# Vector Autoregressions and Causality

https://doi.org/0012-9682(199311)61:6<1367:VAAC>2.0.CO;2-T
p. 1367-1393

Hiro Y. Toda, Peter C. B. Phillips

This paper develops a limit theory for Wald tests of Granger causality in levels vector autoregressions (VAR's) and Johansen-type error correction models (ECM's), allowing for the presence of stochastic trends and cointegration. Earlier work by Sims, Stock, and Watson (1990) on trivariate VAR systems is extended to the general case, thereby formally characterizing the circumstances when these Wald tests are asymptotically valid as $\chi^2$ criteria. Our results for inference from unrestricted levels VAR are not encouraging. We show that without explicit information on the number of unit roots in the system and the rank of certain submatrices in the cointegration space it is impossible to determine the appropriate limit theory in advance; and, even when such information is available, the limit theory often involves both nuisance parameters and nonstandard distributions, a situation where there is no satisfactory statistical basis for mounting these tests. The situation with 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 nuisance parameter dependencies asymptotically and, in some cases that we make explicit, nonstandard limit theory. Both these results are somewhat surprising in the light of earlier research on the validity of asymptotic $\chi^2$ 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 that influence the asymptotics.