Testing for Granger Causality

Several people have asked me for more details about testing for Granger (non-) causality in the context of non-stationary data. This was prompted by my brief description of some testing that I did in my "C to Shining C"  posting of 21 March this year. I have an of example to go through here that will illustrate the steps that I usually take when testing for causality, and I'll use them to explain some of pitfalls to avoid. If you're an EViews user, then I can also show you a little trick to help you go about things in an appropriate way with minimal effort.

In my earlier posting, I mentioned that I had followed the Toda and Yamamoto (1995) procedure to test for Granger causality. If you check out this reference, you'll find you really only need to read the excellent abstract to get the message for practitioners. In that sense, it's rare paper!

It's important to note that there are other approaches that can be taken to make sure that your causality testing is done properly when the time-series you're using are non-stationary (& possibly cointegrated). For instance, see Lütkepohl (2007, Ch. 7).

The first thing that has to be emphasised is the following:

If you are using a Wald test to test linear restrictions on the parameters of a VAR model, and (some of) the data are non-stationary, then the Wald test statistic does not follow its usual asymptotic chi-square distribution under the null.

In fact, if you just apply the test in the usual way, the test statistic's asymptotic distribution involves 'nuisance parameters' that you can't observe, and so it is totally non-standard. It would be very unwise to just apply the test, and hope for the best on the grounds that you have a large sample size.

Of course, testing for Granger (non-) causality is just a specific example of testing some zero restrictions on certain of the parameters in a VAR model, so the warning given above applies here. (Parenthetically, you can't get around the problem by using an LM test or an LR test, either.)

What I'm going to do is:

• Remind you of what we mean by Granger non-causality testing.
• Spell out the steps that are involved in applying the Toda-Yamamoto (T-Y) procedure.
• Illustrate the analysis with a simple example, including some screen-shots from EViews.
• List a few things that you should not do when testing for causality.

First, a simple definition of Granger Causality, in the case of two time-series variables, X and Y:

"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.

Note that in what follows I''ll often refer to the 'levels' of the data. This simply means that the data have not been differenced. The series may be in the original units, or logarithms may have been taken (e.g., to linearize a trend). In either case, I'll talk about the 'levels'.

Now, here are the basic steps for the T-Y procedure:

1. Test each of the time-series to determine their order of integration. Ideally, this should involve using a test (such as the ADF test) for which the null hypothesis is non-stationarity; as well as a test (such as the KPSS test) for which the null is stationarity. It's good to have a cross-check.
2. Let the maximum order of integration for the group of time-series be m. So, if there are two time-series and one is found to be I(1) and the other is I(2), then m = 2. If one is I(0) and the other is I(1), then m = 1, etc.
3. Set up a VAR model in the levels of the data, regardless of the orders of integration of the various time-series. Most importantly, you must not difference the data, no matter what you found at Step 1.
4. Determine the appropriate maximum lag length for the variables in the VAR, say p, using the usual methods. Specifically, base the choice of p on the usual information criteria, such as AIC, SIC.
5. Make sure that the VAR is well-specified. For example, ensure that there is no serial correlation in the residuals. If need be, increase p until any autocorrelation issues are resolved.
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, preferably using Johansen's methodology (based on your VAR) for a reliable result.
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. Now 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. It's essential that you don't include the coefficients for the 'extra' m lags when you perform the Wald tests. They are there just to fix up the asymptotics.
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. That is, a rejection supports the presence of Granger 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."

So, if your data are cointegrated but you don't find any evidence of causality, you have a conflict in your results. (This might occur if your sample size is too small to satisfy the asymptotics that the cointegration and causality tests rely on.) If you have cointegration and find one-way causality, everything is fine. (You may still be wrong about there being no causality in the other direction.) If your data are not cointegrated, then you have no cross-check on your causality results.

