Self-weighted LAD-based inference for heavy-tailed threshold autoregressive models

The least squares estimator of the threshold autoregressive (TAR) model may not be consistent when its tail is less than or equal to 2. Neither theory nor methodology can be applied to model fitting in this case. This paper is to develop a systematic procedure of statistical inference for the heavy-tailed TAR model. We first investigate the self-weighted least absolute deviation estimation for the model. It is shown that the estimated slope parameters are n-consistent and asymptotically normal, and the estimated thresholds are n-consistent, each of which converges weakly to the smallest minimizer of a compound Poisson process. Based on this theory, the Wald test statistic is considered for testing the linear restriction of slope parameters and a procedure is given for inference of threshold parameters. We finally construct a sign-based portmanteau test for model checking. Simulations are carried out to assess the performance of our procedure and a real example is given.