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HB31
.M415
i
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department
of economics
CONDITIONAL INDEPENDENCE IN
SAMPLE SELECTION MODELS
Joshua D. Angrist
96-27
October, 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
CONDITIONAL INDEPENDENCE IN
SAMPLE SELECTION MODELS
Joshua D. Angrist
96-27
October, 1996
-
FEE
Conditional Independence in Sample Selection Models
by,
Joshua D. Angrist
MIT, Hebrew
University, and
NBER*
October 1996
'Thanks go
to
Gary Chamberlain
of the author.
Guido Imbens, Whitney Newey, Paul Rosenbaum, Tom Stoker, and especially to
for helpful discussions and comments. Any errors are, of course, solely the work
Conditional Independence in Sample Selection Models
Econometric sample selection models typically use a linear latent-index with constant coefficients
to
model the selection process and the conditional mean of the regression error
A
feature
common
to
most of these models
is
that the conditional
an invertible function of the selection propensity score,
covariates.
i.e.,
mean
in the selected
sample.
function for regression errors
the probability of selection conditional on
Consequently, conditioning on the selection propensity score controls selection bias, a fact
which underlies much of the recent
literature
on non-parametric and semi-parametric selection models.
This literature has not addressed the question of whether the propensity-score conditioning property
necessarily a feature of sample selection models.
properties that
make
it
mechanism and imposes no
The
if
I
describe the conditional independence
resulting characterization does not rely
on a
structure other than an assumption of independence
error term and the regressors in
determine
In this note,
is
possible to use the selection propensity score to control selection bias in a general
sample selection model.
to
is
random samples. This approach
conditioning on the selection propensity score
is
latent
index selection
between the regression
leads to a simple rule that can be used
sufficient to control selection bias.
Joshua D. Angrist
MIT Department
of Economics
Memorial
50
Drive
Cambridge, MA 02139-4307
ANGRIST@MIT.EDU
Keywords: non-random samples, selection
bias, propensity score
1.
Introduction
Sample selection problems occur
interest in
wages
A
in
both experimental and observational studies. Econometricians'
sample selection models originated with research
for workers to infer the distribution of offered
similar problem occurs
wage equations
also affect
when
wages
in labor
economics using the distribution of
for everyone (Gronau,
1
974; Heckman,
employment
interpreted as a form of selection bias
traditional
974).
instrumental variables that are being used to correct for endogeneity in
status (see, e.g., Angrist, 1995). In experiments, the bias
by controlling for "post-treatment variables" that are correlated with the outcome of
The
1
(Ham and
econometric solution
to
interest
induced
can also be
Lalonde, 1996; Rosenbaum, 1984).
problems of
this
type invokes an assumption of joint
normality for the regression error and the error term in a latent-index selection mechanism. This approach
leads to a
added
maximum
likelihood or two-step selection correction in the form of inverse Mills-ratio terms
to the regression function
of interest (see Heckman, 1976 and 1979).
A
number of
semi-parametric variations on this normal selection model have also been introduced (see,
Powell, 1993, and references in
Newey, Powell, and Walker, 1990 and Heckman,
parametric and semi-parametric formulations have two
is
used
to characterize the
mean of unobserved
covariates and the sample selection rule.
explicitly handled
common
threads.
First,
less restrictive
1990).
a latent index framework
regression error terms conditional on the regression
Second, the problem of selection bias
is
either implicitly or
i.e.,
the
1
This paper describes the conditional independence properties that
selection propensity score to control selection bias in regression estimates.
make
it
possible to use the
The problem studied here
includes applications involving experimental data and instrumental variables as special cases.
is
Both the
by conditioning on an invertible function of the selection propensity score,
probability of selection given covariates.
objective
Ahn and
e.g.,
My
first
a nonparametric formulation of the sample selection problem where the only structure consists
'The propensity-score terminology originates with Rosenbaum and Rubin (1983, 1984), who
introduced the propensity-score method of controlling for confounding in evaluation research.
of the regression equation of
interest
and an independence
on the regression
restriction
The
error.
elementary properties of conditional independence, as discussed by Dawid (1979) and Florens and
Mouchart (1982), are then applied
to this formulation.
