Identification and Estimation Issues in Exponential Smooth Transition Autoregressive Models

Abstract

Exponential smooth transition autoregressive (ESTAR) models have been widely used in the empirical international finance literature. We show that the exponential function used in ESTAR models is ill-suited as a regime weighting function because of two undesirable properties. The first is that it can be well approximated by a quadratic function in the threshold variable whenever the transition function parameter γ, which governs the shape of the function, is ‘small’. This leads to identification issues with respect to the transition function parameter and the slope vector in ESTAR models. The second is that the exponential function becomes an indicator function over the entire range of the threshold variable, except at the point where the threshold variable is equal to the location parameter µ. This results in a high propensity to spuriously overfit a small number of observations around µ, leading to an ‘outlier fitting effect’ of the exponential function. We show the effect of both of these problems on estimation of ESTAR models by means of an empirical replication of the well known study by Taylor et al. (2001), and an extensive simulation exercise, where we vary the magnitude of the threshold parameter as well as the sample size.