STUDENTIZATION IN EDGEWORTH EXPANSIONS FOR ESTIMATES OF SEMIPARAMETRIC INDEX MODELS * by Y Nishiyama and P M Robinson Department of Economics, London School of Economics Contents: Abstract 1. Introduction 2. Theoretical and Empirical Edgeworth Expansions 3. Proof of Theorems A and B Appendix A Appendix B References List of previous papers in this series The Suntory Centre Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics and Political Science Discussion Paper Houghton Street No.EM/99/374 London WC2A 2AE October 1999 Tel.: 0171-405 7686 * Research supported by ESRC Grant R000235892. The second author’s research was also supported by a Leverhulme Trust Personal Professorship. This paper has been prepared for a Festschrift volume in honour of Takeshi Amemiya. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Research Papers in Economics
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STUDENTIZATION IN EDGEWORTH EXPANSIONSFOR ESTIMATES OF
SEMIPARAMETRIC INDEX MODELS *
by
Y Nishiyama and P M RobinsonDepartment of Economics, London School of Economics
Contents: Abstract1. Introduction2. Theoretical and Empirical Edgeworth Expansions3. Proof of Theorems A and BAppendix AAppendix BReferencesList of previous papers in this series
The Suntory CentreSuntory and Toyota International Centresfor Economics and Related DisciplinesLondon School of Economics and PoliticalScience
Discussion Paper Houghton StreetNo.EM/99/374 London WC2A 2AEOctober 1999 Tel.: 0171-405 7686
* Research supported by ESRC Grant R000235892. The second author’s
research was also supported by a Leverhulme Trust Personal Professorship.This paper has been prepared for a Festschrift volume in honour of TakeshiAmemiya.
brought to you by COREView metadata, citation and similar papers at core.ac.uk
The summand of (C)" can be expressed as follows using (iii), , Lemma 3-(b)E(W12-
V1eit 2
nsV1)�0
and (3.17).
35
E(Wjk-
Vjeitb2) � { (t)}n&2E(W12
-
V1eit
2
ns(V1%V2)
)
� { (t)}n&2E W12-
V1eit
2V1
ns(eit
2V2
ns�1�it2V2
ns) �
2it
nsW12-
V1V2
.� { (t)}n&22it
nsE(W12
-
V1V2) � O (�t�2
nhd%2)
Thus, using (3.7) and (B.13),
(C ))) �{ (t)}n&2
s 3�8n(n�1)
n 5/2
2it
nsE(W12
-
V1V2) � O (t 2
nhd%2)
. (B.14)�{ (t)}n&2
s 3O (
�t�n
�t 2
n 3/2hd%2)
Similarly,
. (B.15)(C )))) �{ (t)}n&2
s 3O (
�t�n
�t 2
n 3/2hd%2)
By (B.12), (B.14) and (B.15),
(C ) � (C )) � (C ))) � (C ))))
.�{ (t)}n&3
s 3O
�t�3��t�n
�t 2
n 3/2hd%2�
t 6
n 3hd%2�
�t�5
n 5/2hd%2�t 4��t�3
n 2hd%2
(B.16)
Therefore, by (3.7), (B.1), (B.4), (B.11) and (B.16),
E(b )
2eitb2) � (A)�(B)�(C )
� �{ (t)}n&1 it2nE(W 2
12) � O (�t�
n 2hd%2�
t 2
n 3/2hd%2�
�t�h L
nh d%2)
� { (t)}n&14E(V 3
1 )�8E(W12V1V2)
n 1/2� O (
�t�n)
� { (t)}n&2(it)2
n 1/2{4E(V 3
1 ) � 8E(W12V1V2)} � O (t 2��t�3
n�
t 4
n 3/2)
.� { (t)}n&3 O�t�3��t�
n�
t 2
n 3/2hd%2�
t 6
n 3hd%2�
�t�5
n 5/2hd%2�t 4��t�3
n 2hd%2
(b) Writing, using (3.7) and Lemma 4 of Robinson (1995),
�E(b2meitBm)� C
n 3/2hd%2�m
j'1�E(Vje
itBm)�
36
�C
n 3/2 �n
j'1�m
k'1�E(djVke
itBm)� � �m
j'1�n
k'm%1�E(djVke
itBm)�
�C
n 5/2 �n&1
j'1�n
k'j%1�m
s'1�E(ejkVse
itBm)� � �m
j'1�n
k'j%1�n
s'm%1�E(ejkVse
itBm)�
�C
n 7/2 �n
j'1�n (j)
k<l�m
s'1�E(VjWklVse
itBm)�
� �n
j'1�m (j)
k'1�n (j)
l'k%1�n
s'm%1�E(VjWklVse
itBm)�
, (B.17)� �m
j'1�n&1 (j)
k'm%1�n (j)
l'k%1�n
s'm%1�E(VjWklVse
itBm)�
�E(VjeitBm)� � �E(Vje
it 2
nsVj)E{e
it( 2
ns
n
kújVk % b3&b3m% b2& b2m% b
)
3& b)
3m)
}�
(B.18) E�Vj�� (t)�m&1
for since is independent of .j�1,��� ,m , b3�b3m� b)
2�b)
2m� b)
3� b)
3m V1,��� ,Vm
For , and ,j m k m j� k
�E(djVkeitBm)�
� �E[djVkeit{ 2
ns(Vj%Vk%
n
l'm%1Vl)%b3&b3m%b2&b2m%b
)
3&b)
3m}
]��E{eit 2
ns
m
lúj,kVl}�
. (B.19) E�djVk�� (t)�m&2
For ,j � k m
�E(djVjeitBm)�
� �E[djVjeit{ 2
ns(Vj%
n
k'm%1Vk)%b3&b3m%b2&b2m%b
)
3&b)
3m}
]��E{eit 2
ns
m
kújVk}�
. (B.20) E�djVj�� (t)�m&1 E�djVj�� (t)�m&2
For and ,j m k� m�1
�E(djVkeitBm)�
� �E[djVkeit{ 2
ns(Vj%
n
l'm%1Vl)%b3&b3m%b2&b2m%b
)
3&b)
3m}
]��E{eit 2
ns
m
lújVl}�
. (B.21) E�djVk�� (t)�m&1 E�djVk�� (t)�m&2
For and , similarly to (B.21),j� m �1 k m
(B.22)�E(djVkeitBm)� E�djVk�� (t)�m&2
37
Therefore, by (B.19)-(B.22) and Lemma 4, for all j, k,
(B.23)�E(djVkeitBm)� E�djVk� � (t)�m&2
Similarly to the derivation of (B.23), for any j, k, l, s,
, (B.24)�E(ejkVseitBm)� E�ejkVs� � (t)�m&3
. (B.25)�E(VjWklVseitBm)� E�VjWklVs� � (t)�m&4
Substituting (B.18), (B.23)-(B.25) into (B.17), using ,� (t)� 1
�E(b2meitBm)� C � (t)�m&4×
1
n 3/2hd%2�m
j'1E�Vj� �
1
n 3/2 �n
j'1�m
k'1E�djVk� � �
m
j'1�n
k'm%1E�djVk�
�1
n 5/2 �n&1
j'1�n
k'j%1�m
s'1E�ejkVs� � �
m
j'1�n
k'j%1�n
s'm%1E�ejkVs�
�1
n 7/2 �n
j'1�n (j)
k<l�m
s'1E�VjWklVs� � �
n
j'1�m (j)
k'1�n (j)
l'k%1�n
s'm%1E�VjWklVs�
. (B.26)� �m
j'1�n&1 (j)
k'm%1�n (j)
l'k%1�n
s'm%1E�VjWklVs�
The summations in the square brackets have the following bounds.
by Lemma 1-(d) of NR. (B.27)�m
j'1E�Vj� C m
(B.28)�n
j'1�m
k'1E�djVk� � �
m
j'1E�djVj� � �
n
j'1�m (j)
k'1E�djVk�
by Lemma 4. C (m � mn)
by Lemma 4-(a). (B.29)�m
j'1�n
s'm%1E�djVs� C mn
�n&1
j'1�n
k'j%1�m
s'1E�ejkVs� � �
n&1
j'1�n
k'j%1�m (j,k)
s'1E�ejkVs� � �
m
j'1�n
k'j%1E�ejkVj�
� �m&1
j'1�m
k'j%1E�ejkVk�
(B.30) C (mn 2 � mn � m 2)h &2
38
by Lemma 6, denoting summations excluding�s
(i1,i2,@@@ ,ir) s�i1,i2,��� ,ir .
