An Asymptotic Test for Constancy of the Variance under Short-Range Dependence

Abstract

We present a novel approach to test for heteroscedasticity of a non-stationary time series that is based on Gini's mean difference of logarithmic local sample variances. In order to analyse the large sample behaviour of our test statistic, we establish new limit theorems for U-statistics of dependent triangular arrays. We derive the asymptotic distribution of the test statistic under the null hypothesis of a constant variance and show that the test is consistent against a large class of alternatives, including multiple structural breaks in the variance. Our test is applicable even in the case of non-stationary processes, assuming a locally stationary mean function. The performance of the test and its comparatively low computation time are illustrated in an extensive simulation study. As an application, we analyse data from civil engineering, monitoring crack widths in concrete bridge surfaces.

Detecting changes in the trend function of heteroscedastic time series

“…Apart from these works, to some extent related problems have also been considered by Vogt [32] and by Juhl and Xiao [18]. Note that our procedure could be combined with that of Schmidt et al [30] to yield a test for stationarity of the first two moments. Moreover, it can be employed to evaluate the fit of some estimated mean function, e.g., of some parametric model as considered in Zhang and Wu [39], by subtracting the fitted trend function from the data and applying our test to the residuals.…”

“…Furthermore, our test can be used complementary to the results in Schmidt et al [30] who develop a test for the stationarity of the variance based on Gini's mean difference of the logarithmic local sample variances. The authors too work under the model (1), i.e.…”

“…Since their assumptions are quite similar to those obtained here (see, their Theorem 2.4 and Corollary 2.8), both procedures can be combined to test for stationarity of the first two moments. Moreover, note that in case the test in [30] finds the variance to be stationary, we can apply the simplified version of our test from Theorem 2.15.…”

“…, (j * + 1) n }, we exclude such values of t that are too close to the boundaries of B j * ∪ B j * +1 . A similar procedure has already been used in Wornowizki, Fried and Meintanis [33] and in Schmidt et al [30] to obtain an estimator for structural breaks in the variance. The estimated mean function μ is obtained by taking the sample mean on each segment between two subsequent change points.…”

“…Our test statistic is inspired by the one recently proposed by Schmidt et al [30]. The authors test for stationarity of the variance in an absolutely regular time series via a test statistic based on Gini's mean difference of logarithmic local sample variances.…”