This paper is concerned with the problem of combining a data set that identifies the conditional distribution P (y|x) with one that identifies the conditional distribution P (z|x), in order to identify the regressions E (y|x, middot) identical with [E (y|x, z = j), j element of Z] when the conditional distribution P (y|x, z) is unknown. Cross and Manski (2002) studied this problem and showed that the identification region of E (y|x, middot) can be precisely calculated, when y has finite support. Here we generalize Cross and Manski's result showing that the identification region can be precisely calculated also in the case in which y has infinite support.

Molinari, Francesca

Peski, Marcin

English

Research Papers in Economics

CAE Working Paper #05-09
Generalization of a Result on
“Regression, Short and Long”
                                                   
by
                           
Francesca Molinari
and
Marcin Peski
March 2005
.Generalization of a Result on ￿Regressions, Short and Long￿ ￿
Francesca Molinari
Department of Economics
Cornell Universityy
Marcin Peski
Department of Economics
Northwestern University
March 2005
Abstract
This paper is concerned with the problem of combining a data set that identi￿es the condi-
tional distribution P (yjx) with one that identi￿es the conditional distribution P (zjx); in order
to identify the regressions E (yjx;￿) ￿ [E (yjx;z = j);j 2 Z] when the conditional distribution
P (yjx;z) is unknown. Cross and Manski (2002) studied this problem and showed that the
identi￿cation region of E (yjx;￿) can be precisely calculated, when y has ￿nite support. Here
we generalize Cross and Manski￿ s result showing that the identi￿cation region can be precisely
calculated also in the case in which y has in￿nite support.
Motivation and Results
Applied economists often face the problem that no single data set contains all the variables that
are necessary to conduct inference on a population of interest. When this is the case, they need to
integrate the information contained in di⁄erent samples; for example, they might need to combine
survey data with administrative data (see Ridder and Mo¢ tt (2003) for a survey of the econometrics
of data combination). From a methodological perspective, the problem is that while the samples
being combined might contain some common variables, other variables belong only to one of the
samples. When the data is collected at the same aggregation level (e.g., individual level, household
level, etc.), if the common variables include a unique (and correctly recorded) identi￿er of the
units constituting each sample, and there is a substantial overlap of units across all samples, then
exact matching of the data sets is relatively straightforward, and the combined data set provides
￿We are grateful to the coeditor Paolo Paruolo, an anonymous referee, Maria Goltsman, Nick Kiefer, Tymon Tatur
and Tim Vogelsang for useful comments. Any remaining errors are our own responsibility.
yCorresponding author. 492 Uris Hall, Ithaca NY 14853-7601, USA, fm72@cornell.edu. Financial support from
Northwestern University￿ s Dissertation Year Fellowship is gratefully acknowledged.
1all the relevant information to identify features of the population of interest. However, it is rather
common that there is a limited overlap in the units constituting each sample, or that variables that
allow identi￿cation of units are not available in one or more of the input ￿les, or that one sample
provides information at the individual or household level (e.g., survey data) while the second sample
provides information at a more aggregate level (e.g., administrative data providing information at
the precinct or district level).
This empirical problem can be formalized in the following way. Suppose that each member l
of a population of interest L has an outcome yl in < and covariates (xl; zl) in a space X ￿ Z: Let
the random variables (y;x;z) : L ! < ￿ X ￿ Z have distribution Q(y;x;z): Here X is a ￿nite
