Weighted least absolute deviations estimation for ARMA models with infinite variance

For autoregressive moving average (ARMA) models with infinite variance innovations, quasi-likelihood-based estimators (such as Whittle estimators) suffer from complex asymptotic distributions depending on unknown tail indices. This makes statistical inference for such models difficult. In contrast, the least absolute deviations estimators (LADE) are more appealing in dealing with heavy tailed processes. In this paper, we propose a weighted least absolute deviations estimator (WLADE) for ARMA models. We show that the proposed WLADE is asymptotically normal, is unbiased, and has the standard root-n convergence rate even when the variance of innovations is infinity. This paves the way for statistical inference based on asymptotic normality for heavy-tailed ARMA processes. For relatively small samples numerical results illustrate that the WLADE with appropriate weight is more accurate than the Whittle estimator, the quasi-maximum-likelihood estimator (QMLE), and the Gauss-Newton estimator when the innovation variance is infinite and that the efficiency loss due to the use of weights in estimation is not substantial.