Judging by the number of hits, comments, and questions that I've had in relation to my various posts on testing for Granger (Non-) Causality, this seems to be a topic that a lot of followers find interesting. For instance, see the posts here, here, here, and especially here.
In the comments, and in a large number of related emails that I've received, several questions seem to recur, and I thought it would be worth addressing them in one place - right here, to be specific!
The following discussion relates to the (usual) case where there is the possibility that one or more of the time-series variables under consideration may be non-stationary, and some of the variables may be cointegrated. In such cases we have to be especially careful when we apply tests for Granger causality. The reasons for this, and for adopting a modified testing procedure, such as that proposed by Toda and Yamamoto (1995), or that of Dolado and Lütkepohl (1996) and Saikkonen and Lütkepohl (1996), are laid out in this earlier post, and I won't repeat them here. I'll make the bold assumption that you've done your homework.
- Although the illustrative examples that I used in my previous posts involved just two variables, the methodology I outlined applies equally if we a VAR model for three or more variables. Indeed, the discussion in Toda and Yamamoto (1995) is for the case of an arbitrary number of variables, n.
- In the very special case where we have just two variables, and we are sure that they are cointegrated, then we don't need to use the Toda-Yamamoto (or similar procedure). The usual Wald test will be valid, asymptotically, for testing for Granger non-causality.See Lütkepohl and Reimers (1992). Of course, the outcome of cointegration testing is rarely "definitive", so it's a good idea to us the T&Y procedure even when n = 2 and we think that the two variables may be cointegrated. We may lose a little power with the MWALD test, but the results will still be valid asymptotically.
- The formal analysis in the Toda-Yamamoto paper is for the situation where the maximum order of integration for any of the variables being considered is I(2). In practice, this would not be limiting for economic data. However, Toda and Yamamoto make it perfectly clear that this restriction is just for expository purposes, and their results also hold for higher orders of integration.
- We usually think of the Granger non-causality testing in terms of a Wald test. Indeed, the Toda-Yamamoto / Lütkepohl testing is sometimes referred to as "MWald" (modified Wald) testing. However, all of the issues and asymptotic results apply equally if we are using a Likelihood Ratio Test (LRT) for the restrictions implied by non-causality, rather than a Wald test.
- When employing the Toda-Yamamoto testing procedure, the lag length (k) used for the VAR model must be greater than or equal to the maximum order of integration for the variables in the system. This is unlikely to be a restrictive requirement in practice. Note that k is the selected lag length before the additional (non-tested) lags are added to the equations of the VAR model.
- In practice, the value of k will not be known in advance. There are various ways of selecting k when its value is unknown . Toda and Yamamoto seem to suggest that "asymptotic t-tests" (the statistics for which will be standard normally distributed, for large enough T). In finite samples this would lead to a pre-testing problem, as the successive tests are not statistically independent. Asymptotically, there is no such problem. Most of us would use an information criterion to select k. SIC (BIC) is a good choice as it will be a consistent model selection procedure in this context.
- The Toda-Yamamoto procedure can be applied, without further modification, if it is deemed necessary to include trend or seasonal dummy variables in the VAR specification.
- If we use follow the Toda-Yamamoto analysis when in fact all of the variables under consideration are I(0), we lose some asymptotic efficiency in the estimation of the VAR model, but all of the other asymptotic results are unaffected. So, the modified Wald test (or LRT) will still have the desired asymptotic distribution - namely, chi-square under the null hypothesis of no Granger causality.
Dolado, J. and H. Lütkepohl, 1996. Making Wald tests work for cointegrated VAR systems. Econometric Reviews, 15, 369–38.
Lütkepohl, H. and H. Reimers. 1992. Granger-causality in cointegrated VAR processes. The case of the term structure. Economics Letters, 40, 263-268.
Saikkonen, P. and H. Lütkepohl, 1996. Infinite-order cointegrated vector autoregressive processes. Econometric Theory, 12, 814–844.
Toda, H. Y. and T. Yamamoto, 1995. Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.