Econometric Reviews

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Yet another look at the omitted variable bias
Masayuki Hirukawa, Irina Murtazashvili & Artem Prokhorov
To cite this article: Masayuki Hirukawa, Irina Murtazashvili & Artem Prokhorov (2023)
Yet another look at the omitted variable bias, Econometric Reviews, 42:1, 1-27, DOI:
10.1080/07474938.2022.2157965
To link to this article: https://doi.org/10.1080/07474938.2022.2157965

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Published online: 08 Feb 2023.

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ECONOMETRIC REVIEWS
2023, VOL. 42, NO. 1, 1–27
https://doi.org/10.1080/07474938.2022.2157965

Yet another look at the omitted variable bias
Masayuki Hirukawaa, Irina Murtazashvilib, and Artem ProkhorovcaRyukoku University, Kyoto, Japan; bDrexel University, Philadelphia, PA, USA; cUniversity of Sydney, Sydney,
NSW, Australia; CEBA, St. Petersburg State University, St. Petersburg, Russia; CIREQ, Universite de Montreal,
Montreal, QC, Canada
ABSTRACT

When conducting regression analysis, econometricians often face the situation where some relevant regressors are unavailable in the data set athand. This article shows how to construct a new class of nonparametricproxies by combining the original data set with one containing the missing
regressors. Imputation of the missing values is done using a nonstandard
kernel adapted to mixed data. We derive the asymptotic distribution of theresulting semiparametric two-sample estimator of the parameters of interest and show, using Monte Carlo simulations, that it dominates the solutions involving instrumental variables and other parametric alternatives. An
application to the PSID and NLS data illustrates the importance of our estimation approach for empirical research.

KEYWORDS

Data combination;
endogeneity; instrumental
variable estimation;
omitted variable bias;
two-sample estimation
JEL CLASSIFICATION
CODES

C13; C14; C31

1. Introduction
Omission of relevant variables leads to challenging problems in applied work. If the correlation
between the omitted variable and included regressors is strong, least squares estimates are biased
and inconsistent. This issue has a long history in economics.
A classic example of the missing regressor is an ability measure in Mincer’s (1974) wage
regression, where estimates suffer from the so called “ability bias” unless the regressors include avariable representing ability (see, e.g., Card, 1995, for details). Micro-level data sets such as theCurrent Population Survey (CPS) and the Panel Study of Income Dynamics (PSID) do not generally contain individual test scores that can be used as (a proxy for) ability.
Another example can be found in the study of gender wage gap (e.g., Black et al., 1999;
Zabalza and Arrufat, 1985). Work experience is an important regressor in the wage regression.
Although the General Household Survey (GHS) and CPS contain wages and other predictors,
they do not include a variable for actual work experience.
This article argues that availability of a suitable auxiliary sample offers an underexploited
opportunity. We use the sample to construct a new semiparametric estimator and we study its
properties. In a nutshell, our two-sample estimation procedure is to run the ordinary leastsquares (OLS) after replacing the missing regressor with a nonparametric estimate of its conditional mean, obtained using a nonstandard kernel function tailored for both discrete and continuous data.
Our approach is different from but not unrelated to the work on matched-sample estimators(see, e.g., Abadie and Imbens, 2006, 2011; Hirukawa and Prokhorov, 2018). One key advantage of
CONTACT Masayuki Hirukawa
hirukawa@econ.ryukoku.ac.jp
Faculty of Economics, Ryukoku University, 67 Tsukamotocho, Fukakusa, Fushimi-ku, Kyoto 612-8577, Japan.
Supplemental data for this article can be accessed online at https://doi.org/10.1080/07474938.2022.2157965.
ß 2023 Taylor & Francis Group, LLC

2

M. HIRUKAWA ET AL.

our estimator is that the imputed proxy does not rely on a combined sample constructed via thenearest-neighbor matching (NNM) or another ad hoc matching scheme. Matching-free estimationis also the purpose of some well-known moment-based two-sample methods (e.g., Angrist andKrueger, 1992, 1995; Arellano and Meghir, 1992; Inoue and Solon, 2010; Klevmarken, 1982;
Murtazashvili et al., 2015; Pacini and Windmeijer, 2016). Unfortunately, these approaches are notapplicable in the setting of a linear regression where some regressors are missing in the primary
sample, and two-sample moment-based estimation is infeasible. Our setup is closer to what is
considered by Pacini (2019), Chen et al. (2008) and Graham et al. (2016), who also use an auxiliary sample to impute missing regressors. However, Pacini’s (2019) main focus is on relaxing
identification conditions for two-sample estimation, while Chen et al. (2008) and Graham et al.
(2016) study parametric and semi-parametric efficiency bounds for GMM.
We establish large-sample properties of the proposed semiparametric two-sample estimator. Its
finite-sample properties are examined through extensive Monte Carlo simulations (largely
deferred to an online supplement to save space), where we show that our estimator dominates
the competitors, in particular, instrumental variable (IV) estimators using linear and nonlinear
instruments and a parametric two-sample alternative. In an empirical application, we look for asetting where we are able to obtain estimates based on both the IV estimator and proposed estimator at the same time. We do so for the return to schooling and offer new insights arising fromthe use of IVs based on one sample as well as from the use of proxies based on two samples.
The nonparametric imputation we propose is important in several practical dimensions. First,
having access to a second sample, empirical economists commonly impute the variables that arenecessary for estimation but missing in the first sample. For example, Fang et al. (2008) impute
medical expenditure of Health and Retirement Study (HRS) respondents using the informationfrom the Medicare Current Beneficiary Survey (MCBS), and Flavin and Nakagawa (2008) imputethe square footage of the homes of PSID respondents using data on housing from the American
Housing Survey (AHS). Instead of estimating the conditional mean of the missing variable nonparametrically, these papers adopt a linear projection-based imputation. As a matter of course, if
the functional form turns out to be incorrectly specified, then consistency of the estimators canno longer be guaranteed. Furthermore, estimation errors associated with imputation need to be
explicitly accounted for, as imputed values cannot be treated as error-free and adjustments need
to be made to the standard errors. A remarkable observation is that our nonparametric imputation approach performs no worse even under correct parametric specification. The result on
inconsistency of the linear projection approach and the circumstances under which it still works
may be of independent interest. We explore this in the Monte Carlo simulations of this article
and in the online supplement.
Second, our nonparametric imputation method is important in the context of the literature on
two-step semiparametric regression estimation with a plug-in first-step nonparametrically generated regressor (see, e.g., Hsu et al., 2022; Newey, 2009; Rilstone, 1996; Stengos and Yan, 2001).
The literature explores statistical properties of regression estimators when the regressors that cannot be observed directly are estimated by nonparametric methods, similarly to Pagan’s (1984)classical fully parametric case. However, these approaches are not directly applicable when imputation needs to be based on two samples or when there are no common observations in them.
Finally, we employ a new uniform consistency result for estimators based on a nonstandard
kernel and mixed data, which is of independent interest. A key aspect of the result is compactnessof the support of the kernel function, a feature that often arises in economics and finance either
by construction or as a theoretical requirement. For example, economic and financial variables
such as expenditure and budget shares, unemployment rates and default and recovery rates are
typically bounded from above and below by construction. Theoretical compactness requirements
are used in the context of partial linear regression models (see, e.g., Yatchew, 1997), NNM (see,e.g., Abadie and Imbens, 2006), and estimation of first-price auctions (see, e.g., Guerre et al.,

ECONOMETRIC REVIEWS

32000), to name just a few examples. Interested readers may consult Hirukawa et al. (2022) for
further details.
The remainder of this article is organized as follows. Section 2 describes the model, defines the
proposed estimator, and derives and discusses its convergence properties. Section 3 presents
results of selected Monte Carlo simulations to compare finite-sample properties of the proposed
estimator with the IV and other alternatives. As an empirical example, in Section 4, we apply thetwo-sample estimator to a version of Mincer’s (1974) wage regression. Both in simulations andthe application, we compare our estimator (a) with the IV approach, (b) with an approach where
the missing variable is imputed using a fully parametric method, and (c) with the approach of
Hirukawa and Prokhorov (2018) based on NNM. Section 5 concludes with a few questions for
future research. Technical proofs are provided in the Appendix. The online supplement contains
details of the alternative estimators (a)–(c), comprehensive simulation results for a variety ofdesigns, and additional discussions on the empirical example.
1=2
The article adopts the following notational conventions: kAk ¼ trðA0 AÞ
is the Euclidean
norm of matrix A; 1fg denotes an indicator function; 0pq signifies the p  q zero matrix, whereÐ1
the subscript may be suppressed if q ¼ 1; Bðp, qÞ ¼ 0 yp1 ð1  yÞq1 dy for p, q > 0 is the beta
function; and the symbol > applied to matrices means positive definiteness.

2. Two-sample two-step estimation
2.1. Model
We consider the following linear regression model0
Y ¼ b0 þ X10 b1 þ X20 b2 þ X3I
b3 þ u,(1)

where X1 2 Rd1 , X2 2 Rd2 , X3I 2 Rd3I (the distinction between these regressors will be made clear
shortly). Throughout, we assume that either b1 or b3 is the parameter of primary interest. Let
0 0
Þ 2 Rdþ1 is exogenous and a single random sample ofd :¼ d1 þ d2 þ d3I : When ð1, X10 , X20 , X3I
ðY, X1 , X2 , X3I Þ is available, the OLS estimator of b ¼ ðb0 , b01 , b02 , b03 Þ0 is consistent under the usualassumptions.
Let S 1 denote the data set at hand and let S 2 denote a second data set which will be required
for a two-sample estimation. We assume that S 1 ¼ ðY, X1 , X3 Þ and S 2 ¼ ðX2 , X3 Þ, where X3 :¼
ðX3I , X3E Þ is the vector of common variables across the two samples that are partitioned into
those included (X3I ) and those excluded (X3E ) from regression (1). We note that X2 is missing in
S 1 although it is assumed to be relevant, i.e., b2 6¼ 0: This setup was recently considered by
Pacini (2019) but his focus was on relaxing the identification assumptions involving the joint distribution of ðY, X1 , X2 , X3I Þ:
The distinction between X1 and X3 will become important later. For now, we can think of X3
as variables that are common across S 1 and S 2 , enter as regressors and are used, due to theirpresence in both samples, for imputation of the missing regressors in X2. On the other hand,when a common variable serves merely as a regressor but is not used for imputation, it is classified as X1. The vector X1 can be empty; the vector X3 cannot.
Even though S 1 and S 2 contain common variables X3, this does not mean that they need tohave common observations, i.e., the two samples do not have to overlap. Finally, denote d3 :¼
dimðX3 Þ ¼ dimðX3I Þ þ dimðX3E Þ :¼ d3I þ d3E , where d3 > 0 must be the case, and either d3I or
d3E is allowed to be zero.
An example of this sampling arrangement is when a micro-level data set (e.g., CPS or PSID)
can be complemented by an auxiliary data set which is focused on some important variables thatare not available elsewhere, e.g., a psychometric data set with ability measures.

