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ARTICLE IN PRESS
Journal of Econometrics 141 (2007) 492–516
www.elsevier.com/locate/jeconom
On the second-order properties of empirical
likelihood with moment restrictions
Song Xi Chena,, Hengjian Cuib
a
b
Department of Statistics, Iowa State University, Ames, IA 50011-1210, USA
Department of Statistics and Financial Mathematics, Beijing Normal University, Beijing 100875, China
Available online 22 November 2006
Abstract
This paper considers the second-order properties of empirical likelihood (EL) for a parameter
defined by moment restrictions, which is the inferential framework of the generalized method of
moments. It is shown that the EL defined for this general framework still admits the delicate secondorder property of Bartlett correction. This represents a substantial extension of all the established
cases of Bartlett correction for the EL. An empirical Bartlett correction is proposed, which is shown
to work effectively in improving the coverage accuracy of confidence regions for the parameter.
r 2006 Elsevier B.V. All rights reserved.
JEL classification: C14; C30; C40
Keywords: Bartlett correction; Coverage accuracy; Empirical likelihood; Generalized method of moments
1. Introduction
Generalized method of moments (GMM) introduced by Hansen (1982) are an important
inferential framework in econometric studies. GMM is based on, upon given a model,
some known functions gðX ; yÞ of a random observation X 2 Rd and an unknown
parameter y 2 Rp , where g : Rdþp ! Rr , such that EfgðX ; yÞg ¼ 0 which constitutes
moment restrictions on the relationship between X and y. The power of GMM is in its
allowing rXp, namely the number of moment restrictions (instruments) can be larger than
the number of parameter, which leads to a full exploration of inference opportunities
Corresponding author. Tel.: +1 515 294 2729; fax: +1 515 294 4040.
E-mail address: songchen@iastate.edu (S.X. Chen).
0304-4076/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jeconom.2006.10.006
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493
provided by the given model. There is a vast pool of literatures on GMM. Here, we only
cite the reviews by Andrews (2002), Brown and Newey (2002), Imbens (2002) and Hansen
and West (2002).
Empirical likelihood (EL) introduced by Owen (1988) is a computer-intensive statistical
method that facilitates a likelihood-type inference in a non-parametric or semiparametric
setting. It is closely connected to the bootstrap as the EL effectively carries out the
resampling implicitly. On certain aspects of inference, EL is more attractive than the
bootstrap, for instance, its ability of internal studentizing so as to avoid explicit variance
estimation and producing confidence regions with natural shape and orientation; see Owen
(2001) for an overview of EL. A key property of EL is that the log EL ratio is
asymptotically chi-squared distributed, which resembles the Wilks’ theorem in parametric
likelihood. The Wilks’ theorem was established in the original proposal of Owen (1988) for
the means, in Hall and La Scala (1990) for smoothed function of means, Qin and Lawless
(1994) for parameters defined by moment restrictions and Kitamura (1997) for moment
restrictions with weakly dependent observations.
There have been comprehensive studies of EL in econometrics. Imbens (1997) shows
that the maximum EL estimator of y is a one-step variation of the two-stage GMM
estimator in the over-identified case of r4p, and achieves the same asymptotic efficiency as
the two-stage estimator. Testing is considered in Kitamura (2001) for moments
restrictions, Tripathi and Kitamura (2004) for conditional moment restrictions and Chen
and Gao (2006) for constructing an adaptive and rate-optimal test for a regression model.
Estimation and testing with conditional moment restrictions are studied in Donald et al.
(2003) and Kitamura et al. (2004). They found that EL possesses the attractive features of
avoiding estimating optimal instruments and achieving asymptotic pivotalness. Tilted EL
and other variations are studied in Kitamura and Stutzer (1997), Smith (1997) and Newey
and Smith (2004).
Another key property of the EL is Bartlett correction, which is a delicate second-order
property that implies a simple mean adjustment to the likelihood ratio can improve the
approximation to the limiting chi-square distribution by one order of magnitude and hence
can be used to enhance the coverage accuracy of likelihood-based confidence regions. In
the context of testing hypotheses, the Bartlett correction reduces the errors between the
nominal and actual significant levels of an EL test. Bartlett correction has been established
for EL by DiCiccio et al. (1991) for smoothed functions of means and Chen (1993, 1994)
for linear regression. Baggerly (1998) showed that EL is the only member within the
Cressie–Read power divergence family that is Bartlett correctable. Jing and Wood (1996)
revealed that the exponentially tilted EL for the means does not admit Bartlett correction
as the tilting alters the delicate second-order mechanism of EL. Recently, Chen and Cui
(2006) show that EL is Bartlett correctable in the presence of a nuisance parameter with
just-identified moment restrictions.
In this paper we show that the EL with over-identified moment restrictions is Bartlett
correctable. The finding represents a substantial extension to all the established cases of
Bartlett correction, which all consider just-identified cases in GMM. The establishment of
the Bartlett correction for the just-identified case is easier as the log maximum EL takes a
constant value n logðnÞ (n is the sample size). However, in the over-identified case the
maximum EL is no longer a constant, rather it introduces many extra terms into the log EL
ratio and makes the study of Bartlett correction far more challenging as can be seen from
the analysis carried out in this paper. The establishment of Bartlett correction in this
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S.X. Chen, H. Cui / Journal of Econometrics 141 (2007) 492–516
general case indicates that EL inherits the delicate second-order mechanism of the
parametric likelihood in a much wider situation. This together with the findings of
Imbens (1997), Kitamura (2001) and Newey and Smith (2004) and others suggests that
the EL is an attractive inferential tool in the context of moment restrictions. The
establishment of the Bartlett correction leads to a practical Bartlett correction, which is
confirmed to work effectively for coverage restoration in our simulation studies reported in
Section 4.
The paper is organized as follows. Section 2 provides an expansion for the log EL ratio
for parameters defined by moment restrictions. Bartlett correction and coverage errors
assessment of EL confidence regions are investigated in Section 3. Simulation results are
reported in Section 4, followed by a general discussion in Section 5. All technical details are
left in Appendix A.
2. EL for generalized moment restrictions
Let X 1 ; X 2 ; . . . ; X n be d-dimensional independent and identically distributed random
sample whose distribution depends on a p-dimensional parameter y which takes values in a
compact parameter space Y Rp . The information about y is summarized in the form of
rXp unbiased moment restrictions gj ðx; yÞ, j ¼ 1; 2; . . . ; r, such that E½gj ðX 1 ; y0 Þ ¼ 0 for a
unique y0 , which is the true value of y. Let
gðX ; yÞ ¼ ðg1 ðX ; yÞ; g2 ðX ; yÞ; . . . ; gr ðX ; yÞÞT
and
V ¼ VarfgðX 1 ; y0 Þg.
We assume the following regularity conditions:
ðiÞ V is a r r positive definite matrix and the rank of E½qgðX 1 ; y0 Þ=qy is p;
ðiiÞ for any j; 1pjpp; all the partial derivatives of gj ðx; yÞ up to the third order
with respect to y are continuous in a neighborhood of y0 and are bounded by some
integrable functions; respectively; in the neighborhood;
ðiiiÞ lim supjtj!1 jE½expfitT gðX 1 ; y0 Þgjo1 and EkgðX 1 ; y0 Þk15 o1.
ð2:1Þ
Conditions (i) and (ii) are standard requirements for establishing Wilks’ theorem and
higher-order Taylor expansions of the EL ratio. The first part of Condition (iii) is the
Cramér’s condition on the characteristic function of gðX ; y0 Þ. Requiring a finite 15th
moment is to ensure finite 5th moment for the sign square root of the EL ratio statistic,
which together with Cramer’s condition are needed in establishing the Edgeworth
expansion for the EL ratio statistic.
To facilitate simpler expressions, we transform gðX i ; yÞ to wi ðyÞ ¼ CV 1=2 gðX i ; yÞ where
C is a r r orthogonal matrix such that
qgðX i ; y0 Þ
1=2
CV
E
(2.2)
U ¼ ðL; 0ÞTrp .
qy
Here U ¼ ðukl Þpp is an orthogonal matrix and L ¼ diagðl1 ; . . . ; lp Þ is non-singular. Define
O ¼ ðokl Þpp ¼:UL1 where okl ¼ ukl l1
l . Here no summation over the subscript l is
carried out due to L being a diagonal matrix.
Let p1 ; p2 ; . . . ; pn be non-negative weights allocated to Q
the observations. The
PnEL for y
n
as proposed in Qin and Lawless (1994) is LðyÞ ¼ i¼1 pi subject to
i¼1 pi ¼ 1
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ARTICLE IN PRESS
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and
Pn
i¼1
495
pi wi ðyÞ ¼ 0. Let ‘ðyÞ ¼ 2 logfLðyÞ=nn g. Standard derivations in EL show
‘ðyÞ ¼ 2
n
X
logf1 þ lT ðyÞwi ðyÞg,
i¼1
P
where l ¼ lðyÞ is the solution of n1 ni¼1 wi ðyÞ=ð1 þ lT wi ðyÞÞ ¼ 0.
According to Qin and Lawless (1994), the maximum EL estimator y^ and its
^ are solutions of
corresponding l, denoted as l,
Q1n ðl; yÞ ¼ n
1
n
X
i¼1
Q2n ðl; yÞ ¼ n
1
wi ðyÞ
¼ 0 and
1 þ lT wi ðyÞ
n
X
ðqwi ðyÞ=qyÞT l
i¼1
1 þ lT wi ðyÞ
¼ 0.
(2.3)
(2.4)
^
The log EL ratio is rðyÞ ¼ ‘ðyÞ ‘ðyÞ.
^ respectively. To expand ‘ðy0 Þ,
In the following we develop expansions for ‘ðy0 Þ and ‘ðyÞ,
define
j
j
aj1 ...j k ¼ Efwi1 ðy0 Þ . . . wi k ðy0 Þg and
n
X
j
j
wi1 ðy0 Þ . . . wi k ðy0 Þ aj 1 ...jk .
Aj1 ...jk ¼ n1
i¼1
Here we use aj to denote the jth component of a vector a. Then, it may be shown that
n1 ‘ðy0 Þ ¼ Aj Aj Aji Aj Ai þ 23 ajih Aj Ai Ah þ Aji Ahi Aj Ah þ 23 Ajih Aj Ai Ah
2ajih Agh Aj Ai Ag þ ajgf aihf Aj Ai Ah Ag 12 ajihg Aj Ai Ah Ag þ Op ðn5=2 Þ.
