754 Microeconometrics: Methods and Developments
explainyin the absence of treatment. This more parametric method has the advan-
tage over matching of not requiring common support for the propensity score and
permitting extrapolation beyond just the sample at hand.
The preceding methods rely on the untestable assumption of conditional inde-
pendence, and presume that the dataset is rich with many control variables, since
observables alone are assumed sufficient to control for treatment selection. Should
these conditions fail, which will be the case in many potential applications, the
previous methods are invalid. For example, the OLS estimator in the simple homo-
geneous effects model (14.37) is inconsistent if the treatment indicator variable
is correlated with the error term even after conditioning on regressorsx.Itis
then necessary to allow for treatment to additionally depend on unobservable
individual heterogeneity, with different methods used to control or eliminate the
unobservables.
Panel data fixed effects estimators, possible if data are available for more than
one period, control for unobserved heterogeneity by assuming that only the time-
invariant component is correlated with treatment. A panel data version of the
homogeneous effects model (14.37) is:
yit=αdit+x′itβ+φi+δt+εit, (14.38)where herexitdoes not include a constant and the intercept has both an individual-
specific componentφiand a time-specific componentδt. Assume that treatment
ditis correlated with the unobservableφi, so OLS is inconsistent, but is uncorre-
lated withuit. Thenαcan be consistently estimated by OLS estimation of the first
differences model:
yit=αdit+δt+x′itβ+εit, (14.39)or by estimation of a mean differences model (the fixed effects estimator), sinceφi
has been eliminated. This standard method presumes panel data are available and
is restricted to homogeneous treatment effects.
The related differences-in-differences method is applicable to repeated cross-
sections, as well as panel data. For simplicity, suppose there are just two periods,
sayt=a(after) andt=b(before), and that all individuals are untreated in the
first period and some are treated in the second period. Lety ̄jtdenote the average
outcome for treatment groupj=0, 1 in periodt=a,b. The outcome changes
over time by(y ̄ 1 a−y ̄ 1 b)in the treated group and by(y ̄ 0 a−y ̄ 0 b)in the untreated
group. The differences in these differences provides an estimate of ATET, called the
differences-in-differences estimator. This estimator is the OLS estimator ofαin the
model:
yit=γ+αdi+βet+uit,t=a,b,wherediis a binary treatment indicator andetis a binary time period indicator.
Consistency of this estimator requires strong assumptions regarding the role of
unobservables. In terms of (14.38) it is assumed that treatment selection does not
depend onεitand that, while it may depend onφi, on average plim(φ ̄ja−φ ̄jb)=
- The method can be extended to estimate heterogeneous effectsαATET(x)by