I
believe this approach provides a very general
description of the identification problem in sample selection models.
rule that can be
It
also leads to an easily interpreted
used to check when nonparametric conditioning on the selection propensity score
is
indeed
sufficient to control selection bias.
2.
Motivation
The equation of
y,
where
e,
= X,'P +
y, is
e„
everybody.
(1)
the dependent variable and X,
population.
in the
interest is
The
We
is
the k x
vector of regressors, assumed to be independent of
observe a random sample of n observations on
coefficient vector
expectation function, so that
if
we
is
assumed
did observe
necessary to justify this definition of P, the
selection bias (e.g., Chamberlain, 1986).
to be
y,
for
X„
but
y, is
not observed for
such that
X/P
gives the population conditional
everyone
in the
sample, P could be consistently
Although independence of regressors and errors
estimated by ordinary least squares (OLS).
to
1
full
independence assumption
At the end of the paper
I
show
is
customary
that full
in
is
not
papers on
independence appears
be of fundamental importance for identification strategies that use the selection propensity score to
control selection bias.
Let w, indicate selection status
(i.e.,
of offered wages). The selection problem
that X, affects w, as well as
yj.
is
w=
l
indicates workers in an application
T|, is
y- is
the log
often motivated using a latent index to capture the possibility
In particular, suppose that
w,= irX/6-T^X)]
where
where
an error term that could be correlated with
(2)
e,
but
is
assumed
to
be independent of X,. Then,
we have
E[e,
X„ w,=l] =
|
E[e,
|
X„ X,'5 >
Selection bias arises because the term E[e,
E[e,
=
X,]
|
in
*
tjj
w=l]
X,,
I
0.
is
generally a function of X, even though
random samples.
Parametric solutions to selection problems of this type typically begin by assuming that the error
terms (£„
are jointly normally distributed, so that
ti,)
X„w = l]
E[e,|
where p
(Ej, T|,)
is
the coefficient from a regression of
with X,,
E[tli
where
and
<j)
= pE[v,|x„X,'8>TU
I
we
(3)
e,
on v
Because of normality and joint independence of
.
;
can simplify further:
X,'S > TiJ = -^(X,'5)/a)(X,'5)
are the normal density
=
XCXi'8),
and distribution functions for
x\
.
{
to (1) provides a feasible solution to the selection
can be estimated from a Probit regression of
w
problem
on
(
Critics of traditional econometric selection
point out that these models
combine a wide
X
and/or 8) for which there
selection
model have
is
context because the required parameters
.
(e.g.,
Hartigan and Tukey, 1986;
Little,
1985)
variety of assumptions that are hard to assess and interpret.
on distributional assumptions, functional form
(3
as a regressor
(
models
In particular, the validity of parametric corrections for
on
in this
Adding A^X/8)
sample selection bias clearly turns
restrictions,
rarely, if ever,
and exclusion
any empirical
restrictions
justification.
led econometricians to develop less restrictive
2
(i.e.,
to a large extent
zero restrictions
These features of the normal
models for selection problems. For
example, Newey, Powell, and Walker (1990), Lee (1982), and others have shown that the distributional
assumptions
in these
model can be relaxed
One approach
in a
semi-parametric framework.
to semi-parametric selection corrections begins with the observation that
"Mroz (1987) documents the
also Olsen (1980, 1982).
even
fact that empirical results can be sensitive to these assumptions.
in
See
the absence of parametric distributional assumptions, selection-correction terms typically consist of non-
linear but invertible functions
of the selection propensity score, P[W;=1
models has been noted by Heckman and Robb (1986), and
selection
On
This feature of selection
motivates the semi-parametric estimators for
models discussed by Newey (1988), Robinson (1988), Choi (1992), and Ahn and Powell (1993). 3
the other hand, earlier
work on semi-parametric
propensity-score conditioning property
I
it
ZJ.
|
is
models has not
selection
clarified
whether the
a necessary feature of the selection problem. In the next section,
use simple conditional independence notation to provide a general characterization of models for which
conditioning property holds.
this propensity-score
3.
