Robust estimation for threshold autoregressive moving-average models
Threshold autoregressive moving-average (TARMA) models have gained attention in econometrics, statistics and other fields, due to their ability to describe nonlinear dynamical features, such as chaos, asymmetric cycles and irreversibility, using few parameters. However, neither theory nor suitable estimation methods are currently available when the data presents extremely heavy tails or anomalous observations deviating from the assumed model structure. In this paper, we introduce robust parameter estimation for TARMA models by minimizing a M-estimating objective function with bounded derivative, which ensures certain robustness properties of the resulting estimator. Under mild conditions, we show that the robust estimator for the threshold parameter is super-consistent, while the autoregressive and moving-average parameters estimators are strongly consistent and asymptotically normal. The Monte Carlo study shows that the robust estimator is able to reduce both bias and variance with respect to the ordinary least squares estimator both in finite samples and asymptotically. We apply our methodology to model a panel of time series of commodity prices and the results show the superiority of the robust TARMA specification in terms of predictive capability.