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

We consider the problem of testing for homogeneity of variance in a time series with 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 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 fractionally di erenced process. The point at which the test statistic, using a non-decimated version of the 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 622 to 1284 AD. The test con rms an inhomogeneity of variance at short scales and identi es the change point around 720 AD, which coincides closely with the construction of a new device around 715 AD for measuring these water levels. Some key words: Cumulative sum of squares; Discrete wavelet transform; Fractional di erence process; Variance change.