ð2:5Þ
We use here a convention where if a superscript is repeated a summation over that
superscript is understood. This expansion has the same form as DiCiccio et al. (1991) for
the mean parameter and Chen (1993) for linear regression.
^ in the general case of r4p, two new systems of notations are introduced.
To expand ‘ðyÞ
Let Z ¼ ðl; yÞ, QðZÞ ¼ ðQT1n ðZÞ; QT2n ðZÞÞT , S21 ¼ UðL; 0Þ and S 12 ¼ S T21 . Due to the early
transformation in (2.2),
!
I S 12
qQð0; y0 Þ
S ¼: E
¼
.
S21 0
qZ
Put GðZÞ ¼ S1 QðZÞ. Now we can introduce the notations involving GðZÞ and their
derivatives
!
n
k j
q G ð0; y0 Þ
1X
qk Gj ð0; y0 Þ
j;j 1 ...j k
j;j 1 ...j k
b
¼E
¼
bj;j 1 ...j k
and B
qZj 1 . . . qZj k
n i¼1 qZj 1 . . . qZjk
and the notations involving wi ðyÞ and their derivatives.
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496
gj;j1 ...j l ;k;k1 ...km ;...;p;p1 ...pt
ql wji ðy0 Þ qm wki ðy0 Þ
qt wpi ðy0 Þ
¼E
p1
qy . . . qypt
qyj1 . . . qyj l qyk1 . . . qykm
!
and
C j;j 1 ...jl ;k;k1 ...km ;...;p;p1 ...pt
n
1X
ql wji ðy0 Þ qm wki ðy0 Þ
qt wpi ðy0 Þ
¼
p1
gj;j1 ...jl ;k;k1 ...km ;...;p;p1 ...pt .
n i¼1 qyj1 . . . qyj l qyk1 . . . qykm
qy . . . qypt
^ yÞ
^ is the solution of Gð^ZÞ ¼ 0, by inverting this equation, we derive in
Since Z^ ¼ ðl;
Appendix A.2 that for j; k; l; m 2 f1; 2; . . . ; r þ pg,
Z^ j Zj0 ¼ Bj þ Bj;k Bk 12 bj;kl Bk Bl Bj;k Bk;l Bl þ 12 bk;lm Bj;k Bl Bm þ bj;kl Bk;m Bm Bl
12 bj;kl bk;mn Bm Bn Bl 12 Bj;kl Bk Bl þ 16 bj;klm Bk Bl Bm þ Op ðn2 Þ.
ð2:6Þ
Note that (2.6) contains expansions for l^ j when jpr and for y^ j when j4r, respectively.
Note that
^ ¼2
‘ðyÞ
n X
i¼1
T
^ 1 ½l^ T wi ðyÞ
^ 2 þ 1 ½l^ T wi ðyÞ
^ 3 1 ½l^ T wi ðyÞ
^ 4
l^ wi ðyÞ
2
3
4
þ Op ðn3=2 Þ.
(2.7)
From now on, we fix the ranges of the superscripts a; b; c; d 2 f1; 2; . . . ; r pg,
f ; g; h; i; j 2 f1; 2; . . . ; rg, k; l; m; n; o 2 f1; 2; . . . ; pg and q; s; t; u 2 f1; 2; . . . ; r þ pg. It is shown
in Appendix A.2 by substituting (2.6) into (2.7) that
^ ¼ 2Bj Aj Bj Bj þ 2C i;k Bi Brþk;q Bq þ 1 bj;uq brþk;st gj;k Bu Bq Bs Bt
n1 lðyÞ
2
bj;uq Bu Bq Brþk;s Bs gj;k brþk;uq Bu Bq C i;k Bi Bj Bi Aji 23 ajih Bj Bi Bh
þ 2C j;k fBj Brþk Bj;p Bp Brþk ½2; j; r þ k þ 12 bj;uq Bu Bq Brþk ½2; j; r þ kg
þ gj;kl fBj Brþk Brþl þ Bj Brþk Brþl;q Bq ½3; j; r þ k; r þ l
12 bj;uq Brþk Brþl Bu Bq ½3; j; r þ k; r þ lg C j;kl Bj Brþk Brþl 23 Ajih Bj Bi Bh
Bj;u Bu Bj;q Bq 14 bj;uq bj;st Bu Bq Bs Bt þ bj;uq Bu Bq Bj;s Bs þ 2gj;i;h;k Bj Bi Bh Brþk
þ Bj Bi;q Bq Aji ½2; j; i 12 bj;uq Bu Bq Bi Aji ½2; j; i þ 13 gj;k;lm Bj Brþk Brþl Brþm
þ 2gj;i;l fBj Bi Brþl Bj Bi Brþl;q Bq þ 12 brþl;uq Bj Bi Bu Bq Brþl Bi Bj;q Bq ½2; j; i
þ 12 bj;uq Bu Bq Bi Brþl ½2; j; ig þ 2Bj Bi Brþl C j;i;l ðgj;i;lk þ gj;l;i;k ÞBj Bi Brþl Brþk
þ 2ajih Bj Bi Bh;q Bq ajih bj;uq Bu Bq Bi Bh 12 ajihg Bj Bi Bh Bg þ Op ðn5=2 Þ,
ð2:8Þ
where ½2; j; i indicates there are two terms by exchanging the superscripts i and j, and
½3; j; i; k means three terms such that the three superscripts take turns to occupy the
^ is more complicated than the just-identified case of
position of j. Expansion (2.8) for ‘ðyÞ
^ ¼0
r ¼ p. In that case, all the Bj ¼ 0 from a result established in (A.1), which means ‘ðyÞ
and rðy0 Þ ¼ ‘ðy0 Þ. This is the situations of all the existing studies on Bartlett correction of
^ contains more terms than that of ‘ðy0 Þ, which
the EL. When r4p, the expansion of ‘ðyÞ
increases substantially the difficulty of the second-order analysis.
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497
Combining (2.5) and (2.8), and carrying out further simplifications,
n1 rðy0 Þ ¼ Al Al Akl Ak Al 2Al;pþa Apþa Al þ 23 aklm Ak Am Al þ 2okl C pþa;k Apþa Al
þ ð2akl;pþa gpþa;mn omk onl ÞApþa Ak Al þ Aji ðAhi Aj Ah Bi;q Bq Bj ½2; i; jÞ
þ 2ðal;pþa;pþb gpþa;pþb;k okl ÞApþa Apþb Al þ Bj;u Bj;q Bu Bq þ 2C j;k Bj;q Brþk Bq
gj;kl Brþk Brþl Bj;q Bq 2gj;kl Bj Brþl Brþk;q Bq 2ajih Bj Bi Bh;q Bq
þ 2gj;i;l ðBj Bi Brþl;q Bq þ Brþl Bi Bj;q Bq ½2; j; iÞ þ ðgj;i;lk þ gj;l;i;k ÞBj Bi Brþl Brþk
þ ð14 bj;uq bj;st 12 bj;uq brþk;st gj;k ÞBu Bq Bs Bt þ ðajih bh;uq gj;i;l brþl;uq ÞBj Bi Bu Bq
þ ðgj;kl brþl;uq gi;j;k bi;uq gj;i;k bi;uq ÞBu Bq Bj Brþk þ 12 gj;kl bj;uq Bu Bq Brþl Brþk
13 gj;klm Bj Brþk Brþl Brþm 2gj;i;h;k Bj Bi Bh Brþk þ 12 ajihg Bj Bi Bh Bg
þ C j;kl Bj Brþk Brþl 2C j;i;l Bj Bi Brþl þ 23 Ajih ðBj Bi Bh þ Aj Ai Ah Þ
2ajih Agh Aj Ai Ag þ ajgf aihf Aj Ai Ah Ag 12 ajihg Aj Ai Ah Ag þ Op ðn5=2 Þ.
ð2:9Þ
This expansion leads to the following signed root decomposition:
n1 rðy0 Þ ¼ Rj Rj þ Op ðn5=2 Þ,
where R ¼ R1 þ R2 þ R3 and Ri ¼ Op ðni=2 Þ for i ¼ 1; 2 and 3. By matching Rl1 Rl1 with the
only term Al Al of order n1 in (2.9), we have
Rl1 ¼ Al .
(2.10)
Then we match 2Rl1 Rl2 with the terms of order n3=2 and obtain
Rl2 ¼ 12 Akl Ak Al;pþa Apþa þ 13 aklm Ak Am þ okl C pþa;k Apþa
þ ðakl;pþa 12 gpþa;mn omk onl ÞApþa Ak þ ðal;pþa;pþb gpþa;pþb;k okl ÞApþa Apþb
ð2:11Þ
and Rj1 ¼ Rj2 ¼ 0 for j 2 fp þ 1; . . . ; rg. Similarly, by matching 2Rl2 Rl3 with the rest of the
terms in (2.9) after removing terms contributing to ðRl1 þ Rl2 ÞðRl1 þ Rl2 Þ, the form of Rl3 is
given in Appendix A.2.
d
From (2.9), rðy0 Þ ¼ nAl Al þ op ð1Þ which means that rðy0 Þ ! w2p and leads to an EL
confidence region for y with nominal confidence level 1 a: I a ¼ fyjrðyÞpca g where ca is
the upper a-quantile of w2p distribution.
3. The second-order properties
This section considers the accuracy of the chi-square approximation to the distribution
of the EL ratio statistic rðy0 Þ and improving this approximation by the Bartlett correction.