Conditional independence and sample selection
Conditional independence
defined as follows:
is
Definition 1 (Florens and Mouchart, 1982).
common
Let
and only
Note
A A
2,
denote random variables defined on a
3
probability space with joint probability measure P[A,,
some values of
conditional probability statement for A, given
if
A,,
if
P[A,
|
A A
2,
3]
= P[A,
|
A
3 ],
A A
A
2,
2
and
and
3 ],
A
3
.
let
2
1
A
3
,
A A
2,
Then we write A,
is
transitive.
3]
denote a
JJ
A
2
1
A
3
The conditional independence
concept, although basic and perhaps even obvious, has proven widely useful.
A
|
a.s.
that conditional independence, like independence,
the notation, A, JJ
P[A,
and argued that many important ideas
in statistics
Dawid (1979) popularized
can best be understood
in
these terms. Chamberlain (1982) and Florens and Mouchart (1982) used conditional independence ideas
to explore the relationship
between alternate definitions of "causality"
Educational researchers and
statisticians
in time-series econometrics.
have also discussed conditioning on the selection
propensity score; see, for example, Powell and Steelman (1984) and Wainer (1986).
And
Rosenbaum and Rubin (1983) used
result for evaluation
The
Lemma
1.
R
Let R,, R,,
(i)
R,
UR
(ii)
R,
n
(iii)
R,
II R-4
is
problems with selection on observables.
2
I
R
I
R
in
(i)
and
and
3
common
R,
II R-4
I
(
R
2>
probability space with joint
R3)
R2
R, II
I
(R 3
,
R4)
Florens and Mouchart (1982), presented here in
(1979).
and
be random variables defined on a
R3
I
in
R4
the following are equivalent:
and
3
Dawid
equivalence of
3,
Then
RJ
(R*
Theorem A.l
Lemma 4.3
4
properties of conditional independence can be characterized as follows:
probability measure.
This
conditional independence arguments to establish the propensity-score
(iii)
Dawid
states the equivalence
by using symmetry of
R
2
and
of (ii) and
R4
somewhat simpler
(iii)
notation, and
only, but this also implies the
in part (ii).
3.1 Selection bias
The problem of
selection bias can be described in conditional independence terms with no
structure other than the regression of interest
and the independence
restriction
on
£,.
This restriction
is
stated formally below:
Assumption
e,
1.
X, where
U
Let {(y„ X,')'; i=l,
8,
=
(y,-X,'(3)
4
and
.
.
E&J
.,
n} denote
i.i.d.
observations on a (k+l)xl random vector. Then
X,]=X,'p\
The propensity-score result for evaluation research says that in an experiment where treatment is
randomly assigned conditional on covariates. it is enough to condition on the probability of selection
given covariates to eliminate confounding from any relationship between the covariates and the
treatment assignment.
We
would
first like to
know when
Instead of observing {(y„ X,')'; i=l,
w, indicate selection status as before.
{(w,y„ Wj, X/)'; i=l,
Proposition
lemma imply
Then given Assumption
X,
|
JJ
w,
JJ
w,
implies
X,.
=
If
e,
Then
E[e,
X,
£,
e,
JJ
1,
w,)|
X„
I
and E(e,
I
w = l)
is
w,)
XJ =
JJ
w,
,
X,
w,
|
X,
I
|
e,
e,
JJ
while
s,
iff
e,
R 2 = X;, R
w„
we need
observe
Uw
w,
3
X,
X,
|
JJ
Note
w,.
R =w„
parts
4
e,
JJ
X,
is
implies either
e,
is
a non-trivial
X,.
