Journal of Econometrics 90 (1999) 63—76
The second moment and the autocovariance function of
the squared errors of the GARCH model
Menelaos Karanasos*
Department of Economics, York University Heslington, York Y01 5DD, UK
Received 1 September 1996; received in revised form 1 January 1998
Abstract
Since Bollerslev and Taylor indepedently introduced the GARCH model almost
a decade ago many questions have remained unanswered. This paper addresses two of
them. First, ‘What is the autocovariance structure of the squared errors?’ and second,
‘What is the condition on the parameters of the GARCH (p, q) model in order for the
fourth moment of the errors to exist?’. In Section 2 of this paper we answer the first
question and in Section 3 we answer the second one. In a recent paper Ding and Granger
introduced an extension of the GARCH(1, 1) model which they called the N-component
GARCH(1, 1) model and they mentioned that it can be expressed as a GARCH(n, n)
model. This GARCH(n, n) representation is presented in Section 3. Finally, in Section 3,
we introduce the two component GARCH(n, n) model and we express it as a
GARCH(2n, 2n) model. 1999 Elsevier Science S.A. All rights reserved.
JEL classification: C22
Keywords: Autocovariance; GARCH model; N-component; Fourth moment
1. Introduction
Two of the most common empirical findings in the finance literature are that
the distributions of asset returns display tails heavier than those of the normal
distribution and the squared returns are highly serially correlated. The
aforementioned stylized empirical regularities led some econometricians to
develop models which can accommodate and account for these phenomena.
Engle (1982) introduced the Autoregressive Conditional Heteroscedasticity
* Correspoding author. E-mail: [email protected]
0304-4076/99/$ — see front matter 1999 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 8 ) 0 0 0 3 2 - 3
64
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
(ARCH) model and Bollerslev (1986) (hereafter B), and Taylor (1986), generalized the ARCH to the GARCH model:
N
O
e
&N(0, h ), h " e # b h !j.
RR\
R
R
R\G
H R
G
H
This paper presents exact formulae for the second moments of the squared
errors of the GARCH(p, q) model. B (1988), Ding and Granger (1996) (hereafter
DG) and Karanasos (1996) (hereafter K) give the autocorrelation function of the
squared errors for the GARCH(1, 1) process. In Section 2 we extend these
results to the GARCH(p, q) model.
Moreover, Milh+l (1985) in Theorem 3 of his paper, gives a necessary and
sufficient condition for the existence of the second-order moment of the squared
errors of the ARCH(p) model. B (1986) in Theorem 2 of his paper gives
a necessary and sufficient condition for the existence of the fourth moment of the
errors of the GARCH(1, 1), GARCH(1, 2) and GARCH(2, 1) models. In Section 3 of this paper we present a method for obtaining the unconditional fourth
moment, and hence the kurtosis coefficient of the errors of the GARCH(p, q)
model.
Furthermore, several empirical studies have pointed out the long memory
property (significant positive autocorrelation for many lags) of speculative
returns (e.g. Ding et al., 1993; DG, 1996). Motivated by this empirical result, DG
(1996) considered a new version of the GARCH(1, 1) model which they called
the N-component GARCH(1, 1) model. In this model the conditional variance
of the errors (h ) is a weighted sum of N components h (i"1, 2, N) with
R
GR
w (i"1, 2, N) as weights, respectively. Each component is a GARCH(1, 1)G
type specification. We use the exact form solutions of Sections 2 and 3 to
analyze the N-component GARCH(1, 1) and the two-component GARCH(n, n)
models. This is accomplished (i) by showing that the N-component
GARCH(1, 1) process corresponds to a GARCH(n, n) model, and that the
two-component GARCH(n, n) process is represented by a GARCH(2n, 2n)
model (Lemmas 3.1 and 3.2, respectively), and (ii) by applying the results for the
general GARCH(p, q) model to obtain the second moments of the squared
errors of the N-component GARCH(1, 1) and the two-component GARCH(n, n)
models. Finally, Section 4 concludes.
2. Autocovariance of the GARCH model
Let e follow a GARCH(p, q) process, given by
R
N
O
h "a # a e # b h Ne
R
G R\G
G R\G
R
G
G
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
where
65
NO
O
"a # ae #v ! b v
G R\G
R
G R\G
G
G
NO
O
N “ (1!a*¸)e"a #v ! b v ,
G
R
R
G R\G
G
G
a*Oa* for iOj,
H
a"a #b for i)p, q, a"a for i, p'q and a"b for i, q'p.
