Palgrave Handbook of Econometrics: Applied Econometrics

Andrew M. Jones 559

where 1 denotes treatment and 0 denotes control.^1 The pure treatment effect
cannot be identified because the counterfactual can never be observed: each indi-
vidual is either treated or untreated at a particular point in time so only one of the
potential outcomes can be observed. The outcome that is actually observed can be
written in terms of the potential outcomes:

yit=y^0 it+dit(yit^1 −y^0 it), (12.2)

whereditis an indicator of treatment.
One response to the problem of defining a counterfactual is to concentrate on
the average treatment effect (ATE), comparing the average outcomes between the
treated and controls:
ATE=E

(

yit^1 −y^0 it

)

. (12.3)

When there is heterogeneity in individual responses to the treatment that may
influence the assignment of treatment, for example, when doctors select patients
on the basis of their capacity to benefit, attention is likely to focus on the average
treatment effect on the treated (ATET) rather than the ATE:

ATET=E

(

y^1 it−yit^0 |dit= 1

)

. (12.4)

This is the average effect of treatment for those individuals who would actually
select into treatment.
Moving towards a regression framework, assume that the observed outcome
under the two treatment regimes is given by the general regression model:

yit=fj(xit,ujit),j=0, 1. (12.5)

The vectorxincludes observable factors that influence the outcome and may influ-
ence the assignment of treatment (reflecting “selection on observables”). Theuare
unobservable factors that influence the outcomes and may influence the assign-
ment of treatment (“selection on unobservables”). Formulating the problem in
this way requires the SUTVA (stable unit-treatment value assumption) to hold – an
individual’s potential outcomes and treatments are independent of others in the
population, ruling out spillover and general equilibrium effects. These spillovers
may be important in some health economics applications and the evaluation of
treatment effects would then have to be designed to accommodate them (see
Chandra and Staiger, 2007; Miguel and Kremer, 2004). Using linear functions for
f(.)gives a switching regression model:

yit=x′itβj+ujit,j=0, 1. (12.6)

A simplification of this model, which assumes a homogeneous treatment effect so
that only the intercept varies with treatment, gives the regression function:

yit=x′itβ+ditδ+uit. (12.7)

In this case ATE=ATET=δ.