dimensional real space, and Z is a J￿element ￿nite set. The identi￿cation problem arises from the
fact that the joint realizations of (y;x;z) are not observable. All that the researcher can observe
are realizations from two separate sampling processes, one which draws persons at random from L
and collects realizations of (x;z) but not y, and the other that draws persons at random from L and
collects realizations of (y;x) but not z. Abstracting from statistical considerations, these sampling
processes reveal the conditional probabilities Pr(z = jjx); j 2 Z; and the conditional distributions
P (yjx), but not the conditional distributions P (yjx;z = j); j 2 Z. Features of these conditional
distributions, e.g., conditional mean, conditional quantiles, etc., are the objects that the researcher
would like to identify and estimate; the empirical evidence alone, however, does not allow for point
identi￿cation and point estimation.
The literature on statistical matching has aimed at using the common variable(s) x as a bridge to
create synthetic records containing (y;x;z). As Sims (1972) pointed out,1 the inherent assumption
at the base of statistical matching is that conditional on x; y and z are independent. This conditional
independence assumption is very strong and untestable. While it does guarantee point identi￿cation
of features of the conditional distributions P (yjx;z = j); j 2 Z, it often ￿nds very little justi￿-
cation in practice. Cross and Manski (2002) (CM hereafter) have recently proposed an alternative
approach to the problem of data combination when exact matching is not possible. They suggested
a method to learn features of the conditional expectations E (yjx;￿) ￿ [E (yjx;z = j); j 2 Z];
using the empirical evidence alone and without maintaining any untestable assumption.2
CM showed that the vector of conditional expectations E (yjx;￿) belongs to an identi￿cation
region (that is, a set of values of E (yjx;￿) which are feasible given the maintained assumptions
and the empirical evidence), given by a bounded and convex set. They derived the extreme points
of such an identi￿cation region, showing that they are the expectations of certain J￿vectors of
distributions. When the support of P (yjx) is ￿nite or J = 2; CM were able to characterize
the identi￿cation region fully, as the convex hull of its extreme points. However, for the case in
1See also Okner (1972).
2See Cross (2000) for an investigation of the identifying power of additional assumptions.
2which P (yjx) has in￿nite support and J ￿ 3; CM could not fully characterize the topology of the
identi￿cation region (they did show, however, that the identi￿cation region contains the convex
hull of its extreme points, and is contained in another convex polytope.). This implies that for
empirical problems in which the variable y is continuous (e.g., y is given by capital gains, income,
etc.), CM￿ s results provide the extreme points of the identi￿cation region, but are not conclusive
as to whether the identi￿cation region itself is given by the convex hull of these extreme points.
The purpose of this paper is to fully characterize the topology of the identi￿cation region in
CM. A full characterization allows empirical researchers to precisely calculate the identi￿cation
region of interest, and to estimate it when only a ￿nite sample is available by replacing population
parameters with sample analogs. We show that the identi￿cation region is given exactly by the
convex hull of its extreme points irrespective of the support of P (yjx). The paper is organized as
follows. Section 2 formally states the problem; Proposition 1 presents the main result of the paper,
showing that the identi￿cation region of E (yjx;￿) is closed even when P (yjx) has in￿nite support
and J ￿ 3: Corollary 2 uses well known results of convex analysis to argue that compactness of the
identi￿cation region (established through Proposition 1) implies that the identi￿cation region is, in
