LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises

This paper develops a systematic procedure of statistical inference for the ARMA model with unspecified and heavy-tailed heteroscedastic noises. We first investigate the least absolute deviation estimator (LADE) and the self-weighted LADE for the model. Both estimators are shown to be strongly consistent and asymptotically normal when the noise has a finite variance and infinite variance, respectively. The rates of convergence of the LADE and the self-weighted LADE are $n^{-1/2}$ which is faster than those of LSE for the AR model when the tail index of GARCH noises is in (0,4], and thus they are more efficient in this case. Since their asymptotic covariance matrices can not be estimated directly from the sample, we develop the random weighting approach for statistical inference under this nonstandard case. We further propose a novel sign-based portmanteau test for model adequacy. Simulation study is carried out to assess the performance of our procedure and one real illustrating example is given.