2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130987This 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.Published at http://dx.doi.org/10.1214/009053605000000598 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)Statistics Theory62F05, 62M10 (Primary) 60G10 (Secondary)text