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Finite-population inference via GMM estimator
Kakehi, Haruo(Otsu, Taisuke)
大津, 泰介Keio Economic Observatory Sangyo Kenkyujo
2024
KEO discussion paper No.178 (2024. 3)
10.14991/004.00000178-0001
Since the seminal works by Abadie et al. (2014, 2020), there has been considerable attention on
finite-population inference for various econometric problems. This paper extends the finitepopulation asymptotic approach to the generalized method of moments (GMM) estimator foroveridentified moment condition models. A motivating setup is the situation where researchers
have auxiliary information on some population moments. Under the finite-population asymptotics,
we study asymptotic properties of the GMM estimator, propose asymptotically conservative
variance estimators, and discuss how to select the GMM weight matrix. A simulation study basedon real data for entrepreneurship training and incubation programs illustrate usefulness of the
proposed method.
Technical Reporthttps://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=AA10715861-00000178-0001

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Finite-population inference via GMM estimator

Haruo Kakehi and Taisuke Otsu
March 5, 2024
KEO Discussion Paper No. 178

KEO Discussion Paper No. 178

FINITE-POPULATION INFERENCE VIA GMM ESTIMATOR
HARUO KAKEHI AND TAISUKE OTSU
Abstract. Since the seminal works by Abadie et al. (2014, 2020), there has been considerable
attention on finite-population inference for various econometric problems. This paper extends
the finite-population asymptotic approach to the generalized method of moments (GMM) estimator for overidentified moment condition models. A motivating setup is the situation where researchers have auxiliary information on some population moments. Under the finite-population
asymptotics, we study asymptotic properties of the GMM estimator, propose asymptoticallyconservative variance estimators, and discuss how to select the GMM weight matrix. A simulation study based on real data for entrepreneurship training and incubation programs illustrate
usefulness of the proposed method.

1. Introduction
Since the seminal works by Abadie et al. (2014, 2020), there has been growing interest in
inference methods under finite-population setups accounting for design-based uncertainty. The
design-based perspective for investigating econometric or statistical methods has been prevalent
in randomized experiments (e.g., Neyman, 1923; Rosenbaum, 2002; Freedman, 2008a, 2008b).
However, the literature does not consider the sampling-based uncertainty deriving from notobserving the entire population since it is common to assume that random assignment is the
only source of uncertainty in an experimental setting. On the other hand, extensive statistical
literature exists on finite-population asymptotics, taking into account of the sampling variation(see, Práŝková and Sen, 2009 for an overview) although this body of work omits consideration on
the design-based uncertainty. Abadie et al. (2020) developed an alternative inference frameworkin observational study settings by incorporating both design and sampling-based uncertainty.
While their framework is restricted to the case of linear regression, the recent literature considers
design-based uncertainty in different settings; M-estimators (Xu, 2021), spatial correlation (Xuand Wooldridge, 2022), difference-in-differences (Rambachan and Roth, 2022), and staggered
difference-in-differences (Athey and Imbens, 2022; Roth and Sant’Anna, 2023). Our paper closely
relates to the work by Xu (2021), which extends Abadie et al. (2020) to the M-estimator setting.
In this paper, we extend the above existing results on finite-population inference to the situations where estimands of interest are defined by overidentified moment conditions. In conventional empirical economic analyses using the infinite-population asymptotics, overidentified
moment condition models are ubiquitous, and there is rich literature on applied and theoretical econometric analyses for these models typically using the generalized method of moments1