Since
,}-
|
|
and
=£2,
to rule out
JJ
w,
JJ
X,]
|
w,
with
(i)
X,
X,
w,
|
l
|
|
XJ
[E(e,
I
w,= l)-E( e|
takes on 2 or
more
|
w,=0)]P[w,=l
|
distinct values since E(e,
I
w)
mechanism
fail in
that
well-known
when
e,
X,
is
|
w,
means
independent of the regression error conditional on
is
also a function of X,.
Conversely,
sample, selection status must be related to the dependent variable.
models with homoscedastic errors
(in
which
will be
JJ £,
if
Assumption
Assumption
Of course,
case,
1
X,
I
Thus, any selection
X,.
related to the dependent variable conditional on the regressors causes
for latent-index
not zero
is
w^O).
the selected sample if selection status
in the selected
is
is
e,
t
random sample. The property w,
that selection status
or
XJ.
consistent or unbiased in the selected sample as well as in a
means
JJ
that
which are based on Assumption
JJ
e,
w,
|
that estimators
£,
of the
(iii)
maintained,
e, JJ
e, JJ
and
XJ
w=0) +
E(e,|
not equal to £(£,
The property
we
n},
.,
w,, this implies
cannot be true when P[w =
this
e,
{£, JJ
always implies
X,
|
|
X,} iff
and
e, JJ
Setting R,=e„
the proposition,
E[E(e,
X,
JJ
w,
I
w,
XJ =
= E[E(e,|
But
JJ
£, JJ
|
with
w,
JJ
E,
X„
|
To complete
or both.
|
w,
{£, JJ
.
of support and that P[w,= l
that X, has at least 2 points
Let Q. denote the sample space.
Proof.
.
Let
to fail.
1
Important consequences of the selection process are summarized below:
n}.
.,
.
Suppose
1.
function of X,.
w,
.
a sample selection process causes Assumption
w,
JJ £,
1
1
to
fails
the first point
I
X,
means
that
Ej
and the latent-index error
are general features of
are uncorrelated).
ri;
sample selection problems. In
The proposition shows
that this
and the converse
particular, these properties hold regardless
of the
underlying selection mechanism or model.
Proposition
effects
1
For
applies equally to experimental data.
illustration,
where y^y.o+P and y i0=u+e, denote the counterfactual outcomes
individuals. Let X, be an indicator of randomized treatment assignment.
a simple comparison of means provides an unbiased estimator of (3 in
at risk
By
assume constant treatment
virtue of random assignment,
random samples from
the population
of treatment. But the independence between randomized treatment assignment and counterfactual
outcomes
is
lost
whenever the experiment
is
analyzed conditional on a variable that
outcomes and affected by the treatment assignment. Suppose
experiment, correlated for with
pairs include death
and health
given X, means that
e,
^ w
implies that estimation of
(3
for
yj
some reason
status, or
Xj.
I
If,
as
conditional on
1
is
seems
w
is
that
is
;
is
{
Correlation between
also affected by
necessarily subject to selection bias.
also relevant for
is
IV
estimation.
for
is
independent of
Rosenbaum (1984) makes
e,
Proposition
1
5
IV estimation
are that in cases
£, in
in the target population.
where the sample selection
rule
the selected sample if and only if
conditional on the instruments.
a similar point regarding the consequences of conditioning on "post-
treatment variables" in experiments.
e,
that the instruments (playing the role of X,) are
involves the instruments, the instruments will be independent of
selection status
and
;
For applications involving
independent of the regression error term (defined around an endogenous regressor)
1
X„ then
w
s
instrumental variables, the key identifying assumption
The implications of Proposition
both correlated with
a second outcome variable in the
and wages.
status
likely, v/
w
is
Examples of such correlated outcome
other than X,.
employment
;
Finally, note that Proposition
5
and nontreated
for treated
3.2
The
selection propensity score
The
selection propensity score
the selection propensity score
e(X,)
which
is
= P(w,
used
The
E[y,
Of course,
is
X,),
below:
selection propensity score
is
said to control selection bias if
for this to be
may
E[e,
|
w,= l,
JJ
X,
I
w„
e(X,).
be possible to estimate
e(X,)].
