Supplement to“Yet Another Look at the Omitted Variable Bias”
Masayuki Hirukawa*
Ryukoku University

Irina MurtazashviliDrexel University

Artem Prokhorov
University of Sydney& CEBA & CIREQ

December 6, 2022

1

Alternative Estimators to PILSThis section discusses the definitions and statistical properties of estimators alternative to PILS. The estimators include IV, PARA and three matching-based estimators.

1.1

IV Estimator

One may attempt to complete the estimation using only S1 , for example, in light ofa cost associated with searching for another data source containing S2 . In this case,
X2 in
0
Y = β0 + X10 β1 + X20 β2 + X3I
β3 + u,(1)

is treated as the vector of omitted variables. Accordingly, regression (1) reduces to
0 0
Y = XS0 βS + v, where XS := (1, X10 , X3I) , βS := (1, β10 , β30 )0 and v := u + X20 β2 . The

subscript “S” stands for a short regression.
OLS for the short regression is inconsistent if (X1 , X3I ) and X2 are correlated.
Suppose that S1 additionally includes a vector of instruments Z ∈ RdZ that correlate* e-mail: hirukawa@econ.ryukoku.ac.jp.
 e-mail: im99@drexel.edu.

e-mail: artem.prokhorov@sydney.edu.au.

1with (X1 , X3I ) but not with X2 . Note that X3E is not used as instruments unless
it is part of Z. Then, it is possible to estimate βS consistently on the basis of the
moment restriction
E (ZS v) := E

1
Z





(u + X20 β2 )= 0(dZ +1)×1 .

Assuming dZ ≥ d1 + d3I , consistency of this estimator will be based on strongq X2 and E (X2 )0 β2 = 0. The benefit is a simple estimatorassumptions such as Z −which, in the just-identified case (dZ = d1 + d3I ), can be written as
!
!−1
n
n
X
X
1
1
−1
0
ZS,i XS,i
ZS,i Yi .
β̂IV := Sn,ZS:=
S ,XS n,ZS ,Y
n i=1
n i=1
p

Under suitable regularity conditions, β̂IV → βS and


√ 
d
0−1
n β̂IV − βS → N 0(dZ +1)×1 , Φ−1
ZS ,XS ΥΦZS ,XS ,
0where Υ := E ZS ZS v 2 .

1.2

PARA Estimator

Instead of imputing a nonparametric estimate of g2 (X3 ) in place of the missing regressor X2 in regression (1), Fang, Keane, and Silverman (2008) propose to employ aparametric, linear predictor of X2 . Since Fang, Keane, and Silverman (2008) do not
document statistical properties of PARA, we explore them below.
PARA is a two-step two-sample estimator. Consider the linear projection of X2
on X3 , i.e., proj (X2 | X3 ) := AX3 , where X3 := (1, X30 )0 and A is some d2 × (d3 + 1)
coefficient matrix. In the first step, we estimate A using S2 by means of equationn on
non
by-equation OLS. As a result, n linear predictors X̃2i
= ÃX3ican bei=1

i=1

obtained, where Ã is the OLS estimate of A. Then, in the second step, we run OLS00
for the regression of Y on X̃ := 1, X10 , X̃20 , X3I
using S1 to estimate β. The PARA
2

estimator is then defined asn−1β̂P ARA := Sn,SX̃,X̃ n,X̃,Y

1X
X̃i X̃i0
n i=1

=

!−1

n

1X
X̃i Yi .
n i=1

The first proposition refers to the condition for consistency of PARA. In what
0 0
follows, denote X A := 1, X10 , (AX3 )0 , X3I.
Proposition S1. Suppose that d3E > 0 and that X2 is relevant in regression (1),pi.e., β2 6= 0. Then, under Assumptions 1, 2 and 3(i) of the paper, β̂P ARA → β asn, m → ∞ if and only if E (X1 e0 ) = 0d1 ×d2 , where e := X2 − AX3 is the projection
error.

Proof. By construction, e is orthogonal to X3 , i.e., E (X3 e0 ) = 0. We note that thisis true even if proj ( X2 | X3 ) 6= g2 (X3 ) = E (X2 | X3 ), which would be the case if theconditional expectation function is nonlinear in X3 . A simple manipulation of regres00sion (1) yields Y = β0 +X10 β1 + X̃20 β2 +X3I
β3 +ε, where ε := u+e0 β2 −X30 Ã − A β2is the composite regression error. Also observe that Ã = A + Op m−1/2 = A + op (1).


n

o
Then, E X̃2 ε = E (AX3 ε) + E Ã − A X3 ε = E (AX3 ε) + o (1), and PARA isqconsistent if and only if E XiA εi = 0(d+1)×1 . By Assumption 3(i) of the paper, Ã − 
S1 and E Ã = A, we have E (εi ) = 0, E (AX3i εi ) = 0d2 ×1 , and E (X3Ii εi ) = 0d3I ×1 .
However, E (X1i εi ) = E (X1i e0i ) β2 6= 0d1 ×1 unless E (X1 e0 ) = 0d1 ×d2 . 
Proposition S1 suggests that the distinction between X1 and X3 becomes important. PARA is consistent if X1 is empty. If X1 is non-empty, then consistency of
PARA is not guaranteed and depends on whether or not E (X1 e0 ) = 0d1 ×d2 . We may
be tempted to test the null of E (X1 e0 ) = 0d1 ×d2 before running PARA. However,since X1 and X2 belong to two distinct samples, this task may not be feasible.

3

In the simulation study in Section 2 of this Supplement, we consider the cases inwhich PARA is consistent although X1 is non-empty in regression (1). The proposition below justifies such cases.

Proposition S2. (Consistency of PARA in Simulations) Suppose that X1 is
non-empty in regression (1). If the function h(·) used in Simulation Studies I and IIIof this Supplement is odd, then PARA is consistent.

Proof. We focus only on the design for Simulation Study I to save space and adopt
the notations used there. Let proj ( X2 | X3 ) = AX3 = A0 + A1 X3IC + A2 X3EC +A3 X3ED . Then, since X3IC , X3EC , X3ED , and η2 are zero-mean mutually independent
random variables, the reduced form of X2 implies that all the following statements
are true:
1. A0 = 0, A2 = − (3/4) (H2 + 1), A3 = 2;
2. A1 satisfies A1 X3IC = proj {h (X3IC )| X3IC }; and3. The projection error e = X2 − AX3 = h (X3IC ) − A1 X3IC + η2 .
By Proposition S1 above, PARA is consistent if and only if E (X1 e) = 0. However,it follows from the reduced form of X1 and distributional assumptions on X3IC , X3EC , 2
X3ED , and η1 that E (X1 e) = H2 = E X3I
h (X3IC ) holds. The right-hand side isCzero if and only if h (·) is odd, which establishes the proposition. We conclude this subsection by documenting the limiting distribution of PARA.
The covariance estimate if n/m → κ is applied in Sections 3 and 4 of the paper, as
well as Sections 2 and 3 of this Supplement.

4

Proposition S3. If the same regularity conditions as in Proposition S1 hold, E (X1 e0 ) =
0d1 ×d2 , ΦX A ,X A = E X A X A0 > 0, and ΦX3 ,X3 = E (X3 X30 ) > 0, then, as n, m → ∞,
 √ 



d
−1
−1

n
β̂
−
β
→
N
0
,
Φ
ΨΦ
if n/m → κ ∈ (0, ∞)P ARA(d+1)×1
X A ,X A
X A ,X A

 √ 



d
n β̂P ARA − β → N 0(d+1)×1 , Φ−1Ψ Φ−1
if n/m → 0,
X A ,X A 1 X A ,X A






d

 √m β̂P ARA − β →N 0(d+1)×1 , Φ−1Ψ Φ−1
if n/m → ∞X A ,X A 2 X A ,X A

where Ψ := Ψ1 + κΨ2 , Ψ1 := E X A X A0 v 2 , v := u + e0 β2 , and−10
Ψ2 := ΦX A ,X3 ΦX
E (X3 X30 β20 ee0 β2 ) Φ−1
X3 ,X3 ΦX A ,X3 .
3 ,X3In addition, asymptotic variances Ψ, Ψ1 and Ψ2 can be consistently estimated byP0Ψ̂ := Ψ̂1 + (n/m) Ψ̂2 , where Ψ̂1 := (1/n) ni=1 X̃i X̃i v̂i2 , v̂i := Yi − X̃i0 β̂P ARA is the
PARA residual,−1Ψ̂2 := Sn,X̃,X3 Sm,X
3 ,X3

!
m
1 X
−1
X3j X3j0 β̂P0 ARA,2 ẽj ẽ0j β̂P ARA,2 Sm,X
S0,
3 ,X3 n,X̃,X3
m j=1

ẽj := X2j − ÃX3j is the first-step OLS residual, and β̂P ARA,2 is the PARA estimateof β2 .

Proof. The proof starts from the first-step equation-by-equation OLS estimation.
The estimate of A using S2 is given bym

1 X
X2j X3j0
Ã = Sm,X2 ,X3 Sm,X3 ,X3 =
m j=1

m

1 X
X3j X3j0
m j=1

!−1
.

Substituting X2j = AX3j + ej into this yields
Ã = A +

m
1 X

m j=1

ej X3j0

m
1 X

m j=1

!−1
X3j X3j0

.

(2)


0
0
Now we proceed to the second step. Denoting X̃2 = ÃX3 and X̃ = 1, X10 , X̃20 , X3I,0
we can rewrite the regression model Y = β0 +X10 β1 +X20 β2 +X3I
β3 +u as Y = X̃ 0 β +ε,

where

0

0
ε = v − X30 Ã − A β2 = (u + e0 β2 ) − X30 Ã − A β2 .
5It follows that
−1β̂P ARA = Sn,SX̃,X̃ n,X̃,Y−1−10= β + Sn,S− Sn,S
X̃,X̃ n,X̃,v
X̃,X̃ n,X̃,X30 (Ã−A) β2!−1 ( n
)
n
n

0
X
X
1X
1
1
=β+
X̃i X̃i0
X̃i vi −X̃i X3i0 Ã − A β2 .
n i=1
n i=1
n i=1

Because Ã = A + Op m−1/2 = A + op (1) by (2),(3)

nSn,X̃,X̃ = Sn,X A ,X A + op (1) =1 X A A0p
X X + op (1) → ΦX A ,X A ,n i=1 i i

(4)


0 0
where X A = 1, X10 , (AX3 )0 , X3I. Also, by CLT,√√√nSn,X̃,v = nSn,X A ,v + nSn,X̃−X A ,v
)
(
n
n

 1 X
1 X A
=√
X vi + O
Ã − A √X3i vin i=1 i
n i=1
n

1 X A
=√
X vi + op (1)n i=1 id
→ N 0(d+1)×1 , Ψ1 ,(5)

where Ψ1 = E X A X A0 v 2 .
Moreover,√nSn,X̃,X 0 (Ã−A)0 β2
3√√= nSn,X A ,X 0 (Ã−A)0 β2 + nSn,X̃−X A ,X 0 (Ã−A)0 β2
3
( 3
)
n
n