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498
We first introduce some quantities used to define the accuracy of the chi-square
approximation. Let
1 lkk omm
7 lkm okm
J lo ¼ 14 ðalokk dlo Þ þ 36
a a
36
a a
þ alo;pþa;pþa
al;pþa;pþb ao;pþa;pþb alk;pþa aok;pþa þ 12 okl gm;pþa;k aom;pþa ½2; l; o
þ gpþa;pþb;k ao;pþa;pþb okl ½2; l; o þ 14 gpþa;mn omk onl aok;pþa ½2; l; o
0 0
0
0
þ 14 gpþa;mn gpþa;m n omk onl om k on o 12 gpþa;mn gk;pþa;v omk onl ovo ½2; l; o
okl go;pþa;pþa;k ½2; l; o þ ðgpþa;k;pþa;v gpþa;pþb;k gpþa;pþb;v Þokl ovo
and
ð3:1Þ
5 lkm okm
1 lok kmm
a a
72
a a
12 alok ak;pþa;pþa
K lo ¼ 18 ðalokk þ dlo Þ 72
12 okm gpþa;k;o alm;pþa 23 okm gpþa;k;pþa alom þ 14 oml onk aok;pþa gpþa;mn
þ 12 onl okm ðgpþa;k;o gpþa;n;m gpþa;n;o gpþa;k;m Þ
þ 12 okn ovl omo ðgpþa;mv gpþa;k;n gpþa;kv gpþa;m;n Þ
þ omo ovl okn ðgpþa;km gpþa;n;v gpþa;kv gpþa;n;m Þ
0
0
0 0
0
0
þ 18 om l omo on k onk ðgpþa;mn gpþa;m n gpþa;mm gpþa;nn Þ.
ð3:2Þ
P
1 lkk lmm
a a Þ where Dll ¼ 2K ll þ J ll ml ml and ml ¼ 16 n1 alkk .
Let Bc ¼ p1 ð pl¼1 Dll þ 36
The accuracy of the chi-square approximation is evaluated in the following theorem.
Theorem 1. Under Condition (2.1),
sup jPfrðy0 Þoxg Pðw2p oxÞ þ n1 Bc xf p ðxÞj ¼ Oðn2 Þ,
x2R
where f p ðÞ is the density of w2p distribution.
Theorem 1 indicates that the coverage error of the EL confidence region I a is Oðn1 Þ,
which is the same order as a standard two sided confidence region based on the asymptotic
^ The attractions of the EL confidence region are: (i) there is no need to carry
normality of y.
out any secondary estimation procedure in formulating the confidence region; and (ii) the
shape and the orientation of the region are naturally determined by the likelihood ratio
surface, free of any subjective intervention.
It can be shown by combining the expressions of EðRli Rlj Þ given in Appendix A.3 that
Efrðy0 Þg ¼ pð1 þ Bc n1 Þ þ Oðn2 Þ.
(3.3)
The rationale of the Bartlett correction is to adjust the mean of the EL ratio rðy0 Þ to make
it agreeable with the mean of w2p to the order of n1 . This mean adjustment, originally
proposed by Bartlett (1937), improves the chi-square approximation to the distribution of
the likelihood ratio to Oðn2 Þ. The following theorem formally establishes the Bartlett
correction for the EL ratio rðy0 Þ.
Theorem 2. Under Condition (2.1),
sup jPfrðy0 Þoxð1 þ Bc n1 Þg Pðw2p oxÞj ¼ Oðn2 Þ.
x2R
The theorem shows that the Bartlett correction is maintained by the EL for the situa^ has a
tion of general moment restrictions, despite that r may be larger than p and ‘ðyÞ
rather complex expression. This indicates that the EL is resilient in sharing this delicate
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499
second-order property with a parametric likelihood and the existence of certain internal
mechanism in the EL that resembles that of the parametric likelihood.
The Bartlett factor Bc can have a quite involved expression for a general over-identified
case of r4p due to the lengthy expressions of Dll . However, it admits simpler expression in
two special situations. One is in the situation of just-identified moment restrictions with
r ¼ p. It may be easily checked from (A.14) that
Bc ¼ p1 ð12 allkk 13 alkm alkm Þ,
(3.4)
which is the Bartlett factor obtained in DiCiccio et al. (1991) for smooth function of means
and Chen (1993) for linear regression. The other situation is when r4p, but (i)
Covfgj ðX ; y0 Þ; gpþa ðX ; y0 Þg ¼ 0 for any jpp and apr p and (ii) gj ðx; yÞ ¼ gj ðxÞ does not
depend on y for j ¼ p þ 1; . . . ; r. Assumption (i) means that the first p estimating equations
are uncorrelated with the last r p estimating equations at y0 and (ii) means that the last
r p estimating equations are free of parameters. In this case,
Bc ¼ p1 ð12 allkk þ all;pþa;pþa 13 alkm alkm allk akpþa;pþa alf ;pþa alf ;pþa Þ.
A special case of the second situation is considered in Cui and Yuan (2001) for a quantile
with r ¼ p þ 1.
To practically implement the Bartlett correction in a general situation, either Bc or
bc ¼:1 þ Bc n1 has to be estimated. It is noted that the direct plug-in estimator of bc can be
obtained by substituting all the populations moments involved by their corresponding
sample moments. However, considering the rather lengthy forms of bc , we propose using
the following bootstrap procedure to estimate bc .
Step 1: Generate a bootstrap resample fX i gni¼1 by sampling with replacement from the
^ ¼ ‘ ðyÞ
^ ‘ ðy^ Þ, where ‘ and y^ are,
and compute r ðyÞ
original sample fX i gn
i¼1
respectively, the log EL ratio and the maximum EL estimate based on the resample.
^ . . . ; rB ðyÞ.
^
Step 2: P
For a large integer N, repeat Step 1 B times and obtain r1 ðyÞ;
B
1
b ^
As B
b¼1 r ðyÞ estimates Efrðy0 Þg, a bootstrap estimate of bc is
b^ c ¼ ðBpÞ1
B
X
^
rb ðyÞ.
b¼1
Let Xn ¼ fX 1 ; . . . ; X n g be the original sample. It may be shown by standard bootstrap
arguments, for instance, those given in Hall (1992), that
Eðb^ c jXn Þ ¼ ð1 þ Bc n1 Þf1 þ Op ðn1=2 Þg
(3.5)
pffiffiffi
which means that the bootstrap estimate of bc is n-consistent. Now a practical Bartlett
corrected confidence region is I a;bc ¼ fyjrðyÞpca b^ c g.
The above use of the bootstrap to estimate bc naturally leads ones to think of using the
bootstrap to calibrate directly on the a-quantile of the EL ratio rðy0 Þ. Let c^a be the ath
^ given Xn , namely, Pfr ðyÞo^
^ ca jXn g ¼ a. The quantile
quantile of the distribution of r ðyÞ
^ B where ½ is the integer
can be estimated by the ð½aB þ 1Þth ordered value of frb ðyÞg
b¼1
truncation operator. We will ignore the error of the quantile estimation as it can be made
as small as possible by increasing the number of bootstrap simulation B. Then a direct
bootstrap confidence interval at a nominal level a is I a;bt ¼ fyjrðyÞp^ca g.
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Theorem 3. Under Condition (2.1),
Pfrðy0 Þoca b^ c g ¼ a þ Oðn3=2 Þ and
Pfrðy0 Þoc^a g ¼ a þ Oðn3=2 Þ.
The theorem indicates that the coverage errors of the two confidence regions I a;bc and
I a;bt are at the same order of n3=2 , which is one order of magnitude smaller than the
original EL region I a . However, they are both of larger order than Oðn2 Þ when the true Bc
is used as conveyed in Theorem 2. The underlying reason for achieving a coverage error at
the order of n2 in Theorem 2 is due to a fact that an even/odd order Hermit polynomial is
3=2
an even/odd function, which makes
-order term in the Edgeworth expansion
pffiffiffi the n
vanish. However, when we used a n-consistent estimate of bc in I a;bc , the n3=2 -order term
in the Edgeworth expansion involves no longer just Hermite polynomials. The error in the
estimation of bc has an effect at the order of n3=2 .
4. Simulation results
We report in this section results of two simulation studies which are designed to confirm
the theoretical findings of Bartlett correction of the EL by implementing the proposed
empirical Bartlett correction. For comparison purposes, the bootstrap confidence regions
I a;bt is also evaluated.
In the first simulation study, X 1 ; . . . ; X n are independent and identically Nðy; y2 þ 1Þ
distributed, as considered in an example of Qin and Lawless (1994). The relationship
between the mean and variance leads to moment restrictions: g1 ðX 1 ; yÞ ¼ X 1 y and
g2 ðX 1 ; yÞ ¼ X 21 2y2 1. This is an over-identified case as there are two moment
restrictions and one parameter of interest, i.e. r ¼ 2 and p ¼ 1. Like Qin and Lawless, the
value of y is chosen to be 0 and 1, respectively. The sample size used in the simulation study
is n ¼ 20; 30; 40 and 50, respectively.
In the second simulation study, we consider the following autoregressive panel data
model, which is an example considered in Brown and Newey (2002):
X it ¼ yX it1 þ ai þ it ;
X i0 ¼
ai
þ ei ,
1r
(4.1)
for t ¼ 1; . . . ; 4 and i ¼ 1; . . . ; n, where jyjo1, fit g4t¼1 and ai are mutually independent
standard normal random variables, ei Nð0; ð1 y2 Þ1 Þ and independent of fit g4t¼1 and ai .
Let X i ¼ ðX i1 ; . . . ; X i4 Þ. The moment restrictions after taking time differencing are
g1 ðX i ; yÞ ¼ X i1 ðDX i3 yDX i2 Þ, g2 ðX i ; yÞ ¼ X i1 ðDX i4 yDX i3 Þ and g3 ðX i ; yÞ ¼ X i2 ðDX i3 yDX i2 Þ where DX it ¼ X it X it1 . It is easy to check from model (4.1) that Efgj ðX i ; yÞg ¼ 0.
Hence, there are three constraints and one parameter, i.e. r ¼ 3 and p ¼ 1, another overidentified case. The parameter y, the autoregressive coefficient, is assigned values of 0:5 and
0:9 to obtain different levels of correlations. The sample size is chosen at n ¼ 50 and 100,
respectively. Three confidence intervals are evaluated. They are I a based on the limiting
chi-square distribution, I a;bc via the empirical Bartlett correction and I a;bt based on the
direct bootstrap calibration.
To gain information on the degree of over-identification in the moment restriction on
the confidence intervals, we also carry out simulations for each model with only one
estimating equation, which happens to be the first estimating equation, respectively. These
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correspond to just-identified cases, where the Bartlett factor are given explicitly in (3.4).
Therefore, in additional to the above three confidence intervals, we also evaluate the
theoretical Bartlett corrected interval which replaces b^ c by the real value bc in I a;bc . We are
also able to obtain the real Bc value for the over-identified case in the first simulation
model, hence the true Bartlett corrected interval is performed for all cases in the first
model.