(GMM) (see, e.g., Hall, 2005, for an overview). Therefore, it is of substantial interest to develop the GMM theory for finite-population inference problems. One motivating example is thesituation where the researcher has auxiliary information on the population moments as investigated in Imbens and Lancaster (1994). Under the finite-population asymptotic framework, theproportion of the sample size to the population size is non-negligible and it may be plausiblethat researchers have access to some population moments. Examples include large-scale experiments where sample representativeness is important (Muralidharan and Niehaus, 2017; Duflo
and Banerjee, 2017) and the Integrated Public Use Microdata Series (IPUMS) data, which is the
10% sample of the U.S. Census. In these cases, researchers may incorporate moments based on
the entire states or counties to improve point estimators and associated inference. Furthermore,
the finite-population asymptotic analysis of the GMM estimator and development of feasible
inference methods are open issues in the literature. In contrast to M-estimation problems asstudied in Xu (2021), the GMM requires a weight matrix, and currently there is no guidance on
its choice under the finite-population asymptotics.
This paper studies asymptotic properties of the GMM estimator for overidentified moment
condition models under the finite-population asymptotics. We derive the consistency and asymptotic normality of the GMM estimator. In particular, we find that its finite-population
asymptotic variance takes a different form from the conventional infinite-population asymptotic
variance, which is an analogous finding in just-identified modes (e.g., Abadie et al., 2020; Xu,
2021). Since our asymptotic variance for the GMM estimator involves a component which is notconsistently estimable, we propose two asymptotically conservative variance estimators: one isthe conventional variance estimator and the other is an adapted version of Abadie et al. (2020)’s
variance estimator to the GMM context. Furthermore, we discuss the choice of the GMM weightmatrix under the finite-population asymptotic framework and suggest a feasible data-dependent
weight, where the associated conservative variance estimator shows a desirable property. These
theoretical findings are illustrated by a empirically motivated simulation study by using real dataon entrepreneurship training and incubation programs in North American undergraduates.
The rest of the paper is organized as follows. Section 2 presents our main results. After introducing our basic setup and the GMM estimator in Section 2.1, Section 2.2 presents asymptotic
properties of the GMM estimator under the finite-population asymptotics and studies estimationof the asymptotic variance, and we discuss the choice of the GMM weight matrix in Section 2.3.
Section 3 illustrates the proposed inference method by a simulation study based on the real data
for entrepreneurial activities of minority groups, where auxiliary information on some population
moments are available as in Imbens and Lancaster (1994).
2. Main results
2.1. Setup and estimator. We first introduce our basic setup based on Abadie et al. (2020)
and Xu (2021). For each unit i = 1, . . . , M with population size M , consider the population{Xi , zi , Yi }M
i=1 , where Xi is a vector of assignment variables, zi is a vector of i’s attributions,and Yi is a vector of outcome variables. Our interest is in the effect of Xi rather than zi . For
example, in empirical analysis, Xi refers to a treatment status of an experiment or indicator for2

a policy intervention, while zi includes e.g. age, gender, and socioeconomic status of individuals.
Throughout the paper, we assume that zi is non-random, and the outcomes are written as
Yi = yi (Xi ) for a potential outcome function yi (·). Although these variables depend on the
population size M to conduct asymptotic analysis for M → ∞, we suppress such dependence
to simplify the presentation. From the finite population {Xi , zi , Yi }M
i=1 , we observe a random
sample by Bernoulli sampling. Let Ri be a Bernoulli random variable, which equals to one if iPis sampled, and zero otherwise. Let N = M
i=1 Ri be the sample size, which is random.
This paper is concerned with the situation where the estimand of interest is defined as asolution of overidentified moment restrictions. Let gi (Xi , θ) := g(Xi , zi , yi (Xi ), θ) with a kdimensional vector of moment functions g and a p-dimensional vector of parameters. The esti∗ is defined as a unique solution ofmand of interest θM
M

1 X∗
E[gi (Xi , θM)] = 0,
M

(2.1)

i=1

for k > p, i.e., the moment condition (2.1) is overidentified. Note that the expectation E[·] istaken with respect to the assignment variables Xi . It should be noted that the existing papers
such as Abadie et al. (2020) and Xu (2021) do not cover such a setup.
∗ , this paper focuses on the GMM estimator
As an estimator of θM
(
)′
(
)
M
M
1 X
1 X
θ̂N (WN ) = arg min
Ri gi (Xi , θ) WN
Ri gi (Xi , θ) ,
θ∈Θ
N
N
i=1