 1 X
0


0
1 X A 0Ã − A √=√X X Ã − A β2 + O
X3i X3i0 Ã − A β2
n i=1 i 3i
n i=1
r 

0

√ 
n
−1/2
,
(6)
=
Sn,X A ,X3 m Ã − A β2 + Op mmwhere, by (2),
√



0

m

m Ã − A β2 =1 X
X3j X3j0
m j=1

!−1

m

1 X
√
X3j e0j β2
m j=1


d
−1
0 0
0
→ N 0(d3 +1)×1 , Φ−1
X3 ,X3 E (X3 X3 β2 ee β2 ) ΦX3 ,X3 .
6

Therefore,
0

√ 
d
Sn,X A ,X3 m Ã − A β2 → N 0(d+1)×1 , Ψ2 ,

(7)

where
−1
0
Ψ2 = ΦX A ,X3 ΦX
E (X3 X30 β20 ee0 β2 ) Φ−1
X3 ,X3 ΦX A ,X3 .
3 ,X3

 
q S1 and E Ã = A,
Finally, by Ã −
E

XiA vi0 0 
0
 
A 0
A A0
0
Xi X3i Ã − A β2
= E X X vX3 E Ã − A β2 = 0(d+1)×(d+1)(8)

so that two CLT terms (5) and (6) are asymptotically independent.
In what follows, we consider three different divergence scenarios of (n, m). The
asymptotic analysis in each scenario is based on substituting (4), (5), (7), and (8)into (3). First, if n/m → κ ∈ (0, ∞), then√ n β̂P ARA − β
√

√
−1
= Sn,X̃,X̃
nSn,X̃,v − nSn,X̃,X 0 (Ã−A)0 β2
3
r


0

√
√ 
n
−1
= Sn,X A ,X A + op (1)nSn,X A ,v −S A
m Ã − A β2 + op (1)m n,X ,X3
d
→ N 0(d+1)×1 , Φ−1ΨΦ−1,
X A ,X A
X A ,X A
where Ψ = Ψ1 + κΨ2 . Second, if n/m → 0, then

√

√ 
√
−1
0
n β̂P ARA − β = Sn,X̃,X̃
nSn,X̃,v − nSn,X̃,X 0 (Ã−A) β2
3

−1 √



nSn,X A ,v + op (1)= Sn,X A ,X A + op (1)d−1→ N 0(d+1)×1 , Φ−1ΨΦ
.
X A ,X A 1 X A ,X A

7

Third, if n/m → ∞, then√ m β̂P ARA − β

√
√
−1
0
mS
−
mS
= Sn,
n,X̃,vn,X̃,X30 (Ã−A) β2
X̃,X̃
r

0


√ 
m √
−1
−1/2
= Sn,X A ,X A + op (1)nSn,X A ,v + op (1) + Sn,X A ,X3 m Ã − A β2 + Op m
n


0

√ 
−1
= Sn,X A ,X A + op (1)Sn,X A ,X3 m Ã − A β2 + op (1)


d
−1
−1
→ N 0(d+1)×1 , ΦX A ,X A Ψ1 ΦX A ,X A .
Lastly, it is straightforward to demonstrate consistency of Ψ̂1 , Ψ̂2 and Ψ̂ = Ψ̂1 +(n/m) Ψ̂2 for Ψ1 , Ψ2 and Ψ, respectively. Therefore, the proof for this part is omitted.


1.3

MSOLS, MSII and MSII-FM Estimators

Hirukawa and Prokhorov (2018) propose a set of two-sample estimators to address
imputation biases caused by a practice known as “hot-deck imputation” in Census.
Missing values for non-respondents are imputed using the values from respondents
with similar characteristics without accounting for the imputation bias. Effectively,
the MSII estimator exploits the structure of a surrogate sample obtained by matchingon variables that are common to both respondents and non-respondents.
The MSII estimation starts from obtaining a combined sample by means of NNM
on the observed variables. As before, let S1 = (Y, X1 , X3 ) denote a sample for which
we have all responses and let S2 = (X2 , X3 ) denote the sample which contains thevariable with non-response as well as the observed variables to match on. Hot-deck
imputation uses a distance measure between X3 in S1 and X3 in S2 to obtain amatched value of X2 for S1 . Specifically, for each observation of X3 in S1 , K-NNM
picks out K closest matches of X2 from S2 through finding K closest matches of X3
8

in S2 with respect to the Mahalanobis distance. The resulting combined sample canbe written asS=Yi , X1i , X2j1 (i) , . . . , X2jK (i) , X3Ii , X3Ei

 n
i=1



.

0
0
0
1, X1i, X2j(i), X3Ii0

The MSOLS estimator for the regression of Yi on Xi,j(i) :=,P
where X2j(i) := (1/K) K
k=1 X2jk (i) , generates a non-vanishing, classical measurement
error (or attenuation) bias. The bias is attributed to using X2j(i) as a proxy for X2i ,and the source of attenuation is Σ2 = E (η2 η20 ).
As is well known in the literature on errors-in-variables models, the bias cannot becorrected in general without imposing additional identification conditions. However,in the above two-sample setup it can be corrected analytically with no such extra
conditions because S2 serves as repeated measurements. MSII is a bias-corrected
estimator defined as
!−1
n
n
1X
1
1X
0
Xi,j(i) Xi,j(i)Xi,j(i) Yi ,β̂II :=
− Σ̂
n i=1
K
n i=1
o
n
where Σ̂ := diag 0(d1 +1)×(d1 +1) , Σ̂2 , 0d3I ×d3I , and Σ̂2 is the difference-based varianceestimator that can be obtained by reordering S2 with respect to X3 .
√Notice that MSII can attain n-asymptotic normality when the number of matching variables is one. To overcome the curse of dimensionality in continuous matchingvariables, Hirukawa and Prokhorov (2018) propose a two-step estimator called thefully-modified MSII (MSII-FM) estimator. In its second step, MSII-FM eliminates
the second-order bias due to the so-called matching discrepancy asymptotically bymeans of a polynomial approximation similar to the one studied by Abadie and Imbens (2011) in the context of average treatment effect estimation. It is demonstrated
that MSII-FM can achieve parametric convergence when the number of matchingvariables is no greater than three.
92Additional Monte Carlo Results

This section presents the results from four different simulation designs. First two
studies deal with the cases of both empty and non-empty X1 . The difference is thatwhile the first study considers only linear instruments, the second one covers bothlinear and nonlinear instruments. The third study is based on a somewhat different
design with a wider range of functions h(x) used in the reduced form, and focuses
exclusively on the cases of non-empty X1 . The fourth study is similar to the one
included in the paper, with the difference that X1 is an independent regressor. The
results below indicate superior performances of PILS.

2.1

Simulation Study I

The first simulation study covers the cases of both empty and non-empty X1 . We
begin with the case of non-empty X1 and explain how the data are generated. Thelong regression is in the form of
Y = β0 + β1 X1 + β2 X2 + β3 X3IC + u,(9)where true parameter values are β0 = β1 = β2 = β3 = 1, and β3 is the parameter ofinterest. The short regression after omitting X2 from (9) is
Y = β0 + β1 X1 + β3 X3IC + (u + β2 X2 ) =: β0 + β1 X1 + β3 X3IC + v.

(10)

There are three common variables, namely, an included continuous one X3IC , anexcluded continuous one X3EC and an excluded discrete one X3ED . Also for r ≥ 0
 r
and some function h (·) specified shortly, let Hr := E X3I
h (X3IC ) . Then, X1 , X2
C

10

and Z, the instrument for X3IC , are generated, respectively, as
2
X1 = 1 + X3I+ X3EC + 2X3ED + η1 ,C

X2 = h (X3IC ) −3(H2 + 1) X3EC + 2X3ED + η2 , and4

Z = π (X3IC − 2H1 X3ED ) + ξ,
where π =

(11)
(12)

p
3ρ2 / {4 − ρ2 (4 + 3H12 )} and ρ = corr (X3IC , Z) ∈ {0.1, 0.4}. Notice thatρ = 0.4 roughly mimics the sample correlation coefficient between the endogenous
regressor (educ) and its IV (fatheduc) in Section 4 of the paper.
When X1 is empty in (9), we consider two alternative cases for the pair of themissing regressor X2 and the instrument Z. The first case deals with an additively
separable reduced form of X2 . In this case, the pair of X2 and Z are still (11) and(12), respectively. The second case is based on an additively non-separable reduced
form of X2 . In this case, X2 and Z are specified as
2
X2 = h (X3IC ) X3E+ 2X3ED + η2 , andC
8Z = ζ X3IC − H1 X3ED + ξ,
3

where ζ =p
9ρ2 / {12 − ρ2 (12 + 16H12 )}.

For each case, the function h (·) is specified as one of the following:x[Model A]sin (πx/4)[Model B]
h (x) =,{(5/3) x + 3/2} 1 {x < 0} + {− (4/3) x + 3/2} 1 {x ≥ 0} [Model C]
x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4[Model D]
where φ (·) and Φ (·) are the pdf and cdf of N (0, 1), respectively. Models A and B aremonotone. Linearity of h (·) in Model A does not cause identification failure, because
two common variables (X3EC , X3ED ) are excluded from (9). Model B is nonlinear andodd. Models C and D are non-monotone, and the former is even non-smooth whenx = 0. Inspired by the Monte Carlo design of Horowitz and Spokoiny (2001), ModelD can be viewed as a linear function with a bump.
11

The following distributional assumptions also apply.
(u, η1 , η2 , ξ)0 ∼ N (04×1 , I4 ) .
iid
X3IC , X3EC ∼ U [−2, 2] .

X3ED ∼ Bernoulli (1/2) − 1/2 =−1/2 w.p. 1/2.
1/2 w.p. 1/2

q X3 = (X3IC , X3EC , X3ED ) .
u−
The distributional assumption also implies that values of (H1 , H2 , H3 ) are:
(0, 4/3, 0)[Model A]2(0, 8/π , 0)[Model B](H1 , H2 , H3 ) =
.
(0,
2/9,
−1)
[Model C]
(0, 4/3, −0.9989) [Model D]
The above procedure provides us with two observable samples
S1 = {(Yi , X1i , X3IC i , X3EC i , X3ED i , Zi )}ni=1 and S2 = {(X2j , X3IC j , X3EC j , X3ED j )}m
j=1 .
The complete sample
S ∗ = {(Yi , X1i , X2i , X3IC i , X3EC i , X3ED i , Zi )}ni=1
is the sample that would not be observed in practice. Finally, the matched sampleS=Yi , X1i , X2j1 (i) , . . . , X2jK (i) , X3IC i , X3EC i , X3ED i , Zi

 n
i=1

is constructed via the NNM with respect to X3 = (X3IC , X3EC , X3ED ) as in Hirukawa
and Prokhorov (2018). The NNM is based on the Mahalanobis distance, and thenumber of matches is set equal to K = 1 (single match) that is most commonlychosen. Sample sizes of S1 and S2 are n ∈ {500, 1000, 2000} and m = n/2. For eachcombination of the functional form of h (·) and value of π, we draw 1000 Monte Carlosamples.
We compare the following estimators of β3 : (i) the infeasible OLS estimator for(9) using S ∗ [OLS*]; (ii) the inconsistent OLS estimator for (10) using S1 only [OLSS]; (iii) the IV estimator for (10) using S1 only [IV-S]; (iv) the inconsistent MSOLS
12