Now it's time for our example. As usual, the data are available on the Data page that goes with this blog, and there is an EViews workfile on the Code page. We're going to take a look at the world prices of Arabica and Robusta coffees. Here's a plot of the monthly data from January 1960 to March 2011 - a nice long time series with lots of observations:

It looks as if there may be a structural break in the form of a shift in the levels of the series in 1975. We know that this will affect our unit root and cointegration tests, and it will also have implications for the specification of our VAR model and causality tests. This can all be handled, of course, but rather than getting side-tracked by these extra details, I'll focus on the main issue here, and we'll shorten the sample as follows:

Now let's go through the various steps for the T-Y causality testing procedure. The results to back up what I conclude along the way are in the EViews file, which contains a 'Read_me" text object that gives more explanation.

1. Both of the series are I(1) when we apply the ADF and KPSS tests, allowing for a drift and trend in each series.
2. So, m = 1.
3. We set up a 2-equation VAR model in the levels of the data, including an intercept in each equation.
4. The various information criteria suggest that we should have a maximum lag length of 3  for each variable:

  5.  However, when we then examine the residuals and apply the LM test for serial independence against the alternative of AR(k)/MA(k), for k = 1, ...., 12, we find that there are problems. This serial correlation is removed (at least at the 5% sig. level) if we increase the maximum lag length to p = 6:

This estimated model is also 'dynamically stable':

  6.  Johansen's Trace Test and Max. Eigenvalue Test both indicate the presence of cointegration between the 2 series, at the 10% significance level:

  7.  This last result is not going to affect anything we do.

  8.  As m = 1, we now re-estimate the levels VAR with one extra lag of each variable in each equation.

Here is where we need to be careful if we're going to "trick" EViews into doing what we want when we test for causality shortly. Rather than declare the lag interval for the 2 endogenous variables to be from 1 to 7 (the latter being p + m), I'm going to leave the interval at 1 to 6, and declare the extra (7th.) lag of each variable to be an "exogenous" variable. The coefficients of these extra lags will then not be included when the subsequent Wald tests are conducted. If I just specified the lag interval to be from 1 to 7, then the coefficients of all seven lags would be included in the Wald tests, and this would be incorrect. If I did that, the the Wald test statistic would not have its usual asymptotic chi-square null distribution.

 

  9. & 10. Now we can undertake the Granger non-causality testing:

 
  11. Note that the degrees of freedom are 6 in each part of the above image - that's correct: p = 6. The extra 7th. lag has not been included in the tests.
  12.  From the upper panel of results, we see that we cannot reject the null of no causality from Robusta to Arabica. From the lower panel we see that we can reject the null of no causality from Arabica to Robusta, at the 10% significance level, and virtually at the 5% significance level as well.

In summary, we have reasonable evidence of Granger causality from the price of Arabica coffee to the price of Robusta coffee, but not vice versa.

Some things to watch out for:

• Don't fit the VAR in the differences of the data when testing for Granger non-causality.
• If you are using a VAR model for other purposes, then you would use differenced data if the series are I(1), but not cointegrated.
• If you are using a VAR model for purposes other than testing for Granger non-causality and the series are found to be cointegrated, the you would estimate a VECM model.
• The usual F-test for linear restrictions is not valid when testing for Granger causality, given the lags of the dependent variables that enter the model as regressors.
• Don't use t-tests to select the maximum lag for the VAR model - these test statistics won't even be asymptotically std. normal if the data are non-stationary, and there are also pre-testing issues that affect the true significance levels.
• If you fail to use the T-Y approach (adding, but not testing, the 'extra' m lags), or some equivalent procedure, and just use the usual Wald test, your causality test results will be meaningless, even asymptotically.
• If all of the time-series are stationary, m = 0, and you would (correctly) just test for non-causality in the 'old-fashioned' way: estimate a levels VAR and apply the Wald test to the relevant coefficients.
• The current Wikipedia entry for Granger Causality has lots of things wrong with it. In particular, see the 'Method' & 'Mathematical Statement' sections of that entry

Finally, if you want to check things out some more, I've put a second EViews workfile, relating to the prices of natural gas in Europe and the U.S., on this blog's Code page. In that file you'll find a "Read_Me" object that will tell you what's going on.

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

Lütkepohl, H. (2006). New Introduction to Multiple Time Series Analysis. Springer, Berlin.

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

© 2011, David E. Giles