�m
j'1�n
k'j%1�n
s'm%1E�ejkVs� � �
m
j'1�n
k'j%1�n (j)
s'm%1E�ejkVs� � �
m
j'1�n
k'j%1E�ejkVk�
by Lemma 6. (B.31) C (mn 2 � mn)h &2
�n
j'1�n (j)
k<l�m
s'1E�VjWklVs�
� �n
j'1�n (j)
k<l�m
s'1
(j,k,l)
E�Vj�E�Wkl�E�Vs� � �m
j'1�n (j)
k<lE�V 2
jWkl�
� �n
j'1�m (j)
k<lE�VjWklVk� � �
n
j'1�m (j)
k<lE�VjWklVl�
(B.32) C (mn 3 � mn 2 � m 2n)h &1
by (iii), Lemma 1-(d) of NR, Lemma 4 of Robinson(1995) and Lemma 5.
�n
j'1�m (j)
k'1�n (j)
l'k%1�n
s'm%1E�VjWklVs�
� �n
j'1�m (j)
k'1�n (j)
l'k%1�n
s'm%1
(j,l)
E�Vj�E�Wkl�E�Vs�
� �n
j'm%1�m (j)
k'1�n (j)
l'k%1E�V 2
jWkl� � �n
j'1�m (j)
k'1�n (j)
l'k%1E�VjWklVl�
(B.33) C (mn 3 � mn 2)h &1
by (iii), Lemma 1-(d) of NR, Lemma 4 of Robinson (1995) and Lemma 5.
�m
j'1�n (j)
m<k<l�n
s'm%1E�VjWklVs�
� �m
j'1�n (j)
m<k<l�n
s'm%1
(k,l)
E�Vj�E�Wkl�E�Vs� � �m
j'1�n (j)
m<k<l(E�VjWklVk� � E�VjWklVl�)
(B.34) C (mn 3 � mn 2)h &1
by (iii), Lemma 1-(d) of NR, Lemma 4 of Robinson (1995) and Lemma 5. Therefore, substituting
(B.27)-(B.34) into (B.26), using ,1 m n�1
39
�E(b )
2meitBm)� C � (t)�m&4 m
n 3/2hd%2�
mn
n 3/2�
mn 2
n 5/2h 2�
mn 3
n 7/2h
, (B.35)C1 m
n 1/2h 2� (t)�m&4
the third term in parentheses dominating for sufficiently large n by assumption (ix).
(c) Using (3.7) and Lemma 4 of Robinson(1995), we write
�E(b )
3meitBm)�
C1
n 5/2hd%2�m
l'1�n
s'l%1�E(Wlse
itBm)�
�1
n 5/2 �m
j'1�n
l<s�E(djWlse
itBm)� � �n
j'm%1�m
l'1�n
s'l%1�E(djWlse
itBm)�
. (B.36)�1
n 7/2 �m
j'1�n
k'j%1�n
l<s�E(Wjk
-
WlseitBm)� � �
n
m<j<k�m
l'1�n
s'l%1�E(Wjk
-
WlseitBm)�
Similarly to (B.23)-(B.25), for all j, k, l, s,
, (B.37)�E(WlseitBm)� E�Wls�� (t)�m&2
, (B.38)�E(djWlseitBm)� E�djWls�� (t)�m&3
. (B.39)�E(Wjk-
WlseitBm)� E�Wjk
-
Wls�� (t)�m&4
Substituting (B.37)-(B.39) into (B.36), we have, due to ,� (t)� 1
�E(b )
3meitBm)�
C � (t)�m&4 1
n 5/2hd%2�m
l'1�n
s'l%1E�Wls�
�1
n 5/2 �m
j'1�n
l<sE�djWls� � �
n
j'm%1�m
l'1�n
s'l%1E�djWls�
.�1
n 7/2 �m
j'1�n
k'j%1�n
l<sE�Wjk
-
Wls� � �n&1
j'm%1�n
k'j%1�m
l'1�n
s'l%1E�Wjk
-
Wls�
40
Applying Lemma 4 of Robinson (1995), Lemmas 7 and 8, and (ix),
�E(b )
3meitBm)� C � (t)�m&4 mn
n 5/2hd%3�
mn 2
n 5/2h�
1
n 7/2
mn
hd%3�mn 3
h 3
� C m � (t)�m&4 1
n 3/2hd%3�
1
n 1/2h�
1
n 5/2hd%3�
1
n 1/2h 3
. C m
n 1/2h 3� (t)�m&4
(d) Write, using (3.7) and Lemma 4 of Robinson(1995),
E�b2m�2 C
1
n 3h 2d%4E��
m
i'1Vi�
2 �1
n 3E��
n
i'1�m
s'1diVs�
2 � E��m
i'1�n
s'm%1diVs�
2
�1
n 5E��
n&1
i'1�n
j'i%1�m
s'1eijVs�
2 � E��m
i'1�n
j'i%1�n
s'm%1eijVs�
2
�1
n 7E��
n
i'1�k<l
(i)�m
s'1ViWklVs�
2 � E��n
i'1�m
k'1
(i)
�n
l'k%1
(i)
�n
s'm%1ViWklVs�
2
. (B.40)� E��m
i'1�n&1 (i)
k'm%1�n (i)
l'k%1�n
s'm%1ViWklVs�
2
We show bounds only of some typical terms. Since is an iid sequence with zero mean, dueVi
to Lemma 1-(d) of NR, WritingE��m
i'1Vi�
2� m E�V1�2 C m .