4

M. HIRUKAWA ET AL.

In such cases it is common to instrument for the unobserved variable (see Section 1.1 of theonline supplement for details of IV estimation). However, the IV estimation is unavailable if there
are no valid instruments in the same sample and it may be desirable to avoid the IV estimation
for a host of other reasons discussed in Section 2.5.

n

m
Denote S 1 ¼ S 1n ¼ ðYi , X1i, X3i Þ i¼1 and S 2 ¼ S 2m ¼ ðX2j , X3j Þ j¼1 : We now discuss how weconstruct a proxy for X2 using S 2m :2.2. Additional notation
We start by introducing some additional notation that will help define the semiparametric twosample estimator and derive its asymptotic properties. First, we allow for the vector of common
variables X3 ¼ ðX3I , X3E Þ to consist of both continuous (C) and discrete (D) variables so that d3 ¼
dimðX3 Þ ¼ dimðX3C Þ þ dimðX3D Þ ¼: d3C þ d3D : While d3D ¼ 0 is allowed, d3C > 0 must be thecase for the subsequent asymptotic analysis. This distinction will be needed for the applicationof kernel smoothing to mixed continuous and discrete data. Second, let X3 :¼ X3C  X3D ,where X3C :¼ suppðX3C Þ and X3D :¼ suppðX3D Þ: Third, denote g2 ðX3 Þ :¼ EðX2 jX3 Þ and g2 :¼
X2  g2 ðX3 Þ with Eðg2 jX3 Þ ¼ 0: The conditional expectation function g2 ðÞ and the error term g2
will determine the asymptotic adjustments required to account for the use of nonparametricP
proxies. Finally, for a pair of vectors ðZ, WÞ, let UZ, W :¼ EðZi Wi0 Þ, Sn, Z, W :¼ ð1=nÞ ni¼1 Zi Wi0
Pm
and Sm, Z, W :¼ ð1=mÞ j¼1 Zj Wj0 , where summations for Sn, Z, W and Sm, Z, W are assumed to betaken within S 1n and S 2m , respectively, and where the subscripts n and m emphasize the relevantsample sizes.
2.3. Estimation of b and g2 ðÞWe reformulate the regression (1) as
Y ¼ X 0 b þ ,
0 0
where X :¼ ð1, X10 , g2 ðX3 Þ0 , X3I
Þ and  :¼ u þ g02 b2 :In essence, we wish to obtain an estimator of the unknown function g2 ðX3 Þ using S 2 and construct a proxy for X2 using some distance measure between the common variables X3 in S 1 andS 2 : We will use a kernel function (as opposed to NNM or another matching scheme) toachieve that.
Let ^g 2 ðÞ be some consistent nonparametric estimator of g2 ðÞ: Then, the estimation procedure
we propose consists of the following two steps:

mStep 1: Regard ðX2j , X3j Þ j¼1 in S 2 and fX3i gni¼1 in S 1 as m data points and n design points,nrespectively, and obtain n nonparametric estimates ^g 2 ðX3i Þ i¼1 :0 0^ i :¼ ð1, X1i0 , ^g ðX3i Þ0 , X3Ii
Þ:Step 2: Run OLS for the regression of Yi on X
2
The estimator of b is then
1
^
b
^,Y ¼
^,X
^ Sn, XPILS :¼ Sn, X

n
1X
^ iX
^ 0i
X
n i¼1

!1

n
1X
^ i Yi :X
n i¼1

Because this estimator is the OLS with ^g 2 ðX3i Þ plugged in place of the missing regressor X2i , wecall it the plug-in least squares (PILS) estimator hereinafter.

ECONOMETRIC REVIEWS

5

The part of X3 that is used for imputation but excluded from the original model can be used0uÞ ¼ 0: If g2 ðX3 Þ is not linear in X3E , these restrictions can,to form exclusion restrictions EðX3E
in principle, be used to improve precision of the estimation of b (or to construct specification
tests). However, since our main goal is to use X3E in the construction of a proxy for X2, we do
not pursue the issues of efficiency or specification testing here.
At a first glance, the PILS approach—just like its fully parametric counterpart (see Section 1.2of the online supplement for details)—may be viewed as a variant of generated regressors, whichhave been extensively studied by many researchers. However, two samples and nonparametric
imputation raise new and nontrivial issues, which require careful consideration. We study theissues in the next section.
A remaining task is to deliver a consistent estimator of g2 ðÞ: Taking into consideration thatg2 ðÞ may depend on both continuous and discrete covariates, we choose the kernel regression
smoother for mixed continuous and categorical data proposed by Racine and Li (2004). In thissense, PILS may be viewed as a partial mean estimator of Newey (1994), although Newey (1994)
does not consider the situation where design and data points for kernel smoothing come fromdifferent data sources.
A key part of the estimator is the construction of a multivariate kernel. Let Kðt3C ; x3C , hÞ andLðt3D ; x3D , kÞ denote product kernels for the continuous and discrete components of X3, respectively, where t3 and x3 are data points and design points, and h and k are vectors of bandwidths.
We provide details of how these kernels are constructed in Appendix A.1. Then, the product kernel for X3 isW ðt3 ; x3 , h, kÞ :¼ Kðt3C ; x3C , hÞLðt3D ; x3D , kÞ:It follows that a nonparametric estimator of g2 ðÞ can be defined asPmj¼1 X2j W X3j ; X3i , h, k
 , i ¼ 1, :::, n:^g 2 ðX3i Þ :¼ Pm
j¼1 W X3j ; X3i , h, k
There are many conventional options for the specific univariate kernel to use for continuous
variables due to the requirement for compactness of X3C (see Assumption 2 below). Assumingdwithout loss of generality that the compact set is a d3C -dimensional unit hypercube ½0, 1 3C , wepropose to employ Chen’s (1999) univariate beta kernel with support on ½0, 1 as an attractive
alternative to standard symmetric kernels. Our application of this kernel is motivated by the uniform convergence result on a compact support which we prove in a companion paper (Hirukawa
et al., 2022) and which is, as far as we know, new. This asymmetric kernel is free of the boundary
bias by construction and its shape varies across design points even under a fixed value of thesmoothing parameter b. The latter property implies that the amount of smoothing by this kernelchanges in an adaptive manner.
2.4. Convergence properties of PILS
2.4.1. Regularity conditions
Now we explore asymptotic properties of the PILS estimator as both n and m diverge. For thispurpose, the following regularity conditions are imposed.
Assumption 1. The two random samples ðS 1 , S 2 Þ ¼ ðS 1n , S 2m Þ are drawn independently fromthe joint distribution of ðY, X1 , X2 , X3 Þ with finite fourth-order moments.
Assumption 2. X3C is continuously distributed with a convex and compact support X3C , and its
density is bounded and bounded away from zero on X3C :6

M. HIRUKAWA ET AL.

Assumption 3.
(i) EðujX1 , X3 Þ ¼ 0 and r2u ðX1 , X3 Þ :¼ Eðu2 jX1 , X3 Þ 2 ð0, 1Þ:(ii) ðg2 ??X1 ÞjX3 :(iii) g2 ðÞ is non-constant on X3C if X3E contains at least one continuous variable, and g2 ðÞ isstrictly nonlinear on X3C otherwise.
Assumption 4.
(i) Let f ðÞ be the marginal pdf of X3C : Then, for some integer   2, the th-order derivatives of f ðÞ and f ðÞg2 ðÞ with respect to X3C are continuous and bounded uniformly
on X3C :(ii) UX, X :¼ EðXX 0 Þ > 0:(iii) There exist some constants c 2 ð0, 1Þ and C 2 ½1, 1Þ so that
sup E jjX2 jj2þc jX3 ¼ x3  C:
x3 2X3

Assumption 5. The univariate continuous kernel K is either: (a) a symmetric and bounded kernelÐÐÐfunction of order  that satisfies (i) KðtÞdt ¼ 1, tl KðtÞdt ¼ 0 for l ¼ 1, :::,   1, t  KðtÞdt 6¼
Ð
0, (ii) the first-order Lipschitz condition, and (iii) jtjk jKðtÞjdt < 1 for some k > 2; or (b) thebeta kernel of Chen (1999).
Assumption 6. Sequences of smoothing parameters hp ð¼ hp ðmÞ > 0Þ, bp ð¼ bp ðmÞ > 0Þ and kq ð¼
kq ðmÞ 2 ð0, 1ÞÞ and boundary parameters hp ð¼ hp ðmÞ > 0Þ for p ¼ 1, :::, d3C and q ¼ 1, :::, d3D
satisfy one of the following conditions as m ! 1 : (a) for a symmetric th-order kernel,
Q 3C
hp Þ ! 0; and (b) for the beta kernel, bp , kq , hp ! 0, bp =hp ! 0
hp , kq ! 0 and log m=ðm dp¼1
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Qd3C
! 0:
and log m= m
p¼1 bp hp
The first three assumptions are often encountered in matching-based estimation (see, e.g.,
Hirukawa and Prokhorov, 2018) and are natural in our settings. In particular, it follows from
Assumption 3(i) that even when some part of X3 does not enter regression (1) due to an exclusion restriction, X1 and X3 are exogenous in the full model. Nonlinearity of g2 ðÞ in Assumption
3(iii) is required only when all continuous common variables are included as regressors in (1).
Otherwise, excluded continuous common variables introduce additional randomness in g2 ðÞ,which helps identify b.
Assumption 3(ii) is also standard for the proxy variable literature. It implies thatddg2 jðX1 , X3 Þ ¼ g2 jX3 , where “¼” denotes equality in distribution. It is sometimes possible to use aweaker assumption, e.g., conditional mean independence Eðg2 jX1 , X3 Þ ¼ Eðg2 jX3 Þ (see, e.g.,
Wooldridge, 2010, Section 4.3.2). However, we focus on conditional independence for two reasons. First, it implies orthogonality of g2 and X1, which is a key requirement for consistency of
PILS in Theorem 1. Second, it simplifies the covariance estimation in Section 2.4.3. A similarassumption can be found in Inoue and Solon (2010, Assumption c) and Hirukawa andProkhorov (2018, Assumption 4).
Moreover, this assumption involves quantities that are either unobserved (g2) or unobservedin the same sample (g2 and X1). Shah and Peters (2020) have recently demonstrated that (even if
they were observed) a uniformly valid conditional independence test does not exist. This may
look like a disadvantage of our method. However, as discussed by Wooldridge (2010, p.68), a
conditional independence assumption is routinely made or implied in all proxy-based estimators,

