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+