Testing for parameter constancy in general causal time-series models

We consider a process belonging to a large class of causal models including AR(8), ARCH(8), TARCH(8)... processes. We assume that the model depends on a parameter and consider the problem of testing for change in the parameter. Two statistics and are constructed using quasi-likelihood estimator of the parameter. Under the null hypothesis that there is no change, it is shown that each of these two statistics weakly converges to the supremum of the sum of the squares of independent Brownian bridges. Under the alternative of a change in the parameter, we show that the test statistic diverges to infinity. Some simulation results for AR(1), ARCH(1), GARCH(1,1) and TARCH(1) models are reported to show the applicability and the performance of our procedure with comparisons to some other approaches.