, (B.41)E��n
i'1�m
s'1diVs�
2 C E��m
i'1diVi�
2 � E��m&1
i'1�m
s'i%1diVs�
2 � E��m
s'1�n
i's%1diVs�
2
the first term in parentheses is bounded by
(B.42)mE�d1V1�2 � m(m�1)E�d1V1�E�d2V2� C m 2
due to (iii) and Lemma 4-(b). Since and are iid with zero mean,di Vs
(B.43)E��m&1
i'1�m
s'i%1diVs�
2 � �m&1
i'1E(d 2
i) �m
s'i%1E(V 2
s) C m 2
by Lemma 1-(d) of NR and (A.2) under (i). Similarly, using Lemma 4-(a),
. (B.44)E��m
s'1�n
i's%1diVs�
2 �m
s'1�n
i's%1E(d 2
i)E(V2s) C m n
From (B.41)-(B.44),
.E��n
i'1�m
s'1diVs�
2 C (m 2 � mn)
41
Similarly,
E��m
i'1�n
j'm%1diVs�
2 � �m
i'1E(d 2
i) �n
s'm%1E(V 2
s) C mn .
We next consider
E��n&1
i'1�n
j'i%1�m
s'1eijVs�
2 C E��m&1
s'1�n&1
i's%1�n
j'i%1eijVs�
2 � E��m
i'1�n
j'i%1eijVi�
2
. (B.45)� E��m&1
i'1�m
s'i%1�n
j's%1eijVs�
2 � E��m&1
i'1�m
j'i%1eijVj�
2 � E��m&2
i'1�m&1
j'i%1�m
s'j%1eijVs�
2
Due to (iii), , and Lemma 6, the triple summation terms on theE(eij�i)�E(eij�j)�0 E(Vs)�0
right of (B.45) is . Using Lemma 6 and Hölder's inequality, the secondO (m 3 � m 2n � mn 2)h &d&4
term in (B.45) is
�m
i'1�n
j'i%1E(eijVi)
2 �2�m&1
i'1�m
k'i%1�n
j'k%1E(eijViekjVk)
. (B.46) C [mnE(e12V1)2 � m 2n{E(e13V1)
2E(e23V2)2}1/2] C m 2nh &d&4
Similarly, the fourth term of (B.45) is Using Lemma 5, as above, the termsO (m 3h &d&4) .
