On robust testing for conditional heteroscedasticity in time series models

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We consider robust and powerful tests for autoregressive conditional heteroscedasticity (ARCH) in the residuals from a nonlinear dynamic regression model. We advocate a new robustified autocorrelation function of the squared residuals, which relies essentially on the squared of winsorized robust residuals. This new tool allows us to introduce two new portmanteau-type test statistics which are robust to outliers. Each robust test statistic can be interpreted using a spectral approach. The first portmanteau test is based on a comparison between the estimated spectral density of the squared winsorized residuals and the spectral density under the null hypothesis of no ARCH effect via a quadratic norm. This test statistic consists in a weighted sum of squared robust autocorrelations of the squared residuals and it provides a robust version of the test statistic proposed by Hong and Shehadeh (J. Business Econom. Statist. 17 (1999) 91). Furthermore, it gives a generalized version of the robust Lagrange multiplier test of van Dijk et al. (J. Appl. Econom. 14 (1999) 539). However, since testing for ARCH represents a one-sided problem, we propose a robust testing procedure which takes into account the one-sided nature of the alternative hypothesis. The resulting test statistic is a weighted sum of robust autocorrelations of the squared residuals and it gives a robust version of Hong's (J. Time Ser. Anal. 18 (1997) 253) one-sided test for ARCH effects. One-sided tests for ARCH at individual lags are considered, as complementary tools. Both proposed portmanteau tests have a convenient asymptotic normal distribution under the null hypothesis of no ARCH effect, when the unknown parameters are estimated by a robust and consistent method. The new robust tests rely on the choice of a kernel. Typically, more weight can be given to lower orders of lags and less weight to higher orders of lags, giving more powerful procedures for many time series. Asymptotic arguments suggest that the first test statistic should be more powerful than the second asymptotically, but to exploit the one-sided nature of the alternative hypothesis can be particularly powerful with small samples. That issue is addressed in our Monte Carlo experiments. Using the highly robust S-estimators of regression to estimate robustly the regression parameters, the proposed robust test statistics are studied and compared empirically to many nonrobust and robust tests for ARCH effects, under various scenarios for the occurrence of outliers.