A score-based threshold effect test in time series models

A score-based test statistic is developed to compare a linear ARMA model with its threshold extension. In particular, the focus is on testing the threshold effect in continuous threshold models with no jump at the threshold. Notably, while developed for continuous threshold models, the proposed test remains effective for discontinuous cases. The proposed test does not require fitting the model under the alternative hypothesis, making it computationally more efficient than the quasi-likelihood ratio test. The asymptotic distributions of the score-based test statistic are derived under both the null hypothesis and local alternatives. Simulations indicate that the proposed test has better size than the quasi-likelihood ratio test and demonstrates stronger power compared to the Lagrange Multiplier test. The asymptotic theory of the least square estimation for the continuous threshold ARMA model is further established. An application to the quarterly U.S. civilian unemployment rates data is given.