endogenous selection bias | Robert Pickett

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} = μ_{0} + (μ_{1} - μ_{0}) D_{i} + {v_{i}^{0} + D_{i} (v_{i}^{1} - v_{i}^{0})}

$Y_i = \mu_0 + (\mu_1 - \mu_0)D_i + \{v^0_i + D_i(v^1_i-v^0_i)\}$ where

D_{i}

$D_i$ is binary assignment to treatment for individual

i

$i$ ,

μ_{0}

$\mu_0$ is the expected outcome for control,

μ_{1}

$\mu_1$ is the expected outcome for treated,

μ_{1} - μ_{0}

$\mu_1 - \mu_0$ is the Average Treatment Effect or

δ

$\delta$ .