Likelihood Ratio Testing for Cointegration Ranks in I(2) Models.

This paper presents the likelihood ratio (LR) test for the number of cointegrating and multi-cointegrating relations in the I(2) vector autoregressive model. It is shown that the asymptotic distribution of the LR test for the (multi-) cointegration ranks is identical to the asymptotic distribution of the much applied test statistic based on the Two-Step procedure in Johansen (1995), Paruolo (1996), and Rahbek, Kongsted, and Jørgensen (1999). By construction the LR test statistic is smaller than the non-LR test statistic from the Two-Step procedure as the latter ignores some of the restrictions concerning the hypothesis of I(2), and application of the LR test may change rank selection in empirical work. Based on a study of existing empirical applications and related Monte Carlo simulations we conclude that the LR test has much better size properties when compared to the Two-Step based test. Overall, we propose to use of the LR test for rank determination in I(2) analysis as the Two-Step based statistic was developed as a feasible approximation to the then unobtainable LR test.