estimator for (9) using S [MSOLS]; (v) the MSII-FM estimator for (9) using S and asecond-order polynomial [MSII-FM], along with the first-step MSII estimator [MSII];(vi) the PARA estimator for (9) using S1 and S2 [PARA]; (vii) the PILS estimator for(9) using S1 , S2 and the Epanechnikov kernel [PILS-E]; and (viii) the PILS estimatorfor (9) using S1 , S2 and the beta kernel [PILS-B].
Notice that OLS* is consistent, because E (u) = E (X2 u) = E (X3IC u) = 0.
Moreover, E (v) = 0, E (X3IC v) 6= 0, ρ = corr (X3IC , Z) 6= 0 and E (Zv) = 0, which
implies inconsistency of OLS-S and consistency of IV-S.
When X1 is empty in (9), PARA is consistent by Proposition S1. On the other
hand, when X1 is non-empty, the functional form of the imputed regressor in PARA
matches the reduced form of X2 for Model A only. In reality, however, even if the
reduced form of X2 is misspecified, it is still possible to estimate the regression (9)consistently by PARA. It turns out that under this Monte Carlo design, PARA is
consistent if and only if h(·) is odd, i.e., for both Models A and B, as indicated in
Proposition S2.
Implementing the two PILS estimators requires a choice of the smoothing parameters. We put ĥ = σ̂X (log m/m)0.3 for the Epanechnikov kernel, b̂ = σ̂U (log m/m)0.6
for the beta kernel, and λ̂ = (log m/m)0.6 for the discrete kernel, where σ̂X and σ̂U aresample standard deviations of the continuous common variable in the original scale
X (= X3IC , X3EC ) and in the transformed scale U := (X + 2) /4 ∈ [0, 1], respectively.
These shrinkage rates indeed fulfill the requirements in Section 2.4.4 of the paper.
For each estimator of β3 , the following performance measures are computed: (i)M ean (simulation average of the parameter estimate); (ii) SD (simulation standard
deviation of the parameter estimate); (iii) RM SE (root mean-squared error of theparameter estimate); (iv) SE (simulation average of the standard error); and (v)13CR (coverage rate for the nominal 95% confidence interval). Formulae for standard
errors are as follows: (a) heteroskedasticity-robust standard errors by Eicker (1963)
and White (1980) are calculated for OLS* and IV-S; (b) the one for MSII-FM follows Theorem 4 of Hirukawa and Prokhorov (2018); (c) the one for PILS appears in
Theorem 2; and (d) the one for PARA is given in Proposition S2 above. Because
of inconsistency we do not compute SE or CR of OLS-S and MSOLS. Those of thefirst-step MSII are also omitted due to its nonparametric convergence rate.
TABLES 1-3 HERE

Table 1 presents comprehensive results when X1 is non-empty. The results when
X1 is empty are available in Tables 2 (when X2 has an additively separable reducedform) and 3 (when X2 has a non-separable reduced form).

It can be found that

PILS-B performs well regardless of the models.

2.2

Simulation Study II

In this simulation study, the long and short regression models are still (9) and (10),respectively. As before, the true parameter values are β0 = β1 = β2 = β3 = 1, andβ3 is assumed to be the parameter of interest. The regressor X1 either enters or is
excluded from the regression.

14

15

0.99691.99630.99680.59811.37330.92600.89150.99591.14941.00920.99681.60450.99770.83661.22000.95530.9404
0.99571.0832
0.99300.9967
1.16310.99911.02551.0591
0.98280.97990.95081.02150.9946

0.9974
1.99670.99700.4076
1.37170.91040.87580.7282
1.14420.9873OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.04810.07730.214212.37200.09130.16730.17400.18760.11660.11840.03930.07830.20033.47780.07660.10240.10220.1186
0.0926
0.09200.04280.0746
0.19502.71560.08160.12520.12780.08980.10220.10120.04830.07460.20505.02850.09100.16690.17330.10550.11770.1171

0.04810.99970.214212.3861
0.38280.18980.21380.33030.18540.11900.03940.18090.20033.47790.09680.10380.10420.12840.0951
0.09220.04290.60900.19502.72050.23460.13300.14100.08990.13180.10140.04840.99910.2051
5.04450.38430.18250.20450.10560.1902
0.1175

94%
−
95%
98%
−
−
88%
95%
84%
93%

94%
−
94%
96%
−
−
91%
95%
73%
95%

0.9996
1.60841.00230.94591.22050.98000.97330.9994
1.04680.99610.99982.00041.00040.90281.3686
0.9671
0.95181.00041.08471.0080M ean

0.03361.00170.14600.87540.37410.11530.12240.07270.11470.07730.0334−0.14310.8039−−0.09720.07250.08810.08150.02950.05170.13910.76780.05730.08600.08690.06200.06740.06680.02950.61060.13910.76970.22780.08830.09090.06200.08200.06690.0297−0.13670.7171−−
0.06780.06210.0717
0.0664Model B: h (x) = sin (πx/4)

0.0336
0.05160.14600.86990.06390.11050.11250.07270.07730.0769Model A: h (x) = x(n, m) = (1000, 500)
SD
RM SE
SE

95%
−
96%
96%
−
−
86%
95%
91%
96%

95%
−
95%
95%
−
−
91%
95%
85%
96%

CR

1.00001.60751.00450.95911.2227
0.99370.99061.00291.03301.00190.99981.99971.00420.93761.36840.98680.97951.00351.05691.0108M ean

0.0468−0.203436.0469−−
0.1573−0.11890.12230.0389−0.18935.6981−−0.0737−
0.0944
0.09720.9996
1.16581.00020.98551.0601
0.99130.9896
0.95611.01110.99590.0266
0.05200.13140.65940.05270.06950.06990.08250.06070.06040.02660.17380.13140.65950.07990.07000.0706
0.0935
0.06170.06050.0276−0.13340.6506−−0.0502−0.07100.0664

96%
−
95%
98%
−
−
84%
−
97%
97%

1.0001
1.16671.00411.0076
1.06300.9996
0.99880.95861.01041.0008

94%
−
94%
97%
−
−
91%
−
75%
95%

1.0004
1.99920.9966
0.89131.36340.9556
0.93970.73691.07740.98840.03250.05130.14190.84770.0624
0.10890.11130.13200.07520.07610.03251.00050.14190.85470.36870.11760.1266
0.29440.10790.07700.0332−0.14320.8034−−0.0985−0.08840.0830

95%
−
96%
95%
−
−
90%
−
89%
96%

0.99972.00001.00380.95871.36160.9803
0.97300.7371
1.05100.9959Model D: h (x) = x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4

96%
−
94%
99%
−
−
85%
−
95%
96%

Model C: h (x) = {(5/3) x + 3/2} 1 {x < 0} + {− (4/3) x + 3/2} 1 {x ≥ 0}

0.0419−0.19395.9106−−0.10240.08800.09690.09620.0470−0.2031
14.6315−−0.15280.10290.11860.1188

CR

0.02330.03790.10130.49820.04450.06820.06890.09480.05190.05290.01990.03840.09120.39910.03810.04720.04730.05940.04190.04200.02120.03830.09730.44420.03990.05650.05670.04550.04650.04700.0236
0.03830.10190.50320.04410.07140.07190.05330.0541
0.05490.02331.00080.10130.49990.36440.07100.07400.27950.07280.05310.01990.1711
0.09130.39920.07360.04720.04730.07240.04320.04200.02120.6087
0.09740.44610.22630.05680.05750.0456
0.05710.04710.02361.00040.10200.50700.37110.07260.07480.05340.07850.05590.0235−0.10080.4896−−
0.0655−0.06520.06090.0195−0.09430.4120−−0.0350−0.05320.04950.0210−0.09650.4417−−0.04650.04400.05290.04900.0235−0.10080.4897−−0.0656
0.05140.06500.0602(n, m) = (2000, 1000)
SD
RM SE
SE

95%
−
94%
96%
−
−
94%
−
91%
98%

94%
−
96%
97%
−
−
85%
−
98%
98%

95%
−
95%
97%
−
−
89%
93%
93%
96%

95%
−
94%
96%
−
−
93%
94%
89%
96%

CR

Notes: M ean = simulation average of the parameter estimate; SD = simulation standard deviation of the parameter estimate; RM SE = root mean-squared errorof the parameter estimate; RM SE = root mean-squared error of the parameter estimate; SE = simulation average of the standard error; and CR = coverage rate
for the nominal 95% confidence interval.

M ean

Estimator

Table 1: Simulation Study I: Estimates of β3 in Presence of X1(n, m) = (500, 250)
SE
SD
RM SE

16

1.00222.0011
0.99800.58951.38330.94740.91341.0111
1.1722
1.02761.00191.6091
0.99830.9333
1.22950.97270.95771.00871.1003
1.00741.0014
1.16590.99530.0295
1.06770.99540.98941.00411.03051.00251.0011
1.99910.99510.56771.3786
0.92180.88371.0097
1.1596
0.9974OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.0466
0.07390.20605.34400.09060.16880.17520.1562
0.11360.11570.03830.07480.185826.88210.07590.10190.10180.10270.08960.08920.04090.07340.18944.36190.08070.12430.12730.0906
0.09900.09880.04590.07350.20164.96210.08970.16380.17090.10690.11430.11440.04661.00180.20615.36150.38930.18600.21020.15650.19590.11570.03830.18200.185926.89960.10180.10200.10240.10280.09470.08930.04100.61350.18944.36240.24330.12720.1341
0.09110.14090.0991

0.04601.00380.20164.9791
0.39360.17200.19160.10750.20670.1177

96%
−
96%
97%
−
−
88%
94%
69%
89%

96%
−
95%
95%
−
−
92%
94%
53%
90%

0.03340.05220.14630.93080.06370.10930.11070.07490.07860.07890.03351.00330.14630.9416
0.37650.11500.12200.07490.11900.07970.0334−0.14500.8960−−0.09790.07270.06800.0706

Model A: h (x) = x(n, m) = (1000, 500)
SD
RM SE
SE

1.00141.60990.9996
0.92611.22350.98080.97381.00141.05110.99980.02970.05220.13780.78850.05700.08500.0856
0.06350.06800.06790.02970.61210.13780.79200.23070.08710.08950.06350.08500.06790.0297−0.13770.7627−−0.06800.06220.05590.0574Model B: h (x) = sin (πx/4)1.00112.00190.99640.85751.3711
0.9644
0.94881.00081.08931.0112

M ean

96%
−
95%
97%
−
−
88%
94%
79%
91%

95%
−
95%
95%
−
−
92%
94%
70%
91%

CR

1.00101.61001.0044
0.9512
1.21980.98620.98300.99871.02970.99821.00132.00221.00180.92571.3652
0.97750.97000.99881.05381.0068

M ean

96%
−
97%
99%
−
−
85%
94%
87%
91%

1.0017
1.16811.00011.01581.06370.99530.99391.00051.01481.00020.02740.05350.13080.7846
0.05340.06950.06930.07220.06040.06020.02740.17640.13080.78470.08320.06970.06960.07220.06220.06020.0276−0.13410.7344−−0.05030.07230.05240.0552

95%
−
95%
98%
−
−
86%
96%
90%
93%

1.0006
1.16841.00370.98391.0601
0.99350.99280.99761.00690.99720.0469−0.20629.8155−−0.15600.15050.09520.1025

96%
−
95%
96%
−
−
92%
93%
59%
91%

1.00172.00130.99380.87301.36610.95540.93991.00031.08400.99420.03320.05290.14040.91140.06330.10860.11050.10850.07640.07810.03331.00270.14050.92020.37150.11750.12580.10850.11350.07840.0332−0.14600.8826−−
0.09870.1066
0.06810.0722

