This paper investigates the (conditional) quasi-likelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has nontrivial asymptotic power. The results are based on a new weak convergence of a linear marked empirical process, which is independently of interest. This paper also gives an invertible expansion of the threshold MA models.
"Testing for a linear MA model against threshold MA models." Ann. Statist. 33 (6) 2529 - 2552, December 2005. https://doi.org/10.1214/009053605000000598