory parameter

There are a number of estimators of a long-memory process' long-memory parameter when the parameter is assumed to hold constant over the entire data set, but currently no estimator exists for a time-varying long-memory parameter. In this paper we construct an estimator of the time-varying long-memory parameter that is based on the time-scale properties of the wavelet transform. Because wavelets are localised in time they are able to capture the time-varying statistical properties of a locally stationary longmemory process, and since wavelets are also localised in scale they identify the self-similarity scaling behaviour found in the statistical properties of the process. Together the time and scale properties of the wavelet produce an approximate log-linear relationship between the time-varying variance of the wavelet coefficients and the wavelet scale proportional to the local long-memory parameter. To obtain a least-squares estimate of the local long-memory parameter, we replace the time-varying variance of the wavelet coefficient with the sample variance of the wavelet coefficients computed over the so-called 'cone of influence.' That is, we use only those wavelet coefficients whose time index falls within the support of the wavelet basis function in order to compute the local sample wavelet variance. To test the empirical properties of our estimator we perform a number of Monte Carlo experiments. We find the wavelet-based estimator of the local long-memory parameter to have empirical properties similar to other waveletbased estimators of the long-memory parameter for globally stationary long-memory processes. For processes where the longmemory parameter suddenly changes, the wavelet-based estimator again performs well, only exhibiting an elevated positive empirical bias at points in time right before the long-memory parameter increases, and a negative bias immediately after the change. The wavelet-based estimator of the local long-memory parameter is demonstrated using vertical ocean shear data.