In both simulation studies, the empirical coverage and length of the EL, Bartlett
corrected EL and the direct bootstrap calibrated intervals are evaluated with nominal
coverage levels of 90% and 95%, respectively. The bootstrap resample size B used is 250
and the number of simulation is 1000.
Tables 1 and 2 contain the empirical coverage and the averaged length of the confidence
intervals, which can be summarized as follows. First of all, the need for carrying out the
second-order correction to the EL confidence interval I a is obvious as the original EL
interval has quite severe under coverage for all the cases considered even for a sample size
of 100 for the panel data model. The under coverage is particularly severe when the sample
size is small for the normal mean model Nð1; 2Þ and for the panel data model. These are the
situations where the Bartlett correction is needed. In all the cases considered the empirical
Bartlett correction (I a;bc ) improves significantly the coverage of I a . The restoration of
coverage by the Bartlett correction is impressive when the sample size is small. We also
observed that, as anticipated, the direct bootstrap confidence interval has similar
performance with the Bartlett corrected intervals in most of the cases. However, in the
Table 1
Empirical coverage (in percentage) and averaged length of the EL confidence interval I a , the theoretical Bartlett
corrected (TBC) interval I tbc by using the ture Bartlett factors, the empirical Bartlett corrected (EBC) interval I a;bc
and the direct Bootstrap (BT) calibrated confidence interval I a;bt with X Nðy; y2 þ 1Þ and y ¼ 0
Nominal level
90%
sample size
EL
95%
TBC
EBC
BT
EL
TBC
EBC
BT
ðaÞ With two moment restrictions
20
coverage
84.50
length
0.635
30
coverage
85.75
length
0.543
40
coverage
87.65
length
0.490
60
coverage
87.50
length
0.448
89.50
0.712
89.25
0.602
90.10
0.515
89.50
0.465
90.30
0.835
89.45
0.626
89.80
0.546
89.04
0.425
87.90
0.792
87.32
0.605
88.76
0.542
88.75
0.413
91.20
0.760
91.20
0.672
92.50
0.585
94.00
0.540
94.80
0.847
94.10
0.720
94.80
0.615
95.50
0.560
93.82
0.932
93.50
0.765
95.60
0.627
94.63
0.547
93.40
0.930
93.20
0.761
94.80
0.625
93.87
0.532
ðbÞ With one moment
20
coverage
length
30
coverage
length
40
coverage
length
60
coverage
length
88.70
0.753
89.30
0.613
90.20
0.524
89.40
0.472
90.10
0.865
89.32
0.676
90.10
0.563
89.34
0.452
89.50
0.852
88.75
0.662
89.30
0.557
88.90
0.445
92.60
0.878
93.20
0.718
94.0
0.618
94.10
0.552
94.70
0.912
95.10
0.737
94.50
0.630
94.80
0.567
94.30
0.987
94.20
0.794
94.60
0.645
94.60
0.558
93.80
0.983
94.60
0.806
95.20
0.650
94.30
0.550
restriction
86.50
0.723
86.50
0.595
88.10
0.515
87.80
0.464
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Table 2
Empirical coverage (in percentage) and averaged length of the EL confidence interval I a , the theoretical Bartlett
corrected (TBC) interval I tbc by using the true Bartlett factors, the empirical Bartlett corrected (EBC) interval I a;bc
and the direct Bootstrap (BT) calibrated confidence interval I a;bt with X Nðy; y2 þ 1Þ and y ¼ 1
Nominal level
90%
sample size
EL
95%
TBC
EBC
BT
EL
TBC
EBC
BT
ðaÞ With two moment restrictions
20
coverage
81.40
length
0.661
30
coverage
82.50
length
0.610
40
coverage
85.50
length
0.462
60
coverage
88.50
length
0.412
87.50
0.807
92.00
0.705
89.50
0.531
89.75
0.443
86.45
0.792
89.15
0.683
88.40
0.523
90.35
0.452
85.48
0.783
88.63
0.677
88.90
0.526
89.23
0.440
87.00
0.789
92.50
0.707
91.25
0.590
92.25
0.492
92.10
0.956
95.25
0.794
94.25
0.657
94.40
0.531
91.80
0.924
94.21
0.782
93.70
0.643
94.25
0.535
90.20
0.917
93.60
0.764
93.20
0.635
94.50
0.540
ðbÞ With one moment
20
coverage
length
30
coverage
length
40
coverage
length
60
coverage
length
90.35
1.065
89.20
0.865
89.35
0.734
89.50
0.610
89.75
1.050
89.50
0.869
89.45
0.736
90.15
0.615
90.10
1.063
88.90
0.860
89.90
0.743
90.25
0.619
93.00
1.228
93.50
1.010
94.25
0.862
94.50
0.721
94.20
1.275
94.65
1.036
94.75
0.876
95.15
0.730
94.50
1.283
94.80
1.041
94.60
0.870
95.10
0.726
94.35
1.280
94.40
1.030
94.40
0.867
95.20
0.732
restriction
88.90
1.025
87.80
0.843
88.60
0.720
88.90
0.602
normal mean model, the coverage of the direct bootstrap interval I a;bt is not as good as the
Bartlett corrected interval I a;bc in most of the situations. The robust performance of the
Bartlett corrected interval may be due to the fact that the estimation of the Bartlett factor
bc , which involves simple bootstrap averaging, is less variable than the bootstrap
^ (see also Tables 3 and 4).
estimation of an extreme quantile of the distribution of rðyÞ
When the number of moment restriction is reduced to one, the lengths of all intervals
increase which is expected as the estimation efficiency declines. The theoretical Bartlett
correction is slightly better than I a;bc and I a;bt in terms of coverage accuracy and length.
However, when n is larger (n ¼ 60 in the first study and n ¼ 100 in the second), the three
second-order intervals perform almost the same.
It is noticed that the lengths of the confidence intervals for the panel data model is
comparable with those given in Brown and Newey (2002). Finally, we observed that as the
sample size increases the EL interval I a improves both in its coverage and length, whereas
the improvement on the Bartlett intervals and the bootstrap interval is more in reducing
the length of the intervals.
5. Discussions
The main finding of the paper is that the EL with general moment restrictions are
Bartlett correctable. This is a substantial extension of the previously established cases of
Bartlett correction of EL, including the case of smoothed functions of means by DiCiccio
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Table 3
Empirical coverage (in percentage) and averaged length of the EL confidence interval I a , the theoretical Bartlett
corrected (TBC) interval I tbc by using the true Bartlett factors (with one moment restriction only), the empirical
Bartlett corrected (EBC) EL interval I a;bc and the direct Bootstrap (BT) calibrated confidence interval I a;bt for the
panel data model (4.1) with y ¼ 0:5
Nominal level
90%
sample size
EL
95%
EBC
ðaÞ With two moment restrictions
50
coverage
81.25
length
0.917
100
coverage
84.50
length
0.706
Nominal level
90%
sample size
EL
ðbÞ With one moment
50
coverage
length
100
coverage
length
restriction
85.70
1.382
87.50
1.162
87.30
1.325
89.90
0.875
BT
EL
89.20
1.384
89.80
0.894
EBC
87.10
1.088
90.20
0.818
BT
93.40
1.425
93.80
1.065
94.05
1.504
94.30
1.096
95%
TBC
88.50
1.457
89.75
1.182
EBC
88.75
1.525
88.90
1.175
BT
90.10
1.534
89.50
1.180
EL
TBC
92.25
1.574
89.75
1.302
EBC
94.10
1.631
94.75
1.320
93.82
1.625
94.30
1.335
BT
94.30
1.685
94.50
1.343
Table 4
Empirical coverage (in percentage) and averaged length of the EL confidence interval I a , the theoretical Bartlett
corrected (TBC) interval I tbc by using the true Bartlett factors (with one moment restriction only), the empirical
Bartlett corrected (EBC) EL interval I a;bc and the direct Bootstrap (BT) calibrated confidence interval I a;bt for the
panel data model (4.1) with y ¼ 0:9
Nominal level
90%
sample size
EL
95%
EBC
ðaÞ With two moment restrictions
50
coverage
80.50
length
1.596
100
coverage
81.40
length
1.540
Nominal level
90%
sample size
EL
ðbÞ With one moment
50
coverage
length
100
coverage
length
restriction
86.50
1.849
87.25
1.808
88.45
1.819
89.20
1.623
BT
EL
89.30
1.823
88.80
1.656
EBC
87.30
1.765
89.20
1.726
BT
93.80
1.950
93.50
1.905
94.40
1.936
94.60
1.913
95%
TBC
90.20
1.889
90.20
1.829
EBC
90.30
1.894
90.10
1.820
BT
89.40
1.876
89.40
1.813
EL
92.30
1.923
93.00
1.910
TBC
95.10
1.940
94.80
1.928
EBC
94.70
1.934
94.30
1.920
BT
94.80
1.939
94.70
1.925
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et al. (1991) and the linear regression by Chen (1993). It shows that the Bartlett property of
the EL is still preserved even in the case of over-identification. Although the Bartlett factor
can admit a rather involved expression with over-identified moment restrictions, proving
that the EL is Bartlett correctable provides the theoretical foundation to the proposed
easily implementable empirical Bartlett correction and bootstrap calibration of the EL
ratio.
Although we have focused on the coverage accuracy of the EL confidence regions, the
results of this paper have implications on hypothesis tests. For testing the simple
hypothesis, H0 : y ¼ y0 , a size a EL test rejects H0 if rðy0 ÞXca . The Bartlett corrected EL
test rejects H 0 if rðy0 ÞXb^ c ca . The latter test has more accurate size approximation than the
^ can be used to test over-identification
original EL test. The maximum EL ratio ‘ðyÞ
restrictions EfgðX ; yÞg ¼ 0, as proposed in Qin and Lawless (1994) and Kitamura (2001),
which mirrors the GMM test of Hansen (1982). The expansion given in (2.8) would be
useful in studying the second-order properties of the EL over-identification restrictions
test.
The use of the bootstrap to carry out the Bartlett correction empirically is due to a rather
involved expression for the Bartlett factor. Although it may be expected that the direct
bootstrap calibration would give the same effect as the Bartlett correction, the justification
of the direct bootstrap method inevitably needs those cumulants and the Edgeworth
expansions established in this paper.
