Tests For Nonlinear Cointegration

This paper develops tests for the null hypothesis of cointegration in the nonlinear regression model with I(1) variables. The test statistics we use in this paper are Kwiatkowski, Phillips, Schmidt, and Shin’s (1992; KPSS hereafter) tests for the null of stationarity, though using other kinds of tests is also possible. The tests are shown to depend on the limiting distributions of the estimators and parameters of the nonlinear model when they use full-sample residuals from the nonlinear least squares and nonlinear leads-and-lags regressions. This feature makes it difficult to use them in practice. As a remedy, this paper develops tests using subsamples of the regression residuals. For these tests, first, the nonlinear least squares and nonlinear leads-and-lags regressions are run and residuals are calculated. Second, the KPSS tests are applied using subresiduals of size b. As long as b/T → 0 as T → ∞, where T is the sample size, the tests using the subresiduals have limiting distributions that are not affected by the limiting distributions of the full-sample estimators and the parameters of the model. Third, the Bonferroni procedure is used for a selected number of the subresidual-based tests. Monte Carlo simulation shows that the tests work reasonably well in finite samples for polynomial and smooth transition regression models when the block size is chosen by the minimum volatility rule. In particular, the subresidual-based tests using the leads-and-lags regression residuals appear to be promising for empirical work. An empirical example studying the U.S. money demand equation illustrates the use of the tests.