# Aggregate effects of randomized experiment

Consider the following two-equation model for observational data from a randomized experiment:

(1) $$y_{ij} = \alpha + \beta D_{ij} + \gamma X_{j} + \varepsilon_{ij}$$

(2) $$D_{ij} = \gamma + \delta L_{ij} + \eta X_{j} + \upsilon_{ij}$$

$$y_{ij}$$ is an outcome of interest of project $$i$$ that is part of lottery $$j$$. $$D_{ij}$$ is a randomized binary treatment variable with imperfect compliance allocated via the lottery mechanism. $$X_j$$ is a vector of indicator (dummy) variables for each lottery $$j$$; this makes sure effects are only estimated within a given lottery $$j$$. Because the decision to comply with the treatment is potentially endogenous, one can instrument $$D_{ij}$$ with $$L_{ij}$$, which is the lot number drawn from the lottery. $$L_{ij}$$ is a strong instrument for $$D_{ij}$$, suggesting widespread compliance.

Suppose I am interested in an aggregate outcome $$y_c$$, say on the county-level, that potentially comprises multiple projects $$i$$ from different lotteries $$j$$. That is, I would like to estimates

(3) $$y_c = \alpha + \beta D_c + \gamma X^j_c + \varepsilon_c$$,

where $$D_c$$ is 1 if D_{ij} of any project in any lottery is 1, and 0 otherwise. $$X^j_c$$ is a vector of indicator variables for the number of projects that enter each lottery $$j$$.

Is there an existing literature addressing under which conditions such an "aggregation" still yields meaningful causal effects of $$D$$ on $$y$$?