Category Mostly Harmless Econometrics: An Empiricist’s Companion

The Selection Problem

We take a brief time-out for a more formal discussion of the role experiments play in uncovering causal effects. Suppose you are interested in a causal “if-then” question. To be concrete, consider a simple example: Do hospitals make people healthier? For our purposes, this question is allegorical, but it is surprisingly close to the sort of causal question health economists care about. To make this question more realistic, imagine we’re studying a poor elderly population that uses hospital emergency rooms for primary care. Some of these patients are admitted to the hospital. This sort of care is expensive, crowds hospital facilities, and is, perhaps, not very effective (see, e. g., Grumbach, Keane, and Bindman, 1993)...

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Random Assignment Solves the Selection Problem

Random assignment of Dj solves the selection problem because random assignment makes Dj independent of potential outcomes. To see this, note that

E[yj |dj = 1] – E[Yj|Dj =0] = E[yli IDj = 1] – E[Yoj|Dj =0]

= E[yij |Dj = 1] – E[Yoj|Dj = 1],

where the independence of Yoj and Dj allows us to swap E[Yoj|Dj = 1] for E[Yoj|Dj = 0] in the second line. In fact, given random assignment, this simplifies further to

E [Yij|Dj = 1] – E [Yo j | D j = 1] = E [y ij – Yoj|Dj = 1]

= E [yij – Yoj] .

The effect of randomly-assigned hospitalization on the hospitalized is the same as the effect of hospitalization on a randomly chosen patient. The main thing, however, is that random assignment of Dj eliminates selection bias...

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Regression Analysis of Experiments

Regression is a useful tool for the study of causal questions, including the analysis of data from experiments. Suppose (for now) that the treatment effect is the same for everyone, say Yii — Yoi = p, a constant. With

constant treatment effects, we can rewrite equation (2.1.1) in the form

Y i = a + p D; + Pi;

(2.3.1)

E(Y0i) (Yii – Y0i) Y0i – E(Yoi)

where р; is the random part of Yo;. Evaluating the conditional expectation of this equation with treatment status switched off and on gives

E [Y; ID; = 1] = a + p + E [р; ID; = 1]

E [Y; ID; =0] = a + E [р; |D; =0] ,

so that,

E[Y;|D; = 1] – E[Y;|D; = 0] =

treatment effect

+ E[P;|D; = 1] – E[P;|D; =0] selection bias

Thus, selection bias amounts to correlation between the regression error term, р;, and the regressor, D;. Since

E [P;|D; = 1] ...

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