∗
where Θ is parameter space of θM

i=1

and WN is a k × k weight matrix. The choice of WN isdiscussed in Section 2.3 below.
2.2. Finite-population asymptotic theory. We now study large sample properties of the
GMM estimator θ̂N (WN ) in our asymptotic framework. We impose the following assumptions.
Assumption.
M(1) {Xi }M
i=1 is independent but not necessarily identically distributed. {Ri }i=1 is independentand identically distributed sample of Bernoulli random variables with ρM = P(Ri =M1) satisfying ρM → ρ ∈ [0, 1] as M → ∞. Furthermore, {Xi }M
i=1 and {Ri }i=1 are

independent.
(2) Θ is compact. WN converges in probability to a positive definite matrix W .
1 PM∗∗
limM →∞ M
i=1 E[gi (Xi , θ)] = 0 is uniquely satisfied at θ = θ := limM →∞ θM . gi (x, θ)is continuous at each θ ∈ Θ for almost every x, and supi,M E[supθ∈Θ ||gi (Xi , θ)||4 ] < ∞.
There exists functions h1 (·) and b1i (·) such that limu→0 h1 (u) = 0, supi,M E[b1i (Xi )] < ∞,and ||gi (Xi , θ) − gi (Xi , θ1 )|| ≤ b1i (Xi )h1 (||θ − θ1 ||) for each θ, θ1 ∈ Θ.
(3) θ∗ ∈ int(Θ). gi (x, θ)his continuously
differentiable on int(Θ) for almost every x, and G :=i
∗ )∂gi (Xi ,θM
1 PM
limM →∞ M i=1 E
has full column rank. There exists functions h2 (·) and∂θ′b2i (·) and a neighborhood N around θ∗ such that supi,M E[supθ∈N ||∂gi (Xi , θ)/∂θ′ ||2 ] <∞, limu→0 h2 (u) = 0, supi,M E[b2i (Xi )] < ∞, and ||∂gi (Xi , θ)/∂θ′ − ∂gi (Xi , θ1 )/∂θ′ || ≤b2i (Xi )h2 (||θ − θ1 ||) for each θ, θ1 ∈ N .
3

Assumption (1) is on the sampling framework, which is also employed by Abadie et al. (2020)
PMand Xu (2021). This implies the sample size N =i=1 Ri is random and its expectation
E[N ] = M ρM diverges at the same rate as M → ∞. Assumptions (2) and (3) collect regularity
conditions on the weight matrix WN and the moment function gi . These are natural adaptationsof the conventional GMM theory to our finite-population setup. Assumption (2) is used to derive
the consistency of the GMM estimator, and Assumption (3) contains additional conditions toestablish asymptotic normality.
Under the above assumptions, the asymptotic properties of the GMM estimator θ̂N (WN ) are
obtained as follows.

Theorem 1.
p∗ → 0.
(1) Under Assumptions (1)-(2), it holds θ̂N (WN ) − θM(2) Under Assumptions (1)-(3), it holds√d∗N (θ̂N (WN ) − θM
) → N (0, VGMM (W )),

(2.2)

where
VGMM (W ) = (G′ W G)−1 G′ W (Ω − ρ∆)W G(G′ W G)−1 ,
M

Ω =

1 X
∗
∗ ′
E[gi (Xi , θM)gi (Xi , θM
) ],M →∞ M
lim

i=1
M

∆ =

1 X
∗
∗
E[gi (Xi , θM)]E[gi (Xi , θM)]′ .
M →∞ M
lim

i=1

Theorem 1 (1) says that the GMM estimator θ̂N (WN ) is consistent for the population param∗ , and Theorem 1 (2) derives its asymptotic distribution. Compared to the conventional
eter θM

