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 Winshipâ€™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 \(\delta\).

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