A Flexible Coefficient Smooth Transition Time Series Model

In this paper, we propose a flexible smooth transition autoregressive (STAR) model with multiple regimes and multiple transition variables. We show that this formulation can be interpreted as a time varying linear model where the coefficients are the outputs of a single hidden layer feedforward neural network. This proposal has the major advantage of nesting several nonlinear models, such as, the Self-Exciting Threshold AutoRegressive (SETAR), the AutoRegressive Artificial Neural Network (AR-ANN), and the Logistic STAR models. Furthermore, if the neural network is interpreted as a nonparametric universal approximation to any Borel-measurable function, our formulation is directly comparable to the Functional Coefficient AutoRegressive (FAR) and the Single-Index Coefficient Regression models. The motivation for developing a flexible model is twofold. First, allowing for multiple regimes is important to model the dynamics of several time series, as for example, the behaviour of macro economic variables over the business cycle. Second, multiple transition variables are useful in describing complex nonlinear behaviour and allow for different sources of nonlinearity. A model building procedure consisting of specification and estimation is developed based on statistical inference arguments. A Monte-Carlo experiment showed that the procedure works in small samples, and its performance improves, as it should, in medium size samples. Several real examples are also addressed.