ECONOMETRIC REVIEWS

7and our approach is no different. As we argue in Section 4.3, the assumption is easier to claim
given enough observables in X3.
Along with compactness of X3C in Assumption 2, Assumptions 4–6 are required to establish
weak uniform consistency of the nonparametric regression estimator ^g 2 ðÞ on X3 : Similar conditions can be found, for instance, in Li and Ouyang (2005), Hansen (2008) and Su et al. (2013) fora symmetric kernel, and Hirukawa et al. (2022) for the beta kernel. Notice that the beta kernel isnonnegative so that the order  ¼ 2 is the case in Assumption 4(i).
Strictly speaking, uniform consistency holds on X3 for a symmetric kernel. In contrast, for thebeta kernel, the boundary parameters h1 , :::, hd3C play a key role in uniform consistency of ^g 2 ðÞ:Specifically, uniform consistency in this case is established on SX3C  X3D , where SX3C :¼
Qd3C
d3C(more slowly than the smoothing
p¼1 hp , 1  hp is a compact set expanding to X3C ¼ ½0, 1
parameters b1 , :::, bd3C ) so that SX3C  X3D is expanding to X3 as m ! 1: On the other hand,because the regression estimator ^g 2 ðÞ is a Nadaraya-Watson-type estimator, it suffers from theso-called boundary bias when a symmetric kernel is employed. When the beta kernel is used,^g 2 ðÞ is free of this issue by construction.
We note that Assumption 5(a) allows for higher-order kernels (i.e.,  > 2). In theory, such
kernels could be a remedy for the curse of dimensionality implied by Corollary 1 below. In practice, however, it is well known that higher-order kernels are unlikely to dominate nonnegative
ones for moderate sample sizes, and that the advantage, if any, is typically marginal (see, e.g.,
Marron, 1994; Marron and Wand, 1992). In fact, our simulation results indicate poor finite-sample performance of PILS using higher-order kernels.
2.4.2. Consistency and asymptotic normality of PILS
^ : In particular,
The next two theorems establish consistency and asymptotic normality of b
PILS
^
is
obtainedafter
correcting
for
B
,
the
bias
term
due to kernel
asymptotic normality of b
g2
PILS
smoothing. Each theorem holds regardless of the number of common variables and regardless ofthe divergence patterns in (n, m).
^
Theorem 1. If Assumptions 1–6 hold, then b
PILS ! b as n, m ! 1:p

Theorem 2. If Assumptions 1–6 hold, thenpﬃﬃﬃ^
d
1
XU
n b PILS  b  Bg2 ! N 0ðdþ1Þ1 , V :¼ N 0ðdþ1Þ1 , U1
X, X
X, X
as n, m ! 1, whereBg2 :¼ S1^ , ½g2 ðX3 ÞEf^g ðX3 ÞjX3 g0 b2^,X^ S n, X
n, X
2
!1
n
n


1X
1X
^ iX
^ 0i
^ i g2 ðX3i Þ  E ^g ðX3i ÞjX3i 0 b2
X
X
¼
2
n i¼1
n i¼1
8 P
Pd3D 
d3C >< Opfor a symmetric th-order kernel
p¼1 hp þ
q¼1 kq
P

, and
¼
P
d3C
d3D
>
: Op
for
the
beta
kernel
b
þ
k
p¼1 p
q¼1 q


0
2
X:¼ X1 þ X2 :¼ EðXX  Þ þ E XX 0 b02 g2 g02 b2 :As derived in the proof of Theorem 1, there is the asymptotically negligible bias term Bg2 , due
pﬃﬃﬃ
to kernel smoothing. There are also two n-asymptotically normal terms. One comes from the
1sampling error in regression (1), and it has the asymptotic variance U1
X , X X1 UX , X : The other is

8

M. HIRUKAWA ET AL.

1
due to the approximation error of g2, and its asymptotic variance is U1
X , X X2 UX , X : As a consequence, the asymptotic variance X in Theorem 2 has similar structure to that of the double kernel
estimator by Stengos and Yan (2001, Theorem 1) and the two-step series estimator by Hsu et al.
(2022, Theorem 2.1). A difference from Stengos and Yan (2001) and Hsu et al. (2022) is that thetwo terms are uncorrelated. Each of them can be obtained within two independent samples S 1and S 2 , and thus X is free of covariance terms.

2.4.3. Covariance estimation
Covariance estimation is essential for inference. The problem of estimating V consistently boils^ ,^ 0b
down to proposing consistent estimators of X1 and X2 : For the PILS residual ^ i :¼ Yi  X
i PILSan estimator of X1 isnX^ 1 :¼ 1^ iX^ 0i^ 2i :
X
X
n i¼1

^ 1 can be established in the same manner as in the proofs of Theorems 1 and 2.
Consistency of X^0^
€ 2 :¼ ð1=nÞ Pn X
^ iX
^ 0b
^g ^g 0 b
might work as a consistent estiOn the other hand, Xi 2, PILS 2i 2i 2, PILS

i¼1^
g 2 :¼ X2  ^g 2 ðX3 Þ: In reality, it turns
mator for X2 , where b
2, PILS is the PILS estimator of b2 and ^out to be difficult to obtain f^g 2i gni¼1 because of the absence of X2 in S 1 : Instead, usingdg2 jðX1 , X3 Þ ¼ g2 jX3 yields


X2 ¼ EfXX 0 b02 E g2 g02 jX3 Þb2 g ¼: E XX 0 b02 R2 ðX3 Þb2 :Then, we can alternatively consider the estimatornX^^0^ 2 :¼ 1^^ iX^ 0i bX
X
2, PILS R 2 ðX3i Þb 2, PILS ,
n i¼1

where
^ 2 ðX3i Þ :¼
R

Pm



g 2j ^g 02j W X3j ; X3i , h, k
j¼1 ^
Pm

 ,
j¼1 W X3j ; X3i , h, k

and the nonparametric regression residuals

i ¼ 1, :::, n,m^g 2j m
¼ X2j  ^g 2 ðX3j Þ j¼1 can be obtained within
j¼1^ 2 ðÞ is inspired by the Algorithm in Section 2.4 of Fan and Yao
S 2 : The estimation procedure for R^ 2 , we can see that a similar argument to the proof of Lemma A1 in(1998). To show consistency of X
Appendix A.2 establishes uniform consistency of ^g 2j for g2j , which in turn leads to uniform consistp^ 2 ¼ ð1=nÞ Pn Xi X 0 b0 R2 ðX3i Þb2 þ op ð1Þ !
^ 2 ðX3i Þ for R2 ðX3i Þ: Then, X
ency of R
X2 :
i 2
i¼1
In conclusion, V can be consistently estimated by^ 2 S1^ ^ :^ 1^ ^ :¼ S1^ ^ X^1 þ X^ :¼ S1^ ^ XS
Vn, X , X
n, X , X
n, X , X
n, X , X
pﬃﬃﬃ
2.4.4. n-consistency of PILS
Theorems 1 and 2 jointly imply that although PILS is consistent, its convergence is affected by
the bias term Bg2 generated by kernel smoothing. For a large d3C , the order of magnitude in the^bias term dominates and the convergence rate of b
becomes inferior. It appears that this probPILS

lem is unavoidable in a regression that uses a nonparametric component. A similar phenomenon
arises in the context of imputation bias correction (see, e.g., Hirukawa and Prokhorov, 2018).

ECONOMETRIC REVIEWS

9

pﬃﬃﬃ
^
However, n-consistency of b
PILS automatically holds for small values of d3C : To illustratesuch cases, we modify Assumption 6 in order to control the bias and variance convergence of^g 2 ðÞ more easily.
Assumption 60 . Sequences of the smoothing and boundary parameters satisfy one of the following
conditions as m ! 1 : (a) for a th-order symmetric kernel, there is a bandwidth hð¼ hðmÞ > 0Þ sothat h1 , :::, hd3C / h, k1 , :::, kd3D / h , h ! 0, and log m=ðmhd3C Þ ! 0; and (b) for the beta kernel,
there are a smoothing parameter bð¼ bðmÞ > 0Þ and a boundary parameter hð¼ hðmÞ > 0Þ so thatnob1 , :::, bd3C / b, k1 , :::, kd3D / b, h1 , :::, hd3C / h, b, h ! 0, b=h ! 0, and log m= mðbhÞd3C =2 ! 0:pﬃﬃﬃ^
The corollary below describes the cases in which b
becomes n-asymptotically normal.
PILS

a

2a

Corollary 1. Let h / ð log m=mÞ and b / ð log m=mÞ for some constant a > 0. Also suppose that
one of the following divergence patterns in (n, m) is true: (i) n=m ! j 2 ð0, 1Þ; (ii) n=m ! 0; or
pﬃﬃﬃ ^
(iii) n=m ! 1 and n=m4a ! 0. Then, under Assumptions 1–5 and 60 , as n, m ! 1, nðbPILSdbÞ ! Nð0ðdþ1Þ1 , VÞ holds either (a) if a th-order symmetric kernel is employed and d3C  2  1or (b) if the beta kernel is employed and d3C  3:
Corollary 1 presents an upper bound of the number of continuous common variables for PILS
pﬃﬃﬃ
to be n-consistent. This is recognized as the curse of dimensionality in continuous common
variables, and it can be relaxed, in theory, by using higher-order kernels. In particular, whenemploying a nonnegative kernel (which is either a symmetric pdf or the beta kernel), we must
pﬃﬃﬃ
limit the number of continuous common variables to 3 or less to make PILS n-consistent. To
see this in more detail, note that for the most realistic case n=m ! j, shrinkage rates of thesmoothing parameters are h / ð log m=mÞa and b / ð log m=mÞ2a for some a 2 ð1=4, 1=d3C Þ (seeAppendix A.4). These rates are faster than what yields Stone’s (1982) optimal global rate of convergence, i.e., h / ð log m=mÞ1=ð4þd3C Þ and b / ð log m=mÞ2=ð4þd3C Þ that can be obtained throughpﬃﬃﬃ
balancing the squared bias and variance of ^g 2 ðÞ: This is because to attain n-consistency weshould keep the convergence rate of the dominant bias term in ^g 2 ðÞ sufficiently fast by undersmoothing, or more specifically, by setting the exponent a within the aforementioned range. We
give some examples of h and b that fulfill the rate requirement in Sections 3 and 4.
Finally, it may be the case that the missing regressor X2 can be observed with a measurement
error. This issue occurs, for example, when an ability measure is unreliable in the wage regres^sion. Suppose that we can observe X2 ¼ X2 þ t at best, where t is the measurement error. As
long as EðtjX3 Þ ¼ 0d2 1 , the effect of t is filtered out via kernel smoothing andPm ^j¼1 X2j W X3j ; X3i , h, k^
 ¼ ^g 2 ðX3i Þ þ op ð1Þ, i ¼ 1, :::, ng2 ðX3i Þ :¼ Pm
j¼1 W X3j ; X3i , h, kholds. In the end, consistency of PILS is maintained.
2.5. Comparison of PILS with competing estimators
We conclude this section by comparing PILS with competing estimators. The competing estimators we consider are IV estimators using linear and nonlinear instruments, a fully parametricalternative to PILS (called PARA hereinafter), and the matched-sample indirect inference (MSII)
and fully-modified MSII (MSII-FM) estimators. PARA is often a method of choice (see, e.g., Fanget al. 2008). MSII(-FM) was proposed by Hirukawa and Prokhorov (2018) as an alternative to theinconsistent OLS estimator using the matched sample (called MSOLS hereinafter). Definitions