G
G
G
G
G
G
G
(2.1)
¹heorem 2.1. ¹he autocovariance function of the squared errors of the above
GARCH(p, q) process is given by
NOe z (2/3) f ,
G
GH GH
c"
H
NOe z (2/3) f ,
G
GH GO
where
j)q!1,
(2.2)
j*q,
(a*)H> NO\
G
e "
,
GH “ NO(1!a*a*)“ NO (a*!a*)
J
G J
II$G G
I
f "E(e),
R
(2.2a)
and
H O\J
O
z " b# b b [(a*)J#(a*)\J]
GH
I
I I>J G
G
I
J I
O O\J
# b b [(a*)J#(a*)J\H], b "!1.
I I>J G
G
JH> I
(2.2b)
Observe that, for j"q, the third term on the right-hand side of the above
equation disappears since the lower limit exceeds the upper limit of the double
summation and that the autocovariance function depends on the fourth moment of
the errors which we give in Section 3. 䊐
3. Fourth moment of the GARCH model
Let h follow a GARCH(p, q) process, given by
R
N
O
h "a # a e # b h .
R
G R\G
G R\G
G
G
(3.1)
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M. Karanasos / Journal of Econometrics 90 (1999) 63–76
¹heorem 3.1. ¹he fourth moment of the errors of the above GARCH(p, q) process is
given by f "3a f /(1!d ), where f , and d are given by
a
f "
,
1!N a !O b
G G
G G
N\
N>O\
"1#a #b # [a #a (a #b )] C
N
O
H
N N\H
N\H
H>O\G
H
G
O\
N>O\
# [b #b (a #b )] C
H
O O\H
O\H
HG
H
G
N\O
N>O\
O\N
O>N\
#b a
C
#a b
C ,
O
O>H
H>O\G
N
N>H
HG
H
G
H
H
(3.2)
(3.2a)
N\
d "3a#b#a b #a b # [a #a (a #b )]
N
O
O O
N N
H
N N\H
N\H
H
N>O\
O\
; (c a #b )C
# [b #b (a #b )]
G G\BG
G\BG H>O\G
H
O O\H
O\H
G
H
N>O\
N\O
N>O\
; (c a #b )C #b a
(c a #b )C
G G\BG
G\BG HG
O
O>H
G G\BG
G\BG H>O\G
G
H
G
O\N
N>O\
#a b
(c a #b
)C ,
(3.2b)
N
N>H
G G\BG
G\BG HG
H
G
where C is the ijth element of the matrix C"A\. A is a (p#q!2);
GH
(p#q!2) matrix and consists of four submatrices:
A"
A
A
.
A A
A is a (q!1);(q!1) matrix. Its ijth element is !(b #a #b ), where
G\H
G\H
G>H
b "0 for k'q or k(0, b "!1 and a "0 for k'p or k)0. A is
I
I
a (q!1);(p!1) matrix. Its ijth element is !a , where a "0 for k'p. A is
G>H
I
a (p!1);(q!1) matrix. Its ijth element is !b , where b "0 for k'q. A is
G>H
I
a (p!1);(p!1) matrix. Its ijth element is !(a #b #a ), where a "0
G\H
G\H
G>H
I
for k'p or k(0, a "!1 and b "0 for k)0 or k'q. Moreover, d "q!1
I
G
for i*q, zero otherwise, and c "3 for i*q, one otherwise. Finally, the condition
G
for the existence of the fourth moment is d (1. )
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
67
3.1. The N-component GARCH(1,1) model
In what follows, we examine the N-component GARCH(1, 1) process,
a special case of the GARCH model, introduced by DG (1996):
L
h " w h where h "a e #b h
#c a ,
R
G GR
GR
G R\
G GR\
G G
(3.3)
where c "1 and c "0 for i*2.
G
¸emma 3.1. ¹he above N-component GARCH(1, 1) process can be expressed as
a GARCH (n, n) process given by
L
L L J\
L\J\\H
J\
h " w a e # “
“ (b )(!1)J>w a e
R
G G R\
G G R\J
GH
H
H
H
G
G J H G G \>G $G H
J
L\J\H
J
L
L
“ ( b )(!1)J>h
# “
#w a “ (1!b ).
