Granger-Causality Should be Renamed
It’s not because of an off-color tweet by Sir Clive W. J. Granger before his rise to fame, unlike many modern-day cancellations.
It’s because Granger himself suggested that the term “Granger Causal” is inappropriate to describe the effect observed. He wrote this a decade after his flagship paper (which, matching his humble personality, did not suggest an eponymous name) and four and a half decades later in his Nobel Prize lecture. He, along with the broader statistics community, knew that correlation is not synonymous with causation. Granger-causality should be “Granger-correlation-which-sometimes-exists-because-of-causation”, or alternatively, “temporal correlation.”
As defined, Granger-causality is a very specific definition of causality that does not align with the everyday, philosophical use of the word; hence its prefix. But the core still causes confusion. To see why the implication of causality is inappropriate, let us recall the definition of Granger-causality.
A time series Granger-causes a second when it provides useful information about future values of a second series.¹ In other words, their correlations are shifted in time. For example, the number of unemployment claims may Granger-cause GDP. Previous observations of unemployment claims consistently provide information about upcoming GDP, such that we have a better picture of GDP today if we know what the number of unemployment claims were last month or quarter. It is natural to assume that unemployment claims cause a change in GDP, because they most likely do.
But take another example. If you have knee problems like me, you’ll be able to predict an impending thunderstorm due to pain caused by changes in atmospheric pressure. The time series of joint discomfort precedes and provides information about the time series of storm conditions, thus providing a Granger-causal relationship. Even though the pain precedes the rain, we know that aching knees do not cause a thunderstorm.
Granger-causality isn’t really causality. It’s a relationship shifted in time. One series leading another consistently (or, conversely, lagging the other) only provides information about how the two series move together. In finance this notion is called “co-movement”; in macroeconomics, a “leading indicator”. Both may be jargon but are nonetheless accurate descriptors.
Leading indicator and co-movement are practically synonymous with Granger-causality, so why bother with the formal eponym?
For one, Granger-causality can be bidirectional, implying a feedback mechanism between the two series. Stock prices are a leading indicator of GDP, but GDP is also a leading indicator of stock prices. Intuition prohibits a leading indicator to be led by itself. A more accurate yet less simple statement is that GDP and stock prices have a “bi-directional Granger-causal” relationship.
Precision, not simplicity, is the statistician’s objective for terminology. “Leading indicator” and “co-movement” simply don’t cut it. But if precision is the goal, well, Granger-causality is even further from the mark!
I suggest terminology that is both precise and generalizable: temporal correlation. Temporal reminds us that the series are functions of time, for which the application is intended. Correlation reminds us of the nature of co-occurrence without inferring causation. Best of all, this expression comes courtesy of Granger himself (1977, p. 225).
Temporal correlation does not suggest a direction. Series A may lag series B or series B lag series A, but “A is temporally correlated to B” suggests no more or less information than “B is temporally correlated to A”. Any inference of philosophical causality is avoided.
Moreover, we may be specific about direction by using “lag” and “lead” in place of the generic “temporal”: B may have a lagging correlation to A, but A not a lagging correlation to B.
For a feedback relationship, we may say they have bi-directional correlation.
Temporal correlation can be useful in prediction, regardless of a true causal relationship. Knee pain is temporally correlated to rain and does indeed help forecast an impending storm. We do not care about causation between the two, only the usefulness in prediction.
Econometric literature has seen vast treatment on the difference between Granger-causality and true causal relationships. This notion is not new to anyone in the time series field, but the name can cause newcomers to stumble. James Hamilton suggested describing the relationship as “whether y helps forecast x rather than whether y causes x” (1994, p. 308). Frank Diebold suggests the term “predictive causality”, which differs from the implied “philosophical causality” (2007, p. 230).
Granger himself has written extensively about this subject. Consider his 1980 paper, “Testing for Causality: A Personal Viewpoint”, in which he debates on the merits of each naming convention. Ultimately, he believed that “cause”, like every other word, is a template that is populated with meaning through context. He championed the case against jargon — indeed, a noble cause.
The Gaussian Distribution has a lay name of the “Normal” Distribution. So too should we honor the legacy of Sir Clive W.J. Granger, but use an alternative, common-sense name. “Granger-correlation” has a nice ring to it, but a doesn’t provide enough differentiation. My submission, reflecting Granger’s own, is temporal correlation. That he chose “causality” for its simplicity over “temporal correlation” for its accuracy is a long-lasting irony.
 Informally, series X Granger-causes series Y if
That is, the MSE when conditioned on set X is noticeably lower than the MSE when X is not included.
Technically, X Granger-causes Y if the F-test rejects the null hypothesis Ho = γ_1 = γ_2 = … = γ_p = 0, with these γ coming from the OLS estimates of the model:
such that the lagged x observations’ added information is worth the added complexity. The F statistic will have degrees of freedom p and T-2p-1.
Granger, C. W. J. (1969). “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods.” Econometrica, vol. 37, no. 3, pp. 424–438.
Granger, C. W. J.; Newbold, Paul (1977). Forecasting Economic Time Series. New York: Academic Press. p. 225. ISBN 0122951506.
Granger, Clive W.J. (2003) “Time Series Analysis, Cointegration, and Analysis.” Nobel Lecture. www.nobelprize.org/uploads/2018/06/granger-lecture.pdf.
Granger, C. W. J. (1980) “Testing for Causality: A Personal Viewpoint.” Journal of Economic Dynamics and Control 2, June 1980, pp. 329–352., doi:https://doi.org/10.1016/0165-1889(80)90069-X.
Hamilton, James D. (1994). Time Series Analysis. Princeton University Press. pp. 304–308. ISBN 0–691–04289–6.
Diebold, Francis X. (2007). Elements of Forecasting (4th ed.). Thomson South-Western. pp. 230–231. ISBN 0324359047.
Many thanks are due to Eric Boerman and Cameron McWilliams for their insight and edits. Any errors or oversights that remain are my own. I do not speak on behalf of my employer, the Federal Reserve Board of Governors, or any of its members; all opinions are my own. I deeply admire Sir Clive W.J. Granger and wish no harm to his legacy.