VAR or VECM When Testing for Granger Causality?

It never ceases to amaze me that my post titled "How Many Weeks are There in a Year?" is at the top of my all-time hits list! Interestingly, the second-placed post is the one I titled "Testing for Granger Causality". Let's call that one the number one serious post. As with many of my posts, I've received quite a lot of direct emails about that piece on Granger causality testing, in addition to the published comments.

One question that has come up a few times relates to the use of  a VAR model for the levels of the data as the basis for doing the non-causality testing, even when we believe that the series in question may be cointegrated. Why not use a VECM model as the basis for non-causality testing in this case?

On the face of it, this might seem like a good idea. It's been suggested that as the VECM incorporates the information abou the short-run dynamics, tests conducted within that framework may be more powerful than their counterparts within a VAR model. In fact, however, there's a very good reason for not using a VECM for this particular purpose.

First, let's recall the main message from my earlier post. A simple definition of Granger Causality, in the case of two time-series variables, X and Y is:

"X is said to Granger-cause Y if Y can be better predicted using the histories of both X and Y than it can by using the history of Y alone."

We can test for the absence of Granger causality by estimating the following VAR model:

Y_t = a₀ + a₁Y_t-1 + ..... + a_pY_t-p + b₁X_t-1 + ..... + b_pX_t-p + u_t      (1)

X_t = c₀ + c₁X_t-1 + ..... + c_pX_t-p + d₁Y_t-1 + ..... + d_pY_t-p + v_t      (2)

Then, testing H₀: b₁ = b₂ = ..... = b_p = 0, against H_A: 'Not H₀', is a test that X does not Granger-cause Y.

Similarly, testing H₀: d₁ = d₂ = ..... = d_p = 0, against H_A: 'Not H₀', is a test that Y does not Granger-cause X. In each case, a rejection of the null implies there is Granger causality.

Now, if any of the variables are non-stationary (whether or not they are cointegrated), the usual Wald test statistic for this testing will not have an asymptotic Chi-Square distribution. An easy way to deal with this is to use the following procedure proposed by Toda and Yamamoto (1995) - more details are provided in my previous post:

1. Test each of the time-series to determine their order of integration.
2. Let the maximum order of integration for the group of time-series be m.
3. Set up a VAR model in the levels (not the differences) of the data, regardless of the orders of integration of the various time-series.
4. Determine the appropriate maximum lag length for the variables in the VAR, say p, using the usual methods.
5. Make sure that the VAR is well-specified.
6. If two or more of the time-series have the same order of integration, at Step 1, then test to see if they are cointegrated.
7. No matter what you conclude about cointegration at Step 6, this is not going to affect what follows. It just provides a possible cross-check on the validity of your results at the very end of the analysis.
8. Take the preferred VAR model and add in m additional lags of each of the variables into each of the equations.
9. Test for Granger non-causality as follows. For expository purposes, suppose that the VAR has two equations, one for X and one for Y. Test the hypothesis that the coefficients of (only) the first p lagged values of X are zero in the Y equation, using a standard Wald test. Then do the same thing for the coefficients of the lagged values of Y in the X equation.
10. Make sure that you don't include the coefficients for the 'extra' m lags when you perform the Wald tests.
11. The Wald test statistics will be asymptotically chi-square distributed with p d.o.f., under the null.
12. Rejection of the null implies a rejection of Granger non-causality.
13. Finally, look back at what you concluded in Step 6 about cointegration:

"If two or more time-series are cointegrated, then there must be Granger causality between them - either one-way or in both directions. However, the converse is not true."
(This last piece of information may provide a cross-check on your overall conclusions.)

O.K. - now back to VARs vs. VECMs!

Suppose that at Step 6 we come to the conclusion that the time-series are cointegrated. In general, the presence of cointegration would suggest that we should model the data using a VECM model, rather than using a VAR model. That's modelling the data, though, not testing for Granger non-causality.

Here's the deal.

To get to the point where we are considering using a VECM model as the basis for the causality testing, we had to go through the prior step of testing for cointegration; and only if we rejected the hypothesis of "no cointegration" would we even consider estimating a VECM model. This is a classic example of "preliminary test testing". That is, the framework (model) chosen as the basis for the non-causality test is conditional on the outcome of a previous test - a test for non-cointegration. It's as if the choice between a VAR model and a VECM model (as the framework within which to test for non-causality) is made by flipping a biased coin. (Remember that there are always Type I and Type II errors associated with any classical hypothesis test.)

So, one important question that arises is the following one:

If we first test for non-cointegration, and then (conditional on the outcome of this test) we perform another test, what are the properties of this second test?

You see, the second test (the test for non-causality) will be of one form if we decide to use the VAR, and of a different form if we decide to use a VECM. When we pre-test, the second test is actually a random mixture of two tests. The "actual" test statistic is a weighted sum of the test statistic that would be obtained if we used a VAR model, and the test statistic that would be obtained if  we used a VECM model. And the weights are random, with values that depend on the properties of the prior (non-cointegration) test.

The upshot of all of this is as follows. When we test for no cointegration, then decide on a VAR model or a VECM model, and then apply a Granger non-causality test, the properties of this last test aren't at all what we think they are. They're really messy, and the best way to find out what's going on is to conduct a Monte Carlo experiment.

In particular, there will almost certainly be some distortion in the significance level (and hence the power) of the final test. We may think we're applying the non-causality test at (say) the 5% level, but the true significance level (the actual rate of rejection of  the null hypothesis when this hypothesis is false) may be quite different. And this might (should?) bother us.

[As an aside, if you think that the "size distortion" that can arise from pre-test testing may not be a big deal, then take a look at the results in Table 2 of King and Giles, 1984.]

So, has anyone investigated the issue of the effects of pre-test testing in the case we're interested in here - the case of testing for Granger non-causality, after first testing to to see if there is cointegration, so that we effectively randomize the choice of a VAR or VECM model?

Of course they have! You can take a look at the studies by Toda and Phillips (1994), Dolado and  Lütkepohl (1996), Zapata and Rambaldi (1997), and Clarke and Mirza (2006) for lots of interesting details. I particularly recommend the last of these papers, by my colleague Judith Clarke and her former student, Sadaf Mirza.

Zapata and Rambaldi (1997, p.294) find that the T-Y Wald test is clearly preferred to the likelihood ratio test used in the context of a VECM model, unless the sample size is extremely small. (Would we really want to be going through all of this with a very small sample, especially when cointegration is a long-run phenomenon?)

The big take-home message from this research is very simple:

"We find that the practice of pretesting for cointegration can result in severe overrejections of the noncausal null, whereas overfitting [that's the T-Y methodology; DG] results in better control of the Type I error probability with often little loss in power." (Clarke & Mirza, 2006, p.207.)

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Clarke, J. A. and S. Mirza (2006). A comparison of some common methods for detecting Granger noncausality. Journal of Statistical Computation and Simulation, 76, 207-231.

Dolado, J. J and H. Lütkepohl (1996). Making Wald tests work for cointegrated VAR systems. Econometric Reviews, 15, 369-386.

Toda, H. Y. and P. C. B. Phillips (1994). Vector autoregressions and causality: a theoretical overview and simulation study. Econometric Reviews, 13, 259-285.

Toda, H. Y. and T. Yamamoto (1995). Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.

Zapata, H. O. and A. N. Rambaldi (1997). Monte Carlo evidence on cointegration and causation. Oxford Bulletin of Economics and Statistics, 59, 285-298.

© 2011, David E. Giles