(3
(4)
of any practical use, that must be enough variability
In the literature
fixed.
it
e,
sample because then we have
X„ w,= l] = X/p +
I
fact that
emphasized by the notation
is
conditioning on the selection propensity score controls selection bias,
in the selected
The
the conditional probability of selection given X,.
a function of X,
is
in the definition
Definition 2.
If
|
is
on semi-parametric selection models,
priori exclusion restrictions on
(3
in
X, to identify
this variation is
(3
once e(X,)
obtained by imposing a
and/or on e(X,). See Chamberlain (1986) for a precise statement of the
restrictions required for identification in this context.
Here
I
focus on Definition 2 as a likely starting
point for any possible strategy to identify p.
The
selection
models
Proposition
Proof.
relationship between Definition 2 and other the conditional independence properties of sample
2.
is
outlined in the following proposition:
The following
U
(i)
w,
(ii)
(e„ w,) II X,
(iii)
e,
First
U
X,
X,
|
|
are equivalent:
e„ e(X,)
w„
observe that w,
|
e(X,)
e(X.)
U
X,
|
e(X,) because
P[w = l
8
|
X„
e(X,)]
=
e(X,) and that
£,
TJ
X,
|
e(X,)
because
X
e, ]J
{(i), e,
and e(X,)
s
H
X,
Therefore, since w,
established.
is
e(X,)}
|
U
X,
a function of X,.
»
(ii) <=>
{(iii),
e(X,) and e
|
:
U
lemma
Moreover,
U
w,
X,
X,
1
implies:
e(X,)}.
|
(6)
e(X,) hold automatically in this case, the result
I
is
6
Line
(iii)
of Proposition 2
same
the
is
as Definition 2,
i.e.,
this part
of the proposition consists of
the statement that conditioning on the selection propensity score controls selection bias. Lines
(i)
of the proposition therefore provide equivalent necessary and sufficient conditions for Definition
X
that while the statements w, JJ
w)
stronger statement that (e ,
independence
is
JJ
;
X,
e(X,) automatically hold, line
|
are jointly independent of X, given
;
;
e(X,) and e
|
s
Of
e(X,).
(ii)
and
2.
(ii)
Note
makes
the
course, this joint
not implied by marginal independence. Therefore, one consequence of the if-and-only-if
relationship between
(ii)
and
that conditioning
(iii) is
on the selection propensity score does not necessarily
control selection bias.
The equivalence of
lines (i)
sufficient condition for (i)
Proposition
support of
3.
X„
also has practical consequences because line
given in Proposition
is
Suppose the selection mechanism
easy to
In particular, a
3:
monotonic,
is
status.
(i) is
i.e.,
for any
two values
x', x° in
the
either
P[w = l
P[w,= l
where conditioning on
e, is
1
|
x
j
x', e,]
,
£,]
> P[w=l
< P[w,=
what makes these
of selection controls selection
6
(iii)
models for the mechanism determining selection
verify in the context of specific
weak
and
|
x°, £,], a.s.
or
x°, e,], a.s.,
inequalities
(7)
random. Then conditioning on the probability
bias.
Choi (1992) shows that part
error and the regression error
l
|
(ii)
[i.e.,
of Proposition 2
replacing w, with
[joint
r|,
independence], imposed on a latent-index
in (2)], implies part (iii).
Proof.
Note
=
J
that if e(x')=e(x°),
{P[w,= l
|
x
1
,
e(x'), e,]-P[w,=l
x°, e(x'), e,]}h(e,)de,
|
Given a monotonic selection mechanism, the quantity
to equal zero,
that for a
we must
therefore have
P[w =
l
|
in brackets is
x', e(x'),
£,]=P[w = l
always non-negative. For the integral
This argument shows
x°, e(x'), £,].
|
monotonic selection mechanism,
P[w = l
x
|
1
,
£,]=P[w,= l
|
x°, e,]
whenever
e(x')=e(x°).
