Stability tests in error correction models

This paper derives the asymptotic distribution of the fluctuations test of Ploberger-Kramer-Kontrus (1989) in a reduced rank error correction model with nonstationary variables. We show how the fluctuations test is constructed in this case where the coefficient matrix being tested is singular. The test converges to a sum of tied-down Bessel processes whose critical values are tabulated in Andrews (1993). We also show how the fluctuations test is affected by misspecification of the rank (i.e., by misspecification of the number of stationary relationships in the model). In particular, the test over accepts (over rejects) the null of stability when the true rank is under (over) specified. Because the test is sensitive to correct specification of the rank, we also propose tests for rank stability. The test requires separate treatment for the case that there are more/less cointegrating vectors at some point in time. In this case the test takes the form of the LR/LM test. The LR test is simply a pointwise version of Johansenʹs LR test for rank determination while the LM test is an application of Kwiatkowski-Phillips-Schmidt-Shinʹs (1992) test for stationarity.