Inference for Heavy-Tailed and Multiple-Threshold Double Autoregressive Models

This article develops a systematic inference procedure for heavy-tailed and multiple-threshold double autoregressive (MTDAR) models. We first study its quasi-maximum exponential likelihood estimator (QMELE). It is shown that the estimated thresholds are n-consistent, each of which converges weakly to the smallest minimizer of a two-sided compound Poisson process. The remaining parameters are √n -consistent and asymptotically normal. Based on this theory, a score-based test is developed to identify the number of thresholds in the model. Furthermore, we construct a mixed sign-based portmanteau test for model checking. Simulation study is carried out to access the performance of our procedure and a real example is given.