Testing the constancy of the variance for time series with a trend

The assumption of constant variance is fundamental in numerous statistical procedures for time series analysis. Nonlinear time series may exhibit time-varying local conditional variance, even when they are globally homoscedastic. Two novel tests are proposed to assess the constancy of variance in time series with a possible time-varying mean trend. Unlike previous approaches, the new tests rely on Walsh transformations of squared processes after recentering the time series data. It is shown that the corresponding Walsh coefficients have desirable properties, such as asymptotic independence. Both a max-type statistic and an order selection statistic are developed, along with their asymptotic null distributions. Furthermore, the consistency of the proposed statistics under a sequence of local alternatives is established. An extensive simulation study is conducted to examine the finite-sample performance of the procedures in comparison with existing methodologies. The empirical results show that the proposed methods are more powerful in many situations while maintaining reasonable Type I error rates, especially for nonlinear time series. The proposed methods are applied to test the global homoscedasticity of a financial time series, a well log time series with a non-constant mean structure, and a vibration time series.