Multivariate Granger Causality

[Dedder]

Hello,

I need to test for Granger-Causality in a multivariate (ie. not just in a biariate) VAR.

Lütkepohl (2005, New Introduction to Multiple Time Series, p.49 and 50), Dufour and Renault (2010, Short and long run causality measures: Theory and inference) point out that normal Granger-Causality tests are not reliable any more in a VAR with more than 2 variables.

Example: A trivariate VAR(2) with x_t,y_t,z_t:

y_t might be 1-step noncausal for z_t but it might still be h-step causal for z_t (h>1).

I found the following in a RATS help file:

*Multivariate Granger Causality (test to see if a block of variables is exogenous
*money and output trivariate system,

but that's not what I mean with multivariate Granger-Causality (that's just a normal LR-test).

I didn't find anything on my problem in this forum or the web, but:

Is there any way to implement Granger-Causality tests in a multivariate VAR?



Or asked the other way: I want to implement both short- and long-run restrictions in a 6 variable SVAR. But there's no economic story to tell (eg like money policy has no long-run impact on output), so I must get my restrictions out of the data. I have some zero correlations at t=0 and need to check whether I can use them together with non-granger causality relationships for long-run restrictions. But i cannot use the normal Granger-causality tests as they are valid only in a bivariate VAR.

[TomDoan]

My (unpublished) job market paper was "Defining and Testing Causal Orderings in Multivariate Systems" which showed, in effect, that the only block restrictions of real consequence are the block exogeneity restrictions.

[Dedder]

Thanks for replying me that quickly!

As I am unable to read your paper I can post two papers:

-J.-M. Dufour, A. Taamouti / Journal of Econometrics 154 (2010) 4258: Short and long run causality measures: Theory and inference

and

-M. Eichler / Journal of Econometrics 137 (2007) 334–353: Granger causality and path diagrams for multivariate time series

The first one introduces h-step noncausality whereas the second one has tries a graph approach (which i do not want to do)..

They show that blockwise granger-(non)causality can lead to misleading interpretations

Another source (New introduction to multiple time series, luetkepohl 2005, p.49) is cited below:

"Often we are interested in the causal links between two variables in a higher
dimensional system. In other words, we are interested in analyzing Grangercausality
in a framework where the information set contains more than just the
variables of direct interest. In the bivariate framework when the information
set is limited to the two variables of interest, it was seen that if the 1-step
ahead forecasts of one variable cannot be improved by using the information
in the other variable, the same holds for all h-step forecasts, h = 1, 2, . . . .
This result does not hold anymore if the information set contains additional
variables, as pointed out by Luetkepohl (1993)..."

Regards

Dedder

[TomDoan]

While it's possible to define "h" step (non-)causality, it's a completely pointless concept.

I'll see what I can do about putting together a PDF copy of my original paper. However, the summary is:

...we could define "causality" by x "causes" y if and only if there is no decomposition of U (the set of variables) with a lower triangular moving average representation in which y is a member of the first block and x is a member of the second. Causality can be rejected by verifying that there exists as least one such decomposition, but acceptance requires an examination of all partitions which put y in the first set and x in the second. Because the causal relationship between a pair of variables in a multivariate system depends strongly upon how they are related to the other variables in the system, it is best to think of an entire causal ordering, rather than a few elements of the relation.

[Dedder]

Hello Tom,

I really appreciate your help. This subject is quite new (ie. the papers are not too old) and I am not sure now whether there is a point in going deeper into it or not..

As your paper has not been published, you might not want to put it on the web, but maybe you could send me a longer abstract (or section) via email ?

As you don't know me personally i would understand if you would not agree..

Regards


Dedder

[TomDoan]

Hello Tom,

I really appreciate your help. This subject is quite new (ie. the papers are not too old) and I am not sure now whether there is a point in going deeper into it or not..

As your paper has not been published, you might not want to put it on the web, but maybe you could send me a longer abstract (or section) via email ?

As you don't know me personally i would understand if you would not agree..

Regards


Dedder

I don't have a philosophical problem with posting it on the web. However, this was written in 1982, and all I have now is a printed copy (and not a particularly pretty one). I'll scan it and hope that it's legible.

[TomDoan]

This is a very preliminary editing of this:

Causal Orderings.pdf

(248.62 KiB) Downloaded 1651 times

[Dedder]

Tom,

thank you very much!! I will read your paper ..

By the way, I found you about "Granger Causally Prior" on Chris Sims's website:

http://sims.princeton.edu/yftp/Times05/GCP05.pdf

..."This result was first obtained in unpublished work by
Thomas Doan."

[apollon]

Lutkepohl actually goes on to define the Wald statistic and provides an example (3.6.2) of multivariate Granger causality.
Are we saying this statistic flawed ? If not, does this statistic exist or can it be easily creted in RATS ?

[TomDoan]

In that example, he is doing a test that gives a valid block triangular system:

x 0 0
x x x
x x x

With that type of restriction, y2 and y3 cannot help explain y1 at any horizon. It's hypotheses like

x x 0
x x x
x x x

which, for instance, are what the block F-tests give, that aren't really all that meaningful, since, while y3 doesn't help predict y1 one-step ahead, it can help predict y1 at other horizons since it helps predict y2.

