Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers to causality in the context of time series (or, more generally, stochastic processes). Moreover, reading this nice blog post created additional confusion about how to view this approach.
I'm by no means a person knowledgeable about causality, as my fuzzy understanding of the concept consists of partly common sense, common knowledge, some exposure to latent variable modelling and structural equation modelling (SEM) and reading a bit from Judea Pearl's work on causality – not THE book of his, but more along the lines of an interesting overview paper by Pearl (2009), which for some reason, surprisingly, doesn't mention Granger causality at all.
In this context, I'm wondering whether Granger causality is something more general than a time series (stochastic) framework and, if such, what is its relation (commonalities and differences) to Pearl's causality framework, based on the structural causal model (SCM), which, as far as I understand, is, in turn, based on direct acyclic graphs (DAGs) and counterfactuals. It seems that Granger causality can be classified as a general approach to causal inference for dynamic systems, considering the existence of dynamic causal modelling (DCM) approach (Chicharro & Panzeri, 2014). However, my concern is about whether (and, if so, how) it is possible to compare the two approaches, one of which is based on stochastic process analysis and the other is not.
More generally, what do you think would be a sensible high-level approach – if one is possible – for considering all currently existing causality theories within a single comprehensive causality framework (as different perspectives)? This question is largely triggered by my attempt to read an excellent and comprehensive paper by Chicharro and Panzeri (2014) as well as reviewing an interesting causal inference course at the University of California, Berkeley (Petersen & Balzer, 2014).
References
Chicharro, D., & Panzeri, S. (2014). Algorithms of causal inference for the analysis of effective connectivity among brain regions. Frontiers in Neuroinformatics, 8(64). doi: 10.3389/fninf.2014.00064 Retrieved from http://journal.frontiersin.org/article/10.3389/fninf.2014.00064/pdf
Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3, 96–146. doi:10.1214/09-SS057 Retrieved from http://projecteuclid.org/download/pdfview_1/euclid.ssu/1255440554
Petersen, M., & Balzer, L. (2014). Introduction to causal inference. University of California, Berkeley. [Website] Retrieved from http://www.ucbbiostat.com
Best Answer
Granger causality is essentially usefulness for forecasting: 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. GC has very little to do with causality in Pearl's counterfactual sense, which involves comparisons of different states of the world that could have occurred. So Peeps Granger-cause Easter, but they do not cause it. Of course, the two will overlap in a world where there are no potential causes other than X, but that is not a very likely setting and a fundamentally untestable one. Another less restrictive way they can coincide is, if, conditional on the realised history of Y and X, the next realisation of X is independent of the potential outcomes. This point is made in Lechner, M. (2010), "The Relation of Different Concepts of Causality Used in Time Series and Microeconometrics," Econometric Reviews, 30, 109-127 (WP link), which is written in the potential outcomes framework, rather than Pearl's DAGs.
Addendum: Let me make an implicit assumption more explicit. The crucial ingredient for my claim is that Easter does not have a fixed date. Suppose you knew nothing about Easter and wanted to forecast its date next year. From historical data (history of Y), you can see that Easter takes place in the spring. But can we do better than that? Using Peeps sales or marketing data (X) from near the holiday, we can see that peeps do Grange-cause it since that data is useful for forecasting Easter more precisely.
The corollary is that Christmas trees sales do not Granger-cause Christmas since if you know that Christmas took place on December 25th for centuries (adjusting for various calendar reforms and church schisms), tree sales do not help.