95%
−
96%
95%
−
−
91%
95%
73%
93%

1.00102.00171.00030.93351.35770.9701
0.96280.99651.0477
0.9917

Model D: h (x) = x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4

0.0389−0.1889148.4350−−
0.07320.10210.07320.0779Model C: h (x) = {(5/3) x + 3/2} 1 {x < 0} + {− (4/3) x + 3/2} 1 {x ≥ 0}

0.0420−0.1942
8.1980−−0.10010.08760.07860.08050.0472−0.20469.1768−−0.14880.10230.0947
0.0988

CR

0.02430.03720.10290.49570.04540.07250.07300.07690.05200.05270.02000.0376
0.09640.40510.03800.04830.04830.05130.04150.04140.02150.03700.09910.44290.04090.05770.05790.04430.04590.04570.02410.03690.10380.50010.0456
0.07320.07370.0517
0.0531
0.05310.02431.00240.10290.50010.36060.07840.08190.07700.07050.05330.02000.1726
0.09650.40540.0711
0.04870.04880.05130.0421
0.04150.02150.61120.09920.44560.2236
0.05930.06030.04440.05470.04570.0242
1.00290.10390.5056
0.36800.07660.07950.05170.0756
0.05350.0235−0.1026
0.4930−−0.06620.07540.0491
0.05130.0195−0.09450.4121−−0.0351
0.05110.03770.03920.0210−0.09690.4428−−0.04690.04410.03990.04090.0236−0.10190.4920−−0.06630.05150.04900.0505(n, m) = (2000, 1000)
SD
RM SE
SE

Notes: M ean = simulation average of the parameter estimate; SD = simulation standard deviationof the parameter estimate; RM SE = root mean-squared error of the parameter estimate; SE = simulation average of the standard error; and CR = coverage rate
for the nominal 95% confidence interval.

M ean

Estimator

Table 2: Simulation Study I: Estimates of β3 in Absence of X1 (Separable Case)(n, m) = (500, 250)
SD
RM SE
SE

95%
−
95%
96%
−
−
90%
95%
81%
95%

95%
−
94%
98%
−
−
85%
94%
93%
93%

95%
−
94%
97%
−
−
88%
94%
84%
92%

95%
−
94%
96%
−
−
92%
95%
77%
94%

CR

17

1.00142.32900.9860−0.8925
1.41080.79940.74501.00791.14531.00381.00161.80880.98990.7602
1.32300.91680.8922
1.00511.13781.02071.00141.2176
0.99432.7043
1.06380.97840.96971.00171.02220.99901.00102.32380.9853−1.0251
1.28230.80560.75801.0172
1.07680.9631OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

OLS*
OLS-S
IV-S: ρ = 0.4
ρ = 0.1
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.04460.11180.283844.06850.10330.17660.18680.32100.11940.11780.03830.09240.226052.7949
0.08680.11340.11320.1446
0.0976
0.09640.04420.07690.19373.58080.08700.16020.16620.14350.1086
0.10950.04880.09660.238946.70610.10380.20950.22150.24280.12780.1251

0.0446
1.32850.284244.11500.3006
0.26260.30570.32150.14200.12340.0383
0.23640.226152.8224
0.10770.11550.11720.14460.1001
0.09640.04430.81240.19403.58880.33450.1806
0.19810.14360.17540.11150.04881.33250.239346.7444
0.42370.29010.33780.24290.19350.1252

95%
−
95%
97%
−
−
91%
94%
61%
87%

95%
−
95%
95%
−
−
86%
95%
71%
94%

CR

0.0341
0.06760.1753
1.18630.07100.12260.12510.17330.08140.08240.0342
1.33800.17541.2036
0.38150.17020.18840.17340.10300.0831Model A: h (x) = x
0.0347−0.17331.1256−−
0.11750.16740.08270.0873(n, m) = (1000, 500)
SD
RM SE
SE

1.00181.81320.99800.89161.30990.9486
0.93791.00371.06891.00770.03090.05400.13950.82640.06020.10360.10510.10180.07200.07320.03100.81500.13950.83350.31570.11570.12200.10190.09970.0736

0.0315−0.14130.8225−−0.0893
0.09830.06130.0624Model B: h (x) = sin (πx/4)1.00192.33630.99320.79641.37480.88190.8592
1.00641.06300.9897

M ean

95%
−
95%
96%
−
−
89%
94%
76%
90%

95%
−
95%
95%
−
−
82%
94%
88%
96%

CR

1.00071.81251.00050.93941.3006
0.96870.9636
0.99871.03781.00191.00072.3356
0.99800.90831.35420.93340.92281.00031.03030.9871

M ean

96%
−
96%
99%
−
−
84%
95%
90%
93%

1.00181.22330.99841.0394
1.0566
0.9861
0.9842
1.00251.00880.99690.02740.06650.15480.89350.05950.07300.07230.0996
0.0632
0.06320.02740.23300.15480.89440.08210.07430.07400.09970.06380.06330.0276−
0.16220.8712−−0.05200.10110.05930.0641

95%
−
95%
99%
−
−
85%
96%
94%
95%

1.0006
1.22311.00220.97911.05020.98800.9871
0.99741.0022
0.99360.0451−0.2901
699.8918−−
0.17040.31490.11820.1382

95%
−
96%
97%
−
−
76%
94%
89%
97%

1.00232.3351
0.99240.84811.24170.89260.8752
1.01701.02570.96280.03190.08230.19871.2821
0.06940.10020.1022
0.21860.07480.07690.03191.3376
0.19881.29110.25140.14690.16130.21930.07910.08540.0320−0.20571.2489−−0.09190.21940.0852
0.0956

95%
−
96%
96%
−
−
74%
95%
96%
97%

1.00102.33451.00060.9205
1.21920.93620.92801.0073
1.00640.9650Model D: h (x) = x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4

0.0389−0.2285510.0343−−
0.07930.14300.0831
0.0916

Model C: h (x) = {(5/3) x + 3/2} 1 {x < 0} + {− (4/3) x + 3/2} 1 {x ≥ 0}

0.0445−0.19965.9520−−0.14280.13940.08630.08760.0490−0.2452742.6379−−
0.21070.23850.11400.1225(n, m) = (500, 250)
SD
RM SE
SE

0.02320.05720.14650.69670.0506
0.06440.06470.15270.05190.05280.02000.04660.11980.49820.04170.04930.04930.0711
0.04300.04340.0226
0.03820.10000.46100.04340.06730.06770.06930.04980.05010.02470.04740.1214
0.59850.05120.07690.07760.11630.05650.05690.0232
1.33570.14650.70120.22500.0907
0.09680.15290.05230.06330.02000.22790.11980.49870.06530.05070.05090.07120.0431
0.04390.02260.81340.10000.46500.30370.07420.07690.06930.06250.05020.0247
1.33640.12140.60540.35790.10180.10940.11630.0641
0.05830.0226−0.14480.6817−−
0.05640.1552
0.0617
0.06630.0195−0.11440.4988−−0.03550.07160.04270.04500.0223−0.0996
0.4646−−
0.05870.06990.04390.04450.0246−0.12210.5924−−0.07210.11890.06020.0621(n, m) = (2000, 1000)
SD
RM SE
SE

Notes: M ean = simulation average of the parameter estimate; SD = simulation standard deviationof the parameter estimate; RM SE = root mean-squared error of the parameter estimate; SE = simulation average of the standard error; and CR = coverage rate
for the nominal 95% confidence interval.

M ean

Estimator

Table 3: Simulation Study I: Estimates of β3 in Absence of X1 (Non-Separable Case)

94%
−
95%
95%
−
−
74%
95%
98%
97%

95%
−
94%
97%
−
−
84%
95%
95%
95%

95%
−
95%
96%
−
−
88%
95%
82%
91%

95%
−
95%
96%
−
−
82%
95%
94%
96%

CR

However, the reduced forms of regressors are different. These are
12
X1 = X3IC X3E
X3ED + η1 ,
C
4

π
1
1q
2
X
X2 = X3IC X3EC +25 − X3I
sin
3EC X3ED + η2 , and
C
2√
4
4
√

 p
3 (1 − ρ21 − ρ22 )
4
3
3 5
2
ρ1 Z +
ρ2 Z −+
η2 ,X3IC =
28
3
2
where ρ1 := corr (X3IC , Z), ρ2 := corr (X3IC , Z 2 ), and ρ1 , ρ2 ∈ {0.1, 0.4}. It is worth
remarking that ρ1 = ρ2 = 0.4 roughly mimics the sample correlation coefficients
between the endogenous regressor and its IVs in the empirical section of the paper,i.e., corrd (educ, f atheduc) = 0.4336 and corrd (educ, f atheduc2 ) = 0.4204.
The following distributional assumptions also apply.
(u, η1 )0 ∼ N (02×1 , I2 ) .
iid
X3EC , Z, η2 ∼ U [−2, 2] .


−1/2 w.p. 1/2.
1/2 w.p. 1/2
q (X3EC , X3ED , Z, η2 ) ⇒ u −q X3 = (X3IC , X3EC , X3ED ) .
u−

X3ED ∼ Bernoulli (1/2) − 1/2 =Sample sizes are n ∈ {500, 1000, 2000} and m = n/2. The number of replications
is 1000. The NNM is based on the Mahalanobis distance, and the number of matches
is set equal to K = 1 (single match).
This design is largely inspired by the study on nonlinear instruments of Dieterle and Snell (2016).

Moreover, the specification of X3IC ensures compactnessof supp (X3IC ) and unity of var (X3IC ) regardless of (ρ1 , ρ2 ).
We compare the following estimators of β3 : (i) the infeasible OLS estimator for (9)using S ∗ [OLS*]; (ii) the inconsistent OLS estimator for (10) using S1 only [OLS-S];(iii) the IV estimator for (10) using S1 only, where Z is used as an instrument for
X3IC [IV1-S]; (iv) the IV estimator for (10) using S1 only, where Z 2 is used as an
instrument for X3IC [IV2-S]; (v) the two-step GMM estimator for (10) using S1 only,
where (Z, Z 2 ) are used as instruments for X3IC [GMM-S], along with the initial 2SLS
18

estimator [2SLS-S]; (vi) he inconsistent MSOLS estimator for (9) using S [MSOLS];(vii) the MSII-FM estimator for (9) using S and a second-order polynomial [MSIIFM], along with the first-step MSII estimator [MSII]; (viii) the PARA estimator for(9) using S1 and S2 [PARA]; (ix) the PILS estimator for (9) using S1 , S2 and theEpanechnikov kernel [PILS-E]; and (x) the PILS estimator for (9) using S1 , S2 andthe beta kernel [PILS-B]. Implementation methods for two PILS estimators are thesame as in Simulation Stydy I.
Note that this design yields that E (u) = E (X1 u) = E (X2 u) = E (X3IC u) = 0,
which implies that OLS* is consistent. Moreover, E (v) = E (X1 v) = 0, E (X3IC v) 6=0, and E (v| Z) = 0, which leads to consistency of IV1-S, IV2-S, 2SLS-S, and GMM-S.
TABLES 4-5 HERE

Tables 4 and 5 present the results where X1 enters and is excluded from theregression, respectively. Observe that PARA performs poorly even in the absence ofX1 (= when PARA becomes consistent in theory). Because the reduced form of X2
is highly nonlinear, the first-step linear approximation is likely to be imprecise (ormay be close to lack of identification in a linear form). As a result, PARA tends tohave huge bias and variability as is the case with weak instruments. PILS performs
exceptionally well.

