The Cusum Test for Parameter Change in Regression with ARCH Errors

In this paper, we concentrate ourselves on Inclán and Tiao (1994)'s cusum test in regression models with ARCH errors. The ARCH and GARCH models have long been popular in financial time series analysis. For a general review, see Gouriéroux (1997).Inclán and Tiao (1994)'s cusum test was originally designed for testing for variance changes and allocating their locations in iid samples. Later, it was demonstrated that the same idea can be extended to a large class of time series models (cf. Lee et all, 2003(a)). Also, the variance change test has been studied in unstable AR models (cf. Lee et al. (2003(b)). In fact, Kim, Cho and Lee (2000) considered to apply the cusum test to GARCH(1,1) models taking account of the fact that the variance is a functional of GARCH parameters, and their change can be detected by examining the existence of the variance change. Although this reasoning was correct, it turned out that the cusum test suffers from severe size distortions and low powers. Hence, there was a demand to improve their cusum test. Here, in order to circumvent such drawbacks, we propose to use the cusum test based on the residuals, given as the squares of observations divided by estimated conditional variances. We intend to use residuals since the residual based test conventionally discard correlation effects and enhance the performance of the test. In fact, a significant improvement was observed in our simulation study. Despite the previous work of Lee et al. (2003(b)) also considers a residual cusum test in time series models, the model of main concern was the autoregressive model with several unit roots. In fact, the mathematical analysis of the cusum test heavily relies on the probabilistic structure of the underlying time series model, and the arguments used for establishing the weak convergence result in unstable models are somewhat different from those in ARCH models. Therefore it is worth to investigate the asymptotic behavior of the residual cusum test in ARCH models. Although the present paper was originally motivated to improve Kim, Cho and Lee (2000)'s test in the GARCH(1,1) model, we consider the cusum test in a more general class of models including regression models with infinite order ARCH errors.