A. Colin Cameron 757that when difference-in-difference methods are used with panel data it is critical
that one uses cluster-robust standard errors that cluster on the treatment unit, often
the state, as the treatment regressor is highly correlated over time. Hausman and
Kuersteiner (2008) consider more efficient GLS estimation in this setting. Imbens
and Angrist (1994) introduce LATE and Björklund and Moffitt (1982) and Heck-
man and Vytlacil (2005) introduce MTE. Hahn, Todd and Van der Klaauw (2001)
provide theory for RD methods; Ludwig and Miller (2007) provide a detailed appli-
cation, and Imbens and Lemieux (2008) provide a survey. More recent research
provides distribution theory when a nonparametric component is used and seeks
to extend methods to nonlinear models (for example, Athey and Imbens, 2006),
and to multiple treatments. Brief surveys include Smith (2000), Blundell and Dias
(2002) and Angrist (2008), while lengthier surveys include Heckman, Lalonde and
Smith (1999) and Angrist and Krueger (1999). The forthcoming book by Angrist
and Pischke (2009) focuses on treatment effects methods.
14.5.2 Instrumental variables methods
Consider the linear model:
yi=x′iβ+ui, (14.45)
where Cor[xi,ui] = 0 so that OLS is inconsistent. Assume there exists an instrument
zisuch that Cor[zi,ui]= 0. The IV estimator for a just-identified model, considered
for simplicity, is:
̂βIV=(Z′X)−^1 Z′y. (14.46)
If Cor[zi,ui]= 0 then̂βIVis asymptotically normal with meanβand:
V[̂βIV]=(Z′X)−^1 Z′Z(X′Z)−^1 , (14.47)where=E[uu′|Z]. This estimator is easily extended to overidentified models, and
to nonlinear models as a special case of GMM.
The applied literature has included many creative examples of instrument use.
For example, in earnings–schooling regression a proposed instrument for schooling
is distance to college, as this may be related to college attendance but may not
directly effect earnings. Another possible instrument is birth month, which may
be related to years of schooling as it determines age of school entry and hence years
of schooling before a person reaches the minimum school leaving age.
This interest in the use of IV methods has been somewhat diminished by recog-
nition of the problems that arise when instruments are weakly correlated with the
regressor(s) being instrumented.
A weak instrument is one for which Cor[zi,xi]is small. More precisely, sup-
pose there is one endogenous regressor and several exogenous regressors. Then the
instrument for the endogenous regressor is weak if the correlation between the
endogenous regressor and the instrument is low after partialing out the effect of
the other exogenous regressors. Then it is well known that̂βIVwill be imprecisely
estimated. Two other complications can arise.
First, suppose that Cor[zi,ui]is close to zero rather than exactly zero. Then not
only is the IV estimator inconsistent, but it can be more inconsistent than the