The results established in Theorems 1 and 2 can be extended to independent but not
identically distributed samples, for instance, those arisen in a regression study. We need to
modify a, b and g as follows:
!
n
n
k j
X
X
q G ð0; y0 Þ
j
j
aj1 ...j k ¼ n1
E½wi1 ðy0 Þ . . . wi k ðy0 Þ; bj;j1 ...j k ¼ n1
E
and
qZj1 . . . qZjk
i¼1
i¼1
!
n
l j
m k
t p
X
1
q
w
ðy
Þ
q
w
ðy
Þ
q
w
ðy
Þ
0
0
i 0
i
E
. . . : p1 i
gj;j1 ...j l ;k;k1 ...km ;...;p;p1 ...pt ¼
pt .
j1
jl
k1
km
n i¼1
qy
.
.
.
qy
qy . . . qy qy . . . qy
P
We need also to re-define V n as n1 ni¼1 VarfgðX i ; y0 Þg. These forms of a and V n
were employed in Chen (1993) to establish Bartlett correction for linear regression
where r ¼ p. Conditions (2.1) should be modified to reflect the independent but not
identically distributed nature of data. Similar conditions as those given in Theorem 20.6
of Bhattacharya and Rao (1976) are required. Then, it may be shown that Theorem 1
is true by employing Skovgaard (1981) on transformation of Edgeworth expansions.
Theorem 2 is then a consequence of Theorem 1 as the calculation of the cumulants
follows the same spirits given in Appendix A for independent and identically distributed
samples.
Acknowledgments
The authors thank two referees, Professor Gautam Tripathi and the Editor, Professor
Takeshi Amemiya, for valuable comments and suggestions which improve the presentation
of the paper. The research was supported by a National University of Singapore Academic
Research Grant, a National Science Foundation Grant (SES-0518904) and a RFDP of
China Grant (20020027010).
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Appendix A
We provide some technical details on the log EL ratio rðy0 Þ in Appendix A.2, the sign
root decomposition in Appendix A.3, and the proofs of the two theorems in Appendix A.4.
More details on these derivations can be found in Chen and Cui (2005).
A.1. Basic formulae
We first present some basic formulae which will be used throughout the derivations.
1
Recall that O ¼ ðokl Þpp ¼:UL1 where okl ¼ ukl l1
l . Since GðZÞ ¼ S QðZÞ where
0
1
!
0
0
OT
1 T
1
T
T
I þ S 12 ðS12 S 12 Þ S12 S 12 ðS 12 S 12 Þ
B
C
S1 ¼
¼ @ 0 I rp
0 A,
1 T
1
T
T
ðS 12 S 12 Þ S 12
ðS 12 S 12 Þ
O
0
OOT
it can be checked that
bj;k
j
qG ð0; y0 Þ
¼E
¼ djk
qZk
0
B1
1
A
B
C
B ¼: @ A ¼ S 1
0
Br
and
0
1
0
B
C
¼ @ A2 A.
OA1
Here AT ¼ ðA1 ; . . . Ar ÞT ¼:ðAT1 ; AT2 ÞT , where A1 ¼ ðA1 ; . . . ; Ap ÞT and A2 ¼ ðApþ1 ; . . . ; Ar ÞT
constitute a partition of A. Therefore for positive integers k and a,
Bk ¼ 0
for kpp;
Brþk ¼ okl Al
Bpþa ¼ Apþa
for apr p
and
for kpp.
ðA:1Þ
Let B1 ¼ ðB1 ; . . . ; Br ÞT and B2 ¼ ðBrþ1 ; . . . Brþp ÞT . Since SB ¼ ðAT ; 0Tp1 ÞT , it means that
B1 þ S 12 B2 ¼ A. As S12 ¼ ðgj;k Þrp and from (A.1) we have
gj;k Brþk ¼ Aj IðjppÞ,
(A.2)
where I is the indicator function. Since
ðBp;q ÞðrþpÞðrþpÞ ¼ S1
ðAij Þ
ðC i;l Þ
ðC i;l ÞT
0
!
,
(A.3)
we have S 21 ðBj;k Þrp ¼ ðC k;m ÞTpr and S21 ðBj;rþa Þrp ¼ 0: As S21 ¼ ðgj;k ÞT , these mean
gj;k Bj;l ¼ C l;k
for lpr and kpp and gj;k Bj;rþa ¼ 0.
(A.4)
Furthermore, (A.3) also implies the following which links the Bs;t system with the
Ajm - and the C j;m -systems:
0
1
ðBk;pþb Þ
ðBk;rþl Þ
ðBk;l Þ
B pþa;l
C
B ðB
Þ ðBpþa;pþb Þ ðBpþa;rþl Þ C
@
A
rþk;l
rþk;pþb
rþk;rþl
ðB
Þ ðB
Þ ðB
Þ
0
1
ðomk C pþb;m Þ
0
ðomk C l;m Þ
B
C
pþa;l
pþa;pþb
pþa;l C
ðA
Þ
ðA
Þ
ðC
Þ
¼B
ðA:5Þ
@
A.
ðokm ½onm C l;n Aml Þ ðokm ½onm C pþb;n Am;pþb Þ ðokm C m;l Þ
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In a similar fashion, we can establish the following links between bs;t and ðaj i ; gj;i Þ
systems where t and i contains either single or double superscripts:
bl;pþa;pþc ¼ ool ½gpþc;pþa;o þ gpþa;pþc;o ;
bpþa;pþb;pþc ¼ 2apþa;pþb;pþc ;
bl;pþa;pþn ¼ ool gpþa;on ;
bl;pþm;pþc ¼ ool gpþc;om ,
bpþa;pþm;pþc ¼ gpþc;pþa;m þ gpþa;pþc;m ,
bl;pþm;pþn ¼ 0,
bpþa;pþb;pþn ¼ gpþa;n;pþb þ gpþa;pþb;n ;
bpþa;pþm;pþn ¼ gpþa;mn ,
brþk;pþa;pþc ¼ 2oko aopþa;pþc oko ono ½gpþc;pþa;n þ gpþa;pþc;n ,
brþk;pþa;pþn ¼ oko omo gpþa;mn oko ½go;n;pþa þ go;pþa;n ,
brþk;pþm;pþc ¼ oko ono gpþc;nm oko ½gpþc;o;m þ go;pþc;m ;
brþk;pþm;pþn ¼ oko go;mn .
ðA:6Þ
See Chen and Cui (2005) for details.
A.2. Derivations of (2.8) and (2.9)
We shall expand each term on the right of (2.7). By ignoring terms of Op ðn5=2 Þ, the first
term
"
#
j
n
n
2 j
3 j
T 1 X
j 1 X
k
k
l
k
l
m
qw
ðy
Þ
1
q
w
ðy
Þ
1
q
w
ðy
Þ
0
i 0 ^ ^
i 0
^ ¼ l^ n
l^ n
y y þ
y^ y^ y^
wi ðyÞ
wji ðy0 Þ þ i k y^ þ
k
l
k
l
m
2
6
qy
qy qy
qy qy qy
i¼1
i¼1
j
j k
j k
j k l
j k l
j k l m
¼ l^ Aj þ gj;k l^ y^ þ l^ y^ C j;k þ 12 gj;kl l^ y^ y^ þ 12 l^ y^ y^ C j;kl þ 16 gj;klm l^ y^ y^ y^ .
Similarly, the second term
n
X
T
l^ n1
^ i ðyÞ
^ T l^
wi ðyÞw
i¼1
^ j ^ h 1
¼ll n
n
X
i¼1
(
)
j
2 h
l
qw
ðy
Þ
1
q
w
ðy
Þ
0
0
i
wji ðy0 Þwhi ðy0 Þ þ whi ðy0 Þ i l y^ ½2; j; h þ wji ðy0 Þ
l
2
qy
qy qyk
j i
j i l
1 j i l k
¼ l^ l^ ðAji þ dji Þ þ l^ l^ y^ fðC j;i;l þ gj;i;l Þ½2; j; ig þ l^ l^ y^ y^ fðC j;i;lk þ gj;i;lk Þ½2; j; ig
2
j i l k
j;l;i;k
j;l;i;k
þ l^ l^ y^ y^ fðC
þg
g
j j
j i
j i l
j i l
j i l k
¼ l^ l^ þ l^ l^ Aji þ 2gj;i;l l^ l^ y^ þ 2C j;i;l l^ l^ y^ þ ðgj;i;lk þ gj;l;i;k Þl^ l^ y^ y^ þ Op ðn5=2 Þ.
For the third term
n
T
2 1 X
^ 3 ¼ 2 l^ j l^ i l^ h ðAjih þ ajih Þ þ 2l^ j l^ i l^ h y^ k gj;i;h;k þ Op ðn5=2 Þ.
n
½l^ wi ðyÞ
3
3
i¼1
Finally, n1
Pn
i¼1
T
^ 4 ¼ l^ j l^ i l^ h l^ g ajihg þ Op ðn5=2 Þ.