10

M. HIRUKAWA ET AL.

and statistical properties of these estimators are reviewed in the online supplement. Finite-sample
properties of all the estimators will be assessed in comparison with PILS in the next section.
2.5.1. Comparison with IV
The use of IVs has been associated with at least two problems. First, instruments that are notstrongly correlated with the endogenous variables—weak instruments—lead to large inconsistencies in the IV estimators even if a weak correlation exists between the instruments and the structural equation error, i.e., even if the instruments only slightly violate the validity assumption (see,e.g., Bound et al., 1995). When we are ready to assume that the IVs are valid, the weak instrument problem results in large asymptotic and finite-sample biases and, when many such instruments are available, the distribution of IV estimators is known to deviate substantially from thelarge-sample approximation (see, e.g., Bekker, 1994; Chao and Swanson, 2005).
Second, finite-sample properties of IV estimators are known to be generally poor, especiallywhen the instruments are weak (see, e.g., Flores-Lagunes, 2007). Various bias reduction techniques have had limited success and even very inventive uses of instruments have, for these reasons,been criticized in terms of their validity, strength and finite-sample performance.
Furthermore, valid and strong instruments may be unavailable in the same sample. A numberof ingenious methods have been proposed to combine data from more than one source in suchcases (see, e.g., Angrist and Krueger, 1992, 1995; Arellano and Meghir, 1992; Inoue and Solon,
2010; Klevmarken, 1982; Murtazashvili et al., 2015). However, the two-sample IV estimators
inherit the same problems as their single-sample counterparts (see, e.g., Choi et al., 2018).
The identification strategy for our estimator is different. The central identification conditions
are that g2 ðÞ either depends on additional exogenous variables X3E or is nonlinear. If there areno additional exogenous variables, then identification relies on nonlinearity in the first step. The
weak identification issue, similar to the weak IV, can appear in our setting if g2 ðÞ is linear anddepends very weakly on X3E so that g2 ðÞ is close to being a linear function of X3I only. We consider these cases in the extensive simulation comparisons.
There is a rich literature proposing ingenious methods such as integrated regression functions,basis functions, sieves and local smoothing, to incorporate nonlinearities in an IV estimation basedon conditional moment restrictions (see, e.g., Dominguez and Lobato, 2004; Kitamura et al., 2004;
Lavergne and Patilea, 2013; Mammen et al., 2016). For example, Mammen et al. (2016) provide ageneral “three-step” estimation framework that uses a nonparametric estimator in each step. A fullaccount of nonlinearity would involve one of these methods. However, our use of several functions of
instruments is quite close to how these nonlinearities are commonly handled in practice.
Let Z denote the instruments. The validity of Z as instruments can be checked. Note that X2
admits the reduced form X2 ¼ g2 ðX3 Þ þ g2 ¼ g2 ðX3I , X3E Þ þ g2 : If X3I is non-empty (i.e., somecommon variables are included as regressors) and some elements of Z are relevant, then thoseelements are either a part of X3I or correlated with X3I : As a result, Z and X2 are also correlated,
and the IV estimator of the omitted variable regression becomes inconsistent. Then, we are
tempted to test the null of consistency of IV estimation indirectly by testing the null hypothesis
H0 : g2 ðX3I , X3E Þ ¼ g2 ðX3E Þ a:e: against the alternative H1 : g2 ðX3I , X3E Þ 6¼ g2 ðX3E Þ for a non-empty
set on suppðX3I Þ: Several versions of the test of significance in nonparametric regressions havebeen proposed in the literature. Examples include Fan and Li (1996) and Racine (1997) for continuous regressors and Lavergne (2001) and Racine et al. (2006) for discrete or categorical regressors, to name a few.
2.5.2. Comparison with PARAPARA imputes a parametric, linear predictor of X2. It seems attractive, as it is not subject to thecurse of dimensionality in continuous common variables unlike PILS. However, it requires

ECONOMETRIC REVIEWS

11dimðX3E Þ > 0 for identification because otherwise, the imputed predictor of X2 is a linear function of X3I : In contrast, as in Assumption 3(iii), PILS achieves identification by nonlinearity evenif X3E is empty.
PARA shares similar properties with the two-sample (TS) two-stage least squares (2SLS) estimator of Inoue and Solon (2010). An interpretation of PARA in the context of TS2SLS is thatthe missing regressor X2 in the former plays the role of an endogenous variable in the latter. In
particular, in the special case when X1 is empty, PARA is numerically equivalent to TS2SLS.1
As demonstrated in Proposition S1 of the online supplement, consistency of PARA depends onwhether X1 is present in (1). It necessarily becomes consistent when X1 is empty. When X1 is nonempty, which is the most common setting in practice, consistency of PARA fails if EðX1 X20 Þ 6¼
EðX1 X30 ÞA0 , where A is the coefficient matrix in the linear projection of X2 on X3.2 On the other
hand, consistency of PILS fails if EðX1 X20 X3 Þ 6¼ EðX1 X3 ÞEðX20 jX3 Þ, which is a violation of an
implication of Assumption 3(ii). Since EðX1 X20 Þ 6¼ EðX1 X30 ÞA0 does not imply EðX1 X20 jX3 Þ 6¼
EðX1 X3 ÞEðX20 jX3 Þ, PILS can be consistent when EðX1 X20 Þ 6¼ EðX1 X30 ÞA0 (i.e., PARA is inconsistent)
but still EðX1 X20 X3 Þ ¼ EðX1 X3 ÞEðX20 jX3 Þ (i.e., Assumption 3(ii) is valid). This could be an explanation for the significant differences observed between the PILS and PARA estimates of the return toschooling in Section 4.
2.5.3. Comparison with MSII(-FM)
While PILS and MSII(-FM) are both two-sample semiparametric estimators, the former has threepﬃﬃﬃ
novel features relative to the latter. First, while both MSII(-FM) and PILS are n-consistent andasymptotically normal two-sample estimators, their approaches to restoring consistency differ.
Consistency of MSII(-FM) is established by imputing X2 from S 2 and then eliminating the nonvanishing bias caused by the imputation. The bias correction requires a consistent, nonparametric^ 2 is Op ðn1=2 Þ: As a consequence, the asympestimate of R2 , and that the estimation error in R
totic variance of MSII(-FM) tends to be large and highly complicated because of multiple asymptotically normal terms with the same Op ðn1=2 Þ rate.
In contrast, consistency of PILS comes from directly imputing a consistent estimate ofg2 ðX3 Þ ¼ EðX2 jX3 Þ in place of the missing X2. By construction, it does not require bias correctionthat is a key ingredient in MSII(-FM).3 The asymptotic variance of PILS also comes from two
Op ðn1=2 Þ terms, namely, the sampling error as in OLS and the approximation error from replacing the unobservable g2 with its kernel estimate ^g 2 : Although it is hard to compare asymptotic
variances of PILS and MSII(-FM) analytically, Monte Carlo simulations below indicate that theformer tends to be smaller than the latter.
Second, the asymptotic analysis we develop for PILS explicitly incorporates discrete matchingvariables. This is in contrast to the asymptotic analysis of Hirukawa and Prokhorov (2018), who
do not accommodate discrete variables explicitly but argue that, similarly to the treatment effectliterature (see, e.g., Abadie and Imbens, 2006), the inclusion of discrete matching variables with afinite number of support points does not affect convergence rates of MSII(-FM).
Third, we clarify the role of excluded continuous matching variables for identification.
Hirukawa and Prokhorov (2018) maintain the assumption that all common variables enter the1

We thank an anonymous referee for pointing this out to us.
The coefficient matrix A, as opposed to the one given in Proposition S1 of the online Supplement, is based on theassumption that the linear projection has no intercepts. Our argument below is not affected with or without intercepts inthe projection.
3
From the viewpoint of imputation, the relationship between MSII(-FM) and PILS may be compared to the one between the
NNM-based estimators studied by Abadie and Imbens (2006, 2011) and kernel-based matching estimators studied by Heckman
et al. (1998).
2

12

M. HIRUKAWA ET AL.

regression and are used for both estimation and matching. The price to pay for this assumption
is that we need to impose nonlinearity of the conditional mean of the missing regressor given thematching variables in order to achieve identification. This article relaxes the assumption so thatsome common variables may be employed only for imputation. The existence of such common
variables allows for linearity in the conditional mean.

3. Finite-sample performance
3.1. Monte Carlo setup
In this section we present selected results of an extensive Monte Carlo study comparing finitesample properties of PILS with the alternative estimators. The design of the Monte Carlo study ismeant to reflect the two scenarios depending on whether or not X1 is empty. Due to the limitation of space, we focus on the more realistic case in which X1 is non-empty. Monte Carlo designs
and results when X1 is empty are available in the online supplement.
The Monte Carlo study is designed to mimic important aspects of the return to schoolingapplication in Section 4. These are: (a) allowing for both included and excluded continuous
matching variables (educ and age in the application); (b) maintaining n=m ¼ 2 and considering
the sample sizes of ðn, mÞ ¼ ð2000, 1000Þ; (c) permitting the included continuous matching variable X3IC (educ) to become endogenous when X2 (abil) is omitted; and (d) choosing as an instrument Z a variable that differs from the matching variable (fatheduc) and setting corrðX3IC , ZÞ andcorrðX3IC , Z2 Þ (roughly) equal to 0.4.4 Of primary interest is estimation of b3, the coefficient on
X3IC (educ).
Our design is largely inspired by the study on nonlinear instruments of Dieterle and Snell(2016). Consider the two forms of our regression
ðLÞ Y ¼ b0 þ b1 X1 þ b2 X2 þ b3 X3IC þ u,ðSÞ

Y ¼ b0 þ b1 X1 þ b3 X3IC þ v, v ¼ u þ b2 X2 ,where true parameter values are b0 ¼ b1 ¼ b2 ¼ b3 ¼ 1, and b3 is the parameter of interest. Let
iid

iid

iid

iid

u, g1 Nð0, 1Þ, X3EC , n U ½2, 2, Z U ½0, 1, and X3ED Bernoullið1=2Þ  1=2, where all theserandom variables are mutually independent. Next, X1 and X3IC are generated as
2
X þ g1 , andX1 ¼ 2X3IC X3E
C 3ED

X3IC ¼ 4q1 Z 

1þ2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1  q21 n,

where q1 ¼ corrðX3IC , ZÞ and q1 2 f0:1, 0:4g: The cases with q1 ¼ 0:1 and 0.4 correspond to
weak and strong instruments, respectively. The specification of X3IC ensures compactness ofsuppðX3IC Þ, EðX3IC Þ ¼ EðX3EC Þ ¼ EðX3ED Þ ¼ 0 and VarðX3IC Þ ¼ VarðX3EC Þ ¼ 4=3 regardless of q1.
A straightforward calculation also yields(
pﬃﬃﬃﬃﬃ
0:0968 for q1 ¼ 0:1
15
2
q1 ¼
:
q2 ¼ corr X3IC , Z ¼
4
0:3873 for q1 ¼ 0:4
The missing regressor X2 is generated as one of the following two models:4

Sample correlations between educ and fatheduc and between educ and fatheduc2 are 0.434 and 0.420, respectively.

