This paper provides a theoretical overview of Wald tests for Granger causality in levels vector autoregressions (VAR’s) and Johansen-type error correction models (ECM’s). for VAR models the results for inference are not encouraging. The limit theory typically involves nonstandard distributions and nuisance parameters, and there is no sound statistical basis for testing causality in such a framework. Granger causality tests in ECM’s also suﬀer from nuisance parameter dependencies asymptotically and nonstandard limit theory. But, 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. In consequence, we recommend some operational procedures for conducting Granger causality tests in the important practical case of testing the causal eﬀects of one variable on another group of variables and vice versa.
Toda, Hiro Y. and Phillips, Peter C.B., "Vector Autoregression and Causality: A Theoretical Overview and Simulation Study" (1991). Cowles Foundation Discussion Papers. 1244.