GH
R\J
G
J H GHGH\> H
G
(3.4)
Moreover, if in the N-component GARCH(1, 1) process, Eq. (3.3), the first
k (1)k(n) components are integrated of order one (b "1!a , i"1,2,k),
G
G
then the GARCH(n, n) process will still be stationary. 䊐
Once the N-component GARCH(1, 1) process is expressed as a GARCH(n, n)
process, Theorems 2.1 and 3.1, can be used to obtain the second moment and the
autocovariance function of the squared errors.
3.2. The two-component GARCH (n, n) model
We now consider another version of the GARCH model, the two-component
GARCH(n, n) model:
h "w h #w h , w "1!w ,
R
R
R
L
L
h "a # a e # b h
,
R
G R\G
G R\G
G
G
(3.5)
L
L
h " a e # b h
.
R
G R\G
G R\G
G
G
(3.5a)
68
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
¸emma 3.2. ¹he above two-component GARCH(n, n) model can be expressed as
a GARCH (2n, 2n) process (without loss of generality, we ignore the constant),
given by
L
L H
h " (w a #w a ) e ! (w a
b #w a
b )e
R
H
H R\H
H>\G G
H>\G G R\ H>
H
H G
L L
L H\
! (w a
b #w a
b )e
# [(b #b )
L>H\G G
L>H\G G R\L>H
H
H
H GH
H G
L L
!b b
]h ! b b
h
.
(3.6)
G H\G R\H
G L>H\G R\L>H
H GH
Moreover, if in the two-component GARCH(n, n) process, Eq. (3.5), one component, h or h , is integrated of order one (L b "1!L a ), then the
R
R
H GH
H GH
GARCH(2n, 2n) process, Eq. (3.6), will still be stationary. 䊐
Having expressed the two-component GARCH(n, n) model as
a GARCH(2n, 2n) model, we can use Theorems 2.1 and 3.1 to obtain the second
moment and the autocovariance function of the squared errors.
4. Concluding remarks
Since the observed volatility of an asset return is regarded as a realization of
an underlying stochastic process it is not surprising that so much effort has been
lavished on building models to measure and forecast it. The GARCH model and
its various generalizations have been very popular in this respect and have been
applied to various sorts of economic and financial data sets. However, there
have been relatively fewer theoretical advancements. This paper has contributed
to the theoretical developments in the GARCH literature. In Section 2 we
presented a method for calculating the autocovariance function of the squared
errors of the GARCH(p, q) model. Subsequently, in Section 3 we presented
a method for calculating the fourth moment of the errors of the GARCH(p, q)
model and we considered two extensions of the GARCH model. In particular,
we expressed the N-component GARCH(1, 1) model as a GARCH(n, n) process
and the two-component GARCH(n, n) model as a GARCH(2n, 2n) process.
Acknowledgements
I would like to thank an Associate Editor, and M. Karanassou for helpful
comments and suggestions.
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
69
Appendix A. Proof of Theorem 2.1
We prove the theorem by induction. If we assume that it holds for
a GARCH(p!1, q) process then it will be sufficient to prove that it holds for
a GARCH(p, q) process.
If e is a GARCH(p, q) process given by Eq. (2.1) (for simplicity we will assume
R
that p!1*q) then it can be written as an ARCH(1) process with
a GARCH(p!1, q) error term (x ) given by
R
e"a*e #x where
R
R\
R
N
O
“ (1!a*¸)x "a #v ! b v .
G
R
R
G R\G
G
G
(A.1)
Note that the GARCH(p, q) process is symmetric with respect to the p roots
a*, 2, a* . Hence, the autocovariance will also be symmetric with respect to the
N
p roots.
Case (i): 0)j)q!1. Since x is a GARCH(p!1, q) its autocovariance is
R
given by
cov (x )"+N eL z var(v )"N eL (z !l ) var(v ), if n*q,
L R
G GL GO
R
G GL GH
GH
R
N eL z
var(v )"N eL (z #z >) var(v ), if
R
G GL GH>@
R
G GL GH
GH@
n"j#b, q!1!j*b*1, N eL z var(v ), if
G GL GH
R
n"j, N eL z
var(v )"N eL (z #z \) var(v ), if
G GL GH\@
R
G GL GH
GH@
R
n"j!b, j*b*1,
(A.2)
where the eL and l are given by
GH
GH
(a*)H(a*)N\
G
G
,
eL "
GH “N (1!a*a*)“N
(a*!a*)
J
J G
II$G G
I
(A.2a)
O\H\
l "! b
[(a*)O\H\K!(a*)\O>K]
GH
O\K G
G
K
O\H\ O\H\I>
# b b [(a*)O\H\J>I!(a*)\O>J>I].
I O\J G
G
I
J
(A.2b)
Note that, when j"q!1, the second term in the above equation vanishes
since the lower limit exceeds the upper limit of the summation.
