Testing for a Unit Root against Transitional Autoregressive Models

Abstract

This paper considers the test of a unit root in transitional autoregressive models. In particular, we develop the asymptotic theory of the inf-t test for the null hypothesis of a unit root in a wide class of nonlinear autoregressive models having parameters that are identified only under the alternative of stationarity. Our framework is very general and allows for virtually all potentially interesting models with the threshold, discrete and smooth transition functions. The specifications of shortrun dynamics used in the paper are also fully general, and comparable to those used in the linear unit root models. Most importantly, our asymptotics take it into consideration that the parameter space has a random limit. This is an essential feature of the unit root test in transitional autoregressive models, which has been ignored in the literature. For this very general class of transitional autoregressive models, we show that the inf-t test has well-defined limit distribution depending only upon the transition function and the limit parameter space. The critical values of the test are provided for some of the commonly used models under the conventional specification of the parameter space. Our simulation study shows that the test has good size with the power that is significantly higher than the usual ADF test even for samples of relatively small sizes. We apply the test to various economic time series and find strong evidence for the rejection of random walks in favor of stationary transitional autoregressive models.