On a Threshold Double Autoregressive Model

This article first proposes a score-based test for a double autoregressive model against a threshold double autoregressive (AR) model. It is an asymptotically distribution-free test and is easy to implement in practice. The article further studies the quasi-maximum likelihood estimation of a threshold double autoregressive model. It is shown that the estimated threshold is n-consistent and converges weakly to a functional of a two-sided compound Poisson process and the remaining parameters are asymptotically normal. Our results include the asymptotic theory of the estimator for threshold AR models with autoregressive conditional heteroscedastic (ARCH) errors and threshold ARCH models as special cases, each of which is also new in literature. Two portmanteau-type statistics are also derived for checking the adequacy of fitted model when either the error is nonnormal or the threshold is unknown. Simulation studies are conducted to assess the performance of the score-based test and the estimator in finite samples. The results are illustrated with an application to the weekly closing prices of Hang Seng Index. This article also includes the weak convergence of a score-marked empirical process on the space (Formula presented.) under an α-mixing assumption, which is independent of interest.