Ordinary Least Squares Estimate of the Fractional Di erencing Parameter Using Wavelets as Derived from Smoothing Kernels

This paper develops a consistent OLS estimate of a fractionally integrated processes' di erencing parameter, using continuous wavelet theory as constructed from smoothing kernels. We show that a log-log linear relationship exists between the variance of the wavelet coe cient and the level at which the fractionally integrated processes is smoothed. This linear relationship occurs because the self-simularity property of the fractionally integrated process and the self-similarity of the wavelet causes the smoothing level to continually appear in the wavelet transformation. Since the wavelet coe cient can be interpreted as the kth order details of the series at some level of smoothing, we also show that the above log-log relationship can be derived from the variance of the 1st order derivative of the time series smoothed by a kernel that is well localized in both time and frequency space. Lastly, we derive the asymptotic biasness and variance of the OLS estimate and test our consistent estimate with a number of Monte Carlo experiments.