Title: Partial Identification of Individual-Level Parameters Using Aggregate Data in a Nonparametric Binary Outcome Model

Abstract: It is well known that the relationship between variables at the individual level can be different from the relationship between those same variables aggregated over individuals. This problem of aggregation becomes relevant when the researcher wants to learn individual-level relationships but only has access to data that has been aggregated. In this paper, I develop a methodology to partially identify linear combinations of conditional average outcomes from aggregate data when the outcome of interest is binary while imposing very few restrictions on the underlying data generating process. I construct identified sets using an optimization program that allows for researchers to impose additional shape and data restrictions. I also provide consistency results and construct an inference procedure that is valid with aggregate data, which only provides marginal information about each variable. I apply the methodology to simulated and real-world data sets and find that the estimated identified sets are too wide to be useful, but become narrower as more assumptions are imposed and data aggregated at a finer level is available. This suggests that to obtain useful information from aggregate data sets about individual-level relationships, researchers must impose further assumptions that are carefully justified or seek out data aggregated at the finest level possible.