Statistics Tutorial, Part 8.
Causality Testing

(c) 2017 by Barton Paul Levenson

There's no actual statistical test that can prove one variable causes another. But a sort of limited causality test is possible with certain assumptions. Economists talk about "Granger causality," proposed by C.W.J. Granger in 1969. You look for an effect from past values of another variable. We assume causality only runs from past to present. So if past values of X help explain Y, then X "Granger-causes" Y.

Sims (1972) invented a test for Granger causality. You regress your Y variable on past values of itself. Then you regress it on past values of itself and past values of X. If significantly more variance is accounted for, Bingo! It doesn't absolutely prove X causes Y, but at least it "Granger-causes" it.

"Granger-causality" assumes the earlier phenomenon causes the later one. It might seem obvious that the one which happens first is the cause, but it isn't always true. The late economist David Tobin gave this example: People who listen to weather reports take umbrellas with them when rain is expected. Nearly always, the people take the umbrellas with them first, and the rain happens later. But the decision to carry the umbrellas doesn't cause the rain. The anticipated rain causes the decision, so here the effect precedes the cause.

But let's ignore that. In most cases, cause comes before effect.

"The Sims partial-F test for Granger causality" is a long name, so I'll just call it the Sims test. First you regress your effect variable on its own autoregression terms--perhaps price inflation on inflation one and two years ago. Then you add lagged values of the presumed cause--say, money supply growth one and two years ago. Data from the two regressions can be combined to yield an F statistic (details in Appendix 6). If the F value exceeds the appropriate critical threshold, we say the added variable "Granger-causes" the independent variable--here, money-supply growth Granger-causes inflation (if, in fact, it does).

The question then arises, how long a lag period is appropriate? You can decide this with "information criteria," such as the Akaike Information Criterion (AIC, and an improved version called "second-order AIC," abbreviated AICc), and the Schwarz Bayesian Information Criterion (SIC or SBC). Appendix 6 shows how to calculate both. For every autoregression, you can calculate AIC and SIC values, and the appropriate lag for the variable of interest is found by selecting the lag period that has the lowest AIC or SIC value. Minimize the information criterion and you've found the appropriate lag period.

The AIC and SIC don't always give the same answers. Choose one, tell people which you're using, and run with it.

I'll give an example without showing the calculations. I have annual money supply data back to 1898 and inflation data back to 1930. For each variable, I calculate autoregressions of varying lag periods from 1 to 10 years and calculate the AIC, AICc and SIC. To make all the autoregressions comparable, I start them all in 1940. Thus the period covered is 1940-2004, for N = 65 points.

The lowest value in each column has a star next to it. For inflation, the AIC and AICc recommend a 10-year lag while the SIC recommends 4 years. For money supply growth, the AIC recommends 4 years while the AICc and SIC recommend 2 years.

To keep the results comparable, and because the maximum lag length here is ten years, I did all the Granger tests over the period 1940-2004 (N = 65 years). Here are the results:

In the tests above, only dM causing dP with a four-year lag is significant at the 5% level; the rest are insignificant. Thus we can conclude that money Granger-causes inflation, and not vice versa.