2.3

Simulation Study III

This simulation study, which deals with yet another case in which X1 is non-empty,
was adopted in the previous version of our paper. The data are generated as follows:
Y = β0 + β1 X1 + β2 X2 + u,
19

(13)

20

M ean

1.00082.14380.7353
1.29981.00341.00081.11410.83660.74092.2743
1.06310.90911.00112.05280.77830.98070.99770.99581.33460.83190.73572.0323
1.23430.96371.00222.05410.98820.57431.00211.0014
1.33840.83720.74162.02791.22690.96531.0022
1.95470.98840.98170.99920.99951.41730.82890.75301.94331.2932
1.0082Estimator

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

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

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

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

0.06350.06750.20230.19850.13830.13860.0901
0.23430.2564
1.51880.11020.10680.07030.0635
0.19844.64120.18810.18820.08410.17430.19201.93030.11490.10480.07170.06453.6286
0.19780.18480.18530.08620.17650.19471.99370.11730.10620.0815
0.06102.713416.30640.72380.72270.08650.14020.15154.22680.11570.1016

0.0636
0.95710.2026
0.19930.13830.1386
0.42690.29020.35601.78790.31320.10720.0703
1.05600.19874.66070.18820.1882
0.34860.23850.32202.18690.25430.11040.07171.05483.6354
0.19870.18480.18530.34560.24380.32832.2451
0.26200.11220.08151.14542.726316.3092
0.72380.72270.14320.21530.30014.41470.13180.13630.0623−0.19650.19700.13910.1390−−0.2519−0.09580.09470.0702−
0.19507.90920.18710.1866−−0.2118−0.09930.09740.0700−5.05780.19570.18770.1873−−0.2133−0.1002
0.09750.0818−3.423676.78180.67430.6735−−0.1665−0.09980.0960

94%
−
95%
96%
95%
95%
−
−
97%
−
18%
91%

95%
−
95%
95%
95%
95%
−
−
91%
−
39%
93%

94%
−
94%
96%
96%
95%
−
−
90%
−
37%
92%

95%
−
93%
93%
92%
92%
−
−
74%
−
86%
84%

CR

0.99771.95210.9954
0.99701.00201.0024
1.39860.91770.88652.00571.2054
0.99490.99832.05200.9944
0.23681.00131.00041.31440.91540.87292.00501.1361
0.96330.99762.05200.7066
0.99591.00231.00191.31320.91070.86702.1037
1.14270.96280.9982
2.14390.8340−1.7724
1.00051.00161.08090.91410.87092.11401.00390.9234M ean

0.05751.14470.9152
82.94960.41340.41410.1006
0.11370.15004.63640.07240.10200.0506
1.05304.24300.13440.12980.13020.31920.13920.17292.14490.16450.08090.04971.05300.132219.96070.12840.12860.32010.13310.16622.05640.15710.08010.04500.04890.13270.1352
0.09680.09690.06440.13100.13501.74210.08110.07400.0451
0.95330.13280.1352
0.09680.09690.40380.15470.17642.01160.22080.0742(ρ1 , ρ2 ) = (0.4, 0.4)

0.04970.04630.132119.9461
0.12840.12860.06050.10270.10711.79400.07840.0711(ρ1 , ρ2 ) = (0.4, 0.1)

0.05050.04584.23280.13440.12970.1302
0.06150.10680.1105
1.83920.08180.07190.0441−0.13640.1366
0.0982
0.0982−−0.1328−0.06790.06580.0497−0.1356131.0213
0.13120.1310−−0.1160−0.06950.06750.0497−4.48700.13580.13130.1312−−
0.1179−0.0701
0.0676

0.0581−0.80131618.0879
0.40170.4015−−0.0875−0.06770.0658(ρ1 , ρ2 ) = (0.1, 0.4)

0.05740.04340.900082.9032
0.41340.41410.05980.07450.07644.50050.07230.0674(ρ1 , ρ2 ) = (0.1, 0.1)

(n, m) = (1000, 500)
SD
RM SE
SE

94%
−
96%
94%
96%
96%
−
−
93%
−
18%
93%

96%
−
95%
94%
96%
96%
−
−
90%
−
51%
91%

95%
−
95%
95%
94%
94%
−
−
87%
−
49%
91%

95%
−
94%
94%
93%
92%
−
−
74%
−
94%
78%

CR

1.00071.95291.0002
0.99340.99950.9997
1.39130.95400.9392
1.95201.13980.99341.00112.05321.00060.92081.00271.00221.3056
0.95400.93362.09491.07750.97171.00082.05250.95230.99370.99750.99691.3041
0.95470.93382.12791.08640.97071.00162.14440.95220.92090.99900.9991
1.0677
0.95390.93322.24580.9891
0.9446M ean

0.0316
0.0341
0.0955
0.09350.06950.06960.04550.08250.08381.7781
0.05390.04960.03570.0321
0.09500.45550.09280.0931
0.04280.06490.06601.77930.05230.04820.03490.03180.4421
0.09220.09060.09050.04190.06300.06412.2356
0.05140.04760.04070.03010.43150.43680.2766
0.27720.04190.0481
0.04882.21190.04720.04500.0316
0.95350.0955
0.09370.06950.0696
0.39390.09450.10352.01690.14980.05010.03581.0537
0.09500.46230.09280.09310.30860.0795
0.09362.0892
0.09350.05590.03491.05300.4447
0.09240.09060.09060.30700.0776
0.09212.50400.10050.05590.04071.14480.43410.44400.2766
0.27720.0796
0.06660.08272.53870.04850.07130.0313−
0.09600.09650.06950.0695−−0.0841−0.04750.04600.0352−
0.09540.44090.0926
0.0925−−0.0731−0.04780.04700.0352−0.4302
0.09600.0931
0.0930−−0.0730−0.04850.0471

0.0411−
0.42230.43250.27450.2745−−0.0541−
0.04610.0454(n, m) = (2000, 1000)
SD
RM SE
SE

Table 4: Simulation Study II: Estimates of β3 in Presence of X1(n, m) = (500, 250)
SE
SD
RM SE

95%
−
94%
96%
95%
95%
−
−
94%
−
19%
92%

95%
−
94%
96%
94%
94%
−
−
91%
−
63%
90%

95%
−
95%
96%
95%
95%
−
−
92%
−
55%
90%

95%
−
94%
96%
95%
94%
−
−
81%
−
93%
76%

CR

21

M ean

0.99972.14160.81580.55571.00070.99791.11220.83660.74172.1833
1.05720.90630.99982.05070.4841
0.99351.00531.00381.33380.83190.73622.00471.23550.96010.99942.05080.9860−2.7750
1.00200.9994
1.33790.83450.73712.17851.23300.96320.9996
1.95110.9862
0.99401.00241.00271.41610.83410.75912.03311.29471.0038Estimator

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

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

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

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

0.06170.06890.1945
0.19870.14150.14180.0932
0.23400.2556
1.54800.11300.11120.0696
0.06520.1939101.6185
0.18770.18870.08770.18700.20901.89650.12050.10700.06820.06454.93930.19870.19190.19220.09020.1807
0.19572.32430.12080.10530.07950.06072.90355.9991
0.7826
0.78790.08500.13110.14251.94090.1126
0.0997

0.0617
0.9536
0.19490.19880.14150.14180.42640.28680.3512
1.86110.31570.11130.0696
1.05280.1944101.6886
0.18770.18870.34910.2497
0.33582.23280.26230.11320.0682
1.05274.9662
0.19880.19200.19230.34580.24680.32852.53220.26470.11260.0795
1.14322.90946.01550.7826
0.78790.14070.20950.29502.27310.12630.13680.0623−0.19500.19450.13860.1385−−0.24612.9925
0.09600.09450.0702−0.19413079.0631
0.18640.1860−−0.21683.28660.1001
0.09750.0702−
12.05190.19360.18630.1858−−0.21464.88060.1007
0.09770.0821−3.4401
12.68440.6656
0.6649−−0.16393.20790.1001
0.0963

SE

95%
−
96%
95%
94%
94%
−
−
96%
80%
19%
90%

95%
−
96%
94%
95%
95%
−
−
91%
81%
38%
92%

96%
−
96%
96%
95%
95%
−
−
91%
82%
39%
92%

95%
−
94%
93%
90%
90%
−
−
72%
83%
88%
83%

CR

1.0006
1.95210.9951
0.99681.00181.00181.40260.92040.8875
1.99101.21120.99721.00012.0516
0.99490.87821.00191.00081.31720.91680.87372.17751.13690.96451.00092.05170.8377
0.99641.00291.00201.31510.91680.87272.08781.14420.96431.00022.1426
0.91300.91421.01041.0102
1.08230.91270.86982.22561.00610.9247

M ean

0.05681.14342.15191.66430.39630.39640.10280.11700.15302.48940.07510.10170.04751.05270.93300.13460.13050.13040.32150.13430.16865.84030.16540.08130.0482
1.05250.13411.27910.12930.12950.32300.13090.16422.27990.15900.08200.04220.04770.13470.13580.0976
0.0976
0.06470.12870.13262.24720.07800.07330.0422
0.95330.13480.13580.0976
0.0976
0.40780.15140.17392.45600.22510.0734(ρ1 , ρ2 ) = (0.4, 0.4)

0.04820.04430.13401.27330.12930.1295
0.06120.10110.10491.95220.08090.0739(ρ1 , ρ2 ) = (0.4, 0.1)

0.04750.04470.91880.13460.13050.13040.06340.10550.11065.73810.08110.0731(ρ1 , ρ2 ) = (0.1, 0.4)

0.05680.04182.1501
1.66210.39610.39630.06160.07790.0804
2.16680.07480.0684(ρ1 , ρ2 ) = (0.1, 0.1)

0.0442−0.13640.13690.09840.0983−−0.13354.89880.0681
0.06570.0497−0.13571.01410.13130.1311−−0.11623.43030.06940.06740.0497−0.87950.13610.13170.1315−−0.116524.69090.07020.06750.0581−
2.1972
1.29720.39900.3988−−0.08883.94990.06770.0657(n, m) = (1000, 500)
SD
RM SE
SE

97%
−
95%
95%
95%
95%
−
−
94%
83%
17%
93%

95%
−
95%
95%
94%
94%
−
−
88%
84%
48%
91%

97%
−
94%
96%
95%
95%
−
−
88%
84%
46%
90%

95%
−
94%
94%
94%
94%
−
−
76%
85%
92%
79%

CR

0.9992
1.95180.99600.99220.99720.9973
1.39240.95630.9412
1.9656
1.1396
0.99250.99892.05130.99510.91890.99750.99721.30390.95140.93102.0936
1.07610.97000.99882.05170.9374
0.9912
0.99470.99441.30090.94900.92792.05991.08370.96830.9983
2.14280.92760.91090.9931
0.9931
1.06450.94990.92942.10410.98530.9417

M ean

0.03100.0336
0.0922
0.09730.0661
0.06610.04580.08310.08432.31710.05580.04990.03500.03140.0924
0.46620.08830.08840.04320.0652
0.06632.8626
0.05270.04960.0356
0.03150.43130.09740.0926
0.09270.04420.0651
0.06672.2652
0.05390.0496

