  Threshold autoregressive moving-average (TARMA) models are popular in time
series analysis due to their ability to parsimoniously describe several complex
dynamical features. However, neither theory nor estimation methods are
currently available when the data present heavy tails or anomalous
observations, which is often the case in applications. In this paper, we
provide the first theoretical framework for robust M-estimation for TARMA
models and also 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. Finally, we apply our methodology to a set of commodity price
time series; the robust TARMA fit presents smaller standard errors and leads to
superior forecasting accuracy compared to the least squares fit. The results
support the hypothesis of a two-regime, asymmetric nonlinearity around zero,
characterised by slow expansions and fast contractions.
