Testing for parameter constancy in general causal time series models

We consider a process $X=(X_t{)}_{t\in \mathbb{Z}}$ belonging to a large class of causal models including AR($\mathrm{\infty }$), ARCH($\mathrm{\infty }$), TARCH($\mathrm{\infty }$),… models. We assume that the model depends on a parameter ${\theta }_0\in {\mathbb{R}}^d$ and consider the problem of testing for change in the parameter. Two statistics ${\widehat{Q}}_n^{(1)}$ and ${\widehat{Q}}_n^{(2)}$ are constructed using quasi-likelihood estimator (QLME) 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 local alternative that there is one change, we show that the test statistic ${\widehat{Q}}_n=\text{max}({\widehat{Q}}_n^{(1)},{\widehat{Q}}_n^{(2)})$ diverges to infinity. Some simulation results for AR(1), ARCH(1) and GARCH(1,1) models are reported to show the applicability and the performance of our procedure with comparisons to some other approaches.