Let's consider selection bias again. Suppose you're interested in the causal effect of attending a prestigious private school on future earnings, and you plan to estimate the following model:

Yi = ? + ?Pi + ?i,(eqn. 5)

where Yi measures yearly future earnings in Canadian dollars, Pi is a dummy variable equal to one if respondent i attended private school (treatment group) and zero if attended public school (control group). ?i is the error term.

a)Use potential outcomes notation {Y1i,Y0i} and conditional sample averages to derive the selection bias associated with this treatment-control group comparison.

b)Suppose you collect some observational data and estimate equation (5). The non-randomness of the decision to attend a prestigious private school complicates the causal interpretation of the coefficient of interest. Give two examples of a non-random reason to attend private school that may result in ?(hat)/= ? .

Next consider Table 2 below, which includes observational data on future earnings (including unobserved counterfactual earnings) and non-random treatment status for a sample of 4 people.

person 1 person 2 person 3 person 4 Actual (observed) future earnings (Yi ) $100,000 $115,000 $60,000 $85,000 Potential outcome with private school (Y1i) $100,000 $115,000 $65,000 $35,000 Potential outcome with public school (Y0i) $100,000 $105,000 $60,000 $75,000 Private school attendancelPi) 0 1 0 1