Cointegration & Granger Causality

Today, I had a query from a reader of this blog regarding cointegration and Granger causality. 

Essentially, the email said:

"I tested two economic time-series and found them to be cointegrated. However, when I then tested for Granger  causality, there wasn't any. Am I doing something wrong?"

First of all, the facts:

• If two time series, X and Y, are cointegrated, there must exist Granger causality either from X to Y, or from Y to X, both in both directions.
• The presence of Granger causality in either or both directions between X and Y does not necessarily imply that the series will be cointegrated.

Now, what about the question that was raised?

Truthfully, not enough information has been supplied for anyone to give a definitive answer.

1. What is the sample size? Even if applied properly, tests for Granger non-causality have only asymptotic validity (unless you bootstrap the test).
2. How confident are you that the series are both I(1), and that you should be testing for cointegration in the first place?
3. What is the frequency of the data, and have they been seasonally adjusted? This can affect the unit root tests, cointegration test, and Granger causality test.
4. How did you test for cointegration - the Engle-Granger 2-step approach, or via Johansen's methodology?
5. How did you test for Granger non-causality? Did you use a modified Wald test, as in the Toda-Yamamoto approach?
6. Are there any structural breaks in either of the time-series? These ail likely any or all of the tests that you have performed.
7. Are you sure that you correctly specified the VAR model used for the causality testing, and the VAR model on which Johansen's tests are based (if you used his methodology to test for cointegration)?

The answers to some or all of these questions will contain the key to why you obtained an apparently illogical result.

Theoretical results in econometrics rely on assumptions/conditions that have to be satisfied. If they're not, then don't be surprised by the empirical results that you obtain.

© 2015, David E. Giles