# Question about the conceptual sampling distribution for one time events

in this paper: Do Political Protests Matter? Evidence from the Tea Party Movement*, the authors use rainfall on the day of the tea party protests as a source of plausibly exogenous variation in rally attendance, i.e. an as an instrumental variable.

I have a question conceptually about this - what exactly in a scenario like this does we think about with the sampling distribution? The way I see it there are two potential ways to think about the conceptual sampling distribution:

1. rainfall fell as it did across the U.S. on the day of the protests. Take that distribution as fixed. Now given how rainfall actually fell on that day, we can think of resampling and forming a distribution.

This would also have implications such as checking rainfall on that day and whether its uncorrelated with observables providing strong evidence of the identifying assumptions.

1. Rainfall is a part of the underlying dgp we are modeling with an equation like $$y = \beta Rainfall + \epsilon$$. So it is not just how rainfall happened to fall on that day as fixed, but the idea would be if we hypothetically went back in time and let the day play out again and again, rainfall would fall different ways each time, and this would generated sampling variability. In this case then, what matters is if the geoclimactic determinants of rainfall is a process that 'assigns' rain, and in each iteration/resampling - i.e. going back and starting the day over again- the county assignment of rainfall would be different. If this is the case, then looking at rainfall across time would be important to show that the process generating rain doesn't systematically correlate with determinants of y.

I hope those two ideas made sense, mainly so that someone can correct my logic or point me in the right direction for thinking of these types of things. Are one of the two the 'correct' way of thinking about the sampling distribution?

In the above picture, $$X$$ is the instrument, $$Y$$ the explanatory variable of interest, $$Z$$ the outcome, and $$U$$ the unobserved confounding variable. The idea is that we can use the exogeneity in $$X$$ to induce exogenous variation in $$Y$$ that could explain $$Z$$. So when we consider asymptotic arguments and thereby allow our sample to grow we operate under the framework that we can keep drawing from $$(X,Y,Z)$$ and exploit the covariation that we find.
So if $$X$$ is rainfall we are essentially assuming we are sampling from the population of rainfall. That population distribution is fixed but we only observe some realization of it on the day. So the sampling distribution may not be the same as the population distribution and may vary. As you said, we could think of this as saying that we are seeing the same day repeat over and over again and observing the amount of rainfall and the population of responses. In fact, this kind of counterfactual is exactly what we need to identify some causal effect.
The thing is we need to rely on this DGP for identification. Since exclusion cannot be tested in-sample (it is a property of the population) we need to hope that the DAG above correctly summarizes the joint relationships. Now looking at the DGP we can see that for the identification argument discussed in the last paragraph to hold it is important that the exclusion restriction holds for the population distribution. That is, we can't have an edge connecting $$X$$ and $$Z$$ that does not travel through $$Y$$. That is, we need that exclusion holds for the population of rainfall.