The proposition can then be established by observing
the probability of selection
To
get an idea of
(8)
that (8) implies line
(i)
of Proposition
conditionally independent of X, given e(X,) and
is
how
general this result
is,
constant coefficients and errors independent of X,
2, i.e., that
£,.
note that any latent-index selection mechanism with
is
monotonic.
assumptions about the error distribution.
Monotonicity
models as well. Consider, for example, a
latent index
is
This fact does not depend on other
satisfied
by a wide range of
model with random
less restrictive
coefficients:
w^irx/s.-Ti.x)],
which can be re-written,
w, = lrXi'S'-Tii'x)],
where
8*
=
E[5,]
and
r\*
=
+
r\
X,(8*-8,)
an error term that
is
is
l
independent of
As long
X,.
as the coefficient
8,
has the
not independent of X, even if
same sign
for
all
i,
T),
is
Proposition 3 implies that
conditioning on the selection propensity score controls selection bias in this model.
Finally,
it
is
worth emphasizing the role played by
full
independence
in obtaining the result that
conditioning on the selection propensity score controls selection bias. Suppose that instead of (1)
we have
a heteroscedastic regression model,
y,
where
E,
= X,'P +
is still
[X,'y]e,
=
X,'(3
+ £,
assumed independent of X
(9)
.
;
The heteroscedastic
10
error term,
£,, is
only mean-independent
of
X,.
In this case, conditioning
X
X„w,=l] =
E[y,|
=
SitfxJ w„
Even
if E[e,
converge
w = l,
still
+ [X/YMe,
X,'(3
w = l,
|
X,]
|
is
w,=l,e(X,)].
(10)
a function of X, and not just e(X,).
In other words,
e(X,).
e(X,)]
The
to (3+ye.
mechanism
4.
I
+ E[^| w=l, XJ
'(3
1
Equation (10) means that E[^,
on the probability of selection does not control selection bias because
is
fixed at a
number
e,
estimates in the selected sample are biased and
identification failure occurs in this case in spite of the fact that the selection
satisfies part (i)
of Proposition
2, i.e.,
w,
TJ
X,
|
e(X,).
^,,
Conclusions
This paper lays out some of the basic conditional independence properties of sample selection
models, focusing on those that justify conditioning on the selection propensity score to control selection
bias.
One
reason
I
think this approach to sample selection models
general formulation of key identifying assumptions. Another
for the validity
it
leads to a
is
that
weak
it
leads to a very
sufficient condition
X„ and
£,),
Finally, the results given here are cast in terms of potentially observable
and not inherently unobservable
researchers could in principle construct an experiment that
An
that
valuable
of identification strategies that either explicitly or implicitly involve conditioning on the
selection propensity score.
quantities (w,,
is
is
additional contribution of this paper
may
latent index error terms.
would
test the identifying
This means that
assumptions.
be to help bridge the gap between the approaches
taken by econometricians' and statisticians' to the problem of selection bias. Use of the propensity score
to control
confounding
in evaluation research
statisticians are familiar
with this idea.
was introduced by Rosenbaum and Rubin (1983) and many
In a discussion of econometric
models for program evaluation,
Holland (1989, page 876) notes that econometricians also use the propensity score, but he suggests that
this
use
is
"quite different for the
two approaches."
Similarly,
11
Heckman and Robb (1986) argue
that the
econometric approach to selection models uses the propensity score "in a different way than that advocated
by Rosenbaum and Rubin [1983]."
This paper shows that differences between the role played by the
propensity score in the econometrics and statistics literature
In evaluation research, conditioning
may
not be as great as previously believed.
on the propensity score controls for the bias caused by an association
between treatment status and observed exogenous covariates, while
the propensity score controls for the bias caused by an association
exogenous covariates.
In
in selection
models, conditioning on
between selection status and observed
both cases, conditioning on the propensity score solves a problem of
confounding that arises because of dependence between a
covariates.
12
dummy
endogenous regressor and other
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Date Due
Lib-26-67