Empirical likelihood for special self-exciting threshold autoregressive models with heavy-tailed errors

Abstract

In this paper, we study the empirical likelihood method for special self-exciting threshold autoregressive models. We assume that the parameters of models depend on some positive integer n, the threshold effect diminishes to zero as n increases, and the errors have heavy-tailed distributions. After replacing the indicator function in the model with a smooth function, we obtain a self-weighted and smoothed least absolute deviation estimator for the threshold parameter and the asymptotic normality of the estimator is proved. Then, the confidence intervals for the threshold parameter can be constructed by the normal approximation method. In order to improve the coverage of confidence intervals, we further propose a profile empirical likelihood ratio, and prove that this statistic has the asymptotically standard chi-squared distribution. Therefore, the confidence interval of the threshold parameter can also be constructed by the empirical likelihood method. Simulations and empirical results demonstrate that the confidence interval constructed by empirical likelihood method is superior to that constructed by normal approximation method.