756 Microeconometrics: Methods and Developments
Evaluating at two pointszandz′and subtracting yields the LATE:
αLATE(z)=
E[y|z]−E[y|z′]
p(z)−p(z′). (14.43)
This can be estimated by comparing averages of the outcomeyand treatment indi-
catordat two different values of the instrumentz.Ifzis binary then this estimate
is the same as the IV estimate. The estimate can be extended to heterogeneous
effects, providedp(z)is monotonic inz. Then it differs from IV and will vary with
the points of evaluationzandz′. A more general treatment effect is the marginal
treatment effect (MTE):
αMTE(x,z)=∂E[y|x,Z]
∂Pr[d= 1 |x,Z]∣∣
∣
∣
Z=z,which gives the mean treatment effect for those at the margin of choosing treat-
ment. ATE, ATET and LATE can be shown to be different weighted averages of
MTE.
A final method is regression discontinuity (RD) design. Suppose treatment occurs
when a variablescrosses a threshold ̄s, so thatd= 1 (s>s ̄), and the outcomeyalso
depends ons. For example, a government program to improve school outcomes
may be applied to schools in low-income areas. A method is developed to calculate
a score, and schools with a score below a certain threshold receive the government
program while those with a higher score do not. A complication is that school
outcome will directly depend on this score. The obvious approach is to comparey
for those withsjust less than ̄sto those withsjust greater than ̄s, but this will use
only a small fraction of the data. Instead usêαfrom the least squares regression:
yi=β+αdi+γh(si)+ui, (14.44)whereh(·)is a flexible function that is specified (for example, polynomial) or is esti-
mated by nonparametric methods. Given the discrete nature of the discontinuity
ats ̄it is clear that the method can also be used when effects are heterogeneous and
will estimate ATE=E[αi|si]under mild additional assumptions. Another extension
is to fuzzy designs where the threshold ̄sis not sharp, as some individuals with
s<s ̄are treated and some withs> ̄sare untreated. Intuitively, if a fractionf
of the population in the immediate vicinity of ̄sswitch from untreated to treated
then ATE is estimated byftimes the estimated OLS coefficient ofdin (14.44). This
adaptation is qualitatively similar to that for LATE in (14.43).
The literature on treatment effects is vast. Econometricians have contributed to
the literature on all the preceding methods, and the sample selection, IV and LATE
methods originated in econometrics. Early econometrics papers, that generally did
not explicitly use the current treatment effects framework, include Ashenfelter
(1978), Heckman (1978, 1979), Heckman and Robb (1985), Lalonde (1986) and
Björklund and Moffitt (1982). Heckman, Ichimura and Todd (1997) and Dehejia
and Wahba (1999) emphasize matching methods. Abadie and Imbens (2006) pro-
vide results for inference. Bertrand, Duflo and Mullainathan (2004) demonstrate