0.04150.02940.44270.45990.26070.26080.04410.05070.05142.45970.04840.04630.03100.95240.09230.0976
0.0661
0.06620.39500.09390.10282.51030.15040.05050.0351
1.05180.09250.47320.08840.08850.30700.08130.09573.06440.0926
0.05800.03571.05220.43580.09780.09280.09290.30410.0827
0.09822.50090.0996
0.05880.0416
1.14320.44850.46840.26080.26090.07810.07130.08732.69620.05060.07450.0313−
0.0961
0.09670.06970.0697−−0.08394.08360.04740.04590.0352−
0.09560.44850.09290.0928−−0.0731
5.83100.04770.04690.0352−0.4351
0.0963
0.0934
0.0933−−0.07344.42930.04830.04700.0411−0.43300.44310.27650.2764−−0.05444.7421
0.04600.0454(n, m) = (2000, 1000)
SD
RM SE
SE

Table 5: Simulation Study II: Estimates of β3 in Absence of X1(n, m) = (500, 250)
SD
RM SE

96%
−
96%
95%
96%
96%
−
−
93%
83%
21%
92%

95%
−
96%
96%
96%
96%
−
−
90%
84%
64%
89%

95%
−
96%
95%
95%
95%
−
−
89%
83%
57%
88%

94%
−
95%
96%
95%
95%
−
−
79%
86%
92%
75%

CR

where true parameter values are β0 = β1 = β2 = 1, and β1 is the parameter of interest.
There are two common variables, namely, a continuous one X3EC and a discrete one
X3ED , and both are excluded from the regression (13). The regressors X1 and X2
have the reduced forms
2
X1 = 1 + X3E+ X3ED + (πZ + ξ) and X2 = {h (X3EC ) + X3ED } + η2 ,C

where Z is the instrument for X1 and the parameter π, whose values will be presented
shortly, controls strength of the instrument. In particular, Z is applied to the shortregression after omitting X2 from (13), i.e.,
Y = β0 + β1 X1 + (u + β2 X2 ) =: β0 + β1 X1 + v.
The function h (·) is specified as one of the followings:

x




tanh (x)√ = {exp (x) − exp (−x)} / {exp (x) + exp (−x)}
(3/4) x + 2 − 1
h (x) =
x3 /2 − x2 /2 − 2x − 2/3|x| − 1
x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4(14)[Model A][Model B][Model C].
[Model D][Model E][Model F]

Observe that Models A and F are exactly the same as Models A and D in Simulation
Stydy I, respectively. The following distributional assumptions also apply.
(u, η2 , ξ)0 ∼ N (03×1 , I3 ) .
iid
Z, X3EC ∼ U [−2, 2] .

X3ED ∼ Bernoulli (1/2) − 1/2 =−1/2 w.p. 1/2.
1/2 w.p. 1/2

q X3 = (X3EC , X3ED ) .
u−
The values of π are π ∈ {1.00, 0.70, 0.15}.

It follows from the distributional

assumptions thatrcorr (X1 , Z) =
 0.58 for π = 1.00
0.44 for π = 0.70 .
≈240π 2 + 481 
0.11 for π = 0.15240π 222

In short, π ∈ {1.00, 0.70} and π = 0.15 correspond to strong and weak IV cases, andthe correlation for π = 0.70 is roughly similar to the sample correlation between theendogenous variable (educ) and its instrument (fatheduc) in our empirical example in
Section 4 of the paper.
The above procedure provides us with two observable samples
S1 = {(Yi , X1i , X3EC i , X3ED i , Zi )}ni=1 and S2 = {(X2j , X3EC j , X3ED j )}m
j=1 .
The complete sample
S ∗ = {(Yi , X1i , X2i , X3EC i , X3ED i , Zi )}ni=1
is the sample that would not be observed in practice. Finally, the matched sampleS=Yi , X1i , X2j1 (i) , . . . , X2jK (i) , X3EC i , X3ED i , Zi

 n
i=1

is constructed via the NNM with respect to X3 = (X3EC , X3ED ) as in Hirukawa andProkhorov (2018). The NNM is based on the Mahalanobis metric and the numberof matches K ∈ {1, 2, 4, 8, 16, 32, 64, 128}.

As regards the sample sizes, we choosen ∈ {500, 1000} and put m = n for simplicity. For each combination of the functional
form of h (·) and the value of π, 1000 Monte Carlo samples are drawn.
We compare the following estimators of β1 : (i) the infeasible OLS estimator for(13) using S ∗ [OLS*]; (ii) the inconsistent OLS estimator for (14) using S1 only
[OLS-S]; (iii) the IV estimator for (14) using S1 only [IV-S]; (iv) the inconsistent
MSOLS estimator for (13) using S [MSOLS]; (v) the MSII estimator for (13) using
S [MSII]; (vi) the PARA estimator for (13) using S1 and S2 [PARA]; (vii) the PILS
estimator for (13) using S1 , S2 and the Epanechnikov kernel [PILS-E]; and (viii) the
PILS estimator for (13) using S1 , S2 and the beta kernel [PILS-B]. Implementation
methods for two PILS estimators are the same as in Simulation Stydy I.
23

As in Simulation Studies I and II, OLS* is consistent, because E (u) = E (X1 u) =E (X2 u) = 0.

Moreover, E (v) = 0, E (X1 v) 6= 0, corr (X1 , Z) 6= 0 and E (Zv) =0, which implies inconsistency of OLS-S and consistency of IV-S. Furthermore, as
indicated in Proposition S2, PARA becomes consistent for an odd h(·), i.e., for Models
A and B. Because of inconsistency we do not compute SE or CR of OLS-S andMSOLS. Likewise, we present SE and CR of PARA only when it becomes consistent,i.e., for Models A and B.
TABLE 6 HERE

24

Table 6: Simulation Study III: Estimates of β1n (= m)π

Estimator

OLS*

OLS-S

IV-S

MSOLS

MSII

PARA

PILS-E

PILS-B

0.99750.04270.04280.039593%
0.99750.04760.04770.043592%
0.99790.05420.05430.048392%
1.00200.03070.03080.027893%
1.00220.03430.03440.030692%
1.00210.03870.03870.034091%

0.99890.03180.03180.032195%
0.99920.03510.03510.035396%
1.00000.03940.03940.039495%
1.00080.02270.02270.022795%
1.00090.02500.02500.025096%
1.00080.02780.02790.027996%

1.00690.03470.03540.038997%
1.00880.03890.03990.042797%
1.01180.04450.04610.047497%
1.00680.02470.02560.027596%
1.00810.02760.02880.030296%
1.0096
0.03140.03280.033695%

1.00500.03450.03490.039397%
1.00650.03860.03920.043298%
1.0091
0.04430.04520.048097%
1.00550.02460.02520.027796%
1.00650.02750.02830.030596%
1.00770.03120.03220.033896%

1.00690.03470.03540.039197%
1.00880.03890.03990.042997%
1.01190.0446
0.04620.047796%
1.00670.02480.02570.027795%
1.00800.02780.02890.030495%
1.00950.0316
0.03300.033895%

1.00510.03450.03490.039597%
1.0066
0.03870.03930.043498%
1.00920.04440.04540.048397%
1.00560.02470.02530.027996%
1.00660.02770.02840.030696%
1.00780.03150.03240.034196%

Model A: h (x) = x
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99840.02240.02250.022395%
0.99850.02450.02460.024595%
0.99920.02700.02710.027295%
1.00030.01590.01590.015895%
1.00040.01750.01750.017495%
1.00060.01930.01930.019395%

1.0611
0.04250.0744−−1.07400.04700.0876−−
1.09200.05290.1061
0.0536−
1.0636
0.03130.0709−−1.07650.03460.0840−−1.09400.03820.1014−−

0.9956
0.07370.07380.073895%
0.99340.10630.10650.106295%
0.81162.72552.73205.591799%
1.00050.04980.04980.051996%
1.0006
0.07120.07120.074496%
1.00410.41370.41370.453498%

1.02270.03730.0436−−
1.02780.04130.0498−−1.03530.04680.05860.0476−
1.02620.02730.0378−−1.03140.03020.0436−−1.03830.03360.0509−−Model B: h (x) = tanh (x) = {exp (x) − exp (−x)} / {exp (x) + exp (−x)}
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99840.02240.02250.022495%
0.99850.02450.02460.024595%
0.99910.02710.02710.027295%
1.00030.01590.01590.015895%
1.00040.01750.01750.017495%
1.00070.01930.01940.019395%

1.06140.03610.0712−−
1.07440.03970.0843−−
1.09250.04460.1027−−1.0636
0.02670.0690−−1.07660.02930.0820−−1.09400.0321
0.0993−−0.99550.06430.06440.064995%0.99330.0927
0.09290.093495%
0.88181.76351.76753.457199%
1.00060.04400.04400.045696%
1.00070.06290.06290.065496%
1.00370.36480.36480.398198%

1.03420.03510.0490−−1.04170.03880.0570−−1.05250.04380.0684−−
1.03730.02570.0453−−
1.04490.02830.0531−−1.05500.03130.0633−−25

0.99600.04430.04450.041594%
0.99560.04970.04990.045994%
0.9956
0.05700.0571
0.051593%
1.00130.03150.03160.028793%
1.00140.03540.03540.031892%
1.00120.04010.04010.035692%

0.99900.03210.03210.032496%
0.99930.03540.03550.035796%
1.00020.03990.03990.039895%
1.00100.02290.02290.022995%
1.00100.02530.02530.025296%
1.00100.02820.02820.028295%

Table 6: (cont’d)n (= m)π

Estimator

OLS*

OLS-S

IV-S

MSOLS

Model C: h (x) = (3/4)
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99840.02240.02250.022394%
0.99850.02460.02460.024595%
0.99910.02710.02710.027295%
1.00030.01590.01590.015895%
1.00040.01750.01750.017495%
1.00060.01930.01930.019396%

1.04240.03350.0541−−
1.0516
0.03700.0634−−1.06440.0416
0.0767−−1.04440.02460.0508−−1.05340.02700.0599−−1.06560.02970.0720−−0.99580.05990.06010.060196%
0.99380.08640.08670.086596%0.96031.22621.22682.243999%
1.00040.04070.04070.042396%
1.00050.05820.05820.060596%
1.0054
0.3357
0.33580.3662
98%

√

MSII

PARA

PILS-E

PILS-B

0.97920.03210.0383−−0.97520.03550.0433−−0.97010.03990.0498−−0.9811
0.02280.0296−−
0.97690.02530.0342−−0.97090.02830.0406−−1.00530.03490.03530.039397%
1.00680.03910.03970.043297%
1.00950.04490.04590.048096%
1.00550.02480.02540.027896%
1.00650.02780.02850.030696%
1.0076
0.03170.0326
0.034095%

1.00250.03470.03480.039998%
1.00350.03890.0391
0.043998%
1.00530.04480.0451
0.048897%1.00360.02470.02500.028196%
1.00420.02770.02800.030996%
1.00490.03150.03190.034396%

0.81850.03620.1851−−0.78100.03980.2226−−0.73030.04470.2734−−0.82020.02540.1816−−
0.78270.02810.2191−−0.73120.03180.2706−−1.00210.03530.03530.039698%
1.00310.03970.03980.043698%
1.00500.04580.0461
0.048697%
1.00360.02500.02530.028096%
1.00420.02810.02840.030896%
1.00500.03200.03240.034396%