ECONOMETRIC REVIEWS

13

Model A:X3IC
þ X3EC þ X3ED þ g2 , and
5q1
!
X3IC
2
g2 ¼ E
Z ¼  ð2Z  1Þ:
5q1
5
X2 ¼

Model B:
1
X3IC 3 1
ppﬃﬃﬃﬃ
X3EC X3ED þ g2 ,
þ sin
4
2
12 C 2q1()

1
X3IC 3
1
1
3
ð
Þ

1
2Z

1
, and
g2 ¼  pﬃﬃﬃﬃ E
Z ¼  pﬃﬃﬃﬃ ð2Z  1Þ þ
2q1
q21
12 C
12 C
8
4
1

þ 4:
C¼
2
105 15q1 3q1

X2 ¼


pﬃﬃﬃﬃ
The constant C for Model B is the variance of E ðX3IC =ð2q1 ÞÞ3 jZg: Hence, the term 12 C in
the denominator of g2 is intended to make Varðg2 Þ invariant to q1. Also observe that X2 is linear
in X3 ¼ ðX3IC , X3EC , X3ED Þ for Model A. This guarantees consistency of PARA. On the other hand,
X2 is nonlinear and even additively non-separable in X3 for Model B. As a result, PARA becomesinconsistent in this case.
The above procedure provides us with two observable samples

n


:
S 1 ¼ ðYi , X1i , X3IC i , X3EC i , X3ED i , Zi Þ i¼1 and S 2 ¼ ðX2j , X3IC j , X3EC j , X3ED j Þ m
j¼1
The complete sample

nS  ¼ ðYi , X1i , X2i , X3IC i , X3EC i , X3ED i , Zi Þ i¼1

is the sample that would not be observed in practice. Finally, the matched sample

S ¼ Yi , X1i , X2j1 ðiÞ , :::, X2jK ðiÞ , X3IC i , X3EC i , X3ED i , Zi ni¼1
is constructed via the NNM with respect to X3 ¼ ðX3IC , X3EC , X3ED Þ as in Hirukawa andProkhorov (2018). The NNM is based on the Mahalanobis distance, and the number of matches
is set equal to K ¼ 1 (single match) that is most commonly chosen. Sample sizes of S 1 and S 2 aren 2 f1000, 2000g and m ¼ n=2: The number of replications is 1000.
There are three options to estimate b3 consistently. The first option is to estimate regression(S) using S 1 only. While EðvÞ ¼ EðX1 vÞ ¼ 0 holds, EðX3IC vÞ 6¼ 0 (i.e., X3IC is endogenous in (S))
is the case and OLS for (S) is inconsistent. However, EðvjZÞ ¼ 0 and q 6¼ 0: Hence, we can estimate b3 consistently by using Z and its functions as instruments for X3IC : The remaining two
options rely on both S 1 and S 2 to estimate regression (L). Option two is to construct thematched sample S from S 1 and S 2 and run MSII or MSII-FM using S: Option three is toemploy PILS using both S 1 and S 2 :Based on the above estimation strategies, we compare the following estimators of b3: (i) the
infeasible OLS estimator for regression (L) using S  [OLS];5 (ii) the inconsistent OLS estimator
for regression (S) using S 1 only [OLS-S]; (iii) the IV estimator for (S) using Z as an instrumentfor X3IC [IV1-S]; (iv) the IV estimator for ðSÞ using Z2 as an instrument for X3IC [IV2-S]; (v) the
two-step GMM estimator for ðSÞ using ðZ, Z2 Þ as instruments for X3IC [GMM-S], along with the
OLS is consistent, because EðuÞ ¼ EðX1 uÞ ¼ EðX2 uÞ ¼ EðX3IC uÞ ¼ 0:

5

14

M. HIRUKAWA ET AL.

initial 2SLS estimator [2SLS-S]; (vi) the inconsistent MSOLS estimator for (L) using S with K ¼ 1[MSOLS]; (vii) the MSII-FM estimator for (L) using S with K ¼ 1 and second-order polynomial[MSII-FM], along with the first-step MSII estimator [MSII]; (viii) the PARA estimator for (L)using S 1 and S 2 [PARA]; (ix) the PILS estimator for (L) using S 1 , S 2 and the Epanechnikov kernel KE ðtÞ ¼ ð3=4Þð1  t2 Þ1fjtj  1g [PILS-E];6 and (x) the PILS estimator for (L) using S 1 , S 2
and the beta kernel [PILS-B].
Implementing the two PILS estimators requires a choice of the smoothing parameters. We use^^ U ð log m=mÞ0:6 for the beta kernel, and^ X ð log m=mÞ0:3 for the Epanechnikov kernel, ^b ¼ r
h¼r
0:6
^
^ X and r^ U are sample standard deviations of the
k ¼ ð log m=mÞ for the discrete kernel, where r
continuous common variable in the original scale Xð¼ X3IC , X3EC Þ and in the transformed scale
U :¼ ðX þ 2Þ=4 2 ½0, 1, respectively. These shrinkage rates fulfill the requirements in
Section 2.4.4.
For each estimator of b3, the following performance measures are computed: (i) Mean (simulation average of the parameter estimate); (ii) SD (simulation standard deviation of the parameter
estimate); (iii) RMSE (root mean-squared error of the parameter estimate); (iv) SE (simulation
average of the standard error); and (v) CR (coverage rate for the nominal 95% confidence interval). Formulae for standard errors are as follows: (a) heteroskedasticity-robust standard errors byEicker (1963) and White (1980) are calculated for OLS and IV-S; (b) the formula for MSII-FM
follows Theorem 4 of Hirukawa and Prokhorov (2018), (c) the formula for PILS appears in
Theorem 2; and (d) the formula for PARA is given in Proposition S3 of the online supplement.
Because of inconsistency we do not compute SE or CR of OLS-S and MSOLS. Those of the firststep MSII are also omitted due to its nonparametric convergence rate. Likewise, we present SE
and CR of PARA only when it is consistent, i.e., for Model A.
We also consider a variation to the above design which examines how the estimators behave
near identification failure. Identification of two-sample estimators heavily relies on the functional
form of g2 ðÞ and existence of excluded matching variable X3E : In particular, identification failure
occurs if g2 ðÞ reduces to a linear function of the included matching variable X3I only.
Therefore, we allow for weaker identification by changing how we generate X2:
Model A0 :
X2 ¼

X3IC
þ wðX3EC þ X3ED Þ þ g2 ,5q1

Model B0 :
X2 ¼

1
X3IC
pﬃﬃﬃﬃ
12 C 2q1

3

1
p
X3EC X3ED þ g2 ,
þ w sin
42

where the weight w 2 f1:0, 0:1g controls the strength of identification in two-sample estimators.
We fix q1 ¼ 0:4, ðn, mÞ ¼ ð2000, 1000Þ and the number of replications is 1000. All other aspectsof the simulation setup remain unchanged. Therefore, w ¼ 1.0 corresponds to the results for q1 ¼
6

In addition to the second-order Epanechnikov kernel, we consider the fourth and sixth-order Epanechnikov kernels by Hansen(2005), where their functional forms are
KE, 4 ðtÞ ¼

15
7
175
33
1  t2 KE ðtÞ and KE, 6 ðtÞ ¼
1  6t2 þ t4 KE ðtÞ,
8
3
64
5

respectively. However, our early simulation results indicate huge finite-sample biases from these estimators, and thus are
not reported.

ECONOMETRIC REVIEWS

15

Table 1. Monte Carlo results.
ðn, mÞ ¼ ð1000, 500ÞEstimator

Mean

SD

RMSE

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99862.98020.74610.70761.02071.02091.12650.90580.82030.99901.02881.00130.05590.04501.2942
1.59970.92470.9251
0.06240.08050.0827
0.06120.07200.06370.05591.98071.31891.62620.92490.92530.14110.12390.19780.06120.07750.0637

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99991.42030.99930.99891.00181.00191.0166
0.99520.98610.99981.00941.00130.02940.04510.11630.11980.11650.11670.03130.03170.03170.03100.03540.03180.02940.42280.11630.11980.11650.11680.03540.03210.03470.03100.03670.0318

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99861.43340.93590.9204
0.99780.99741.09070.86110.8204
1.4632
1.00870.96070.06020.02880.42590.53610.37170.37210.05980.11020.11852.81220.07020.07270.06020.43440.43060.54200.37170.37210.10870.17730.21522.85010.07070.0827

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99901.10770.99790.9971
0.99870.99871.04040.98540.98091.13241.01920.99920.03710.02830.06870.07090.06890.06900.03700.05380.05530.90970.04140.04170.03710.11130.06870.07100.06890.0691
0.05480.05570.05850.91920.0456
0.0418ðn, mÞ ¼ ð2000, 1000Þ

SE

CR

Mean

Model A (q1 ¼ 0:1)
0.056095%
1.0020––2.9807
1.166892%0.90191.331693%
0.89210.901988%
1.01150.901288%
1.0112––
1.0938––0.94670.106667%
0.9052
0.061195%
1.00090.077395%
1.00240.080499%
0.9963Model A (q1 ¼ 0:4)
0.029394%
1.0012––1.42040.116596%0.99700.120295%
0.99650.116195%
0.99820.116096%
0.9982––
1.0152––0.99870.030892%
0.99380.030795%
1.00100.032791%
1.00350.031595%
1.0012Model B (q1 ¼ 0:1)
0.061395%
0.9996––1.43430.420797%0.98200.480196%
0.97940.362395%
1.00670.361994%
1.0064––
1.0663––0.92220.130981%0.9079––1.52600.075195%
0.99260.089698%
0.9636
Model B (q1 ¼ 0:4)
0.037396%
1.0005––
1.10840.070396%
1.00040.072695%
1.00020.070296%
1.00080.070195%
1.0008––1.0378––0.9917
0.053493%
0.9897––
1.11480.048794%
1.01100.064098%
0.9992

SD

RMSE


SE

CR

0.03970.03140.62120.66510.56290.5631
0.04340.05040.0511
0.0426
0.04620.04370.03981.98090.62890.67380.56300.56320.10340.07340.10770.0426
0.04630.04390.0397–0.61710.64570.57440.5741––
0.06290.0432
0.05330.0550

95%
–
94%
93%
92%
92%
–
–
72%
96%
98%
99%

0.02040.03110.08000.08320.07980.07980.02190.0221
0.0221
0.02140.02360.02190.02050.42160.08010.08320.07990.07980.02660.0221
0.02290.02140.02390.02190.0207–0.08210.08480.08200.0819––0.02160.02170.02260.0221

96%
–
96%
96%
96%
96%
–
–
94%
96%
93%
95%

0.04470.01990.24320.25930.22990.22990.04400.0616
0.06304.1731
0.04870.05070.04470.43470.24390.26020.23000.23000.0796
0.09920.1116
4.20610.04920.06240.0433–0.24320.25370.23300.2329––0.0676–
0.05230.0583

95%
–
97%
97%
96%
96%
–
–
77%
–
97%
95%

0.02710.01900.0496
0.05110.04940.04940.02650.03620.03662.02420.02980.03070.02710.11010.0496
0.05110.04940.04940.04620.03710.03802.02750.03180.03070.0263–0.0496
0.05120.04950.0495––0.0355–0.03550.0423

