Design-Based Uncertainty for Quasi-Experiments
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Social scientists are often interested in estimating causal effects in settings where all units in the population are observed (e.g. all 50 US states). Design-based approaches, which view the treatment as the random object of interest, may be more appealing than standard sampling-based approaches in such contexts. This paper develops a design-based theory of uncertainty suitable for quasi-experimental settings, in which the researcher estimates the treatment effect as if treatment was randomly assigned, but in reality treatment probabilities may depend in unknown ways on the potential outcomes. We first study the properties of the simple difference-in-means (SDIM) estimator. The SDIM is unbiased for a finite-population design-based analog to the average treatment effect on the treated (ATT) if treatment probabilities are uncorrelated with the potential outcomes in a finite population sense. We further derive expressions for the variance of the SDIM estimator and a central limit theorem under sequences of finite populations with growing sample size. We then show how our results can be applied to analyze the distribution and estimand of difference-in-differences (DiD) and two-stage least squares (2SLS) from a design-based perspective when treatment is not completely randomly assigned.
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