VECTOR AUTOREGRESSIONS AND CAUSALITY

This paper develops a limit theory for Wald tests of Granger causality in levels vector autoregressions (VAR's) and Johansen-type error corr ection models (ECM's), allowing for the presence of stochastic trends and cointegration. Earlier work by Sims, Stock, and Watson (1990) on t rivariate VAR systems is extended to the general case, thereby formall y characterizing the circumstances when these Wald tests are asymptoti cally valid as chi2 criteria. Our results for inference from unrestric ted levels VAR are not encouraging. We show that without explicit info rmation on the number of unit roots in the system and the rank of cert ain submatrices in the cointegration space it is impossible to determi ne the appropriate limit theory in advance; and, even when such inform ation is available, the limit theory often involves both nuisance para meters and nonstandard distributions, a situation where there is no sa tisfactory statistical basis for mounting these tests. The situation w ith regard to the use of causality tests in ECM's is also complex but more encouraging. Granger causality tests in ECM's also suffer from nu isance parameter dependencies asymptotically and, in some cases that w e make explicit, nonstandard limit theory. Both these results are some what surprising in the light of earlier research on the validity of as ymptotic chi2 criteria in such systems. In spite of these difficulties , Johansen-type ECM's do offer a sound basis for empirical testing of the rank of the cointegration space and the rank of key submatrices th at influence the asymptotics.