Testing for Nonlinear Adjustment in Smooth Transition Vector Error Correction Models

The smooth transition autoregressive (STAR) model was proposed by Chan and Tong (1986) as a generalization of the threshold autoregressive (TAR) model, and since then it has attracted wide attention in the recent literature on the business cycles and the equilibrium parity relationships of commodity prices, exchange rates, and equity prices. Economic behavior is affected by asymmetric transaction costs and institutional rigidities, and thus a large number of studies - for example, Neftci (1984), Terasvirta and Anderson (1992), and Michael, Nobay, and Peel (1997) - have shown that many economic variables and relations display asymmetry and nonlinear adjustment. One of the most crucial issues in models of this kind is testing for the presence of nonlinear adjustment with the null of linearity. Luukkonen, Saikkonen, and Terasvirta (1988) expanded the transition function and proposed the variable addition tests as the tests of linearity against smooth transition nonlinearity, and the tests have been used in many empirical studies. However, the test statistics are based on the polynomial approximation, and the approximation errors may affect statistical inference depending on the parameter values of transition rate and location. Furthermore, the tests are not directly related to the smooth transition model, and thus we cannot retrace what causes the rejection of linearity. This paper considers the direct tests for nonlinear adjustment, which are based on the exact specification of smooth transition. The smooth transition model entails transition parameters, which cannot be identified under the null hypothesis. However, the optimality issue in the smooth transition model has not been treated extensively. The optimality issue regarding unidentified parameters has been developed by Davies (1987), Andrews (1993), and Hansen (1996). Hansen (1996) particularly considered the optimality issue in threshold models. The threshold parameter cannot be identified under the null hypothesis, and as a result the likelihood ratio statistic has the nonstandard distribution. The smooth transition model generalizes the threshold model, and thus this paper develops the appropriate tests and the associated distribution theory based on the optimality argument. Many empirical studies have found evidence on the presence of stochastic nonlinear dependence in equilibrium relations such as purchasing power parity. For example, Michael, Nobay, and Peel (1997), considering the equilibrium model of real exchange rate in the presence of transaction costs, found strong evidence of nonlinear adjustment, which conforms to the exponential smooth transition model. There exists a huge literature, and it is growing in this area. However, the econometric methods and the formal theory have been limited. This paper proposes the tests for nonlinear adjustment in the smooth transition vector error correction models, and thereby fills the deficiency in the literature. One technical difficulty is to estimate the smooth transition model. As noted by Haggan and Ozaki (1981) and Terasvirta (1994), it is difficult to estimate the smooth transition parameters jointly with the other slope parameters. The gradient of the transition parameter forces its estimate to blow up to infinity; thus, we cannot depend on the standard estimation algorithm. Our tests are based on the Lagrange multiplier statistic, which can be calculated under the null hypothesis. Therefore, our tests are easy to implement and thus useful. This paper finds that our tests have the asymptotic distribution, which is based on the Gaussian process. However, the asymptotic distribution depends on the nuisance parameters and the covariances are data-dependent; thus, the tabulation of asymptotic distribution is not feasible. This paper suggests the bootstrap inference to approximate the sampling distribution of the test statistics. Simulation evidence shows that the bootstrap inference generates moderate size and power performances