94%
–
95%
96%
95%
95%
–
–
94%
–
96%
98%

0:4 in the original setup. Because the purpose is to check how the quality of two-sample estimates
deteriorates as w shrinks, we only report Mean, SD and RMSE for each estimator.
3.2. Results
Table 1 presents the results from alternative specifications of X2. It is immediately clear that
OLS outperforms all other estimators as would be expected. However, OLS is infeasible, and its

16

M. HIRUKAWA ET AL.

Table 2. Estimates of b3 with shrinking w:
w ¼ 1.0
Estimator

Mean

SD

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

1.00121.4204
0.99700.99650.99820.9982
1.01520.9987
0.99381.00101.00351.00120.02040.03110.08000.08320.07980.07980.02190.0221
0.0221
0.02140.02360.0219

OLS
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

1.0005
1.10841.00041.00021.00081.00081.03780.99170.9897
1.11481.01100.99920.02710.01900.0496
0.05110.04940.04940.02650.03620.03662.02420.02980.0307w ¼ 0.1
RMSE
Model A0 (q1 ¼ 0:4)
0.02050.42160.08010.08320.07990.07980.02660.0221
0.02290.02140.02390.0219Model B0 (q1 ¼ 0:4)
0.02710.11010.0496
0.05110.04940.04940.04620.03710.03802.02750.03180.0307

Mean

SD

RMSE

1.00121.42090.99980.99961.00111.00111.29220.94600.90451.0013
1.14691.05400.04350.01940.05480.05690.05470.05470.04440.16910.18720.08190.06250.07080.04350.42140.05480.05690.05470.05470.29550.17750.21010.08190.15970.08901.0004
1.10841.00041.00021.00081.00081.06290.96470.96001.06651.02901.00280.03170.01900.04950.05110.04940.04930.03090.08150.08401.16490.03750.04240.03170.11010.04950.05110.04940.04930.07010.08880.0931
1.16680.04740.0425

performance can be used only as a benchmark. Instead, realistic comparisons can be madebetween the other estimators.
Performance of IV1-S and IV2-S is largely governed by strengths of instruments, while hugebiases and variability due to weak instruments are ameliorated by a larger sample size. While performances of 2SLS-S and GMM-S improve due to additional moment restrictions, their efficiencies do not surpass those of two PILS estimators.
The instrument strength also influences the performance of the two-sample estimators. MSIIFM is a prominent example. The estimator is severely biased and unstable for each of two weak
instrument cases.
The performance of PARA depends heavily on specification of X2. PARA performs exceptionally well in the more favorable case (i.e., Model A), as expected. On the other hand, the estimator
generates substantial bias and variability for Model B. The poor performance may be attributed
to identification failure due to difficulty in the linear projection of the product of periodic andbinary random variables.
The two PILS estimators appear to be more robust against changes in the strength of instruments and/or specification of X2. An inspection reveals that PILS-B outperforms PILS-E in termsof RMSE, except in the case of Model B with q1 ¼ 0:1:
Table 2 presents the results for the variation addressing near identification failure. Qualities of
OLS, OLS-S and all IV-based estimates remain unchanged throughout this table, as expected. In
contrast, performances of two-sample estimators are influenced by both the functional form ofg2 ðÞ and the value of w.
For Model A0 , g2 ðÞ reduces to a linear function of the included matching variable X3IC as w !
0, and identification failure is expected. Indeed, as w decreases, the finite-sample distribution of
MSII-FM appears to be off-centered from the true value of 1 and highly dispersed. A potential
explanation is that as w decreases, excluded matching variables ðX3EC , X3ED Þ lose their importancein estimating correction terms for non-negligible and matching-discrepancy biases. Nonetheless,ECONOMETRIC REVIEWS

17using the full set of X3 ¼ ðX3IC , X3EC , X3ED Þ makes their estimates quite imprecise due to additional
noises, whereas in reality it may suffice to estimate the terms by using X3IC only. Likewise, thefinite-sample distribution of PARA for Model A0 also tends to be dispersed with w. This may be
attributed to near multi-collinearity in the regression. The proxy of X2 is almost a linear functionof X3IC for a very small w, and imputing it as an extra regressor makes PARA unstable. Largebiases in PILS-E and B are thought to be due to the same identification failure.
On the other hand, g2 ðÞ for Model B0 collapses to a cubic function of X3IC for w ! 0: In thiscase, the role of ðX3EC , X3ED Þ is less important for identification, and the performance of all twosample estimators is only marginally affected by w. In particular, PILS-B seems to be more robust
against changes in w than any other two-sample estimator.

4. An application: Estimation of return to schooling
4.1. Earnings function
Estimation of the causal link between education and earnings has been a focus of labor economists over the last several decades. Card (2001) suggested that endogeneity of education in theearnings equation might at least in part be responsible for the continuing interest in uncovering
the causal effect of education on labor market outcomes. As Griliches (1977) discusses in detail,empirical labor economists have long speculated that the primary reason why education is
endogenous when estimating the returns to schooling is some omitted explanatory variable(s)—unobservable factors that influence education such as ability and motivation—that are likely tohave a direct effect on individual earnings and wages. The aim of this section is to estimate therate of return to additional schooling using alternative estimators available to us. These include
the traditional one-sample OLS and linear and nonlinear IV estimators as well as MSII, PARA,
and PILS. A more comprehensive comparison can be found in the online supplement.
Following the classical framework of the human capital earnings/wage function of Mincer(1974) we assume additivity and wish to use US data in order to estimate the causal effect of education on earnings using the following model:log ðearningsÞ ¼ b0 þ b1 education þ b2 ability þ b3 experience þ b4 experience2
þ b5 married þ b6 black þ b7 south þ b8 urban þ u,(2)

where log ðearningsÞ is the natural logarithm of the individual’s total annual labor income, education is the individual’s completed years of education, ability is the individual’s ability/skills (withzero mean), experience is work experience of the individual, married is an indicator for whether
the individual is married, black is an indicator for whether the individual is black, south is anindicator for whether the individual currently lives in the southern geographical region, urban is
an indicator of the individual’s urban residence while growing up,7 and u is an idiosyncratic error.
The main issue with estimating Eq. (2) is that econometricians are typically unable to observe
the individual’s skills. Because of that, the (one-sample) OLS estimator of the return to schooling—b1—from Eq. (2) where ability is excluded is likely to suffer from an “ability bias.” Two textbook solutions to this problem are (i) to find within the same data set a valid proxy for theunobservable skills and use OLS estimation, and (ii) to find within the same data set a valid
instrumental variable for the individual’s educational level and use the IV approach. Since in addition to these two approaches, we advocate using two-sample estimation, we would like to be able
to apply in practice the one- and two-sample approaches to the same applied task.
7

The publicly available part of the PSID survey that we use does not provide information on current urban residence.

18

M. HIRUKAWA ET AL.

Table 3. Sample characteristics.
Variable

Mean

Std. Dev.
Panel A: The PSID sample

Min.

MedianMax.

Earnings
log (Earnings)
Education
Experience
Married
Black
South
Urban
Age
South while growing up
IQ score
Father’s education

8763.24
8.84
12.15
19.57
0.89
0.26
0.41
0.28
38.57
0.46
9.49
9.26

5968.39
0.80
3.09
13.17
0.31
0.44
0.49
0.45
13.48
0.50
2.28
3.06

30
3.40
5
0
0
0
0
0
17
0
0
0

7900
8.97
12
18
1
0
0
0
37
0
10
8

70000
11.16
17
68
1
1
1
1
86
1
13
17

Education
Married
Black
South
Urban
Age
South while growing up
KWW score

12.98
0.63
0.34
0.53
0.58
28.47
0.55
32.48

1
0
0
0
0
24
0
4

13
1
0
1
1
28
1
33

18
1
1
1
1
34
1
56

Panel B: The NLS sample
3.20
0.48
0.48
0.50
0.49
3.14
0.50
8.42

Notes: ðn ¼Þ2430 individuals in the PSID sample. ðm ¼Þ1102 individuals in the NLS sample. Demeaned IQ score and KWW
score are used.

4.2. Constructing the samples
To estimate earnings and wage equations for the US, labor economists frequently use suchmicro-data sets as CPS, PSID and the National Longitudinal Survey (NLS). While CPS has alarger sample and is more representative of the entire demographic composition of the US population than the other two surveys, it does not typically collect important wage determinants suchas actual work experience and ability. The PSID data set does provide information on actual
work experience but, similar to CPS, generally does not collect data on ability. NLS routinely
reports ability measures and contains data on actual and potential work experience but it is lessrepresentative of the US labor force. Given the features of these surveys, PSID and NLS seem
most suitable as our main and auxiliary data sets, respectively.
We employ the 1972 wave of PSID as our main data set. We choose this wave so that we canestimate Eq. (2) using not only IV and PILS but also using a proxy for unobserved skills and ability. While PSID generally does not contain any measures of unobserved ability, the 1972 wave isamong the few PSID waves that do include an ability measure.
In the 1972 wave, the PSID respondents were administered a particular assessment of abstract
thinking—the Lorge-Thorndike Intelligence Test (LTIT). In essence, this is a test of verbal
skills—a sentence completion test—that Veroff et al. (1971, p.26) describe “as a feasible, reasonably valid assessment of what psychologists have labeled intelligence.” Furthermore, Veroff et al.
(1971, p.26) advise that this test “seems to correlate well with most different kinds of tests ofintelligence, well enough to suggest using it singly without going to multiple measurement.” LTIT
administered to the PSID respondents contained 13 questions, where each question received apoint for the correct response. Thus, the score for the entire sentence completion test can range
from zero (the worst outcome) to 13 (the best outcome). We call the variable containing the total
test score IQ score, and we demean it.
In addition to the information on the individual’s ability discussed above, we also gather information on the individual’s actual work experience, completed years of education, age, whether

ECONOMETRIC REVIEWS

19

the person is black, married, and lives in the southern geographical region.8 Furthermore, we
include a dummy variable for whether the person grew up in an urban area and for whether the
person grew up in the southern region. Finally, in our PSID sample, we also get information on
the completed years of education by the individual’s father. The last variable—father’s educational
level—is used as an instrument for the individual’s educational attainment when no measure ofability is available.
Our main sample contains 2430 men who reported positive labor income in 1971. Panel A of
Table 3 reports summary statistics for the variables in our PSID sample. The sample correlation
between an individual’s education and the father’s education is 0.434.
To exploit the two-sample approaches, we need a second sample where the ability measure isavailable and/or potentially more reliable. We employ the CARD data set provided with
Wooldridge (2013). This data set contains observations from the 1977 wave of the Young Men
Cohort of NLS. We exploit observations with positive wages in 1976.
As an ability measure we employ the results of the “Knowledge of the World of Work”(KWW score) test that the NLS respondents were administered during their interviews in 1966.
The KWW score from NLS is arguably a better measure of unobserved ability than the IQ scorefrom the 1972 wave of PSID. A higher KWW score indicates higher intelligence. We demean thismeasure before using it to estimate Eq. (2).
Panel B of Table 3 provides summary statistics for the NLS sample of m ¼ 1102 respondents
when education, married, black, south, and urban are used as the included common variables,
X3I , and age (the individual’s age, in years) and south while growing up (an indicator for whether
the person resided in the southern region while growing up) are used as the excluded common
variables, X3E : Note that experience and experience2 , which are not observed in the NLS sample,
represent X1 in regression (2). The summary statistics for experience are provided in Panel A of
Table 3.
4.3. Estimation approaches
We report estimation results in Table 4.9 Columns (1)–(3) report the parametric estimation
results based on just the PSID sample. Columns (1) reports the OLS estimates using the ability
proxy from the 1972 waive. Column (2) reports the OLS results that do not account for unobserved ability. In most real-life settings, the results in column (1) are infeasible and those in column (2) are subject to the “ability bias.” Column (3) provides the IV estimates where the father’s
education is used as an IV for an individual’s educational level.
Columns (4)–(7) report two-sample estimates based on both the PSID and NLS samples.
Columns (4), (5), and (7) contain the results based on semiparametric two-sample approaches,
while column (6) reports the only estimates based on a fully parametric two-sample procedure.
Specifically, column (4) contains the OLS estimates when the two samples are combined using
hot-deck imputation and no bias correction is applied. As discussed by Hirukawa and Prokhorov(2018), this estimator is inconsistent. Column (5) provides the MSII-FM results that account for
the imputation biases. In essence, the MSII approach is a bias-corrected version of the MSOLS
approach. Column (6) exhibits the PARA results as yet another two-sample estimator. It is worthemphasizing that PARA is not guaranteed to be consistent in this case. Finally, column (7)reports the PILS results using the beta kernel, chosen on the basis of the simulation results inprevious section.
8

