## Tuesday, May 27, 2014

### Questions About Granger Causality Testing - The Fine Print

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, herehere, 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 anLütkepohl  (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.

Now, in no particular order, here are some supplementary points that seem to be in order.
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
My final recommendation - read the Toda-Yamamoto paper! It's extremely well written.

References

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.

1. An earlier comment was lost. I am a professor of philosophy and would like to understand this article but the writing is too arcane and I am not willing to look up every term in the hope the reading will be worthwhile. I would really appreciate clarity for the non-specialist.

2. Can you give an example of an I(3), or greater, variable? I have a tough enough time getting referees to believe variables are I(2).

1. I can;t think of an economic variable that would be I(3) or higher. Sometimes log(prices) are found to be I(2), so that the inflation rate is I(1).

3. Dear professor Dave Giles,

My name is Tay Shin and I’m a recent graduate from UC Berkeley. First of all, I sincerely appreciate for your great explanation on TY procedure. It helped me to understand deeply. However, I do have some questions.
Suppose I have more than two variables, would it be appropriate to do Granger test on full linear model or on separate linear models? To be specific, for full linear model that I was thinking (with three variables) :
Step1. Build a model: Mod1 dependent : X1 independent : (lagged) X1 X2 X3
Mod2 dependent : X2 independent : (lagged) X1 X2 X3
Mod3 dependent : X3 independent : (lagged) X1 X2 X3
Step2. Wald test : X2 does not Granger causes X1 (using Mod1)
X3 does not Granger causes X1 (using Mod1)
X1 does not Granger causes X2 (using Mod2)
X3 does not Granger causes X2 (using Mod2)
…(and so on)
For separate bivariate models, it is just same as the previous post on your blog.
It would be greatly appreciated if you could help with this problem.

Best
Tay

1. Tay thanks for the comment. You should use a full model - that is, a 3-equation VAR, and then test the sets of lagged variables (with appropriate modification along the T-Y lines if any series is non-stationary), If you're using EViews, then all sets of the restrictions tests will come up automatically.

2. As far as I know, T-Y (1995) and Classical G (1969) are (PAIRWISE) BIVARIATE G-causality tests and therefore cannot be used in systems with more than 2 variables.

In systems with more than 2 variables (when all or some is stationary), in order to find G-causalities among variables, one must use CONDITIONAL or PARTIAL G-causality tests (the latter supersides the former) via the followings:
1. MVGC (Barrett and Anil K. Seth; 2014 May) or
2. FIAR (Roelstraete; 2012). I recommend 0.5 version since VAR order can be passed mutatis mutandis to Bootstrapping problemlessly.

The case of (possibly some of the variables are) non-stationary variables in systems with 2 variables is, I think, really ugly. Windowing technique of "HESSE 2003 Windowing The use of time-variant EEG Granger causality for inspecting" should be used; afaik.

Erdogan CEVHER
The Ministry of Sci. Ind. and Tech of Turkey

3. Sorry but this is rubbish. Read the original papers - they specifically allow for more than 2 variables.

4. Dear Prof. Giles, in my above comment, I had unintentionally not written "MORE THAN" phrase (though you understood what I wanted to mean and reply accordingly): The correct one must be:
"...The case of (possibly some of the variables are) non-stationary variables in systems with "MORE THAN 2" variables is, I think, really ugly....".

The classical G-causality (1969) does NOT detect SPURIOUS G-causalities in case of more than 2 variables. Here are some mentionings:
1. "DING CHEN BRESSLER 2006 Granger Causality: Basic Theory and Application to Neuroscience". p.10-11 Look at the figure at page 11. Assume there are two distinct models with REAL (ACTUAL) G-causalities: 1st model: (Y -> Z -> X) and 2nd model: (Y-->Z-->X) and (Y-->X). The classical G-causality test (1969) canNOT distinguish 1st model and 2nd model and concludes that in EACH model that the G-causality structure is (Y-->Z-->X) and (Y-->X). That is to say, for the 2nd model, classical G-causality test (1969) CORRECTLY finds the G-causalities among variables. However, for the 1st model, classical G-causality test (1969) WRONGLY asserts there is a connection (Y-->X) as well though there is NO such connection. This is called as SPURIOUS G-causality and first analysis of this goes back to Geweke (1982), and later Geweke(1984).

2. Anil K. SETH explains this situation in http://www.scholarpedia.org/article/Granger_causality
The following is from the above link:
"Figure 2: Two possible connectivities that canNOT BE DISTINGUISHED by PAIRWISE analysis. Adapted from Ding et al. (2006)."

This MULTIVARIATE extension, sometimes referred to as ‘CONDITIONAL’ G-causality (Ding et al. 2006), is extremely useful because REPEATED PAIRWISE analyses among MULTIPLE variables can sometimes give MISLEADING results. For example, a REPEATED BIVARIATE analyses would be UNABLE TO DISAMBIGUATE the two connectivity patterns in Figure 2. BY CONTRAST, a CONDITIONAL/MULTIVARIATE analysis would infer a causal connection from X to Y only if past information in X helped predict future Y ABOVE AND BEYOND those signals mediated by Z . Another instance in which CONDITIONAL G-causality is valuable is when a single source drives two outputs with different time delays. A BIVARIATE analysis, BUT NOT A MULTIVARIATE analysis, would FALSELY INFER a causal connection from the output with the shorter delay to the output with the longer delay.

3. "Lionel BARNETT, Anil K. SETH; May 2014; The MVGC Multivariate Granger Causality Toolbox: A New Approach to Granger-causal Inference" p.4:
The UNCONDITIONAL G-causality statistic introduced above has the UNDESIRABLE characteristic that if there are joint (possibly historical) dependencies between X and Y and a third set of variables, Z say, then SPURIOUS CAUSALITIES may be reported. Thus, for instance, IF there is NO direct causal influence Y -> X but there are (possibly lagged) dependencies of X and Y on Z then a SPURIOUS Y --> X causality may be reported. These SPURIOUS causalities MAY BE ELIMINATED by “CONDITIONING OUT” the common dependencies – provided they are available in the data. If, however, there are dependencies on unknown (exogenous) or unrecorded (latent) variables, then it will in general be impossible to eliminate entirely their potentially confounding efffect on causal inference, although attempts have been made to mitigate their impact (e.g. “PARTIAL” G-causality (GUO et al.,2008)).

5. 4. "GUO SETH KENDRICK ZHOU FENG 2008; Partial Granger Causality: Eliminating Exogenous Inputs and Latent Variables" p.1-2

The basic Granger causality described in Appendix A.1 is APPLICABLE ONLY TO BIVARIATE time series. In MULTIVARIATE (MORE THAN TWO) situations, one time series can be connected to another time series in a DIRECT or an INDIRECT manner, raising the important question of whether there exists a (direct) causal influence between two time series when the influence of other time series are taken into account. In this case, REPEATED BIVARIATE analysis can be MISLEADING. For example, one time series may FALSELY APPEAR TO CAUSE ANOTHER if they are both influenced by a third time series but with different delays. To address this issue, Geweke introduced CONDITIONAL Granger causality (Geweke, 1984), as recently reviewed in (Chen et al., 2006; Ding et al., 2006). CONDITIONAL Granger causality is based on a straightforward
expansion of the autoregressive model to a MULTIVARIATE case including all measured variables. In this case, one variable causes a second variable if the prediction error variance of the first is reduced after including the second variable in the model, with all other variables included in both cases. The formalism for CONDITIONAL Granger causality is given in Appendix A.2.

Critically, the ability of CONDITIONAL Granger causality to deal with indirect interactions depends on being able to measure all relevant variables in a system. Often, this is not possible, and both environmental (exogenous) inputs and unmeasured latent variables can confound accurate causal influences. For example, in our experimental data recorded from the inferotemporal (IT) cortex of sheep, every measured neuron receives common exogenous inputs from the visual cortex and feedback from the prefrontal cortex. Moreover, even with advanced MEA techniques, we are only able to record a tiny subset of interacting neurons in a single area and there are bound to be latent variables. Hence controlling for exogenous inputs and latent variables is a critical issue when applying Granger causality to experimental data. In this article, inspired by the definition of the partial correlation in statistics, we introduce a novel definition of PARTIAL Granger causality to confront exactly this problem.

Prof. Giles, could we please look at the above referenced information (Geweke 1982, Geweke1984, Ding2006, Guo2008, Barnett2014) as well as the original articles G1969 and TY1995....

AFAIK, the correct measuring strength is something like:
Pairwise G-causality test(Classical 1969 and TY1995) < Conditional G-causality test < Partial G-causality test < Harmonic G-causality test < Global G-causality test...

6. (((3))) Dear Giles, look at the below post of an anonymous user (http://davegiles.blogspot.com.tr/2011/04/testing-for-granger-causality.html); I think the referees's answer nicely presents that T-Y1995 procedure is BIVARIATE as well as classical G1969.
.....................................
"AnonymousOctober 18, 2012 at 3:31 AM": Dear Professor Giles, I just had a referee report concerning a paper submission. I followed your methodology for Granger causality (E.C.: He is referring to T-Y procedure). I am focused on the relationship between 2 VARIABLES for the US and EU AND have a number of OTHER VARIABLES as controls.(E.C.: 2 VARIABLES+OTHER VARIABLES=MULTIVARIATE (>2))

I presented the results of the procedure you described concerning Granger-causality on pairwise tests. However the referee states that "IT IS KNOWN THAT IT IS NOT OPTİMAL TO TEST FOR CAUSALITY IN A BIVARIATE MODEL, PARTICULARLY IF THERE IS AN AUXILIARY VARIABLE THAT INFLUENCES THE OTHER TWO VARIABLES IN THE BIVARIATE SYSTEM".
............
By the way, I am on the more severe side than the referee: The referee says "...not optimal...."; However, it is a fact that PAIRWISE G-causality tests may reveal
1. existence of G-causalities when in fact actually there is NO such G-causalities (as in Ding2006)
2. non-existence of G-causalities when in fact there IS G-causalities.

The referee should have rejected more severely by taking into account the above 2 facts (to concretize: 55% perhaps is not to optimal 100%; maybe at that stage of the wrong-analysis, the degree of true covering G-causalities was 40%; think that "non-existence of G-causalities when in fact there IS G-causalities" )

4. Dear Professor, my question is i have three variable with l(1) and three with l(0). Do you think using Toda and Yamamoto can i check causality for all the variables together or i should do the analysis for each variables l(0) with variables from l(1).

1. You should use all of them together in one model.

5. Thank you professor

6. Prof. Dave Giles, I applied Toda-Yamamoto causality test via Eviews. According to P-value, and Chi-square test results, there is no causal relationship between variables. But, based on t-test, there exist unidirectional causality. can I interpret this result according to t-test result? Please answer me as soon as possible. Thank you.

7. No, you can't.

1. Thank you Professor.

8. Dear Professor, on the premise that Cointegration implies Granger Causality (GS)..If I have 10 variables that are found to be cointegration but after using the T-M GS procedure, I find that some but not all the variables at least a unidirectional relationship, do I conclude there is a problem? I understand that for two variables that are cointegrated, at least one must GS the other..However, things appear to be silent for 10 or 20 variables. Please provide some insights. Would be well appreciated..To put things in perspective, let's say X1, X2, X3 and X4 are found to be cointegrated, but X3 does not GS X4 neither does X4 GS X3..Howver, the rest variables at least GS each, is this a problem? Is there any inconsistency when put side by side with the fact that they are cointegrated? Please provide guidance.