½l^ wi ðyÞ
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507
We then have for a; b; c; d 2 f1; 2; . . . ; r pg, f ; g; h; i; j 2 f1; 2; . . . ; rg, k; l; m; n; o 2
f1; 2; . . . ; pg and q; s; t; u 2 f1; 2; . . . ; r þ pg that
^ ¼ 2Bj Aj Bj Bj þ 2Bj;q Bq ðAj þ Bj Þ bj;uq Bu Bq ðAj þ Bj Þ
n1 lðyÞ
2Bj;u Bu;q Bq ðAj þ Bj Þ þ bu;qs Bj;u Bq Bs ðAj þ Bj Þ bj;uq bu;st Bq Bs Bt ðAj þ Bj Þ
Bj;uq Bu Bq ðAj þ Bj Þ þ 13 bj;uqs Bu Bq Bs ðAj þ Bj Þ þ 2bj;uq Bu;s Bs Bq ðAj þ Bj Þ
þ 2gj;k fBj;q Bq Brþk þ ½12 bj;uq Bu Bq Brþk þ Bj;u Bu;q Bq Brþk
12 bu;qs Bj;u Bq Bs Brþk bj;uq Bu;s Bq Bs Brþk þ 12 bj;uq bu;st Bq Bs Bt Brþk
þ 12 Bj;uq Bu Bq Brþk 16 bj;uqs Bu Bq Bs Brþk 12 bj;uq Bu Bq Brþk;s Bs ½2; j; r þ k
þ Bj;u Bu Brþk;q Bq þ 14 bj;uq brþk;st Bu Bq Bs Bt g
þ 2C j;k fBj Brþk Bj;q Bq Brþk ½2; j; r þ k þ 12 bj;uq Bu Bq Brþk ½2; j; r þ kg
þ gj;kl fBj Brþk Brþl þ Brþk Brþl Bj;q Bq ½3; j; r þ k; r þ l
12 Bj Brþk brþl;uq Bu Bq ½3; j; r þ k; r þ lg C j;kl Bj Brþk Brþl
þ 13 gj;klm Bj Brþk Brþl Brþm Bj;u Bu Bj;q Bq 14 bj;uq bj;st Bu Bq Bs Bt
þ bj;uq Bu Bq Bj;s Bs Bj Bi Aji þ Bj Bi;q Bq Aji ½2; j; i 12 bj;uq Bu Bq Bi Aji ½2; j; i
þ 2gj;i;l fBj Bi Brþl Bj Bi Brþl;q Bq þ 12 brþl;uq Bj Bi Bu Bq Brþl Bi Bj;q Bq ½2; j; i
þ 12 bj;uq Bu Bq Bi Brþl ½2; j; ig þ 2Bj Bi Brþl C j;i;l ðgj;i;lk þ gj;l;i;k ÞBj Bi Brþl Brþk
23 ajih Bj Bi Bh þ 2ajih Bj Bi Bh;q Bq ajih bj;uq Bu Bq Bi Bh 23 Ajih Bj Bi Bh
þ 2gj;i;h;k Bj Bi Bh Brþk 12 ajihg Bj Bi Bh Bg þ Op ðn5=2 Þ.
Applying (A.1) and (A.4), it may be shown that the third to the 18th terms on the righthand side cancel each other and the application of (A.4) simplifies the 20th term. Keeping
all the other terms, we have (2.8).
Now bringing in the expansion for ‘ðy0 Þ in (2.5) we have
n1 rðy0 Þ ¼ ðAj þ Bj ÞðAj þ Bj Þ Aji ðAj Ai Bj Bi Þ 2C j;k Bj Brþk 2gj;i;l Bj Bi Brþl
þ 23 ajih ½Aj Ai Ah þ Bj Bi Bh þ gj;kl Bj Brþk Brþl þ Aji ðAhi Aj Ah Bi;q Bq Bj ½2; i; jÞ
bj;uq ½C j;k Brþk Brþk;s Bs gj;k þ Bj;s Bs Aji Bi Bu Bq 2ajih Bj Bi Bh;q Bq
þ Bj;u Bj;q Bu Bq þ 2C j;k Bj;q Brþk Bq gj;kl Brþk Brþl Bj;q Bq
2gj;kl Bj Brþl Brþk;q Bq þ 2gj;i;l ðBj Bi Brþl;q Bq þ Brþl Bi Bj;q Bq ½2; j; iÞ
þ ð14 bj;uq bj;st 12 bj;uq brþk;st gj;k ÞBu Bq Bs Bt þ 12 gj;kl bj;uq Bu Bq Brþl Brþk
þ ðgj;kl brþl;uq gi;j;k bi;uq gj;i;k bi;uq ÞBu Bq Bj Brþk 2gj;i;h;k Bj Bi Bh Brþk
þ ðgj;i;lk þ gj;l;i;k ÞBj Bi Brþl Brþk 13 gj;klm Bj Brþk Brþl Brþm
þ 12 ajihg Bj Bi Bh Bg þ ðajih bh;uq gj;i;l brþl;uq ÞBj Bi Bu Bq
þ C j;kl Bj Brþk Brþl 2C j;i;l Bj Bi Brþl þ 23 Ajih ðBj Bi Bh þ Aj Ai Ah Þ
2ajih Agh Aj Ai Ag þ ajgf aihf Aj Ai Ah Ag 12 ajihg Aj Ai Ah Ag þ Op ðn5=2 Þ.
ðA:7Þ
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S.X. Chen, H. Cui / Journal of Econometrics 141 (2007) 492–516
Now the terms appeared on the third line of the above equation cancel each other. To
appropriate this, by applying the relationships implied by (A.5) together with (A.1) and (A.4),
bj;uq ½C j;k Brþk Brþk;s Bs gj;k þ Bj;s Bs Aji Bi Bu Bq
¼ ½bj;uq C j;k Brþk bl;uq gl;k Brþk;pþa Bpþa bl;uq gl;k Brþk;rþm Brþm
bl;uq Alpþa Bpþa bpþb;uq Apþbpþa Bpþa þ bl;uq Bl;pþa Bpþa
þ bpþb;uq Bpþb;pþa Bpþa þ bpþa;uq Bpþa;rþm Brþm Bp Bq
¼ ½bj;uq C j;k Brþk bl;uq ðBl;pþa Al;pþa ÞBpþa bl;uq C l;m Brþm
þ bl;uq ðBl;pþa Al;pþa ÞBpþa bpþb;uq Apþbpþa Bpþa þ bpþa;uq Bpþb;pþa Bpþa
bpþa;uq C pþa;m Brþm Bu Bq ¼ 0.
Applying again (A.5), we can express the terms appeared in the first two lines of (A.7) by
Al Al Akl Ak Al 2Al;pþa Apþa Al þ 23 aklm Ak Am Al þ 2okl C pþa;k Apþa Al
þ ½2akl;pþa gpþa;mn omk onl Apþa Ak Al þ 2½al;pþa;pþb gpþa;pþb;k okl Apþa Apþb Al
which leads us to (2.9).
A.3. Expansion for R3
We subtract Rl2 Rl2 from all the terms appeared in line 4 and below in (2.9). Fortunately
all the terms which do not have Al appeared cancel out with those appeared in Rl2 Rl2 .
Hence, the remaining terms can be written as 2Rl1 Rl3 .
The pursuit for an expression of R3 is done by repeatedly employing formulae (A.5) and (A.6)
as well as (A.1), (A.2) and (A.4). For instance, the terms appeared in the fourth line of (2.9)
Aji Ahi Aj Ah þ Bj;u Bj;q Bu Bq 2Aji Bi;q Bq Bj þ 2C j;k Bj;q Brþk Bq
¼ Akl Akm Am Al þ 2Akl Ak;pþa Apþa Al þ 2Al;pþa Apþa;pþb Apþb Al
þ Ak;pþa Al;pþa Ak Al þ Apþal Apþbl Apþa Apþb þ oml onl C pþa;m C pþb;n Apþa Apþb
þ onl okm C pþa;n C pþa;k Am Al 2omk C pþb;m Apþa;k Apþb Apþa
2omn okl C n;k C pþa;m Apþa Al 2okl C pþa;k Apþa;pþb Apþb Al
2oml okn C pþa;k C pþa;m An Al ,
and the terms in the fifth line
gj;kl Brþk Brþl Bj;q Bq 2gj;kl Bj Brþl Brþk;q Bq
¼ gm;kl Bpþa Brþk Brþl Bm;pþa gpþb;kl Bpþa Brþk Brþl Bpþb;pþa
gpþb;kl Brþk Brþl Brþn Bpþb;rþn 2gpþb;kl Bpþa Bpþb Brþl Brþk;pþa
2gpþa;kl Bpþa Brþl Brþm Brþk;rþm
¼ gm;kl okn olo ovm C pþa;v Apþa An Ao þ gpþb;kl okn olo Apþb;pþa Apþa An Ao
þ gpþb;kl okn olo omv C pþb;m An Ao Av 2gpþb;kl okn olo ovn C pþa;v Apþa Apþb Ao
þ 2gpþb;kl okn olo An;pþa Apþa Apþb Ao þ 2gpþa;kl okv oln omo C v;m Apþa An Ao ,
and so on for the other terms.
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509
Let us define
5 lkm nm k n
Rl31 ¼ 38 Alm Akm Ak þ 13 Alkm Ak Am 12
a A A A
5 knm lm k n
12
a A A A þ 49 alkn aomn Am Ak Ao 14 alknm Am Ak An ,
Rl32 ¼ Alk;pþa Apþa Ak þ Al;pþa;pþb Apþa Apþb 12 okn oml C pþa;km Apþa An
okl C pþa;pþb;k Apþa Apþb þ 12 okm C pþa;k Alm Apþa þ 12 Alk Ak;pþa Apþa
þ Alpþa Apþa;pþb Apþb þ 12 Alpþa Apþak Ak 12 okm onl C pþa;n C pþa;k Am
okl omn C n;k C pþa;m Apþa okl C pþa;k Apþa;pþb Apþb ,
Rl33 ¼ 12 ½gm;vo ovn ool okm 23 anml okm C pþa;k Apþa An þ 12 gpþb;ko okn ool omv C pþb;m An Av
þ ½12 gpþb;ko okn ool alnpþb Apþb;pþa Apþa An þ ovk ½onl ðgk;pþa;n þ gpþa;k;n Þ
alkpþb 12 gpþb;mn oml onk C pþa;v Apþa Apþb þ gpþa;ko oon oml okv C v;m Apþa An
þ ½12 gpþb;mk oml okn alnpþb An;pþa Apþa Apþb ,
Rl34 ¼ gpþa;pþb;n ono oml C o;m Apþa Apþb apþa;pþb;pþc Al;pþc Apþa Apþb
þ ½ðgpþc;pþa;n þ gpþa;pþc;n Þonl 2al;pþa;pþc Apþc;pþb Apþa Apþb
þ ðgpþc;pþa;o þ gpþa;pþc;o Þoon okl C pþc;k Apþa An alk;pþa Am;pþa Ak Am
þ apþa;pþb;pþc oml C pþc;m Apþa Apþb 23 alkm Am;pþa Apþa Ak
½32 ako;pþa þ 14 gpþa;mn omk ono Alo Apþa Ak 2ak;pþa;pþb Alpþb Apþa Ak
½am;pþa;pþb þ gpþa;pþb;k okm Alm Apþa Apþb ,
0
0
0
0
0
0 0
0
Rl35 ¼ ½12 alk;pþa amn;pþa 18 ol l om m on n ok k gpþa;l m gpþa;n k Am An Ak
0 0
0
0
þ ½2apþa;kf almf alkm;pþa 13 almn ðakn;pþa 12 gpþa;m n om k on n Þ
0
0
0
0
0 0
0 0
0 0 0
12 om m ok k ol l ðol v gpþa;m l gv;l k 13 gpþa;m k l ÞApþa Ak Am
12 ½3alk;pþa;pþb þ 23 aklv ðav;pþa;pþb 12 gpþa;pþb;n onv Þ
0 0
0
0
þ ðakvpþa 12 gpþa;mn omk onv Þðalvpþb 12 gpþb;m n om l on v ÞApþa Apþb Ak
þ ½al;pþa;f apþb;pþc;f al;pþa;pþb;pþc ðalk;pþc 12 gpþc;mn oml onk Þ
ðak;pþa;pþb gpþa;pþb;v Þovk Apþa Apþb Apþc
and
0
0
0
Rl36 ¼ f2al;pþa;f ampþbf þ almf apþa;pþb;f þ om m on l ½12 ook ovk gpþa;om gpþb;vn
0
0
0
0
0
12 ðgpþc;pþa;m þ gpþa;pþc;m Þðgpþc;pþb;n þ gpþb;pþc;n Þ
0
0
0
0 0
oko gpþb;n k ðgo;m ;pþa þ go;pþa;m Þ 12 apþa;pþb;pþc gpþc;m n
0 0
0 0
0
12 oko go;m n gpþa;pþb;k þ 12 gpþa;pþb;m n þ 12 gpþa;n;pþb;m gApþa Apþb Am .
