Causality in microeconometrics versus granger causality in time-series econometrics 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. 
 A: I fully agree with Andy, and I was actually thinking of writing something similar, but then I started to wonder myself about this topic. I think we all agree that Granger causality itself really has not much to do with causality as understood in the potential outcomes framework, simply because Granger causality is more about time precedence than anything else. However, suppose there is a causal relationship between $X_t$ and $Y_t$ in the sense that the former causes the latter, and suppose that this happens along a temporal dimension with a lag of one period, say. That is, we can easily apply the potential outcomes framework to two time series and define causality in this way. The issue then becomes: while Granger causality has no "meaning" for causality as defined in the potential outcomes framework, does causality imply Granger causality in the time series context? 
I have never seen a discussion on this, but I think if you or any researcher wants to make a case for this, you need to impose some additional structure. Clearly, the variables need to react sluggishly, i.e. the causal relationship here must not be simultaneous but defined with a lag. Then, I think, it might be reassuring to not reject Granger causality. While this is clearly no evidence in favour of a causal relationship, if you were to claim such, then I would take the GNC test as subjective evidence. 
A: Say you have two vectors
$$
\begin{align}
F_{1,t} &= (y_t, y_{t-1}, y_{t-2},...) \newline
F_{2,t} &= (y_t, z_t, y_{t-1}, z_{t-1},...)
\end{align}
$$
Then $z_t$ does not Granger cause $y_t$ if $E(y_t | F_{1,t-1}) = E(y_t | F_{2,t-1})$, i.e. $z_t$ cannot help to forecast $y_t$. 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):
$$Y_{ist} = \gamma_s + \lambda_t + \beta D_{s,t} + X'_{ist}\pi + \epsilon_{ist}$$
where in this example the indices $i$, $s$ and $t$ are for restaurants, states and time, whereas $D_{st}$ is a dummy equal to one for control group restaurants after the treatment. Given that $D_{st}$ changes at different times in different states, you can test whether past $D_{st}$ matter in predicting the outcome whilst future $D_{st}$ 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:
$$Y_{ist} = \gamma_s + \lambda_t + \sum^{M}_{m=0}\beta_{-m} D_{s,t-m} + \sum^{K}_{k=1}\beta_{+k} D_{s,t+k} + X'_{ist}\pi + \epsilon_{ist}$$
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.
