Testing for Multiple Structural Changes in Cointegrated Regression Models

This paper considers issues related to testing for multiple structural changes in cointegrated systems. We derive the limiting distribution of the Sup-Wald test under mild conditions on the errors and regressors for a variety of testing problems. Our asymptotic results show that as long as the intercept is allowed to change across regimes, inference is possible even if we allow stationary variables in the regression. We also find that including stationary regressors whose coefficients are not allowed to change does not affect the limiting distribution of the tests under the null hypothesis. We propose a procedure that allows one to test the null hypothesis of, say, changes, versus the alternative hypothesis of changes. This sequential procedure is useful in that it permits consistent estimation of the number of breaks present. When the regression is spurious, we show that the procedure tends to select the maximum number of breaks allowed. This feature helps distinguish a cointegrated model from a purely spurious one. Our simulation experiments show that in the presence of serial correlation in the errors, the commonly used LM tests suffer from the important problem of nonmonotonic power in finite samples. In fact, in certain cases, power can go to zero as the magnitude of the break(s) increase. We propose a modified Wald test based on a new estimator of the long run variance which uses information under both the null and alternative hypotheses. The proposed test is able to mitigate size distortions associated with the usual Wald test while maintaining monotonic power.