infinite-population asymptotics, the main difference is presence of the additional term “ρ∆” inthe asymptotic variance. Letting VC (W ) = (G′ W G)−1 G′ W ΩW G(G′ W G)−1 be the asymptotic
variance of the GMM estimator under the conventional infinite-population asymptotics, andVA (W ) = ρ(G′ W G)−1 G′ W ∆W G(G′ W G)−1 be an additional component, the finite-population
asymptotic variance can be written as VGMM (W ) = VC (W ) − VA (W ). Since ∆ is positive
semi-definite, VA (W ) is also positive semi-definite and VGMM (W ) is always smaller than the conventional variance VC (W ) in the matrix sense (denoted by VGMM (W ) ≤pd VC (W )). Although
VC (W ) can be consistently estimated (as shown below), the component ∆ and thus the variance
VGMM (W ) cannot be consistently estimable in general.
As in Abadie et al. (2020) and Xu (2021), we propose conservative estimators for VGMM (W ).
The first variance estimator is a consistent estimator of the conventional variance VC (W ), thatisV̂C (WN ) = (Ĝ′ WN Ĝ)−1 Ĝ′ WN Ω̂WN Ĝ(Ĝ′ WN Ĝ)−1 ,4where Ĝ = N1

PM

1 PM
′
′
i=1 Ri ∂gi (Xi , θ̂N (WN ))/∂θ and Ω̂ = N
i=1 Ri gi (Xi , θ̂N (WN ))gi (Xi , θ̂N (WN )) .

The second variance estimator is constructed by estimating a lower bound for ∆, that isˆZ = 1∆N

M
X

Ri P̂ ′ zi zi′ P̂ ,

i=1

P
PM
′ −1
where P̂ = ( M
i=1 Ri zi zi ) ( i=1 Ri zi gi (Xi , θ̂N (WN ))). Then the second variance estimator isˆ Z )WN Ĝ(Ĝ′ WN Ĝ)−1 .
V̂Z (WN ) = (Ĝ′ WN Ĝ)−1 Ĝ′ WN (Ω̂ − ρ∆The asymptotic properties of these variance estimators are obtained as follows.
Theorem 2.
p(1) Under Assumptions (1)-(3), V̂C (WN ) → VC (W ).
1 PM∗′(2) In addition to Assumptions (1)-(3), assume that limM →∞ M
i=1 E[gi (Xi , θM )]zi exists,
P
M
1
′
ˆ
limM →∞ M
i=1 zi zi exists and non-singular, and supi,M ||zi || < ∞. Then ∆Z converges

in probability to a positive semi-definite matrix ∆Z such that ∆Z ≤pd ∆.
(3) VGMM (W ) ≤pd VZ (W ) ≤pd VC (W ), where VZ (W ) = (G′ W G)−1 G′ W (Ω−ρ∆Z )W G(G′ W G)−1 .
Since the proof of this theorem is similar to the ones in Xu (2021, Theorems 2.2 and 3.1), it
is omitted. Theorem 2 (1) says that the conventional variance estimator V̂C (WN ) is still consistent for the variance component VC (W ) under the finite-population asymptotics. Theorem 2 (2)guarantees conservativeness of the second variance estimator V̂Z (WN ) for VGMM (W ). Theorem
2 (3) clarifies the relationships of the limits of the variance estimators. Although we cannot
consistently estimate the asymptotic variance VGMM (W ) of the GMM estimator, we can provide asymptotically conservative estimators V̂C (WN ) and V̂Z (WN ). We recommend to use thesecond estimator V̂Z (WN ) under the finite-population asymptotic framework because it is lessconservative than the first one V̂C (WN ).
2.3. Choice of GMM weight. The asymptotic analysis in the previous subsection focuses on
the case where the weight matrix WN for the GMM estimation is given. Given the different formof the asymptotic variance VGMM (W ) from the one under the conventional infinite-population
asymptotics, it is interesting to investigate the choice of the weight matrix under the current
setup.
First of all, the variance VGMM (W ) is minimized by Wopt = (Ω − ρ∆)−1 in the matrix sense.
However, due to the component ∆, a consistent estimator of Wopt is not available in general.
Motivated by the discussion in the previous subsection, we can consider two feasible weights,
ˆ Z )−1 . Theorems 1 and 2 implyΩ̂−1 and (Ω̂ − ρ∆
√
√

d

∗
N (θ̂N (Ω̂−1 ) − θM
) → N (0, VGMM (Ω−1 )),d∗ˆ Z )−1 ) − θM
N (θ̂N ((Ω̂ − ρ∆
) → N (0, VGMM ((Ω − ρ∆Z )−1 )).

Although these asymptotic variances are not directly comparable, we can see that
VGMM ((Ω − ρ∆Z )−1 ) ≤pd VZ ((Ω − ρ∆Z )−1 ) ≤pd VZ (Ω−1 ) ≤pd VC (Ω−1 ).
5

(2.3)

∗ , we recommend to
Based on these relationships, to conduct inference on the parameters θMˆ Z )−1 ) combined with the asymptotic variance estimator
use the point estimator θ̂N ((Ω̂ − ρ∆ˆ Z )−1 ).
V̂Z ((Ω̂ − ρ∆3. Numerical illustration
In this section, we conduct a simulation study based on real economic data. We employ thedataset studied by Lyons and Zhang (2017) and Xu (2021) on entrepreneurship training andincubation programs in North American undergraduates between 2011 and 2015. Lyons andZhang (2017) originally analyzed the entrepreneurial activities of minority groups using datafrom 179 finalists out of 188 who accepted the program and 156 finalists out of 166 who did not.
As in Xu (2021), we consider this data as the entire finite-population and analyze the impact ofentrepreneurial activities.
The dataset contains two outcomes for entrepreneurial activities: short-term and ongoing/longrun. The short-term activity is set to one if the finalist has founded/co-founded a startup or
worked for a venture capital firm or a startup after the program but is no longer working at astartup. The ongoing/longer-run startup activity takes one if the finalist is currently working
with a startup. For these binary outcomes, we estimate the linear probability model (LPM) andprobit model by gender and outcome types. For each analysis, we include observable fixed attributes: location, program interview scores, prior entrepreneurial experience, study major, yearof study, university ranking, and year and interviewer dummies. However, as described by Lyons
and Zhang (2017) and Xu (2021), unobservable attribute differences between those who accepted
the program and those who did not may cause self-selection into the program. Therefore, wetake the same empirical strategy and discuss mainly the magnitude of the standard errors rather
than point estimates since our concern is the standard errors.
We conduct a simulation study using this data. In each Monte Carlo replication, we draw asubsample from the population by the sampling ratio ρ = 0.8 (in our preliminary simulations,
we considered different values of ρ, but the results are similar). We compare the M-estimatorwith the associated standard error in Xu (2021) with our GMM estimator with the associatedstandard errors (i.e., V̂C (WN ) and V̂Z (WN ) in the previous section). For the GMM, as in Imbens
and Lancaster (1994), we use the average outcomes by the treatment status in the population asadditional moments, i.e., we calculate the average probabilities of short-term or ongoing activity
for accepted and unaccepted finalists and incorporate them as additional information in the
GMM estimation. We repeat this procedure 1000 times.
Table 1 reports the Monte Carlo averages and standard errors of the M-estimates θ̂M and GMM
ˆ Z )−1 ) using the weights Ω̂−1 and (Ω̂ − ρ∆ˆ Z )−1 , respectively. It
estimates θ̂(Ω̂−1 ) and θ̂((Ω̂ − ρ∆reports the point estimates and the standard errors of the program treatment effect for male andfemale groups by using the linear probability and probit models. From the left panel of Table 1,
we can see that the averages of the estimates are very close for both the M- and GMM estimatesacross all cases, and that program participation has a more considerable impact on females thanmales on both short-term and ongoing/long-term outcomes. Also the right panel of Table 1
shows that the Monte Carlo standard errors of the GMM estimators are smaller than the ones of6

Table 1. Monte Carlo averages and standard errors
Monte Carlo averageMonte Carlo standard error
M-estimator
GMM
M-estimator
GMM
−1
−1
−1
ˆ
ˆ Z )−1 )θ̂Mθ̂(Ω̂ ) θ̂((Ω̂ − ρ∆Z ) )θ̂Mθ̂(Ω̂ ) θ̂((Ω̂ − ρ∆Short term/LPM
FemaleMale
Ongoing/LPM
FemaleMale
Short term/Probit
FemaleMale
Ongoing/Probit
FemaleMale

0.16446
0.138540.164020.13868

0.16433
0.13869

0.043270.024100.03592
0.01363

0.03625
0.01366

0.277220.099590.276540.10023

0.276390.10004

0.052850.028890.04382
0.01687

0.044100.01704

1.097110.57978

1.13677
0.56989

1.14327
0.56995

0.373020.11001

0.344640.062810.34966
0.06299

1.281170.28762

1.28771
0.282101.287100.28186

0.29879
0.07961

0.259400.04836

0.261200.04863

the M-estimators, and that the GMM standard errors are similar for the both weights. In termsof the standard errors, efficiency gains of the GMM estimators over the M-estimator are around6.26-17.08% for females and 38.91-43.44% for males.
It should be noted that the Monte Carlo standard errors of the M- and GMM estimators are
estimating (squared root of) the asymptotic variances, which cannot be estimated consistency
under the finite-population asymptotics. In Table 2 below, we present the Monte Carlo averagesof the asymptotically conservative standard errors: (i) Eicker-Huber-White standard error of theM-estimator, (ii) Xu’s (2021) finite-population standard error of the M-estimator, (iii) EickerHuber-White standard error based on V̂C (Ω̂−1 ), (iv) finite-population standard error based onˆ Z )−1 ). Based on theV̂Z (Ω̂−1 ), and (v) finite-population standard error based on V̂Z ((Ω̂ − ρ∆ˆ Z )−1 ).
analysis in Section 2.3, our recommendation is (v) with the GMM estimator θ̂((Ω̂ − ρ∆
Table 2. Monte Carlo averages of conservative standard errors
M-estimator(i)(ii)
Short term/LPM
FemaleMale
Ongoing/LPM
FemaleMale
Short term/Probit
FemaleMale
Ongoing/Probit
Female
Male

(iii)

GMM
(iv)

(v)

0.11259 0.09851 0.09024 0.08242 0.06483
0.06657 0.06402 0.03824 0.03715 0.03431
0.13746 0.12020 0.11090 0.10280 0.08074
0.07899 0.07604 0.04465 0.04321 0.03991
0.63175 0.56193 0.60853 0.54802 0.43037
0.27185 0.26397 0.16572 0.16140 0.14906
0.54720 0.49208 0.51811 0.48900 0.37355
0.21433 0.20790 0.12501 0.12134 0.11207

Our findings are summarized as follows. Here we focus on the estimates for females althoughsimilar comments apply to the ones for males. First, we confirm that these asymptotically7

conservative standard errors do not fall below the corresponding Monte Carlo standard errors
in Table 1. The conventional standard errors in (i) for the M-estimators and (iii) for the GMM
are around 69.36-160.20% and 76.56-153.08% larger than the (infeasible) Monte Carlo standard
errors in Table 1, respectively. On the other hand, the finite-population standard errors (ii), (iv),and (v) are around 50.64-127.66%, 59.01-129.45%, and 23.08-83.08% larger than the (infeasible)
Monte Carlo standard errors in Table 1, respectively. Based on these results, we can also seethat the finite-population standard errors can alleviate the overestimation of the conventional
standard error. For example, improvements by using the suggested finite-population standard
error (v) over the conventional one (iii) are around 27.19-29.27%. Second, if we compare thefinite-population standard error (ii) for the M-estimator with the recommended one (v) for the
GMM estimator, we can see that (v) is smaller than (ii) by 23.41-34.18%. This result showsthat using the additional moments can also help to reduce the feasible but conservative standard
errors. Finally, Tables 1 and 2 indicate that significance of the coefficients may change dependingon the standard errors. For example, if we look at the estimates and standard errors for females
in the first rows of Tables 1 and 2, inferences based on (i)-(iii) tend to accept the null of no
significance, but (iv)-(v) tend to reject at 5% significance level.

8Appendix A. Mathematical appendix
A.1. Proof of Theorem 1.
A.1.1. Proof of Part (1). It is sufficient to verify the conditions in Newey and McFadden (1994,
Theorem 2.1). Their condition (i) is satisfied due to uniqueness of θ∗ and positive definiteness of
W in Assumption (2). Their condition (ii) (i.e., compactness of the parameter space) is directly
imposed.
To verify their conditions (iii) and (iv), it is sufficient to show that
M

sup
θ∈Θ

M

1 X
1 X
p
Ri gi (Xi , θ) − limE[gi (Xi , θ)] → 0,
M →∞ M
N
i=1

(A.1)

i=1

1 PM

and limM →∞ M

1 PM
i=1 E[gi (Xi , θ)] is continuous at each θ ∈ Θ. The continuity of limM →∞ M
i=1 E[gi (Xi , θ)]follows by the dominated convergence theorem and Jensen’s inequality due to the conditions on
gi (Xi , θ) in Assumption (2). For (A.1), we first note that Abadie et al. (2014, Lemma A.2)
implies the pointwise convergence
M

M

i=1

i=1

1 X
1 X
p
Ri gi (Xi , θ) − limE[gi (Xi , θ)] → 0,
M →∞ MN

for each θ ∈ Θ.

Then Newey (1991, Corollary 2.2) combined with the Lipschitz condition in Assumption (2)implies the uniform convergence in (A.1).
Since we verify all the conditions of Newey and McFadden (1994, Theorem 2.1), the conclusion
follows.
A.1.2. Proof of Part (2). Let ĜN (θ) = N1PMi=1 Ri∂gi (Xi ,θ). By the consistency of θ̂N (WN ) and∂θ′

Assumption (3) (θ∗ ∈ int(Θ) and differentiability of gi (x, θ)), the estimator θ̂N (WN ) satisfies thefirst-order condition(′ĜN (θ̂N (WN )) WN)
M
1 X
Ri gi (Xi , θ̂N (WN )) = 0,
N
i=1

∗ and
with probability approaching one. By expanding gi (Xi , θ̂N (WN )) around θ̂N (WN ) = θM
∗ , we obtainsolving for θ̂N (WN ) − θM

√

M

1 X
∗
∗
N (θ̂N (WN ) − θM
) = [ĜN (θ̂N (WN ))′ WN ĜN (θ̃N )]−1 ĜN (θ̂N (WN ))′ WN √Ri gi (Xi , θM
),
N i=1
p

p

∗ . Since θ̂ (W )−θ ∗ → 0 and θ̃ −θ ∗ →where θ̃N is a point on the line joining θ̂N (WN ) and θM
N
N
N
M
M

0, it is sufficient for the conclusion to show that∂gi (Xi , θ)
sup ĜN (θ) − E
∂θ′
θ∈N

p

→ 0,

(A.2)

M

1 X
d
∗
√
Ri gi (Xi , θM
) → N (0, Ω − ρ∆),N i=1
for some neighborhood around θ∗ .
9

(A.3)

For (A.2), Abadie et hal. (2014,A.2) under Assumption (3) implies the pointwisei Lemmap∂gi (Xi ,θ)convergence ĜN (θ) − E→ 0 for each θ ∈ N . Then Newey (1991, Corollary 2.2)
∂θ′combined with the Lipschitz condition in Assumption (3) implies the uniform convergence in(A.2). For (A.3), it follows directly from Abadie et al. (2020, Lemma A.1) under Assumptions(2)-(3). Therefore, the conclusion follows.

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Graduate School of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345,
Japan.
Email address: haruokakehi@keio.jp
Department of Economics, London School of Economics, Houghton Street, London, WC2A
2AE, UK, and Keio Economic Observatory (KEO), 2-15-45 Mita, Minato-ku, Tokyo 108-8345,
Japan.
Email address: t.otsu@lse.ac.uk

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