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 $y$?


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