0.99570.03500.03530.040098%
0.99530.03940.03960.044197%
0.99530.04540.04570.049197%
0.99890.02490.02500.028196%
0.99850.02800.02800.031096%
0.99790.03180.03190.034596%

x+2−1

1.03050.03350.0453−−
1.03730.03690.0525−−1.04690.0416
0.0627−−1.03310.02440.0411−−1.03980.02680.0480−−1.04880.0296
0.0571−−

0.9926
0.0615
0.06190.052395%
0.99130.07040.07100.058796%0.99000.08110.0817
0.066596%
1.00010.03320.03320.030594%
0.99980.03740.03740.034093%
0.99930.04270.04270.038493%

Model D: h (x) = x3 /2 − x2 /2 − 2x − 2/3
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99860.02270.02280.022594%
0.99870.02490.02500.024795%
0.99930.02760.02760.027495%
1.00020.01590.01590.015995%
1.00030.01750.01750.017594%
1.00050.01950.01950.019595%0.8841
0.04230.1233−−0.86080.04580.1465−−0.82970.05020.1775−−0.88520.02840.1182−−0.86180.03100.1416−−0.82990.03440.1735−−0.99760.07970.07970.076293%
0.99720.11490.11500.109893%
1.29194.1777
4.18788.337199%
1.0001
0.0526
0.0526
0.053695%
1.00040.07520.07520.076895%
1.03130.48770.48870.467298%

26

0.95740.03820.0572−−
0.94900.0422
0.0662−−0.93800.04770.0782−−0.96020.02700.0481−−
0.95170.03000.0568−−0.93990.03380.0689−−1.0001
0.04350.04350.040293%
1.00070.04880.04880.044592%
1.00200.0562
0.05620.049892%
1.00300.03130.03140.028492%
1.00340.03510.03530.031492%
1.00360.03980.04000.035192%

Table 6: (cont’d)n (= m)π

Estimator

OLS*

OLS-S

IV-S

MSOLS

MSII

PARA

PILS-E

PILS-B

0.96660.13970.1436
0.117397%0.93080.46230.46750.378696%
0.97281.17351.17381.912495%
0.98930.0656
0.06650.064096%
0.98300.0896
0.09120.086296%0.96620.15140.15510.137994%

1.17700.03340.1801−−
1.2166
0.03700.2198−−1.2726
0.04160.2758−−1.17850.02390.1801−−1.21790.02640.2195−−1.27310.02920.2747−−1.01160.03980.04140.048298%
1.01640.04670.04950.056097%
1.02620.0582
0.06380.067795%
1.00810.02770.02880.034298%
1.01070.03270.03440.039998%
1.0151
0.0411
0.04380.048696%

1.00610.03970.04020.049599%
1.0091
0.04670.0476
0.057698%
1.01530.05860.06060.070097%
1.00480.02780.02820.034899%
1.00620.03290.03350.040698%
1.00850.04150.04240.049797%

1.00480.03530.03560.040198%
1.00640.03980.04030.044398%
1.00930.04610.04700.049697%
1.00570.02520.02590.028397%
1.00680.02850.02930.031397%
1.00830.0326
0.03370.035096%

1.00180.03520.03530.040798%
1.00280.03970.03980.045097%
1.00470.04600.04620.050497%
1.00370.02530.02550.028697%
1.00440.02850.02880.031697%
1.00520.0326
0.03300.035497%

Model E: h (x) = |x| − 1
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99810.02370.02380.023996%0.99810.02640.02640.026696%
0.99870.02980.02990.030296%
1.00040.01720.01720.016994%
1.00060.01930.01930.018995%
1.00090.02170.02170.021495%

1.22820.03320.2306−−1.27530.03650.2777−−1.33970.04080.3421−−1.22970.02410.2310−−1.27670.02630.2779−−1.34040.02850.3416−−

0.99390.06200.06230.062895%0.99050.08960.09010.090696%
0.90131.32401.32761.915498%
0.99970.04230.04230.044196%
0.99920.06050.06050.063296%
0.96110.37240.37450.408697%

1.1647
0.03620.1686−−1.20550.04050.2094−−1.26580.04630.2698−−1.16740.02560.1694−−1.20800.02850.2099−−1.26750.0316
0.2693−−Model F: h (x) = x + (5/τ ) φ (x/τ ) − (5/2) {Φ (2/τ ) − 1/2} , τ = 3/4
500

1.00

0.70

0.15

1000

1.00

0.70

0.15

M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR
M ean
SD
RM SE
SE
CR

0.99860.02310.0231
0.022794%
0.99870.02540.02550.025095%
0.99940.02830.02830.027995%
1.00000.01640.01640.016194%
1.00000.01810.01810.017894%
1.00020.02020.02020.019895%

0.81150.04450.1937−−0.77340.04890.2318−−0.72220.0546
0.2831−−0.81440.03280.1885−−0.77630.03610.2266−−0.72430.04000.2785−−Results are presented in Table 6.

0.99800.08110.08120.081795%
0.99780.11700.11700.117696%0.96012.65602.65635.267999%
1.00150.05650.05650.057596%
1.00260.08090.08090.082496%
1.06410.62290.62620.604298%

0.94130.03880.0703−−0.92920.04310.0829−−0.91300.0491
0.0999−−0.94470.02840.0621−−0.9326
0.03160.0745−−0.91550.0356
0.0917−−1.0001
0.04460.0446
0.041293%
1.00070.05030.05040.045992%
1.00220.05840.05840.051892%
1.00300.03180.03200.029292%
1.00340.03600.03620.032492%
1.00380.04140.04150.036691%0.74680.03490.2556−−0.69480.03820.3076−−0.62460.04260.3778−−0.74900.02460.2522−−0.69690.02710.3044−−0.62540.03030.3758−−For MSII and MSOLS we report only the

results for K = 1 (single match), because those for K ≥ 2 are qualitatively similar.
27

Moreover, since the regressor X1 depends on π in this design, the results for all
estimators, not only for IV-S, vary with the strength of the instrument, unlike thedesign in Section 3 of the paper. It is indicated that PILS-B again works well for awide variety of models.

2.4

Simulation Study IV

This study is a variation of what ended up in the paper – it uses the setup of Section
3.1 of the paper, except that X1 is handled differently. In the original setup, theregressor X1 is merely orthogonal to the composite error term v. Here, we generateiid

X1 as X1 ∼ N (0, 1), independently of all random variables in the design.

Noticethat the data generation of all other variables is exactly the same as before. It follows
q (u, X2 , Z) that the conditional moment restriction E (v| X1 , Z) = 0 holds
from X1 −for (S). Results for this setup are presented in Table 7.
TABLE 7 HERE

3

More on the Empirical Application

In this section, a wider range of estimators for the return to schooling application
including nonlinear IV are presented. Subsequently, we argue that if the PSID ability
measure is error-ridden, OLS* is upward-inconsistent suggesting that PILS is likely tobe the only estimator that delivers the closest estimate to the true population value.
We conclude this section by conducting sensitivity analysis in PILS to the choices ofsmoothing parameter values.

28

Table 7: Simulation Study IV: Estimates of β3 under Independent X1
Estimator

M ean(n, m) = (1000, 500)
SD
RM SE
SE

CR

M ean

(n, m) = (2000, 1000)
SD
RM SE
SE

CR

Model A (ρ1 = 0.1)
OLS*
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99862.98020.75110.70981.02251.0227
1.12660.90610.82060.99901.02891.00140.05570.04501.2663
1.61960.92720.9271
0.06220.08030.08250.06090.07180.06350.05571.98071.29051.64530.92740.92740.14100.1236
0.19750.06090.07740.06350.0560−1.15171.34110.90170.9010−−0.1065
0.06110.07420.0764

95%
−
92%
92%
88%
88%
−
−
67%
95%
94%
98%

1.00192.98060.90480.8952
1.01381.0135
1.09380.94680.90531.00091.00250.99630.03970.03150.6176
0.66300.55930.55950.04340.05050.0511
0.0426
0.04620.04380.03981.98090.62480.67120.55950.55970.10340.07330.10760.0426
0.04630.04390.0397−0.61530.64400.57290.5727−−0.06290.04320.05080.0522

95%
−
94%
93%
92%
92%
−
−
72%
97%
97%
98%

0.02040.03120.08000.08320.07990.07980.02190.0221
0.02200.02140.02370.02190.02050.42150.08010.08330.07990.07980.02660.0221
0.02290.02140.02390.02190.0207−0.08200.08480.08190.0819−−0.02160.02170.02230.0218

96%
−
95%
96%
96%
96%
−
−
94%
95%
93%
95%

0.04470.01990.24230.25840.22930.22940.04400.06170.0631
4.15980.04870.05070.04470.43480.24290.25910.22950.22950.07970.09910.1115
4.19270.04930.06240.0433−0.24270.25310.23260.2325−−0.0676−0.05000.0539

95%
−
97%
97%
96%
96%
−
−
76%
−
96%
92%

0.02720.01900.04950.05110.04940.04940.0266
0.03620.03672.01880.02980.03070.02710.11010.04950.05110.04940.04940.04620.03720.03812.02220.03180.03070.0263−0.0496
0.05120.04950.0495−−0.0355−0.03250.0367

94%
−
95%
96%
95%
95%
−
−
94%
−
95%
97%

Model A (ρ1 = 0.4)
OLS*
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99991.42041.00020.99981.00281.00281.0166
0.99530.98610.99981.00951.00140.02930.04510.11630.11970.11640.11670.03120.0316
0.03170.03090.03530.03170.02930.42280.11630.11970.11640.11670.03530.03200.03460.03090.0366
0.03170.0292−0.11630.12000.11600.1159−−0.03080.03070.03240.0310

94%
−
96%
95%
96%
96%
−
−
92%
95%
91%
95%

1.00121.4204
0.99730.99680.99850.99841.0151
0.9987
0.99381.00101.00351.0012Model B (ρ1 = 0.1)
OLS*
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.99861.43350.93780.92170.9991
0.9987
1.09100.86180.8211
1.46201.00890.96100.06010.02880.41950.5396
0.36480.36510.05990.11020.11862.84090.07020.07280.06020.43450.42410.54520.36480.36510.10900.17670.21472.87820.07080.0826

0.0613−0.41680.48210.36070.3603−−0.1308−0.07180.0812

95%
−
96%
96%
94%
94%
−
−
81%
−
95%
96%

0.99961.43430.98300.9805
1.0076
1.0073
1.06640.92240.9081
1.5242
0.99280.9637

Model B (ρ1 = 0.4)
OLS*
OLS-S
IV1-S
IV2-S
2SLS-S
GMM-S
MSOLS
MSII
MSII-FMPARA
PILS-E
PILS-B

0.9991
1.10770.99830.99740.99910.99911.04050.9856
0.98111.13361.01940.99940.03690.02820.06830.0706
0.0686
0.0687
0.03690.0536
0.05510.90790.04130.0417

0.03690.11140.06840.07070.0686
0.06870.05480.05550.05830.91770.0456
0.04170.0373−0.07020.07250.07010.0700−−0.0533−0.04530.0554