We follow the 1979 NLS of Youth definition of the southern region. The southern region includes Alabama, Arkansas,
Delaware, District of Columbia, Florida, Georgia, Kentucky, Louisiana, Maryland, Mississippi, North Carolina, Oklahoma, South
Carolina, Tennessee, Texas, Virginia, and West Virginia.
9
GAUSS codes implementing the IV and two-sample estimators as well as the data sets are available at https://sites.google.
com/site/artembprokhorovv0/papers/SimulationAndApplicationFiles.zip.

20

M. HIRUKAWA ET AL.

Table 4. Estimation results.

Education
Experience
Experience2
Ability
Married
Black
South
Urban
Data combination?
Sample size: n
m

ð1Þ
OLS
0.0635
ð0:0056Þ
0.0809
ð0:0043Þ
–0.0017
ð0:0001Þ
0.0313
ð0:0078Þ
0.3717
ð0:0539Þ–0.1302
ð0:0323Þ
–0.0921
ð0:0288Þ
0.1363
ð0:0282Þ
No
2430
–

ð2Þ
OLS-S
0.0718
ð0:0053Þ
0.0818
ð0:0043Þ
–0.0017
ð0:0001Þ–ð  Þ
0.3793
ð0:0536Þ
–0.1741
ð0:0316Þ
–0.0983
ð0:0287Þ
0.1499
ð0:0284Þ
No
2430
–

ð3Þ
IV-S
0.0953
ð0:0159Þ
0.0830ð0:0045Þ
–0.0017
ð0:0001Þ–ð  Þ
0.3844
ð0:0536Þ
–0.1249
ð0:0426Þ
–0.0814
ð0:0314Þ
0.1278
ð0:0332Þ
No
2430
–

ð4Þ
MSOLS
0.0727
ð0:0059Þ
0.0826
ð0:0049Þ
–0.0017
ð0:0001Þ
–0.0012
ð0:0027Þ
0.3799
ð0:0535Þ
–0.1849
ð0:0393Þ–0.0979
ð0:0286Þ
0.1538
ð0:0297Þ
Yes
2430
1102

ð5Þ
MSII-FM
0.0685
ð0:0080Þ
0.0766
ð0:0065Þ
–0.0016
ð0:0001Þ
0.0029
ð0:0081Þ
0.3777
ð0:0535Þ
–0.1504
ð0:0806Þ
–0.0989
ð0:0289Þ
0.1404
ð0:0390Þ
Yes
2430
1102

ð6Þ
PARA
–0.2850
ð0:2281Þ
0.1170ð0:0095Þ
–0.0017
ð0:0001Þ
0.2444
ð0:1514Þ
0.2226
ð0:1319Þ
–0.1819
ð0:1146Þ
–0.1372
ð0:1135Þ
0.0138
ð0:1273Þ
Yes
2430
1102

ð7Þ
PILS
0.0568
ð0:0087Þ
0.0818
ð0:0043Þ
–0.0017
ð0:0001Þ
0.0094
ð0:0045Þ
0.3958
ð0:0543Þ
–0.1776
ð0:0316Þ
–0.1042
ð0:0287Þ
0.1557
ð0:0282Þ
Yes
2430
1102

Notes: The dependent variable is the log of total annual labor earnings. Education, married, black, south, and urban are usedas the included common variables, and age and south while growing up are used as the excluded common variables. The(demeaned) IQ score and KWW score variables are used as ability measures in the PSID and NLS samples, respectively.

The two-sample estimators use five variables common to both samples—education, married, black,south, and urban—as included variables and two variables—age and south while growing up—as
excluded variables. Note that since in our case both the main (PSID) and auxiliary (NLS) samples
contain information on the individual’s educational level, we treat education as a part of X3. We
choose not to employ experience as a common variable because experience in PSID represents actual
experience, while experience in NLS represents potential experience, which is quite different. As aconsequence, our entire set of common variables contains education, married, black, south, urban,
age, south while growing up, where education and age are treated as continuous.
In the context of Assumption 3(ii), the PILS results assume that experience and ability are related only
through education, married, black, south, urban, age, and south while growing up. This seems plausible
given the weak empirical link between experience and ability and the long list of observables we condition
on. In the proxy literature, this is known as redundancy or ignorability of the proxy variable in the structural equation, akin to ignorability of selection in selectivity models (see, e.g., Wooldridge, 2010, p.67).
The NNM used by MSOLS and MSII-FM adopts a single match (K ¼ 1) based on the
Mahalanobis metric. We average the (demeaned) KWW score for ties in our second (NLS) sample and assign this average as a unique value of the ability measure to the respondent with thegiven values of our common variables. As a consequence, m ¼ 1102 respondents remain in ourS 2 : A second-order polynomial is used in MSII-FM.
Our choice of a kernel for PILS reflects the favorable performance of the beta kernel in simulations.
For PILS, each continuous common variable X is converted from its original scale to a variable U :¼
ðX  mX Þ=ðMX  mX Þ 2 ½0, 1, where MX and mX are maximal and minimal values in the pooled
sample constructed from observations of X in S 1 and S 2 : The reason for using the maximum and minimum of the pooled sample is that the ranges of education and age are quite different between S 1 and^ U ð log m=mÞ0:6 and ^k ¼ ð log m=mÞ0:6 for the beta and discrete kernels, respectively,
S 2 : We use ^b ¼ r^ U is the sample standard deviation of a converted continuous common variable U in S 2 :10
where r10
One potential concern about PILS is how sensitive it is to the choice of smoothing parameter values in nonparametricestimation of g2 ðÞ: It is confirmed that even after their original values are doubled or cut by half, the estimation results are
qualitatively similar. Detailed results are available in the online supplement.

ECONOMETRIC REVIEWS

21

4.4. Empirical findings
It is immediately clear from Table 4 that the PARA estimation results substantially differ fromthe rest. First, contrary to the corresponding OLS estimate, the estimate of the rate of return toeducation is negative. Second, the PARA estimate of the coefficient on ability is expectedly positive but extraordinarily large when compared to the OLS estimate. Third, PARA estimates arequite unstable in that the standard error of each coefficient estimate is much larger than those of
the corresponding estimates from other methods in general. Because it is hard to provide a meaningful interpretation to the PARA results, we exclude them from further discussion. One lesson
from the PARA estimates is that parametric imputation methods for missing regressors should be
used with caution. In this respect, MSII(-FM) and PILS adopt nonparametric imputation methods, and thus they are more robust than PARA.
Table 4 shows that the signs of coefficient estimates on all the regressors except ability are as
expected and most of the estimates are significant. The MSOLS estimator yields negative and insignificant estimates of the ability effect, whereas the MSII-FM estimator results in a positive and insignificant
estimate. Interestingly, OLS and PILS are the only two estimators that produce positive (as one would
expect) and statistically significant estimates of the ability effect. Finally, we note that the PILS standard
errors tend to be smaller than those of MSII-FM, as predicted in a previous section.
Next we focus on the estimates of the rate of return to education. The estimates reported in Table 4
provide evidence that the father’s education is unlikely to be a valid instrument for an individual’s education. This is because if the father’s education were a valid IV, the OLS-S and MSOLS estimators of b1
would be upward-inconsistent. However, we observe that the IV estimate is the largest in magnitude.
Furthermore, if this instrument is valid and there is no measurement error in the PSID ability measure,then OLS and IV are both consistent. However, we see that the OLS and IV estimates are noticeably
different, suggesting again that the two estimators are unlikely to both be consistent.
The infeasible OLS – OLS—is the second smallest (conceding in magnitude only to PILS). In
addition, if there is no measurement error in the PSID ability measure then OLS is consistent,
but OLS-S and MSOLS are still upward-inconsistent. This possibility seems to fit well with ourestimates in Table 4. Indeed, the feasible one-sample estimate of b1 with omitted ability—OLSS—is between IV and infeasible OLS. In fact, the MSOLS and MSII-FM estimates of b1 are also
between the one-sample IV and infeasible OLS estimates. Therefore, there are grounds to viewthe infeasible OLS as a benchmark result in our analysis.
Importantly, the only estimate of b1 that is smaller than the infeasible OLS is PILS. We notethat PILS is also the second closest (in absolute value) to the infeasible OLS estimate, while thefirst closest is MSII-FM. This would be expected given that only MSII-FM and PILS are the consistent approaches if the instrument is invalid. Furthermore, the possibility that PILS (and not
OLS) is closer to the true return to education is not unrealistic. If the PSID ability measure iserror-ridden, OLS is upward-inconsistent suggesting that PILS is likely to be the only estimatorthat delivers the closest estimate to the true population value.11

5. Concluding remarks
When some regressors are unavailable for a regression analysis, econometricians often resort toIV estimation. They typically make a considerable effort to find and justify valid instruments forthe regressors that are suspected to be endogenous due to their possible correlation with theomitted regressors. In this article, we have explored two-sample alternatives to this conventional approach.
11

The online supplement presents a detailed discussion on this claim.

22

M. HIRUKAWA ET AL.

We have developed the PILS estimation procedure for such models. The procedure first uses
an auxiliary sample to obtain a nonparametric estimator of the conditional mean of the regressor
which is missing in the main sample. Variables that are common in both samples are used in theconditioning set. In the second step of the procedure, we simply run OLS using a plug-in estimate of the conditional mean.
We establish asymptotic normality of PILS. Attractive finite-sample properties of PILS are confirmed in a Monte Carlo study. In particular, simulations demonstrate numerically that PILS isgenerally more efficient than IV and it dominates other available estimators, e.g., MSII(-FM), in
terms of mean squared error. The results also indicate superior finite-sample properties of PILS
using the beta kernel.
However, in order for PILS to attain the parametric rate of convergence, the number of continuous common variables must be three or less (provided that nonnegative kernels are
employed). In this respect, the curse of dimensionality plays out similarly to the NNM-based estimation considered by Hirukawa and Prokhorov (2018).
A natural extension is to adopt a dimension reduction method when using multiple common
variables. This can be done using a sparsity-based algorithm, e.g., lasso of the first step, or using
a propensity score algorithm. We leave these ideas for future research.

