Abstract: This paper investigates a quasi-likelihood ratio (LR) test for the thresholds in buffered autoregressive processes. Under the null hypothesis of no threshold, the LR test statistic converges to a function of a centered Gaussian process. Under local alternatives, this LR test has nontrivial asymptotic power. A bootstrap method is proposed to obtain the critical value for the LR test. Simulation studies and an example are given to assess the performance of the test. The proof here is not standard and can be used in other non-linear time series models.

Key words and phrases: AR(p) model, bootstrap method, buffered AR(p) model, likelihood ratio test, marked empirical process, threshold AR(p) model.