Palgrave Handbook of Econometrics: Applied Econometrics

A. Colin Cameron 755

grouping onxand then calculating within each group the four relevant averages
ofy.
Sample selection models explicitly specify a distribution for the unobservables.
These introduce a latent variable to explain treatment choice, where the latent
variable includes an unobserved component (or error) that is correlated with the
error in the outcome equation. A linear model that permits heterogeneous effects
and selection on unobservables is:

y 1 i=x′iβ 1 +u 1 i

y 0 i=x′iβ 0 +u 0 i

d∗i=z′iγ+vi, (14.40)

wheredi=1 if the latent variabled∗i >0, anddi=0 otherwise. A homogeneous
effects version restrictsy 1 i=y 0 iaside from a difference ofαin the intercept. Under

the assumption that(u 0 i,u 1 i,vi)are joint normal (withσv^2 = 1 ), some algebra
yields:

E[y 1 i|x,d∗> 0 ]=x′iβ 1 +σ 1 vλ(z′iγ),

E[y 0 i|x,d∗≤ 0 ]=x′iβ 0 −σ 0 vλ(−z′iγ), (14.41)

whereλ(z′γ)=φ(z′γ)/#(z′γ)is an inverse Mills ratio term, withφ()and#()denot-
ing the standard normal density and distribution functions, andσjv=Cov[uji,vi].
From (14.41) consistent estimates ofβ 1 andσ 1 vcan be obtained by OLS estima-
tion for the treated sample ofy 1 onxandλ(z′̂γ), wherêγis obtained by probit
regression ofdonz. Similarly, OLS regression for the untreated sample ofy 0 onx
and−λ(−z′̂γ)gives consistent estimates ofβ 0 andσ 0 v. These estimates can then
be used to estimate:

αATET(x,z)=x′i(β 1 −β 0 )+(σ 0 v−σ 1 v)λ(z′iγ).

The fundamental weakness of this sample selection approach is its reliance on
distributional assumptions. These assumptions can be modified and relaxed, but
even then the assumptions are still felt to be too strong.
Yet another method for control for selection on unobservables is instrumental
variables estimation. Returning to the homogeneous effects model (14.37), the
problem is that the regressordis correlated with the erroru. Assuming there is an
instrumentzthat does not belong in the model, so E[u|x,z]=0, but is correlated
with the treatment indicatord, the treatment effectαcan be consistently estimated
by IV regression ofyonxanddwith instrumentsxandz.
A related method is the local average treatment effect (LATE) estimator. Begin
with the homogeneous effects model with dependence onxdropped for simplicity,
so that:
yi=β+αdi+ui. (14.42)

Assume there is an instrumentzwith E[u|z]=0 and definep(z)=Pr[d= 1 |Z=z]=
E[d|Z=z]. Then:
E[y|z]=β+αp(z).