The effects of innovational outliers and additive outliers in cointegrated vector autoregressive models are examined and it is analyzed how outliers can be modelled with dummy variables. A Monte Carlo simulation illustrates that additive outliers are more distortionary than innovational outliers, and misspecified dummies may distort inference on the cointegration rank in finite samples. These findings question the common practice in applied cointegration analyses of including unrestricted dummy variables to account for large residuals. Instead it is suggested to focus on additive outliers, or to test the adequacy of a particular specification of dummies prior to testing for the cointegration rank. The points are illustrated on a UK money demand data set.

This paper presents the likelihood ratio~LR! test for the number of cointegrating relations in the I~2! vector autoregressive model+ It is shown that the asymptotic distribution of the LR test for the cointegration ranks is identical to the asymptotic distribution of the much applied test statistic based on the two-step estimation procedure in

It is well known that the finite‐sample properties of tests of hypotheses on the co‐integrating vectors in vector autoregressive models can be quite poor, and that current solutions based on Bartlett‐type corrections or bootstrap based on unrestricted parameter estimators are unsatisfactory, in particular in those cases where also asymptotic χ2 tests fail most severely. In this paper, we solve this inference problem by showing the novel result that a bootstrap test where the null hypothesis is imposed on the bootstrap sample is asymptotically valid. That is, not only does it have asymptotically correct size, but, in contrast to what is claimed in existing literature, it is consistent under the alternative. Compared to the theory for bootstrap tests on the co‐integration rank (Cavaliere, Rahbek, and Taylor, 2012), establishing the validity of the bootstrap in the framework of hypotheses on the co‐integrating vectors requires new theoretical developments, including the introduction of multivariate Ornstein–Uhlenbeck processes with random (reduced rank) drift parameters. Finally, as documented by Monte Carlo simulations, the bootstrap test outperforms existing methods.

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We characterize the restrictions imposed by the minimal I(2)-to-I(1) transformation that underlies much applied work, e.g. on money demand relationships or open-economy pricing relationships. The relationship between the parameters of the original I(2) vector autoregression, including the coefficients of polynomially cointegrating relationships, and the transformed I(1) model is characterized. We discuss estimation of the transformed model subject to restrictions as well as the more commonly used approach of unrestricted reduced rank regression. Only a minor loss of efficiency is incurred by ignoring the restrictions in the empirical example and a simulation study. A properly transformed vector autoregression thus provides a practical and effective means for inference on the parameters of the I(2) model.