Category Mostly Harmless Econometrics: An Empiricist’s Companion

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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Making Regression Make Sense

’Let us think the unthinkable, let us do the undoable.

Let us prepare to grapple with the ineffable itself,

and see if we may not eff it after all.’

Douglas Adams, Dirk Gently S Holistic Detective Agency (1990)

Angrist recounts:

I ran my first regression in the summer of 1979 between my freshman and sophomore years as a student at Oberlin College. I was working as a research assistant for Allan Meltzer and Scott Richard, faculty members at Carnegie-Mellon University, near my house in Pittsburgh. I was still mostly interested in a career in special education, and had planned to go back to work as an orderly in a state mental hospital, my previous summer job...

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