involving in (B.40) are so by (ix)ViWklVs O (m 4 � m 3n � m 2n 2 � mn 3)h &d&2 ,
E�b2m�2 C
nhd%2
m
n 2hd%2�
C
n 3(m 2 � mn)
�C
n 5(m 3 � m 2n � mn 2)h &d&4
�C
n 7(m 4 � m 3n � m 2n 2 � mn 3)h &d&2
. C m (1
n 3h 2d%4�
1
n 2)
(e) The derivation is similar using Lemma 4 of Robinson (1995), Lemmas 7 and 8. As in (d), we can
show
E�b )
3m�2 C
n 5h 2d%4mnh &d&2 �
C
n 5(m 3 � m 2n � mn 2)h &d&2
�C
n 7(m 4 � m 3n � m 2n 2 � mn 3)h &3d&6
. C m
n 4h 3d%6
42
(f) Write
. (B.47)�Eb ))
3 eitBm� � �EQ W e
itBm� �EQ1W eitBm� � �EQ2W e
itBm�
By (3.7),
�E(Q1W eitBm)�
C
n 7/2��n&1
j'1�n
k'j%1�n&1
l'1�n
s'l%1E{(Vj�Vk)Wjk�Vj
-
�Vk-
}WlseitBm�
6C
n 7/2�n&3
j'1�n&2
k'j%1�n&1
l'k%1�n
s'l%1E�{(Vj�Vk)Wjk�Vj
-
�Vk-
}Wls�� (t)�m&4
�6C
n 7/2�n&2
j'1�n&1
k'j%1�n
s'k%1E�{(Vj�Vk)Wjk�Vj
-
�Vk-
}Wks�� (t)�m&3
�C
n 7/2�n&1
j'1�n
k'j%1E �{(Vj�Vk)Wjk�Vj
-
�Vk-
}Wjk�� (t)�m&2
C n 1/2E�{(V1�V2)W12�V1-
�V2-
}W34�� (t)�m&4
�C
n 1/2E�{(V1�V2)W12�V1
-
�V2-
}W13�� (t)�m&3
. (B.48)�C
n 3/2E�{(V1�V2)W12�V1
-
�V2-
}W12�� (t)�m&3
Using (i), (iii), Lemmas 1-(d), 4 of NR, (A.1) and Lemma 4 of Robinson (1995), the first expectation
of (B.48) is bounded by
.C E{(�Y1���Y2��1)�W12�}E�W34� C h &2
Using (i), Lemmas 1-(d), 4 of NR, (A.1) and Lemma 4 of Robinson (1995), the second expectation of
(B.48) is bounded by
C E{(�Y1���Y2��1)�W12��W13�}
C E{(�Y1���Y2��1)�W12� E(�W13� �1)}
. C h &1E{(�Y1���Y2��1)(�Y1��1)�W12� }
Similarly to Lemma 1 and (A.5), so that the aboveE�Y1W12��E�Y21W12��E�Y1Y2W12� � O (h
&1)
quantity is . The third expectation of (B.48) is bounded byO (h &2)
43
C E�V1W2
12� � E�V1-
W12� C (h &d&2 � h &1) � O (h &d&2)
due to Lemmas 1-(d), 4 of NR, Lemma 4 of Robinson (1995) and Lemma 2. Therefore,
.�E(Q1W eitBm)� C(
n 1/2
h 2�
1
n 3/2hd%2)� (t)�m&4 C n 1/2
h 2� (t)�m&4
The second term of (B.47) is bounded by, using (3.7),
C
n 9/2��n
r'1�n&1(r)
j'1�n (r)
k'j%1�n&1
l'1�n
s'l%1E(VrWjkWlse
itBm)�
C
n 9/2�n&4
r'1�n&3
j'r%1�n&2
k'j%1�n&1
l'k%1�n
s'l%1E�VrWjkWls�� (t)�m&5
�C
n 9/2�n&3
r'1�n&2
j'r%1�n&1
k'j%1�n
s'k%1E�VrWjkWks�� (t)�m&4
�C
n 9/2�n&3
r'1�n&2
j'r%1�n&1
k'j%1�n
s'k%1E�VrWjkWrs�� (t)�m&4
�C
n 9/2�n&2
r'1�n&1
j'r%1�n
k'j%1E�VrWjkWjk�� (t)�m&3
�C
n 9/2�n&2
r'1�n&1
j'r%1�n
k'j%1E�VrWjkWrk�� (t)�m&3
C n 1/2E�V1�E�W23�E�W45�� (t)�m&5
�C
n 1/2(E�V1�E�W23W24� � E�V1W14�E�W23�)� (t)�m&4
�C
n 3/2(E�V1�E�W23�
2 � E�V1W23W13�)� (t)�m&3
C (n 1/2
h 2�
1
n 1/2h 2�
1
n 3/2hd%2)� (t)�m&5
by (i), (iii), Lemmas 1-(d), 4 of NR, Lemma 4 of Robinson (1995) and Lemma 1. Then apply (ix).
44
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