It may be shown after some algebra that Rl3 ¼
P6
l
i¼1 R3i .
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510
A.4. Proof of the Theorems
The proof p
offfiffiffi Theorem 1 is divided into two parts. In the first part, we derive the
cumulants of nR. In the second part, we establish an Edgeworth expansion for the signed
root which then leads to an Edgeworth expansion for the EL ratio rðy0 Þ.
Cumulants of the signed root R: Since the cumulants of order higher than four are
of Oðn2 Þ or smaller, we only need to derive the first four cumulants. As the first and the
third cumulants are easier to derive than the second and the fourth, we present them first.
From (2.10) and (2.11), and the fact that Rj3 is the product of four zero-mean averages,
we have
EðRl1 Þ ¼ 0;
EðRl2 Þ ¼ n1 ml
and
EðRj3 Þ ¼ Oðn2 Þ,
where ml ¼ 16 n1 alkk . Therefore, the first-order cumulant is
cumðRl Þ ¼ n1 ml þ Oðn2 Þ.
(A.8)
The joint third-order cumulants
cumðRl ; Ro ; Rv Þ ¼ EðRl Ro Rv Þ EðRl ÞEðRo Rv Þ½3; l; o; v þ 2EðRl ÞEðRo ÞEðRv Þ
¼ EðRl1 Ro1 Rv1 Þ þ EðRl2 Ro1 Rv1 Þ½3; l; o; v EðRl2 ÞEðRo1 Rv1 Þ½3; l; o; v
þ Oðn3 Þ.
We note that
EðRl1 Ro1 Þ ¼ n1 dlo
and
EðRl2 Ro1 Þ ¼ n2 ½12 ðalokk dlo Þ alo;pþa;pþa þ 13 alkm aokm þ okl gpþa;o;pþa;k
þ ðalk;pþa 12 omk onl gpþa;mn Þaok;pþa þ ðal;pþa;pþb okl gpþa;pþb;k Þao;pþa;pþb .
Write Rl2 ¼ Rl21 þ Rl22 where Rl21 ¼ 12 Akl Ak þ 13 aklm Ak Am and Rl22 contains the rest of
the terms appeared in (2.11). We have
EðRl21 Þ ¼ 16 n1 alkk ,
EðRl22 Ro1 Rv1 Þ ¼ EðRl22 Þdov þ Oðn3 Þ ¼ Oðn3 Þ and
EðRl21 Ro1 Rv1 Þ ¼ n2 ð16 alkk dov 13 alov Þ þ Oðn3 Þ.
Thus,
EðRl2 Ro1 Rv1 Þ ¼ EðRl2 ÞEðRo1 Rv1 Þ 13 EðRl1 Ro1 Rv1 Þ þ Oðn3 Þ,
(A.9)
which means that
cumðRl ; Ro ; Rv Þ ¼ Oðn3 Þ.
(A.10)
To compute the second cumulants, we have to derive the expectation Rl2 Rl2 which
involves 21 terms. After deriving the expectation of each term and combine them as given
in Chen and Cui (2005), we have
EðRl2 Ro2 Þ ¼ n2 J lo þ Oðn3 Þ,
where J lo is defined in (3.1).
(A.11)
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511
We also need to compute EðRl2 Ro1 Þ þ EðRl3 Ro1 Þ. It may be shown that, with the remaining
terms of Oðn1 Þ,
lkm okm
1 lok kmm
n2 E½Rl31 Ro1 ¼ 58 alokk 38 dlo 29
a
72
a a
,
72 a
n2 EðRl32 Ro1 Þ ¼ 52 alo;pþa;pþa þ alk;pþa aok;pþa þ 12 alok ak;pþa;pþa þ al;pþa;pþb ao;pþa;pþb
þ 12 alo;pþa akk;pþa 12 oko oml gpþa;km;pþa okl gpþa;pþa;k;o
þ 12 okm ðgpþa;k;o alm;pþa þ gpþa;k;pþa alom Þ 12 onl oko gpþa;n;pþa;k
12 okm onl ðgpþa;n;m gpþa;k;o þ gpþa;n;o gpþa;k;m Þ þ alo;pþa apþa;pþb;pþb
okl omn ðgn;k;pþa gpþa;m;o þ gn;k;o gpþa;m;pþa Þ
okl ðgpþa;k;pþb ao;pþa;pþb þ gpþa;k;o apþa;pþb;pþb Þ,
n2 EðRl33 Ro1 Þ ¼ alk;pþa aok;pþa alopþb apþa;pþa;pþb þ 12 ovo onl okm gm;vn gpþa;k;pþa
13 okm gpþa;k;pþa aoml þ 12 oko onl gpþb;kn apþa;pþa;pþb
þ 12 okn ovl omo gpþb;kv gpþb;m;n þ 12 oko onl omv gpþb;kn gpþb;m;v
ovk alk;pþa gpþa;v;o þ ovk onl ðgk;pþa;n þ gpþa;k;n Þgpþa;v;o
þ 12 oml okn gpþa;mk aon;pþa þ ono oml okv gpþa;kn gv;m;pþa ,
n2 EðRl34 Ro1 Þ ¼ 4al;pþa;pþb ao;pþa;pþb 53 alom am;pþa;pþa 72 alk;pþa aok;pþa
alo;pþa akk;pþa alo;pþa apþa;pþb;pþb þ oml onk gk;m;o gpþa;pþa;n
þ onl ðgpþb;pþa;n þ gpþa;pþb;n Þao;pþa;pþb
þ ono okl ðgpþb;pþa;n þ gpþa;pþb;n Þgpþb;k;pþa þ oml gpþb;m;o apþa;pþa;pþb
14 omo onk gpþa;mn alk;pþa okm gpþa;pþa;k alom ,
n2 EðRl35 Ro1 Þ ¼ 12 alo;pþa akk;pþa þ alk;pþa aok;pþa 32 alo;pþa;pþa 13 alok ak;pþa;pþa
þ 16 onv alov gpþa;pþa;n 12 alk;pþa aok;pþa þ 14 omo onv alvpþa gpþa;mn
0
0
0
0
0
0 0
0
18 ol l om o on k ok k gpþa;l m gpþa;n k
0
0
0 0
38 omo onv om l on v gpþa;mn gpþa;m n ,
n2 EðRl36 Ro1 Þ ¼ 2alk;pþa aok;pþa þ 2al;pþa;pþb ao;pþa;pþb þ alok ak;pþa;pþb
0
0
0
0
þ alo;pþa apþa;pþb;pþb þ 12 om o on l omk ovk gpþa;mm gpþa;vn
0
0
0
0
0
0
12 om o on l ðgpþb;pþa;m þ gpþa;pþb;m Þðgpþb;pþa;n þ gpþa;pþb;n Þ
0
0
0
0
0
om o on l okm ðgm;m ;pþa þ gm;pþa;m Þgpþa;n k
0
0
0
0
0 0
0
0
0 0
12 om o on l apþa;pþa;pþb gpþb;m n 12 om o on l okm gm;m n gpþa;pþa;k
0 0
0
þ 12 om o on l ðgpþa;pþa;m n þ gpþa;n;pþa;m Þ,
n2 EðRl2 Ro1 Þ ¼ 12 ðalokk dlo Þ alo;pþa;pþa þ 13 alkm aokm þ al;pþa;pþb ao;pþa;pþb
þ alk;pþa aok;pþa þ okl gpþa;o;pþa;k 12 omk onl gpþa;mn aok;pþa
okl gpþa;pþb;k ao;pþa;pþb .
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In summary,
EðRl2 Ro1 Þ þ EðRl3 Ro1 Þ ¼ n2 K lo þ Oðn3 Þ;
ðA:12Þ
where K lo is defined in (3.2).
In light of (A.11) and (A.12), we have
cumðRl ; Ro Þ ¼ n1 dlo þ n2 Dlo þ Oðn3 Þ,
(A.13)
where
Dlo ¼ K lo ½2; l; o þ J lo ml mo .
(A.14)
The joint fourth-order cumulants of R is
cumðRl ; Rk ; Rm ; Rt Þ ¼ EðRl Rk Rm Rt Þ EðRl Rk ÞEðRm Rt Þ½3; l; k; m; t
EðRl ÞEðRk Rm Rt Þ½4; l; k; m; t þ 2EðRl ÞEðRk ÞEðRm Rt Þ½6
6EðRl ÞEðRk ÞEðRm ÞEðRt Þ
t
l k m t
l k m t
¼ EðRl1 Rk1 Rm
1 R1 Þ þ EðR2 R1 R1 R1 Þ þ EðR3 R1 R1 R1 Þ
t
l k
m t
þ EðRl2 Rk2 Rm
1 R1 Þ½6 EðR1 R1 EðR1 R1 Þ½3; l; k; m; t
t
l k
m t
EðRl2 Rk1 ÞEðRm
1 R1 Þ½12 EðR3 R1 ÞEðR1 R1 Þ½12
t
l
k m t
EðRl2 Rk2 ÞEðRm
1 R1 Þ½6 EðR2 ÞEðR1 R1 R1 Þ½4
t
l
k
m t
EðRl2 ÞEðRk2 Rm
1 R1 Þ½12 þ 2EðR2 ÞEðR2 ÞEðR1 R1 Þ½6
þ Oðn4 Þ.
