Abstract
The aim of this paper is to address the problem of variance break detection in time series in wavelet domain. The maximal overlapped discrete wavelet transform (MODWT) decomposes the series variance across scales into components known as the wavelet variances. We introduce all scale wavelet coefficients based test statistic that allows detecting a break in the homogeneity of the variance of a series through changes in the mean of wavelet variances. The statistic makes use of the traditional CUSUM (cumulative sum) based test designed to test for a break in the mean and constructed using cumulative sums of the square of wavelet coefficients. Under moments and mixing conditions, the test statistic satisfies the functional central limit theorem (FCLT) for a broad class of time series models. The overall performance of our test statistic is compared to the traditional Inclan [8] test statistic. The effectiveness of our statistic is supported by good performances reported in simulations and is as reliable as the traditional statistic. Our method provides a nonparametric test procedure that can be applied to a large class of linear and non linear models. We illustrate the practical use of our test procedure with the quarterly percentage changes in the Americans personal savings data set over the period 1970-2016. Both statistics detect a break in the variance in the second quarter of 2001.
Highlights
The test problem that addresses the issue of change in the variance homogeneity of time series has received considerable attention in the literature
We limit ourselves to test statistics based on the cumulative sums of squares of wavelet coefficients and take a different approach to investigate the null hypotheses of no break in the variance
In order to test for a change in the mean a class of commonly used test statistics is based on the cumulative sums of squares (CUSUMs) process
Summary
The test problem that addresses the issue of change in the variance homogeneity of time series has received considerable attention in the literature. We are motivated by the fact that the sample variance of a time series can be decomposed into components known as the wavelet variances each of which is associated with a particular scales as in [17]. Mathematics and Statistics 8(4): 430-436, 2020 demean and remove trends from a time series These attractive characteristics give us a motivation in this work to explore the cumulative sums of the squared wavelet coefficients which is a time dependent series designed to track the build up over time of the sample variance across scales up to a fixed level.
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