Testing for Homogeneity of Variance in Time Series : Long Memory , Wavelets and the Nile

We consider the problem of testing for homogeneity of variance in a time series that has long memory structure. We demonstrate that a test whose null hypothesis is designed to be white noise can in fact be applied, on a scale by scale basis, to the discrete wavelet transform of long memory processes. In particular, we show that evaluating a normalized cumulative sum of squares test statistic using critical levels appropriate for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractional diierence process. The point at which the test statistic, using the maximal overlap discrete wavelet transform, achieves its maximum value can be used to estimate the time of the unknown variance change. We apply our proposed test statistic on a time series of Nile River yearly minimum water levels covering the years 622 to 1284 AD. The test connrms an inhomogeneity of variance at short scales and identiies the change point around 720 AD, which coincides closely with the construction of a new device in 715 AD for measuring Nile River water levels.