ðA:15Þ
In the above, ½6 means six terms by choosing two superscripts from a total of four; and
½12 implies 12 terms by choosing l from the four superscripts first and then choosing k
from the rest of the three. From now on, we will just use, say, ½6 without the details of the
superscript rotations to save space.
From (A.9), we immediately have
t
k m t
k
m t
4
EðRl2 ÞfEðRk1 Rm
1 R1 Þ½4 þ EðR2 R1 R1 Þ½12 2EðR2 ÞEðR1 R1 Þ½6g ¼ Oðn Þ
which means the sum of the last three terms in (A.15) is negligible.
To facilitate easy expressions, let us define
t1 ¼ alkmn ;
t2 ¼ dlk dmn þ dlm dkn þ dln dkm ,
t3 ¼ alkm anoo þ alkn amoo þ almn akoo þ akmn aloo ,
t4 ¼ alko amno þ almo akno þ alno akmo ,
t5 ¼ akmn al;pþa;pþa þ almn ak;pþa;pþa þ alkn am;pþa;pþa þ alkm an;pþa;pþa ,
t6 ¼ alk;pþa amn;pþa þ almo;pþa akn;pþa þ aln;pþa akm;pþa .
It is relatively easy to show that
t
l k
m t
3
4
EðRl1 Rk1 Rm
1 R1 Þ EðR1 R1 ÞEðR1 R1 Þ½3; l; k; m; t ¼ n ðt1 t2 Þ þ Oðn Þ.
(A.16)
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Derivations in Chen and Cui (2005) show that
t
l k
m t
EðRl2 Rk1 Rm
1 R1 Þ½4 EðR2 R1 ÞEðR1 R1 Þ½12
¼ n3 fool ðgpþa;o;k amn;pþa þ gpþa;o;m akn;pþa þ gpþa;o;n akm;pþa Þ½4
0
0
0 0
6t1 þ 2t2 16 t3 þ 23 t4 ½om l on k gpþa;m n amn;pþa ½6g þ Oðn4 Þ,
ðA:17Þ
t
l k
m t
EðRl2 Rk2 Rm
1 R1 Þ½6 EðR2 R2 ÞEðR1 R1 Þ½6
0
0
0 0
¼ n3 f3t1 t2 þ 16 t3 59 t4 þ ½om m on l gpþa;m n akn;pþa ½6
12 ½ool ðgpþa;o;m akn;pþa þ gpþa;o;n akm;pþa Þ
þ ook ðgpþa;o;m aln;pþa þ gpþa;o;n alm;pþa Þ½6
0
0
0
0
0
0
þ ½ok l om k ðgpþa;m ;n gpþa;k ;m þ gpþa;k ;n gpþa;m ;m Þ½6
0 0
0
0
0
0
12 gpþa;m n ½ðool on k þ ook on l Þðom n gpþa;o;m þ om m gpþa;o;n Þ½6
0
0
0
0
0
0
0
0
0 0
þ 14 ½ol l om m ðon k ok n þ on n ok k Þgpþa;l m gpþa;n k ½6g þ Oðn4 Þ
ðA:18Þ
and
t
l k
m t
EðRl3 Rk1 Rm
1 R1 Þ½4 EðR3 R1 ÞEðR1 R1 Þ½12
0
0
0
0
0
0
¼ n3 f2t1 19 t4 þ ½on l ok k ðgpþa;k ;n gpþa;n ;m þ gpþa;n ;n gpþa;k ;m Þ½6
0 0
0
0
0
0
þ 12 ½gpþa;m n ðool on k þ ook on l Þðom n gpþa;o;m þ om m gpþa;o;n Þ½6
0
0
0
0
0
0
0
0
0 0
14 ½ol l om m ðon k ok n þ on n ok k Þgpþa;l m gpþa;n k ½6g þ Oðn4 Þ.
ðA:19Þ
Combining (A.16)–(A.19) it may be shown that
cumðRl ; Rk ; Rm ; Rt Þ ¼ Oðn4 Þ.
(A.20)
Edgeworth expansion for rðy0 Þ: We first derive an Edgeworth expansion for the
distribution of n1=2 R. Let kj be the jth order joint cumulant of n1=2 R. From (A.8), (A.13),
(A.10) and (A.20),
k1 ¼ n1=2 m þ Oðn3=2 Þ;
k3 ¼ Oðn3=2 Þ;
k2 ¼ I p þ n1 D þ Oðn3 Þ,
k4 ¼ Oðn2 Þ,
where I p is the p p identity matrix, m ¼ ðm1 ; . . . ; mp ÞT with ml ¼ 16 alkk and D ¼ ðDlo Þpp .
Let
Ū A ¼ ðA1 ; . . . ; Ar ; A11 ; . . . ; Arr ; A111 ; . . . ; Arrr ÞT ,
Ū C ¼ ðC 1;1 ; . . . ; C 1;p ; . . . ; C r;1 ; . . . ; C r;p ; C 1;1;1 ; . . . ; C r;r;p ÞT
and Ū ¼ ðU TA ; U TC ÞT is a vector of centralized means. From (2.10), (2.11) and the expansion
for R3 given in Appendix A.3, the signed square root n1=2 R can be expressed as a smooth
function of U, namely there exists a smooth function h such that n1=2 R ¼ hðŪÞ. We can
then use the results given in Bhattacharya and Ghosh (1978) to formally establish
Edgeworth expansion for the distribution of n1=2 R under Condition (2.1). In particular, let
B be a class of Boreal sets in Rp satisfying
Z
fðvÞ dv ¼ OðÞ; # 0,
sup
B2B
ðqBÞ
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where qB and ðqBÞ are the boundary of B and -neighborhood of qB, respectively.
A formal Edgeworth expansion for the distribution function of n1=2 R is
Z
1=2
supPðn R 2 BÞ pðvÞfðvÞ dv ¼ Oðn3=2 Þ,
B2B
B
where pðvÞ ¼ 1 þ n1=2 mT v þ 12 n1 fvT ðmmT þ DÞv trðmmT þ DÞg, fðvÞ is the p-dimensional
standard normal density, and trðÞ is the trace operation for square matrices.
Let H ¼ ðhij Þpp ¼:mmT þ D. By the symmetry of fðvÞ we have
Pf‘ðbÞoca g ¼ Pfðn1=2 RÞT ðn1=2 RÞoca g þ Oðn2 Þ
Z
¼
pðvÞfðvÞ dv þ Oðn2 Þ
1=2
kvkoca
(
)
Z
p
X
X
1
¼ Pðw2p oca Þ þ n1
hii ðv2i 1Þ þ
hij vi vj fðvÞ dv
1=2
2
kvkoca
iaj
i¼1
þ Oðn2 Þ
ðA:21Þ
¼ a Bc ca f p ðca Þn1 þ Oðn2 Þ,
Pp
Pp
where Bc ¼ i¼1 hii ¼ l¼1 ðml ml þ Dll Þ. We note in particular that the remainder term in
(A.21) is Oðn2 Þ rather than Oðn3=2 Þ. This is due to a fact that an even/odd order Hermit
polynomial is an even/odd function. This completes the proof.
Proof of Theorem 2. Based on the Edgeworth expansion given in (A.21),
Pf‘ðbÞoca ð1 þ n1 Bc Þg
¼ Pfw2p oca ð1 þ n1 Bc Þg Bc ca f p fca ð1 þ n1 Bc Þgn1 þ Oðn2 Þ
¼ a þ Oðn2 Þ
since the chi-square distribution satisfies
Pfw2p oca ð1 þ n1 Bc Þg ¼ Pðw2p oca Þ þ Bc ca f p ðca Þn1 þ Oðn2 Þ
and f p fca ð1 þ n1 Bc Þg ¼ f p ðca Þ þ Oðn1 Þ. This proves the theorem.
&
pffiffiffi
Proof of Theorem 3. Note that the bootstrap estimate for bc ¼ 1 þ Bc n1 is n-consistent.
Applying the delta method (Bhattacharya and Ghosh, 1978) on top of the derivation in the
proof of Theorem 2, we have
Pf‘ðbÞoca b^ c g ¼ Pf‘ðbÞoca ð1 þ Bc n1 Þg þ Oðn3=2 Þ
¼ a þ Oðn2 Þ þ Oðn3=2 Þ ¼ a þ Oðn3=2 Þ.
ðA:22Þ
p
ffiffi
ffi
We note here that the Oðn3=2 Þ term is entirely due to using a n-consistent estimator of
bc . This completes the first part of the theorem.
^ It can be shown that its expression is
Let R be the sign root decomposition of r ðyÞ.
given by replacing in R, the sign root of rðyÞ: (i) all the population moments by their sample
averages based on wn and (ii) all A, B and C’s by their corresponding quantities based on
the bootstrap resample fX i gni¼1 . As shown in Hall (1992), the cumulants of R have forms
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515
which essentially replace all those population moments by their sample counterparts based
on the original sample w. These lead to, for any x 2 R,
^
jwn g ¼ Pðw2p ox Þ p1 B^ c x f p ðx Þ þ Op ðn2 Þ,
Pfr ðyÞox
(A.23)
where B^ c is a version of Bc by replacing all the population moments by their sample
counterparts. It may be shown that under the conditions of the theorem,
B^ c ¼ Bc þ Op ðn1=2 Þ.
^ ca jw g ¼ a, we can carry out a Cornish–Fisher type expansion of
As c^a satisfies Pfr ðyÞo^
n
c^a based on (A.23) and obtain
c^a ¼ ca ð1 þ Bc n1 Þ þ Op ðn3=2 Þ.
This together with the Edgeworth expansion for rðyÞ given in the proof of Theorem 1
implies that
PfrðyÞo^ca g ¼ PfrðyÞoca ð1 þ Bc n1 Þg þ Op ðn3=2 Þ
¼ a þ Oðn3=2 Þ
by repeating the derivation of (A.22). In the first equation above, we use again the delta
method whose validity is given in Bhattacharya and Ghosh (1978). &
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