Identification and Estimation Issues in Exponential Smooth Transition Autoregressive Models

Abstract

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 . 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.

Long monthly temperature series and the Vector Seasonal Shifting Mean and Covariance Autoregressive model

“…This manifests itself in the very large standard deviations of the estimates of these parameters, see again Tables 9-28. Buncic (2019) has an illustrative discussion of this case. Analogously to the previous case, one could replace the estimated function by a quadratic trend but this has not been done because the estimation algorithm has nonetheless converged after …xing the value of the location parameter.…”

Reinvestigating the Oil Dependency of the GCC Countries’ Stock Market: A Regime-Switching Cointegration Approach

“…Furthermore, negative oil changes do not affect Bahrain and Saudi Arabia. However, Buncic (2019) argued that the exponential function is "ill-suited as a regime weighting function in the smooth transition models" and requires a small transition function parameter. Otherwise, the exponential regime weighting function behaves like a dummy variable.…”