explicating what he calls the ‘‘Rubin model,’’ Holland gives a very
revealing illustration of how the first two tasks of Table 1 are con-
flated by one leading figure in the statistical treatment effect litera-
ture. Holland claims that there can be no causal effect of gender on
earnings. Why? Because we cannot randomly assign gender. This
confused statement conflates the act of definition of the causal effect
(a purely mental act) with empirical difficulties in estimating it (Steps 1
and 2 in my Table 1). This type of reasoning is prevalent in statistics.
As another example of the same point, Rubin (1978, p. 39)
because a randomization cannot in principle be performed.
defined by a randomization. Issues of definition and identification are
confused. A recent paper shows that this fallacy is alive and well in
statistics. A paper by Berk, Li, and Hickman (2005) makes the same
error as Rubin and Holland. Sobel is correct in saying that population
treatment parameters can be defined abstractly. However, that point
was not made in the statistical treatment effect literature. It is made in
I agree with Sobel that the act of definition is logically separate
my paper. We both agree that a purely mental act can define a causal
effect of gender. That is a separate task from identifying it. What is
odd is that he states his agreement with my position and that of the
econometrics literature as a disagreement. And he fails to accurately
Parenthetically, my title ‘‘Scientific Causality’’ was motivated by
understand the ‘‘causes of effects’’ and the statistical treatment effect literature.
Understanding the causes of effects is an essential activity for prediction and
forecasting—problems P2 and P3 in my paper.
‘‘Without treatment definitions that specify actions to be performed
ments.’’ (Rubin 1978, p. 39).
The LATE parameter of Imbens and Angrist (1994) is defined by an
and Vytlacil (2001b, 2005, 2006b) define the LATE parameter abstractly and
separate issues of definition of parameters from issues of identification. Imbens
and Angrist (1994) use instrumental variables as surrogates for randomization.
REJOINDER: RESPONSE TO SOBEL
to the myth that causality can only be determined by randomization,
and that glorifies randomization as the ‘‘gold standard’’ of causal
4. THE ROY MODEL, THE SWITCHING MODEL AND THE
Sobel repeats an assertion made by Rubin: that I, and other econo-
mists, ‘‘started using the Rubin model in the 1980s.’’
‘‘Rubin model’’ is in fact a version of an econometric model developed
by Roy (1951). It is also a version of the switching regression model of
Quandt (1958, 1972). That model contains both a framework for
potential outcomes (Y
) and also a choice of treatment rule.
There was no explicit discussion of the treatment assignment rule in
until very recently.
Heckman and Honore´ (1990) present a comprehensive analysis
extend it (see also Heckman 2001, and Heckman and Vytlacil
2006a,b). Unlike the statisticians, Pearl (2000) is forthright about his
own debt to the economics literature in the distinction between ‘‘fix-
ing’’ and ‘‘conditioning,’’ which is central to his work on causality. See
Haavelmo (1943) for the source of Pearl’s ‘‘do’’ operator.
As noted in my essay, and in Heckman (1992), self selection provides
See Rubin (2000).
One cannot find any explicit analysis of treatment selection rules in the
statistical literature (Neyman 1923; Rubin 1978; Holland 1986; Rubin 1986) other
than the randomized-nonrandomized dichotomy previously discussed.
Sobel cites Rosenbaum (2002) for use of such rules. As previously
does not go deeper, nor does he consider how the form of the treatment assign-
ment rule affects the choice of an appropriate estimator. That point is developed
in Heckman and Robb (1985, 1986).
Lewis (1963) is an early pioneering analysis of counterfactuals in
comes and a decision rule (treatment assignment rule). In its simplest
version, the treatment indicator variable is D
might assign treatment on the basis of which therapy has the best
outcome. A student may decide to go to college vs. stopping at high
school based on which option has the highest income. The Roy model
is a version of the competing risks model of biostatistics.
Rubin, as does the Thurstone (1927) model of counterfactual utilities
of choices developed in mathematical psychology.
More general versions of this model developed in econometrics
allow agents to be partially informed about (Y
) when they make
their decisions and to allow for more general costs. In the generalized
Roy model, D
information set, C is the cost of moving from ‘‘0’’ to ‘‘1’’ where ‘‘0’’
is the initial state, and g is a general preference function for the agent
making the treatment decision. In the original Roy model C
I ¼ ðY
Þ and g ¼ (Y
). The general form of this model
allows analysts to distinguish objective from subjective evaluations
of treatments and ex ante and ex post versions of both. See Carneiro,
Hansen, and Heckman (2001, 2003), Cunha and Heckman (2006a),
Cunha, Heckman, and Navarro (2005, 2006), Heckman and Vytlacil
(2006b) and Heckman and Navarro (2006) for more general analyses.
In my 1974 paper, Y
is the market wage of a woman. Y
to work (D
¼ 1) if the market wage is greater than the reservation
a model for hours of work. Willis and Rosen (1979) use this model
is college earnings and Y
is high school earnings. They allow
for costs C. D
¼ 1 (a person goes to college) if Y
À C > 0
À C ! 0)).
There is a huge literature starting in
See Heckman (1987) where this link is established. Versions of the
See Gronau (1974) for a closely related model.
They assume perfect certainty. See Cunha and Heckman (2006a) for a
version of this model with uncertainty as well as additional features.
REJOINDER: RESPONSE TO SOBEL