752 Microeconometrics: Methods and Developments
Note that (14.26) and (14.27) imply:
yi=diy 1 i+( 1 −di)y 0 i=y 0 i+αidi. (14.28)Since only one ofy 1 iandy 0 iare observed,αiis not observable. Instead, the goal
is to estimate population averages ofαi, notably the average treatment effect (ATE):
αATE=E[αi], (14.29)and the average treatment effect on the treated (ATET):
αATET= E[αi|di= 1 ]. (14.30)These are conceptually quite different quantities. ATET gives the average gain in
earnings for a person who actually receives training. ATE gives the earnings gain
averaged across those who did and those who did not receive the training.
The evaluation problem can be illustrated by decomposing ATET into two
terms as:
αATET={E[y 1 i|di= 1 ]−E[y 0 i|di= 0 ]}−{E[y 0 i|di= 1 ]−E[y 0 i|di= 0 ]}. (14.31)A naive estimate ofαATETuses just the first term. But this ignores the second term,
a selection term that arises if the treated and untreated are different in that, on
average, they would have different untreated outcome. Methods differ according
to whether this selection term can be solely controlled for by regressors, or whether
it additionally depends on unobservables.
Given regressorsx, similar average effects can be defined, now varying with
regressors. The ATE is:
αATE(x)=E[αi|Xi=x]. (14.32)
and the ATET is:
αATET(x)=E[αi|Xi=x,di= 1 ]. (14.33)
Treatment effects are called heterogeneous if these quantities vary with the eval-
uation pointx, and are called homogeneous ifαATE(x) =αATET(x) = α.In
practice, researchers usually report estimates of the population measuresαATE=
E[αATE(x)]andαATET =E[αATET(x)]that average across individuals with differ-
ent characteristics. For example, individual-level estimates of̂αATE(xi)lead to
̂αATE=N−^1
∑N
i= 1 ̂αATE(xi).
A critical simplifying assumption, discussed further below, is that selection is on
observables only. Assuming conditional independence, outcomes are independent
of treatment after conditioning on regressors, so that:
f(yji|xi,di= 1 )=f(yji|xi,di= 0 )=f(yji|xi),j=0, 1. (14.34)This assumption of exogenous selection of treatment (givenx)is often written as
y 0 i,y 1 i⊥di|xiand has several other names, including unconfoundedness and
ignorability. For some purposes it can be weakened to apply to onlyy 0 ior to