96%
−
96%
95%
96%
95%
−
−
93%
−
93%
98%

29

1.0005
1.10841.00051.00031.00091.00091.03780.99170.98971.11681.01110.9992

3.1

Additional Results

In addition to the results reported in Section 4 of the paper, the following IV-based
estimation methods are applied to the short regression with abil omitted:
1. Just-identification cases:(a) The IV for educ is f atheduc [IV1-S]; and(b) The IV for educ is f atheduc2 [IV2-S].
2. Over-identification cases where IVs for educ are (f atheduc, f atheduc2 ):(a) Two-stage least squares [2SLS-S]; and(b) Two-step generalized method of moments with 2SLS-S used as the initial
estimate [GMM-S].
Table 8 presents the extended estimation results. Both f atheduc and f atheduc2
appear to be valid and strong IVs, because the value of the J-statistic from GMM-S
is 1.5154 (df = 1) and corrd (educ, f atheduc) = 0.4336, and corrd (educ, f atheduc2 ) =0.4204. Totally different coefficient estimates and large SEs of PARA may be again
due to poor linear approximation in the first step.
TABLE 8 HERE

3.2

Discussion of Inconsistencies of the Estimators

Here we discuss the estimators we consider in the empirical section in light of thedifferent assumptions applicable to the earnings equation and to the PSID and NLS

30

Table 8: Estimation Results
Education
Experience
Experience2
Ability
M arried
Black
South
U rban(1)
OLS*
0.0635(0.0056)0.0809(0.0043)−0.0017(0.0001)
0.0313(0.0078)
0.3717(0.0539)−0.1302(0.0323)−0.0921(0.0288)
0.1363(0.0282)(2)
OLS-S
0.0718(0.0053)0.0818(0.0043)−0.0017(0.0001)−(−)
0.3793(0.0536)−0.1741(0.0316)−0.0983(0.0287)
0.1499(0.0284)(3)
IV1-S
0.0953(0.0159)
0.0830(0.0045)−0.0017(0.0001)−(−)
0.3844(0.0536)−0.1249(0.0426)−0.0814(0.0314)0.1278(0.0332)(4)
IV2-S
0.0908(0.0163)0.0828(0.0044)−0.0017(0.0001)−(−)
0.3835(0.0537)−0.1342(0.0436)−0.0846(0.0314)0.1320(0.0334)(5)
2SLS-S
0.0944(0.0159)
0.0830(0.0045)−0.0017(0.0001)−(−)
0.3843(0.0536)−0.1267(0.0427)−0.0821(0.0314)0.1286(0.0332)(6)
GMM-S
0.0954(0.0159)
0.0826(0.0044)−0.0017(0.0001)−(−)
0.3907(0.0534)−0.1220(0.0425)−0.0829(0.0314)0.1272(0.0332)(7)
MSOLS
0.0727(0.0059)
0.0826(0.0049)−0.0017(0.0001)−0.0012(0.0027)
0.3799(0.0535)−0.1849(0.0393)−0.0979(0.0286)
0.1538(0.0297)(8)
MSII-FM
0.0685(0.0080)
0.0766(0.0065)−0.0016(0.0001)
0.0029(0.0081)0.3777(0.0535)−0.1504(0.0806)−0.0989(0.0289)0.1404(0.0390)(9)PARA−0.2850(0.2281)0.1170(0.0095)−0.0017(0.0001)
0.2444(0.1514)0.2226(0.1319)−0.1819(0.1146)−0.1372(0.1135)
0.0138(0.1273)

Data Combination?
No
No
No
No
No
No
Yes
Yes
Yes
Sample Size:
n
2430
2430
2430
2430
2430
2430
2430
2430
2430
m
−
−
−
−
−
−
1102
1102
1102
Notes: The dependent variable is the log of total annual labor earnings. Education, married, black, south, and urban areused as the included common variables, and age and south while growing up are used as the excluded common variables.
The (demeaned) IQ score and KW W score variables are used as ability measures in PSID and NLS samples, respectively.
The value of the J-statistic from GMM-S is 1.5154 (df = 1).

samples. To save space, we focus on a reduced version of modellog (earnings) = β0 + β1 education + β2 ability + β3 experience + β4 experience2+ β5 married + β6 black + β7 south + β8 urban + u,
where we ignore all the covariates except for the ones of interest.
As a starting point, consider the following model:log (earnings) = β0 + β1 education + β2 ability + u.

(15)

The probability limit of the OLS estimator of β1 from equation (15) where ability isunobserved isplim β̂1OLS = β1 + β2n→∞cov(ability, education).
var(education)(16)

If we assume that cov(ability, education) > 0 and β2 > 0, β̂1OLS for equation (15)with unobserved ability is upward-inconsistent.

31

(10)
PILS
0.0568(0.0087)
0.0818(0.0043)−0.0017(0.0001)
0.0094(0.0045)
0.3958(0.0543)−0.1776(0.0316)−0.1042(0.0287)
0.1557(0.0282)
Yes
2430
1102

Second, if there is no ability measure available in our sample, we can employ IV
estimation of the wage equation (15) and it will produce a consistent estimator ofβ1 as long as the instrument used – father’s education – is a valid instrument for
education.
However, if our instrument has a direct effect on the individual’s wage, the IV
estimator of equation (15) with unobservable ability will be inconsistent. Indeed, if
the individual’s wage is determined by:log (earnings) = γ0 + γ1 education + γ2 f ather0 s education + u,then the relation between β1 (from equation (15) without the ability variable) and γ1and γ2 can be obtained following the omitted variable formula:
β1 =

λσf
cov(f ather0 s education, log (earnings))= γ1 + γ2,0cov(f ather s education, education)
σewhere λ = corr(f ather0 s education, education), σf2 = var(f ather0 s education), and
σe2 = var(education). In this case, the probability limit of the IV estimator of β1
from equation (15) where ability is excluded from the equation isplim β̂1IV =n→∞cov(f ather0 s education, log (earnings))
σf(1 − λ2 )σf
=
γ
+γ
=
β
+γ
.
1
2
1
2
cov(f ather0 s education, education)
λσe
λσe
(17)

If we assume that 0 < λ < 1 and γ2 > 0, β̂1IV for equation (15) with unobserved
ability is upward-inconsistent.
Third, if there is measurement error in ability under the classical errors-in-variables
assumption that the correlation between the true unobserved variable and its measurement error is zero, the OLS estimator of β1 from equation (15) with an error-ridden
measure of ability will be inconsistent. Let
ability = ability ∗ + ,
32

where ability ∗ is the unobserved error-free ability, ability is the observed mismeasured
ability, and  is the measurement error. We can show thatσ 2 σea − σa2 σeplim β̂1OLS = β1 + β2  2 2,
2σe σa − σea
n→∞

(18)

where σ2 = var(), σe2 = var(education), σa2 = var(ability), σea = cov(education, ability),and σe = cov(education, ). If we assume that σea > 0 and the correlation between
the individual’s educational level and the measurement error in his ability is zero (i.e.,σe = 0), thenplim β̂1OLS = β1 + β2n→∞σ2 σea,σe2 σa2 (1 − ρ2ea )(19)

where ρea = corr(education, ability). Thus, β̂1OLS from equation (15) with mismeasured ability is upward-inconsistent.
Finally, note that Hirukawa and Prokhorov (2018) provide a measurement error
interpretation for the MSOLS estimator of the earnings equation when ability isunobserved in (15). Thus, we expect the MSOLS estimator of β1 from (15) withmissing ability to be also upward-inconsistent following our argument above for thecase of mismeasured ability.
Based on the above calculations we can make the following conclusions. First, if
we use a valid instrument to estimate β1 in (15) where ability is missing, then thisIV estimator of β1 is consistent while the OLS estimators of β1 with mismeasured oromitted ability (including the MSOLS estimator) are upward-inconsistent. Second, if
we think there is no measurement error in the ability measure which is available tous, then the OLS* estimator of β1 is consistent, while the OLS-S and MSOLS are stillnot. Third, when the instrument for education is valid and ability is missing fromequation (15) then it is, in general, unclear which one of the three (four if we count
MSOLS) upward inconsistencies considered above will be larger and which smaller.
33

3.3

Sensitivity Analysis in PILS Estimation

A concern on PILS estimation is how sensitive it is to the choice of smoothing parameter values in nonparametric estimation of g2 (·).

The benchmark PILS estimatesin Section 4 are based on b̂ = σ̂U (log m/m)0.6 and λ̂ = (log m/m)0.6 for the beta
and discrete kernels, respectively, where σ̂U is the sample standard deviation of aconverted continuous common variable U in S2 . Numerical values of the smoothing
parameters are b̂1 = 0.0090 for education (included continuous common variable),b̂2 = 0.0022 for age (excluded continuous common variable), and λ̂ = 0.0481 for all
binary common variables.
We multiply each smoothing parameter value by {1/2, 2/3, 1, 3/2, 2} and reestimate the regression model. Table 9 indicates how the PILS estimates change from
the benchmark case (“ b̂1 , b̂2 , λ̂ multiplied by 1”). It can be found that even after
the original smoothing parameter values are doubled or cut by half, the estimation
results are qualitatively similar.
TABLE 9 HERE

34

Table 9: Sensitivity of PILS to Smoothing Parameter Valuesb̂1 , b̂2 , λ̂ multiplied by

1/2

2/3

1

3/2

2

Education

0.04190.05290.05680.05700.0582(0.0086)(0.0070)(0.0087)(0.0103)(0.0186)

0.09430.07320.08180.08160.0814(0.0095)(0.0044)(0.0043)(0.0043)(0.0044)−0.0022−0.0014−0.0017−0.0017−0.0017(0.0004)(0.0001)(0.0001)(0.0001)(0.0001)

0.00960.00880.00940.00940.0088(0.0036)(0.0033)(0.0045)(0.0057)(0.0105)

0.3971

0.38150.39580.39380.3912(0.0618)(0.0510)(0.0543)(0.0545)(0.0564)−0.1864−0.2120−0.1776−0.1768−0.1757(0.0353)(0.0287)(0.0316)(0.0316)(0.0317)−0.1078−0.1122−0.1042−0.1027−0.1013(0.0320)(0.0262)(0.0287)(0.0286)(0.0287)

0.13100.13170.15570.1556

0.1550(0.0335)(0.0269)(0.0282)(0.0282)(0.0303)

Experience

Experience2

Ability

M arried

Black

South

U rbanReferences[1] Dieterle, S. G., and A. Snell (2016): “A Simple Diagnostic to Investigate Instrument Validity and Heterogeneous Effects When Using a Single Instrument,”
Labour Economics, 42, 76 - 86.
[2] Eicker, F. (1963): “Asymptotic Normality and Consistency of the Least Squares
Estimators for Families of Linear Regressions,” Annals of Mathematical Statistics,
34, 447 - 456.

35

[3] Fang, H., M. P. Keane, and D. Silverman (2008): “Sources of Advantageous Selection: Evidence from the Medigap Insurance Market,” Journal of Political Economy, 116, 303 - 350.
[4] Hirukawa, M., and A. Prokhorov (2018): “Consistent Estimation of Linear Regression Models Using Matched Data,” Journal of Econometrics, 203, 344 - 358.
[5] Horowitz, J. L., and V. G. Spokoiny (2001): “An Adaptive, Rate-Optimal Testof a Parametric Mean-Regression Model Against a Nonparametric Alternative,”
Econometrica, 69, 599 - 631.
[6] White, H. (1980): “A Heteroskedasticity-Consistent Covariance Matrix Estimatorand a Direct Test for Heteroskedasticity,” Econometrica, 48, 817 - 838.

36