statistics and it is this literature that influences econometric analyses
of causal models.
The ‘‘Rubin model’’ is thus a version of this classical econo-
metric model without explicit specification of the decision model for
choice of treatment. Sobel claims that statisticians such as
Rosenbaum (2002) are now using the framework developed by econ-
omists some 30 years before. If true, this is a welcome acquisition
from economics by statistics.
As previously noted, Sobel also claims that Rubin explicitly
discusses treatment assignment rules. In fact, Rubin relies on the
crutch of randomization to define his models and only vaguely
describes other assignment rules as ‘‘not randomized.’’
23
One needs
to look to Gronau (1974) and Heckman (1974, 1976, 1979, 1978) for
explicit development of selection models with explicit treatment
assignment and selection rules. Had Rubin understood the general
selection model, he would not have advocated matching or balancing
to overcome nonrandomized assignment as he does in his 1974 and
1978 papers. Heckman and Navarro (2004), Heckman and Vytlacil
(2006b), and Heckman, Urzua, and Vytlacil (2006) show the bias that
arises from using matching when a general selection rule characterizes
treatment choice.
5. COMPARISON OF ESTIMATORS
My analysis in Section 4 was only intended to illustrate the basic point
that each evaluation estimator makes assumptions. Sobel misinter-
prets this section as my attempt to write an exhaustive survey instead
of my attempt to illustrate some points from the earlier part of the
paper. In Section 4, I focus attention on certain mean treatment effect
parameters because of their familiarity and simplicity. Heckman,
LaLonde, and Smith (1999) and Heckman and Vytlacil (2006b) pre-
sent comprehensive surveys of the econometric approach.
Sobel confines his discussion to a few mean treatment para-
meters, ignoring the range of parameters introduced in the earlier
23
Rubin (1978) discusses the distribution of treatment assignment rules
in a general way but never develops their properties in the systemic, formal way it
is developed in economics.
150
HECKMAN
sections of my essay. His discussion is selective and he seizes on small
points to make objections to my paper. He misses key developments
in the econometrics literature that show that in models with hetero-
geneous responses, IV and selection models are closely related
(Heckman and Vytlacil 2005; Heckman, Urzua, and Vytlacil 2006).
I use separability in my analysis of selection models in Section
4 only to simplify the exposition. Matzkin (2006) presents a compre-
hensive discussion of nonparametric identifiability in nonseparable
selection (and other) models.
His contrast between matching and control functions on this
issue is specious and ignores an entire recent semiparametric literature
in econometrics (see Heckman and Vytlacil 2006a; Matzkin 2006).
Selection models do not require normality, separability or standard
exclusion restrictions in order to be identified.
He takes out of context my claim that control functions are
more general than matching. That claim is made under a series of
assumptions about the separable selection model and was not
intended as a general characterization.
The recent semiparametric literature by Heckman (1980, 1990),
Powell (1994) and Carneiro, Hansen, and Heckman (2003) does not
rely on normality or functional form assumptions. On this point Sobel
inaccurately characterizes the econometrics literature. Even in the
absence of distributional assumptions, no exclusions are needed to
identify the Roy model, contrary to his claims. In his notation, a
model with Z
¼ X can be identified using curvature restrictions with-
out any exclusion of variables. See Heckman and Honore´ (1990).
24
The key idea underlying the control function approach intro-
duced in Heckman (1980) and in Heckman and Robb (1985, 1986) is
to model the relationship between the unobservables in the treatment
choice equation and the unobservables in the outcome equations
rather than to assume they are independent given a specified set of
variables as is done in the matching literature. Sobel inaccurately
compares matching and selection estimators in terms of the number
of assumptions invoked by each method and not in terms of their
strong implications.
24
Sobel relies on a flawed survey by Vella (1998) which does not accu-
rately portray the econometric selection literature.
REJOINDER: RESPONSE TO SOBEL
151
Matching assumes that, on average, the marginal person and
average person with the same observed conditioning variables
respond the same to treatment (TT
¼ ATE ¼ TUT). It assumes that
the analyst knows the right conditioning set and uses it. Selection
models allow for variables that produce conditional independence
invoked in matching to be unobserved by the analyst (see Carneiro,
Hansen, and Heckman 2003). Sobel’s analysis of IV also ignores the
entire body of recent econometric work which establishes what instru-
mental variables estimate in the general nonseparable case (see
Heckman and Vytlacil 1999, 2001a, 2006b). I now turn to that work.
6. THE UNIFYING ROLE OF THE
MARGINAL TREATMENT EFFECT
Sobel has evidently not read my 2001 Nobel Lecture or my work with
Vytlacil (1999, 2001a,b, 2005, 2006a,b). Had he done so he would
not claim that ‘‘sociologists will usually be more interested in
Treatment on the Treated (TT) or ACE (Average Causal Effect) than
the Marginal Treatment Effect (MTE).’’ In rereading my essay, I now
realize it was a mistake for me not to discuss my work with Vytlacil in
my paper.
Vytlacil and I establish that the marginal treatment effect
(MTE) is a device that unifies the evaluation literature. From knowl-
edge of the MTE, analysts can interpret what IV estimates as well as
the commonly used treatment effects, OLS and matching estimators
as a different weighted average of the marginal treatment effect.
Under the assumptions clearly stated in our papers, we establish
that all treatment effects and all estimands (probability limits of IV,
matching, OLS, control function estimators) can be expressed as
weighted averages of the MTE with known weights, i.e., weights
that can be estimated from the sample data. Letting MTE(x, u) be
the MTE for a given value of X
¼ x (observables) and U ¼ u
(unobservables), we may write the estimand or treatment effect j
given x, Á
j
(x) as
Á
j
ðxÞ ¼
ð
a
b
MTE
ðx; uÞ!
j
ðx; uÞdu
ð1Þ
152
HECKMAN
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