Lutkepohl's Wald test is asymptotically equivalent to the LR test shown in the RATS manual (varcause.rpf).

[apollon]

So if my understanding is correct, his Wald test is valid and similar to Rats LR test.
To be more specific, varcause (which is the same program I was referring above) is doing this:

system(model=unrestricted)
variables loggdp unemp logi
lags 1 to 4
det constant logp{1 to 4} logm2{1 to 4}
end(system)

So it's testing whether logp and logm2 affect loggdp, unemp and logi. What I am interested in finding out is
if unemp and logi cause loggdp. How can I program that ?

thanks!

[TomDoan]

So if my understanding is correct, his Wald test is valid and similar to Rats LR test.
To be more specific, varcause (which is the same program I was referring above) is doing this:

system(model=unrestricted)
variables loggdp unemp logi
lags 1 to 4
det constant logp{1 to 4} logm2{1 to 4}
end(system)

So it's testing whether logp and logm2 affect loggdp, unemp and logi.

That's not quite accurate. Theoretically, you would reject if at least one of the two (logp or logm2) affects at least one of the three (loggdp, unemp or logi).

What I am interested in finding out is if unemp and logi cause loggdp. How can I program that ?

thanks!

That's not a well-formed hypothesis since it leaves out two of the variables. Where do the two nominal variables fit it? If what you want is a causal ordering in which the two nominals don't affect the three reals and the unemp and logi also don't affect loggdp (as a joint hypothesis), you can do that, since it specifies a block triangular representation for the system.

[dacanoo]

Hello Tom,

I understood that the best way to test Granger Causality in a multivariate VAR is with block exogeneity tests.
But in varcause example what is being tested is the exogeneity of real variables, and I dont understand very well where goes causality. The model is something like

rrrnn
rrrnn
rrrnn

versus

rrr00
rrr00
rrr00
?

But causality shouldn't be testing something like

xxx00
xxx00
xxx00
xxxxx
xxxxx
?

I understand that logp and logm are assumed exogenous because they are nominal variables, but if this cannot be assumed, how to include the other equations in the block exogeneity test?

What I'm trying to do is test Granger Causality in a VECM, where for some variables the hypothesis of zero coefficient of the error term is rejected, so I cannot indlude these varaibles as deterministic. Besides I think that block exogeneity tests are no longer valid because of the error correction term, by which a variable can affect another, even if its lags don't do it.
I've thinked about test block exogeneity but adding the restriction of zero coefficient (not zero alpha but zero beta) in the cointegrating equation for the variable tested, is this valid?

[TomDoan]

Hello Tom,

I understood that the best way to test Granger Causality in a multivariate VAR is with block exogeneity tests.
But in varcause example what is being tested is the exogeneity of real variables, and I dont understand very well where goes causality. The model is something like

rrrnn
rrrnn
rrrnn

versus

rrr00
rrr00
rrr00
?

But causality shouldn't be testing something like

xxx00
xxx00
xxx00
xxxxx
xxxxx
?

I understand that logp and logm are assumed exogenous because they are nominal variables, but if this cannot be assumed, how to include the other equations in the block exogeneity test?

The LR test for the zero block in your five variable system can be computed by looking only at the three equations that are restricted under the null.

What I'm trying to do is test Granger Causality in a VECM, where for some variables the hypothesis of zero coefficient of the error term is rejected, so I cannot indlude these varaibles as deterministic. Besides I think that block exogeneity tests are no longer valid because of the error correction term, by which a variable can affect another, even if its lags don't do it.
I've thinked about test block exogeneity but adding the restriction of zero coefficient (not zero alpha but zero beta) in the cointegrating equation for the variable tested, is this valid?

That would be a very ugly joint hypothesis (zero on the lagged differences combined with zero on the beta's). Even if you could figure out how to estimate that, the resulting test statistic would have a non-standard distribution. You should probably just do a standard exogeneity test on the VAR form and bootstrap the p-value. See the discussion in http://www.estima.com/forum/viewtopic.php?f=4&t=19.

[bok1234]

Dear Mr.Doan,


Let me ask you 2 things about multivariate Granger Sims causality test(Multi-GS test), especially focusing on varcause.rpf.

[Question 1] On Sat Apr 21, 2007, you replied as follows; "The test is whether the nominal variables enter the real equations. The null is that they don't, which means that the real variables don't respond to nominal shocks - they form an exogenous block."

I guess that the 2nd 'they' in the last sentence means 'real variables.' Therefore, according to the result of varcause.rpf, a block of real variables do not secure exogeneity. This means that 'a bundle of nominal variables' DO GRANGER CAUSE 'a bundle of real variables'. And finally, this is "the one-side test(nominal variables -> real variables)", not including a test of 'real variables->nominal variables' Am I right?


[Question 2] In your reply Fri Jul 16, 2010. You referred that unit root could result in some problems. If differenced data series does not have unit root anymore and they get to input into the VAR system(varcause.rpf, for example), the probable problems that you referred would be solved. Is this right?