Robust Estimation for Threshold Autoregressive Moving-Average Models

Threshold autoregressive moving-average (TARMA) models extend the popular TAR model and are among the few parametric time series specifications to include a moving average in a nonlinear setting. The state dependent reactions to shocks is particularly appealing in Economics and Finance. However, no theory is currently available when the data present heavy tails or anomalous observations. Here we provide the first theoretical framework for robust M-estimation for TARMA models and study its practical relevance. Under mild conditions, we show that the robust estimator for the threshold parameter is super-consistent, while the estimators for autoregressive and moving-average parameters are strongly consistent and asymptotically normal. The Monte Carlo study shows that the M-estimator is superior, in terms of both bias and variance, to the least squares estimator, which can be heavily affected by outliers. The findings suggest that robust M-estimation should be generally preferred to the least squares method. We apply our methodology to a set of commodity price time series; the robust TARMA fit presents smaller standard errors and superior forecasting accuracy. The results support the hypothesis of a two-regime non-linearity characterized by slow expansions and fast contractions.