I understand the causality as used in microeconomics (in particular IV or regression discontinuity design) and also the Granger causality as used in time-series econometrics. How do I relate one with the another? For example, I have seen both approach being used for panel data (say N=30, T=20). Any reference to the papers in this regard would be appreciated.
Say you have two vectors
Then zt does not Granger cause yt if E(yt|F1,t−1)=E(yt|F2,t−1), i.e. zt cannot help to forecast yt. So the term Granger “causality” is somewhat misleading because if a variable A is useful in forecasting another variable B this does not imply that A actually causes B. See for instance the discussion in Hansen (2014) (p. 319).
As a stupid example, in the morning just before the sun rises the rooster will crow. If you run a Granger causality test on a series of rooster crows and sun rises, you will find that the rooster’s crow causes the sun to rise. But then this can’t really be truly a causal relationship. The reason I labeled this example as “stupid” is provided in the neat comment by Hao Ye. The example is useful to illustrate why an event may Granger cause another but not actually cause it in the sense that microeconometricians understand causation.
Causality in microeconometrics is mainly based on the potential outcomes framework by Donald Rubin (see Angrist, Imbens and Rubin (1996)). From the question it seems that you have read Mostly Harmless Econometrics, so I assume you are familiar with what kind of causal effects the different methods like IV, difference-in-differences, matching, or regression discontinuity designs estimate. Either way, there is no direct link between these microeconometric methods of estimating causal effects and Granger causality for the simple fact that Granger causality is not really causality.
In recent applications of difference-in-differences (DiD) the idea of Granger causality is used in assessing whether there are anticipatory or lagged effects of the treatment. For the usual DiD model that you can find in Mostly Harmless Econometrics (chapter 5, p. 237):
where in this example the indices i, s and t are for restaurants, states and time, whereas Dst is a dummy equal to one for control group restaurants after the treatment. Given that Dst changes at different times in different states, you can test whether past Dst matter in predicting the outcome whilst future Dst do not. The idea is that if there are anticipatory effects, the estimated treatment effect in the usual DiD setting will under-estimate the total effect. Likewise the fading out of a treatment effect over time might be interesting. You can assess this by including K leads and M lags which will capture anticipatory and lagged treatment effects, respectively, in the model:
An application of this is provided in your textbook on the following pages using a study by Autor (2003) who assessed the anticipatory/lagged effects of increased employment protection on firms’ use of temporary workers.
This idea picks up the argument made in coffeinjunky’s answer. When we can already credibly make the point that there is a causal effect, we can use the idea of Granger causation to further explore the effect like Autor (2003) does. It cannot be used to prove it though.