Fully modified vector autoregressive inference in partially nonstationary models

The likelihood ratio (LR) test for ranks is commonly used to test for the number of cointegrating relationships among multivariate time series. The distribution of the LR test in partially nonstationary models is nonstandard and contains nuisance parameters in the presence of non-iid errors and misspecified lags. In contrast, I show that under less-restrictive assumptions on the errors, the fully modified (FM) vector autoregressive (VAR) rank test has a chi-squared distribution for the null of cointegration but is degenerate for the null of no cointegration, unlike its LR counterpart, which is well defined for both nulls. It turns out that augmenting the VAR by an exogenous I(0) variable solves the degeneracy problem. The procedure can also be applied to testing for Granger causality and is in fact a generalization of Toda-Yamamoto's procedure of augmenting the VAR with additional lagged I(1) variables. Unlike the LR test for ranks or Toda-Yamamoto's test for causality, the FM tests do not require correct specification of the number of lagged variables in the VAR nor does it require that errors be iid.