Identification and Estimation Issues in Exponential Smooth Transition Autoregressive Models

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