Identification and estimation issues in exponential smooth transition autoregressive models

Exponential smooth transition autoregressive (ESTAR) models are widely used in the international finance literature, particularly for the modelling of real exchange rates. We show that the exponential function is ill-suited as a regime weighting function because of two undesirable properties. Firstly, it can be well approximated by a quadratic function in the threshold variable whenever the transition function parameter, which governs the shape of the function, is "small". This leads to an identification problem with respect to the transition function parameter and the slope vector, as both enter as a product into the conditional mean of the model. Secondly, the exponential regime weighting function can behave like an indicator function (or dummy variable) for very large values of the transition function parameter. This has the effect of "spuriously overfitting" a small number of observations around the location parameter. We show that both of these effects lead to estimation problems in ESTAR models. We illustrate this by means of an empirical replication of a widely cited study, as well as a simulation exercise.