fact, given by the convex hull of its extreme points.
Proofs and Discussion
By the Law of Total Probability:
P (yjx) =
X
j2Z
P (yjx;z = j)Pr(z = jjx) (1)
and by the Law of Iterated Expectations:
E (yjx) =
X
j2Z
E (yjx;z = j)Pr(z = jjx)
Assume that E (yjx) exists and that Pr(z = jjx) > 0 8j 2 Z. Following CM￿ s notation, let ￿
denote the set of all probability distributions on <, and let ￿x denote the set of all J-vectors of
distributions on < that satisfy (1). That is,
￿
 jx;j 2 Z
￿
2 ￿x if and only if
P (yjx) =
X
j2Z
 jx￿xj (2)
where ￿xj ￿ Pr(z = jjx) > 0 8j 2 Z. Let  jx (A) = P (y 2 Ajx;z = j), A ￿ <. The identi￿cation
region of E (yjx;￿) is
Dx =
￿￿Z
yd jx;j 2 Z
￿
:
￿
 jx;j 2 Z
￿
2 ￿x
￿
CM show that Dx is a bounded and convex set. Below we show that Dx is a closed set.
3Proposition 1 Suppose that E (yjx) exists and that ￿xj > 0 8j 2 Z. Then Dx is a closed set.
Proof. The proof proceeds in two steps. In all that follows we will suppress the subscript x and
the conditioning on x for ease of notation, and we will show the following: Suppose that there is a
sequence ￿n 2 D such that ￿n ! ￿; then ￿ 2 D. As a starting point, notice that there is a sequence
of elements of ￿,  n = ( n
1; n
2;:::; n
J) 2 ￿; such that
￿
E n
1y;E n
2y;:::;E n
Jy
￿
= ￿n 2 D; where
E jy will denote
R
yd j in all that follows.
Step 1: The sequence f ng contains a convergent subsequence f nmg with  nm =)  ￿ 2 ￿.
As a consequence of (2), for any set A ￿ Y ,
￿j j (A) ￿ P (y 2 A) 8j 2 Z (3)
Since P (y) is tight, for any " > 0 we can ￿nd a compact set K" s.t. P (y 2 Kc
") < "; where Kc
" is the
complement of K". This implies that for any " > 0 we can ￿nd a compact set K" s.t.  j (Kc
") < "
￿j
8 j 2 ￿j; and hence the set ￿j is tight. Therefore, by Prohorov￿ s theorem,
￿
 n
j
￿
contains a
subsequence which converges weakly to a probability measure. Denote  n
j ￿
n
 nm
j
o
=)  ￿
j:
Since the linear relation in (2) is continuous in the  j￿ s, we have that
P
 ￿
j￿j = P (yjx) and
 ￿ =
￿
 ￿
j;j 2 Z
￿
2 ￿:
Step 2.
￿
E ￿
1y;E ￿
2y;:::;E ￿
Jy
￿
= ￿, hence ￿ 2 D.
Step 2 implies that ￿￿ ￿
￿R
y ￿ d ￿
j(y)
￿
j=1;::;J 2 D. We will complete the proof by showing that
￿ = ￿￿. To this end, we only need to show that for any j 2 f1;::;Jg, the distributions
n
 nm
j
o
are
uniformly integrable. By (3), for any ￿ 2 <
Z
jyj￿￿
jyjd n
j ￿
R
jyj￿￿ jyjdP
￿j
; 8j 2 Z; 8n:
Therefore
0 ￿ lim
￿!1
sup
n
Z
jyj￿￿
jyjd n
j ￿
1
￿j
lim
￿!1
Z
jyj￿￿
jyjdP = 0;
where the second limit converges since E (y) exists. This establishes uniform integrability of the
distributions
n
 nm
j
o
; j 2 Z, and completes the proof.
￿
Corollary 2 Suppose that E (yjx) exists and that ￿xj > 0 8j 2 Z. Then Dx is given by the convex
hull of its extreme points.
Proof. CM showed that Dx is a bounded convex set. We showed that Dx is closed. Hence Dx
is a compact convex set. Standard results of convex analysis (e.g., Rockafellar (1970), Theorem
18.5 and Corollary 18.5.1) insure that a compact convex set in <J is the convex hull of its extreme
points, hence the result follows.
￿
4References
Billingsley, P. (1999): Convergence of Probability Measures. John Wiley and Sons, New York.
Cross, P. J. (2000): ￿Three Essays in Nonparametric Identi￿cation,￿Ph.D. thesis, University of
Wisconsin￿ Madison.
Cross, P. J., and C. F. Manski (2002): ￿Regressions, Short and Long,￿Econometrica, 70(1),
357￿ 368.
Okner, B. A. (1972): ￿Constructing a New Microdata Base from Existing Microdata Sets: The
1966 Merge File,￿Annals of Economic and Social Measurement, 1, 325￿ 362.
Ridder, G., and R. Moffitt (2003): ￿The Econometrics of Data Combination,￿ Chapter for
the Handbook of Econometrics, Vol. 6.
Rockafellar, R. T. (1970): Convex Analysis. Princeton University Press, Princeton, New Jersey.
Sims, C. A. (1972): ￿Comments and Rejoinder (On Okner (1972)),￿ Annals of Economic and
Social Measurement, 1, 343￿ 345, 355￿ 357.
5

Generalization of a Result on "Regression, Short and Long"

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