# endogenous selection bias

## Regression and Potential Outcomes

Introduction What follows is an example I worked up when trying to figure out how to communicate potential outcomes in a regression framework to students graphically. This discussion is derived from Morgan and Winship’s 2015 book1, especially pages 122-123. The goal is to represent the potential outcomes framework using standard regression notation, and then to discuss endogenous selection bias using this framework. Math Define $$Y_i = \mu_0 + (\mu_1 - \mu_0)D_i + \{v^0_i + D_i(v^1_i-v^0_i)\}$$ where $$D_i$$ is binary assignment to treatment for individual $$i$$, $$\mu_0$$ is the expected outcome for control, $$\mu_1$$ is the expected outcome for treated, $$\mu_1 - \mu_0$$ is the Average Treatment Effect or $$\delta$$.