Acknowledgments
The authors would like to thank the editor, an associate editor and two anonymous referees as well as the participants of CFE, IAAE, iCEBA, SETA, and a seminar at Durham University for their constructive comments andsuggestions.

Funding
Financial support through grants from Japan Society for the Promotion of Science (M. Hirukawa, Project No.
19K01595) and the Russian Science Foundation (A. Prokhorov, Project No. 16-18-10432) for various and nonoverlapping parts of this research is gratefully acknowledged.

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Appendix A. Technical proofs
A.1.

Construction of product kernel for estimating g2 ðÞ

We employ a continuous univariate kernel for each of d3C continuous variables X3C, p , p ¼ 1, :::, d3C : For simplicity, suppose that X3C, p is smoothed by a univariate symmetric kernel KðÞ and bandwidth hp. Then, the productkernel for X3C, p is

ECONOMETRIC REVIEWS

Kðt3C ; x3C , hÞ ¼

d3C
Y
1

h
p¼1 p

K

25

t3C, p  x3C, p,
hp

where t3C :¼ ðt3C, 1 , :::, t3C, d3C Þ, x3C :¼ ðx3C, 1 , :::, x3C, d3C Þ and h :¼ ðh1 , :::, hd3C Þ are vectors of data points, design
points and bandwidths, respectively.
Next, we construct a kernel for the discrete component. While a variety of discrete kernels can be applied forthis component, our focus is on those given by Aitchison and Aitken (1976). In what follows a product of theirdiscrete kernels is exclusively considered. Each of d3D discrete variables X3D, q , q ¼ 1, :::, d3D is assumed to takerq ð 2Þ different values, i.e., X3D, q 2 f0, 1, :::, rq  1g: In addition, each discrete variable is classified into either
unordered or ordered, because the kernels employed for the two types of categorical variables differ slightly. The
univariate discrete kernel for an unordered variable is
l t3D, q ; x3D, q , kq :¼(

1

if t3D, q ¼ x3D, q

kq =ðrq  1Þif t3D, q 6¼ x3D, q,

where t3D, q , x3D, q and kq 2 ð0, 1Þ are the data point, design point and bandwidth, respectively. The univariate discrete kernel for an ordered variable takes the form of‘ t3D, q ; x3D, q , kq :¼

rq
t3D, q  x3D, q!
r  t x
jt3D, q x3D, q j
1  kq q j 3D, q 3D, q j kq:

If there are q1 ð d3D Þ unordered discrete variables, then the product kernel for all d3D discrete variables is given
by
9
( q1
)8 d
3D
Y 
 < Y

=
Lðt3D ; x3D , kÞ ¼
l t3D, q ; x3D, q , kq
‘ t3D, q ; x3D, q , kq ,:q¼q1 þ1;
q¼1
where t3D :¼ ðt3D, 1 , :::, t3D, d3D Þ, x3D :¼ ðx3D, 1 , :::, x3D, d3D Þ and k :¼ ðk1 , :::, kd3D Þ:
For the continuous component, we also apply an alternative nonstandard kernel. Taking compactness of X3C
into account (see Assumption 2), we employ the beta kernel by Chen (1999) in place of the univariate symmetrickernel. For a design point x 2 ½0, 1 and the smoothing parameter b, the beta kernel is defined asð1xÞ=b
tx=b ð1  tÞ
 1 t 2 ½0, 1 :KBðx, bÞ ðtÞ ¼ ðÞ
B x=b þ 1, 1  x =b þ 1

It turns out that convergence properties of nonparametric estimators based on this kernel family have not been
established. In a companion paper (Hirukawa et al., 2022), we provide a careful proof of weak and strong uniform
convergence with rates of nonparametric estimators smoothed by the beta kernel.

A.2.

Proof of Theorem 1

The proof requires a lemma on uniform consistency of the kernel regression estimator ^g 2 ðÞ: Let g2k ðÞ be the kthelement of g2 ðÞ for k ¼ 1, :::, d2 : In order to differentiate the continuous kernels used in the nonparametric estimag B2k ðÞ,tion of g2k ðÞ, we denote the estimates obtained using th-order symmetric and beta kernels by ^g S
2k ðÞ and ^S
Brespectively. Then, the lemma below establishes weak uniform consistency with rates of ^g 2k ðÞ and ^g 2k ðÞ:

26

M. HIRUKAWA ET AL.

Lemma A1. If Assumptions 1–6 hold, then, for k ¼ 1, :::, d2 , as m ! 1,

0
1
d3C
d3D
n
o
X
X
S

@
hp þ
kq A,
sup E ^g 2k ðx3 Þ  g2k ðx3 Þ ¼ O

x3 2X3p¼1

q¼1

0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1
n
o
log m
S
S
sup ^g 2k ðx3 Þ  E ^g 2k ðx3 Þ ¼ Op @Qd3C A,m p¼1 hpx3 2X3
0
1
d3C
d3D
n
o
X
X
E ^g B2k ðx3 Þ  g2k ðx3 Þ ¼ O@ bp þ
kq A, andsupx3 2SX3C X3D

p¼1

q¼1

0vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1
u
n
o
log m
u
B
B
C:
^g 2k ðx3 Þ  E ^g 2k ðx3 Þ ¼ Op B
sup@t qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A
Qd3C
x3 2SX3C X3Dmbhp¼1 p pA.2.1.

Proof of Lemma A1

First two statements on ^g S
2k ðÞ are implied by Theorem 2.1 of Li and Ouyang (2005). Remaining two statements on^g B2k ðÞ are established by Theorem 7 of Hirukawa et al. (2022) with rn ¼ Oð1Þ due to uniform boundedness frombelow of f ðÞ:
䊏

A.2.2.

Proof of Theorem 1

To save space, we concentrate only on the case with the beta estimator ^g B2 ðÞ: The PILS estimator admits the
expansion
1
1
^
0
b
^ ,  þ Sn, X^ Sn, X
^ Sn, X^,X^,X
PILS ¼ b þ Sn, X^ , fg ðX Þ^g B ðX Þg b2

30
¼ b þ S1
^ Sn, X^,X
^ , ½g ðX ÞEf^g B ðX ÞjX g b
n, X
2

3

3

2

3

2

3

2

(A1)

2

1
0
þ S1
^   Sn, X^,X^,X^ Sn, X,
^ Sn, X^ , ½^g B ðX3 ÞEf^g B ðX3 ÞjX3 g b2 ,
n, X
2
2

where the third and fourth terms on the right-hand side can be interpreted as those of the sampling error inregression (1) and the approximation error of g2, respectively. It follows from Lemma A1 that Bg2 :¼
Pd3C
Pd3D
1=2
0
S1
Þ by the central^,X^ Sn, X
p¼1 bp þ
q¼1 kq Þ: Each of the last two terms is Op ðn^ , ½g2 ðX3 ÞEf^g B ðX3 ÞjX3 g b2 ¼ Op ðn, X
2limit theorem (CLT); see the proof of Theorem 2 for more details. Therefore,
0
1
d3C
d3D
X
X
p
^
@
bp þ
kq A þ Op ðn1=2 Þ þ Op ðn1=2 Þ ! b
b PILS ¼ b þ Op
p¼1

q¼1

as n, m ! 1:

A.3.

䊏

Proof of Theorem 2

We continue to focus on the case with the beta estimator ^g B2 ðÞ: Short-hand notations such as g2i ¼

g2 ðX3i Þ, Ei ð^g B2i Þ ¼ E ^g B2 ðX3i ÞjX3i g, and W j:i ¼ WðX3j ; X3i , b, kÞ also apply. Then, by Lemma A1 and CLT,p

Sn, X,^ X^ ¼ Sn, X, X þ op ð1Þ ! UX, X > 0, and

(A2)

n
X

pﬃﬃﬃ


1
d
Xi i þ op ð1Þ ! N 0ðdþ1Þ1 , X1
nSn, X^ ,  ¼ pﬃﬃﬃ
n i¼1

(A3)

for the sampling error part. Moreover,
Sn, X^ , ½^g B ðX ÞEf^g B ðX ÞjX g0 b ¼ Sn, X, ½^g B ðX ÞEf^g B ðX ÞjX g0 b þ Sn, X^ X, ½^g B ðX ÞEf^g B ðX ÞjX g0 b
2

3

2

3

3

2

2

3

2

3

3

2

2

3

2

3

3

2

for the approximation error part, where the second
term is of a smaller order of magnitude than the first term.
PPm~ Bi :¼ m
Also observe that ^g B2i  Ei ð^g B2i Þ ¼ ~g B2i  Ei ð~g B2i Þ þ ð~g B2i  g2i Þ þ g2i , where u
j¼1 uj W j:i =j¼1 W j:i for u 2
:
Then,
by
a
similarargument
to
Proposition
9ofStengosandYan(2001)
and
CLT,
,
g
g
f 2 2gECONOMETRIC REVIEWS

n
pﬃﬃﬃ


1 X
d
nSn, X^ , ½^g B ðX ÞEf^g B ðX ÞjX g0 b ¼ pﬃﬃﬃ
Xi g02i b2 þ op ð1Þ ! N 0ðdþ1Þ1 , X2 :
3
3
3
2
2
2
n i¼1

Finally, it is straightforward to see that for every i and k,"#  0 0BB
^ k ^g  Ek ^g^ i i X¼ 0ðdþ1Þðdþ1Þ :
b2
E X
2k
2k
The result is established by substituting (A2)–(A5) into (A1) and rearranging it.

A.4.

27

(A4)

(A5)
䊏

Proof of Corollary 1

Again to save space, we provide the proof for the case with the beta estimator ^g B2 ðÞ: To control the convergence
rate of h, let h / bd=d3C for a sufficiently small d > 0 so that b=h ! 0 can be established. Because Bg2 ¼
pﬃﬃﬃ ^
pﬃﬃﬃ ^
Op ðbÞ þ op ðn1=2 Þ, nðbnðb PILS  bÞ þ op ð1Þ holds if
PILS  b  Bg2 Þ ¼
nnb2 ¼ Oð log mÞ4a m14a ! 0:mThis is the case if one of the following conditions is true: (i) n=m ! j and a > 1=4; (ii) n=m ! 0 and a > 1=4; or(iii) n=m ! 1, n=m4a ! 0 at a polynomial rate and a > 1=4: It also follows from Assumption 60 (b) that()
1aðd3C þdÞ
log m
log m
¼O
!0
d =2
m
mðbhÞ 3C^
shouldbe the case. This holds if 1  aðd3C þ dÞ > 0 or a < 1=ðd3C þ dÞ < 1=d3C : Therefore, for b
PILS to bep
ﬃﬃﬃ䊏n-consistent, a must satisfy 1=4 < a < 1=d3C : This range is non-empty if d3C  3: