Empirical likelihood inference for monotone
index modelTaisuke Otsu, Keisuke Takahata and Mengshan Xu
July 23, 2019
KEO Discussion Paper No. 149

KEO Discussion Paper No. 149

EMPIRICAL LIKELIHOOD INFERENCE FOR MONOTONE INDEX MODELTAISUKE OTSU, KEISUKE TAKAHATA, AND MENGSHAN XU
Abstract. This paper proposes an empirical likelihood inference method for monotone index
models. We construct the empirical likelihood function based on a modified score function
developed by Balabdaoui, Groeneboom and Hendrickx (2019), where the monotone link function
is estimated by isotonic regression. It is shown that the empirical likelihood ratio statistic
converges to a weighted chi-squared distribution. We suggest inference procedures based on an
adjusted empirical likelihood statistic that is asymptotically pivotal, and a bootstrap calibration
with recentering. A simulation study illustrates usefulness of the proposed inference methods.

1. Introduction
Single index models are widely used in statistics since they compromise interpretability of index
coeﬃcients in the parametric part and flexibility of regression modeling in the nonparametricpart (see, ch. 8 of Li and Racine, 2007, for a review). Many estimation methods have been
proposed for single index models, such as the semiparametric least squares estimator (Härdle,
Hall and Ichimura, 1993; Ichimura, 1993), M-estimator (Klein and Spady, 1993), and averagederivative estimator (Powell, Stock and Stoker, 1989). Although these estimation methods havedesirable theoretical properties under certain regularity conditions, they typically require some
nonparametric smoothing method to evaluate the unknown link function, which involves tuning
parameters, such as bandwidth and series length parameters, and the optimal choices of them
are substantial (theoretical and practical) problems.
The monotone single index model, in which monotonicity is imposed on the link function,
has been studied in recent years. Balabdaoui, Durot and Jankowski (2016) showed that theleast square estimator of a monotone single index model generally converges at the cube root
rate, but its asymptotic distribution is still unknown. The main diﬃculty for deriving theasymptotic distribution of the least square estimator arises from the non-diﬀerentiability of theobjective function; in a monotone single index model, the link function, which is an infinitedimensional nuisance parameter, is generally estimated by a nonparametric approach such as
isotonic regression, while the index part is parametrically modeled as a linear combination of thecovariates. Then the derivative of the objective function with respect to the index coeﬃcients is
intractable due to the non-smoothness of the estimated nuisance parameter.
To overcome this issue, Groeneboom and Hendrickx (2018) developed a score-type estimator
for the current status model, which is a special case of monotone single index models. Theirapproach is based on the estimating equation which is the same as the first-order condition of
the least square estimator except that it ignores the derivative of the estimated link function.
1

They provedpn-consistency and asymptotic normality of their estimator without any tuning

parameter. Their result was extended to general monotone single index models by Balabdaoui,pGroeneboom and Hendrickx (2019), where they derived n-consistency and asymptotic normality for the parametric component and an n1/3 / log n convergence rate for the nonparametricestimator of the link function.
Although the score estimation approach is remarkable, the main drawback is that it requires
smoothing parameters to estimate the asymptotic variance to implement hypothesis testing andinterval estimation. Because the estimating function in the score-type approach is dependent on
the estimated link function, some conditional expectation is involved in the asymptotic variance.
Besides, the partial derivative of the link function is also included in the asymptotic varianceeven though the estimated link function is not smooth. Therefore, smoothing methods, suchas the kernel smoothing, are employed to estimate such quantities, which require us to selectmultiple smoothing parameters and make statistical inference cumbersome.
To address this problem, we propose an empirical likelihood inference method based on thescore-type approach for monotone index models. We show that the empirical likelihood statistic
based on the estimating equation of Balabdaoui, Groeneboom and Hendrickx (2019) converges
in distribution to the weighted chi-squared distribution. Even in our empirical likelihood approach, the conditional expectation as mentioned above appears in the asymptotic distribution.
To circumvent selection of smoothing parameters, we adapt the bootstrap calibration method
proposed by Hjort, McKeague and van Keilegom (2009) to our context. Because of the estimating equation with the estimated nuisance parameter plugged-in, a classical naive bootstrap
method is not asymptotically valid. Hjort, McKeague and van Keilegom (2009) provided a modified bootstrap method by recentering and reweighting to deal with such a situation. Combining
the empirical likelihood and modified bootstrap methods, our approach provides a simple andtheoretically justified method for statistical inference in monotone single index models.
The remainder of this paper is organized as follows. Section 2 presents our basic setup,methodology, and theoretical results. In Section 3, we conduct a small simulation study toillustrate the proposed method. All proofs are contained in the appendix.
2. Main result
We closely follow the setup and notation of Balabdaoui, Groeneboom and Hendrickx (2019)(hereafter BGH). Consider the monotone index modelY =00 (X ↵0 ) + ✏,E[✏|X] = 0,(1)

where Y is a scalar response variable, X is a d-dimensional vector of covariates, ✏ is an error
term, ↵0 is a k-dimensional vector of parameters, and

0 : R ! R is an unknown monotone

increasing function. For identification, we assume that ↵0 belongs to the d-dimensional unit
sphere Sd 1 = {↵ 2 Rd : ||↵|| = 1}. We are interested in conducting statistical inference (i.e.,interval estimation and hypothesis testing) on ↵0 based on the empirical likelihood approach.

Let S : Rd 1 ! Sd 1 be a parameterization such that for each ↵ in a neighborhood of ↵0

on Sd 1 , there exists a unique2 Rd 1 which satisfies ↵ = S( ). To motivate the score-type2

approach of BGH, we tentatively assume that0 is known.

the least square estimation of 0 ish(1)
E J( 0 )0 X 0 (X 0 S( 0 )){Y
where

(1)
0 is the derivative ofan estimator of0.

of

0 and J(

The population score equation for0
0 (X S( 0 ))}

i

(2)

= 0,

) is the Jacobian of S( ). Thus, it is natural to construct

0 by taking an empirical counterpart of (2) and inserting estimators for

(1)
0 and

However, when we estimate

0 by the isotonic regression method, the resulting estimator(1)
0 is typically discontinuous and it is not clear how to evaluate the derivative 0 without

introducing smoothing parameters. To address this issue, BGH and Groeneboom and Hendrickx(2018) considered the modified population score equation⇥⇤
0
E J( 0 )0 X{Y
0 (X S( 0 ))} = 0.

(3)

In particular, for point estimation of ↵0 , BGH proposed to solve the following score-type equationn

1X ˆ0J( ) Xi {Yinˆ ˆ(X 0 S( ˆ))} = 0,

(4)

i

i=1

with respect to ˆ, and estimate ↵0 by ↵
ˆ = S( ˆ), where for given , ˆ is obtained by the isotonic
regression
ˆ = arg min

2M

n
X
i=1

{Yi

(Xi0 S( ))}2 ,(5)

and M is the set of monotone increasing functions defined on R.

In this paper, we employ the score-type equation in (3) as a moment function and propose thefollowing empirical likelihood statistic`( 0 ) =
where

2 max
n

{pi }i=1nXlog(npi )

s.t.

i=1

n
X

pi = 1,i=1ĝi ( ) = J( )0 Xi {YinX

pi ĝi ( 0 ) = 0,

(6)

i=1

ˆ (X 0 S( ))}.
i

By the Lagrange multiplier argument, its dual form is obtained as`( 0 ) = 2nX

log(1 + ˆ 0 ĝi ( 0 )),

(7)

i=1

where the Lagrange multiplier ˆ solvesn1X
ĝi ( 0 )= 0.
n
1 + ˆ 0 ĝi ( 0 )

(8)

i=1

In practice, we use the dual representation in (7) to implement statistical inference. To study
the asymptotic properties of the empirical likelihood statistic `( 0 ), we impose the following
assumptions. Let k·k be the Euclidean norm and B(a0 , A) = {a : kaaround a0 of radius A.

Assumption.
3

a0 k  A} be a ballA1: {Yi , Xi }ni=1 is an iid sample generated by (1). The support X of X is convex with anonempty interior, and X ⇢ B(0, R) for some R > 0. The Lebesgue density of X has abounded derivative on X . There exist positive constants c and C such that E[|Y |m |X =x]  cm!C m 2 for all integers m

2 and almost every x 2 X .

0 is monotone increasing and there exists K0 > 0 such that | 0 (u)|  K0 for allu 2 {x0 ↵0 : x 2 X }.

A2:

These assumptions are adaptations of Assumptions A1-A6 in BGH. Compared to BGH, ourassumptions are simpler because we do not need to control the behavior of the score function
outside the true parameter ↵0 = S( 0 ). Assumption A1 is on the distribution form of the data.
The support condition in A1 may be relaxed by assuming X to follow a sub-Gaussian distribution. The moment condition in A1, which is analogous to BGH’s A6, is required to guarantee
max1in |Yi | = Op (log n) to control the entropy of a class of score functions. Assumption A2 ison the true link function

0 . Compared to BGH which considers point estimation, we only needto impose boundedness, which is a mild requirement.
Under these assumptions, our main result is presented as follows.
Theorem 1. Under Assumptions A1-A2, it holdsd`( 0 ) ! Z 0 V Z,
where Z ⇠ N (0, ⌃) with ⌃ = J( 0 )0 E[✏2 (X

E[X|X 0 S( 0 )])(X

E[X|X 0 S( 0 )])0 ]J( 0 ) andV = J( 0 )0 E[✏2 XX 0 ]J( 0 ).

Remark 1. This theorem says that the empirical likelihood statistic `( 0 ) is not asymptotically pivotal and converges to a weighted chi-squared distribution w1 21,1 + · · · + wd 1 21,d 1 ,where w1 , . . . , wd 1 are the eigenvalues of ⌃ 1 V and

2 ,...,
1,1

2
1,d 1 are independent

2 ran1

dom variables. This lack of asymptotic pivotalness is caused by the mismatch in the asympP
totic variance ⌃ of the score function p1n ni=1 ĝi ( 0 ) and the limit V of the sample variancePV̂ = n1 ni=1 ĝi ( 0 )ĝi ( 0 )0 . In the literature of empirical likelihood, weighted chi-squared limiting distributions often emerge when the score (or moment) functions involve estimated nuisanceparameters (e.g., Qin and Jing, 2001; Xue and Zhu, 2006; Hjort, McKeague, and van Keilegom,2009).
Remark 2. One way to conduct statistical inference based on `( 0 ) is to estimate the criticalvalues of w1 21,1 + · · · + wd 1 21,d 1 based on some estimators of ⌃ and V . Based on (13), V isconsistently estimated by V̂ . On the other hand, ⌃ can be estimated by
ˆ = J( 0 )0 1⌃nwhere ✏ˆi = Yi

n
X
i=1

✏ˆ2i {Xim̂(Xi0 S( 0 ))}{Xim̂(Xi0 S( 0 ))}J( 0 ),

ˆ (X 0 S( 0 )) and m̂(·) is a nonparametric estimator of m(·) = E[X|X 0 S( 0 ) = ·].
0i

An alternative way for statistical inference is to adjust the empirical likelihood statistic `( 0 )to recover the asymptotic pivotalness. Based on Rao and Scott (1981) (see also Xue and Zhu,
4

2006), the above theorem implies`A ( 0 ) =d1
d`( 0 ) !
1ˆ
trace(⌃ V̂ )

2
d 1.

(9)

Then the confidence region of ↵0 = S( 0 ) can be obtained by {S( ) : `A ( )  qa }, where qa isthe (1a)-th quantile of the2
d 1 distribution.

Remark 3. A drawback of the asymptotic inference method presented in the previous remark
is that it requires a selection of a tuning parameter to implement the nonparametric estimatorm̂(·). In order to obtain an inference procedure which is free from tuning parameters, we adapt
the bootstrap method of Hjort, McKeague, and van Keilegom (2009) as follows.
(1) Based on the original sample {Yi , Xi }n , compute ˆ as in (4), and then computei=1

Mn ( ˆ) =

1
n

n
X

n

ĝi (S( ˆ)),

V̄ =

i=1

1X
ĝi (S( ˆ))ĝi (S( ˆ))0 .
ni=1(2) Draw {Yi⇤ , Xi⇤ }ni=1 from the original sample {Yi , Xi }ni=1 with equal weights.
pute1Mn⇤ ( ˆ) =n

where ˆ⇤ˆ = arg min 2M

Pn

n
X
i=1

J( ˆ)0 Xi⇤ {Yi⇤⇤
i=1 {Yi

Then com-ˆ⇤ (X ⇤0 S( ˆ))},ˆi(Xi⇤0 S( ˆ))}2 .

(3) The bootstrap counterpart of `( 0 ) is given by`⇤ = n{Mn⇤ ( ˆ)Mn ( ˆ)}0 V̄

1{Mn⇤ ( ˆ)Mn ( ˆ)}.

(10)

Under the additional assumptions A3-A5 in the appendix, the validity of this bootstrap approximation is obtained as follows.
Theorem 2. Under Assumptions A1-A5, it holds
sup |P ⇤ {`⇤  t}t 0p

P0 {`( 0 )  t}| ! 0,

where P ⇤ is the bootstrap distribution conditional on the data.
3. Simulation
We conduct a simulation study to investigate the finite sample performance of the proposed
inference methods. We consider the following data generation process:Y=0
0 (X ↵0 ) + ✏,✏ ⇠ N (0, 1),

30 (u) = u ,p↵0 = (1, 1, 1)0 / 3

X ⇠ N (0, I3 ),where I3 is the 3⇥3 identity matrix. We consider sample sizes n = 100, 500, 1000. The number ofMonte Carlo replications is 1000. We consider two testing methods discussed in Remarks 2 and 3.
For the adjusted statistic in (9), we estimate m(·) = E[X|X 0 S( 0 ) = ·] by the Nadaraya-Watson

estimator, and choose the bandwidths based on the expected Kullback-Leibler cross-validationp(Hurvich, Simonoﬀ and Tsai, 1998). To test the null hypothesis H0 : ↵0 = (1, 1, 1)0 / 3, we
calculate the test statistic (9) and compare it with the 95 percentile of the52
d 1 distribution.

Table 1. Rejection frequencies (%)
Adjusted
N A1 A2 A3
100 4.7 4.9 6.1 8.7
500 4.2 7.5 15.9 51.1
1000 7.4 14.8 31.5 86.1
n

Bootstrap
N A1 A2
8.1 8.3 9.0
6.6 10.0 18.1
5.6 18.2 34.9

A3
13.9
53.3
87.8

↵
ˆ1
mean s.d.
0.577 0.0528
0.576 0.0166
0.577 0.0113

For the bootstrap-calibrated test statistic (10), we compute ˆ as in BGH (the computer codeis available at Groeneboom’s website), and generate 499 bootstrap samples, and calculate thebootstrap counterpart `⇤ in (10).
Table 1 presents the rejection frequencies of the above empirical likelihood tests for the nullpp
H0 : ↵0 = (1, 1, 1)0 / 3 when the true values of ↵0 are (N) ↵0 = (1, 1, 1)0 / 3, (A1) ↵0 =ppp
(1.03, 1, 1)0 / 1.032 + 2, (A2) ↵0 = (1.05, 1, 1)0 / 1.052 + 2, and (A3) ↵0 = (1.10, 1, 1)0 / 1.102 + 2.
(N) is for the size properties, and (A1)-(A3) are to evaluate power properties.
The column “ ↵ˆ 1 ” reports the Monte Carlos averages and standard deviations of the firstpelement of the BGH estimator ↵ˆ . It shows that the mean is close to the truth, ↵01 = 1/ 3 '0.577, while the standard deviation becomes smaller with the sample size. From the columns(N), we can see that both the adjusted and bootstrap empirical likelihood tests have reasonablesize properties. Both tests become powerful as the sample size increases and the true values of↵0 are more distinct from the null values (i.e., from A1 to A3). Also, we find that overall thebootstrap test rejects slightly more often than the adjusted test.
Overall, our simulation results are encouraging.

6

Appendix A. Mathematical appendix
A.1. Proof of theorem 1. Here we denote ĝ0i = ĝi ( 0 ), S0 = S( 0 ), and J0 = J( 0 ).
Note that (i) X has bounded support (by Assumption A1), (ii) max |Yi | = Op (log n) (byAssumption A2 and Lemma 7.1 of Balabdaoui, Durot and Jankowski, 2016), and(iii) sup| ˆ (x0 S0 )| = Op (log n) by Lemma 8 of the supplementary material of BGH (hereafterx2X

0

BGH-supp). Combining these results, it holds(11)

max |ĝ0i | = Op (log n).

1inThus, an expansion of (8) around ˆ = 0 using the same argument in Owen (1991, proof of
Theorem 2) based on (11) implies
" n
# 1
n
X
1
1X
0
ˆ=
ĝ0i ĝ0i
ĝ0i + op (n 1/2 ).
n
n
i=1

(12)

i=1

A second-order expansion of (7) around ˆ = 0 using (12) yields
" n
#
n
X
X
0
0
0
`( 0 ) = 2 ˆĝ0i ˆ
ĝ0i ĝ ˆ + op (1)
0i

i=1

n

=

1 X
p
ĝ0i
n
i=1

!0 "

i=1

n

1X
0
ĝ0i ĝ0i
n
i=1

# 1

Then it is enough for the conclusion to show that

n

1 X
p
ĝ0i
n
i=1

!

+ op (1).

n

1X
0 p
ĝ0i ĝ0i! V = J00 E[✏2 XX 0 ]J0 ,
n

(13)

1
p
n

(14)

i=1
n
X
i=1

d

ĝ0i ! N (0, ⌃).

We first show (13). Decompose
" n
#
" n
#
n
1X
1X 2
1X
0
0
0
0
0
0
2
0
ˆ
ĝ0i ĝ0i = J0
✏ i X i X i J0 + J0
{ 0 (Xi S0 )0 (Xi S0 )} Xi Xi J0
n
n
n
i=1
i=1
i=1
" n
#
X
2
+J00
✏i { 0 (Xi0 S0 ) ˆ 0 (Xi0 S0 )}Xi Xi0 J0 .
(15)
n
i=1

By the law of large numbers, the first term of (15) converges to V . By Mukerjee (1988, Theorem3.3), we have the uniform consistencypˆ (x0 S0 )| !
0.
0

sup | 0 (x0 S0 )

x2X

(16)

Therefore, both the second and third terms of (15) converge to zero. Combining these results,
we obtain (13).
We now show (14). Let Pn be the empirical measure of {Xi , Yi }ni=1 , P0 be the true measureof (X, Y ), andE[X|x0 S0 ] = E[X|X 0 S0 = u] evaluated at u = x0 S0 .

7

Decompose
n

1X
ĝ0i
n

=

J00

=

J00

i=1

ˆ

x{y

ˆ (x0 S0 )}dPn (y, x)0ˆ{x E[X|x0 S0 ]}{y ˆ 0 (x0 S0 )}dPn (y, x)ˆ+J00 {E[X|x0 S0 ] Ēn (x0 S0 )}{y ˆ 0 (x0 S0 )}dPn (y, x)ˆ0+J0 Ēn (x0 S0 ){y ˆ 0 (x0 S0 )}dPn (y, x):= J00 (I + II + III),
where

8
>
>
E[X|x0 S0 = ⌧i,S0 ]if><Ēn (u) = E[X|x0 S0 = s]if>>>
:E[X|x0 S = ⌧0
i+1,S0 ] if

0 (u) >
0 (s) =
0 (u) <

ˆ (u) for all u 2 (⌧i , ⌧i+1 ),0ˆ (s) for some s 2 (⌧i , ⌧i+1 ),0(17)ˆ (u) for all u 2 (⌧i , ⌧i+1 ),0and ⌧i,S0 is the sequence of jump points of ˆ 0 . By the definition of Ēn (x0 S0 ), it holds III = 0(see, (C.10) in BGH-supp).
For II, decompose
II

=

ˆ

{E[X|x0 S0 ] Ēn (x0 S0 )}{y ˆ 0 (x0 S0 )}d(Pn P0 )(y, x)ˆ0+ {E[X|x0 S0 ] Ēn (x0 S0 )}{y0 (x S0 )}dP0 (y, x)ˆ0+ {E[X|x0 S0 ] Ēn (x0 S0 )}{ ˆ 0 (x0 S0 )0 (x S0 )}dP0 (y, x)(18):= IIa + IIb + IIc .

The same argument as in pp. 19-20 of BGH-supp guarantees IIa = op (n 1/2 ) and IIb =op (n 1/2 ). For IIc , using (C.11) of BGH-supp and Proposition 4 of BGH, we haveˆ02
kIIc k  C { ˆ 0 (x0 S0 )0 (x S0 )} dP0 (y, x)= Op ((log n)2 n 2/3 ) = op (n 1/2 ),for some C > 0. Therefore, we obtain
II = op (n 1/2 ).
For I, decompose
I

=

ˆ

0
{x E[X|x0 S0 ]}{y0 (x S0 )}dPn (y, x)ˆ+ {x E[X|x0 S0 ]}{ 0 (x0 S0 ) ˆ 0 (x0 S0 )}dPn (y, x):= Ia + Ib .

8

(19)

From pp. 21-22 of BGH-supp, we can show that Ib = op (n 1/2 ). Therefore,
ˆ
n
1X
0
0
1/2
ĝ0i = J0 {x E[X|x0 S0 ]}{y)0 (x S0 )}dPn (y, x) + op (n
n
i=1

n

= J00

1X
{Xi
n

E[Xi |Xi0 S0 ]}✏i + op (n 1/2 ),

i=1

(20)

and the central limit theorem implies (14). Therefore, the conclusion is obtained.

A.2. Proof of Theorem 2. Based on Hjort, McKeague and van Keilegom (2009), it is suﬃcientfor the conclusion to show thatP

V̄ !0 J00 E[✏2 XX 0 ]J0 ,pdn{Mn⇤ ( ˆ) Mn ( ˆ)} ! N (0, ⌃),

(21)
(22)

where ˆ is obtained by solving (4). For the validity of bootstrap, we add the following assumptions.
A3: There exists

0
0 > 0 such that the mapping u 7! E[Y |X ↵ = u] is monotone increasingon I↵ = {z 0 ↵, z 2 Z} for each ↵ 2 B(↵0 , 0 ).

A4: For all

6=

0 with S(

almost surely.
(1)⇥) 2 B(↵0 , 0 ), Cov ( 0)0 J( )0 X,

0
0 (S( 0 ) X)|S(

A5: J00 E[ 0 (X 0 ↵0 )V ar(X|X 0 ↵0 )]J0 is non-singular.
By BGH, it can be shown that under A1-A5, ˆ is consistent andnormal. LetV̄(u) = E[Y |X 0 S(

p

n( ˆ

⇤
)0 X =
6 00 ) is asymptotically) = u]. For (21), note that

"

#
n
X
1
= J( ˆ)0
Xi {✏i + ˆ(Xi0 S( ˆ)) ˆ ˆ(Xi0 S( ˆ))}2 Xi0 J( ˆ) + op (1)
n
i=1
( n
)
1X 2
0
0
= {J0 + op (1)}✏i Xi Xi + op (1) {J0 + op (1)},ni=1where the first equality follows from0ˆˆˆ (x S( )) = Op (0 ) for almost every x
ˆ(by p. 26 and Lemma 17 of BGH-supp) and the consistency of , and the second equality follows
from (16), A3, and the consistency of ˆ. Thus, by the law of large numbers, we obtain (21).
00 (x S( 0 ))We now prove (22). Note that Mn⇤ ( ˆ)Mn ( ˆ) = Mn⇤ ( ˆ) by (4). Let P̂n be the empiricalmeasure of the bootstrap resample. Decompose
ˆ
⇤ ˆ
0
ˆ
Mn ( ) = J( ) {x E(X|x0 S( ˆ))}{y ˆ⇤ˆ(x0 S( ˆ))}dP̂nˆ0ˆ+J( ) {E(X|x0 S( ˆ)) Ēn⇤ (x0 S( ˆ))}{y ˆ⇤ˆ(x0 S( ˆ))}dP̂n
ˆ
0
ˆ
+J( )
Ēn⇤ (x0 S( ˆ)){y ˆ⇤ˆ(x0 S( ˆ))}dP̂n:= I ⇤ + II ⇤ + III ⇤ ,(23)9where Ēn⇤ (·) is defined similarly to (17) with respect to ˆ⇤ˆ. Again, we have III ⇤ = 0 by thedefinition of Ēn⇤ (·). For II ⇤ , similar to (19) and p. 3481 of Groeneboom and Hendrickx (2017)(GH hereafter), we have II ⇤ = oPM (n 1/2 ), where PM is defined in p. 3450 of GH.
For I ⇤ , decomposeI⇤=J( ˆ)0ˆ{x E(X|x0 S( ˆ))}{y ˆ⇤ˆ(x0 S( ˆ))}d(P̂n Pn )ˆ0ˆ+J( ) {x E(X|x0 S( ˆ))}{y ˆ ˆ(x0 S( ˆ))}dPnˆ0ˆ+J( ) {x E(X|x0 S( ˆ))}{ ˆ ˆ(x0 S( ˆ)) ˆ⇤ˆ(x0 S( ˆ))}dPn:= Ia⇤ + Ib⇤ + Ic⇤ .
1/2 ) imply
For Ib⇤ , (4) and pp. 19-20 of BGH-supp combined with ˆ
0 = Op (nˆ⇤0ˆIb = J( )x{y ˆ ˆ(x0 S( ˆ))}dPnˆ0ˆJ( ) {E(X|x0 S( ˆ)) Ēn (x0 S( ˆ))}{y ˆ ˆ(x0 S( ˆ))}dPn= op (n 1/2 ).
For Ic⇤ , (6.21) in GH and pp. 21-22 of BGH-supp yieldˆ⇤0ˆ
Ic = J( ) {x E(X|x0 S( ˆ))}{ ˆ(x0 S( ˆ)) ˆ⇤ˆ(x0 S( ˆ))}dPnˆ0ˆ+J( ˆ)0 {x E(X|x0 S( ˆ))}{ ˆ ˆ(x0 S( ˆ))ˆ (x S( ))}dPn= op (n 1/2 ).
Finally, for Ia⇤ , we haveIa⇤

= J( ˆ)0ˆ{x

E(X|x0 S0 )}{y= J( ˆ)0ˆ{x

E(X|x0 S0 )}✏d(P̂n00 (x S0 )}d(P̂nPn ) + oPM (n 1/2 + ( ˆ0 ))Pn ) + oPM (n 1/2 ),where the first equality follows from a similar argument to (6.25) in GH, and the second equality
1/2 ).
follows from a rearrangement and ˆ
0 = Op (nCombining these results, we haveMn⇤ ( ˆ)Mn ( ˆ) = J( ˆ)0ˆ{x

E(X|x0 S0 )}✏d(P̂nPn ) + oPM (n 1/2 ).

Comparing this and (20), the central limit theorem yields (22). Therefore, the conclusion follows.

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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.
E-mail address: t.otsu@lse.ac.uk
Faculty of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan.
E-mail address: kei_t_12@keio.jp
Department of Economics, London School of Economics, Houghton Street, London, WC2A
2AE, UK.
E-mail address: M.Xu8@lse.ac.uk

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