An empirical study on the parsimony and descriptive power of TARMA models

In linear time series analysis, the incorporation of the moving-average term in autoregressive models yields parsimony while retaining flexibility; in particular, the first order autoregressive moving-average model, ARMA(1,1) is notable since it retains a good approximating capability with just two parameters. In the same spirit, we assess empirically whether a similar result holds for threshold processes. First, we show that the first order threshold autoregressive moving-average process, TARMA(1,1) exhibits complex, high-dimensional, behaviour with parsimony, by comparing it with threshold autoregressive processes, TAR( p ), with possibly large autoregressive order p . Second, we study the descriptive power of the TARMA(1,1) model with respect to the class of autoregressive models, seen as universal approximators: in several situations, the TARMA(1,1) model outperforms AR( p ) models even when p is large. Lastly, we analyze two real world data sets: the sunspot number and the male US unemployment rate time series. In both cases, we show that TARMA models provide a better fit with respect to the best TAR models proposed in literature.