70
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
In addition, z is given by Eq. (2.2b) and z > is given by
GH
GH@
z
O\H\
T
" !b # b b
[(a*)LO\T\LH>@
GH@
O\T
J O\T\J
G
T
J
>
!(a*)O\H\T],
G
(A.3)
where n "1 for v)q!j!b!1, n "!1 for v*q!j!b, n "1 for
v)q!j!b!1, n "0 for v*q!j!b, 1)j)q!2, 1)b)q!1!j,
and z \ is given by
GH@
z
O>@\H>
T
" !b # b b
GH@
O\T
J O\T\J
T
J
\
;[(a*)O>@\H>T!(a*)LO\T\LH],
G
G
(A.4)
where n "1 for 0)v)q!j!1, n "!1 for v*q!j, n "1 for
0)v)q!j!1, n "0 for v*q!j, and 2)j)q!1, 1)b)j!1.
Note that in Eqs. (A.3) and (A.4), when v"0, the second summation term
vanishes since the lower limit exceeds the upper limit of the summation operator.
In Eq. (A.2) we express the z , z
and z
as functions of the z coeffiGO GH\@
GH>@
GH
cients.
The covariance between e and x
is given by
R
R\ H>T
H>T\
e "a*e #x " (a*) Gx # (a*) H>T>Gx
Ncov (e,
R
R\
R
R\G
R\H>T>G
R
G
G
H>T\
x
)" (a*)Gcov
(x )# (a*)H>T>G cov (x ).
(A.5)
R\H>T
H>T\G R
G R
G
G
Substituting Eq. (A.2) into the above equation, and after some algebra, we get
N
eL z
G GH (a*) H>T
cov (e , x
)" R R\ H>T
(a*!a*) G
G G
N
1
a*
G
#(a* ) H>T eL z
!
G GH 1!a*a* a*!a*
G G
G
N
(a*)O>H>T
(a*)T>H\O>!(a*)T>H\O>
! eL l
#n G
GO GH 1!a*a*
a*!a*
G
G
G
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
71
O\\H N
#(a*)T eL
z (a*)H>I
GH>I GHI> IT> G
H N
#(a*)T eL
z [n (a*)I#(a*)H\I]
GH\I GHI\ I G
TO\\H N
#(a*)T
eL
z [(a*)\I
GH>I GHI> I
G
#(a*)H>I] var(v ),
R
(A.6)
where n "1 for v'q!j!1, n "0, otherwise, and n "0 for k"j, n "1,
otherwise.
The jth autocovariance of e (0)j)q!1) is given by
R
e " (a*)Gx
Ncov (e)" (a*)Gcov(e, x
).
R\H
R\H\G
H R
R R\H\G
G
G
(A.7)
Substituting Eq. (A.6) into the above equation, and after some algebra, we get
where
N
cov (e)" e z #(a*) Hf var(v ),
H R
R
GH GH
G
(A.8)
1
H N
f"
eL
z [n (a*)I\H#(a*)H\I]
GH\I GHI\ 1!(a*)
I G
O\\H T N
# (a*)TeL
z [(a*)\I\H#(a*)H>I]
GH>I GHI> T I G
(a*)O\H O\\H N
# eL
z [(a*)\I\H#(a*) H>I]
GH>I GHI> 1!(a*)
I G
O\\H O\\H N
# (a*)TeL
z (a*)H>I
GH>I GHI> T IT> G
N
eL z
1
a*
G GH
G
#
!
1!(a*) 1!a*a* a*!a*
G
G
G
N
(a*)O
(a*)O\Ha*
G
! eL l
#
GO GH (1!a*a*)[1!(a*)] (a*!a*)(1!a*a*)
G
G
G
G
(a*)O\H>
!
,
(a*!a*)[1!(a*)]
G
where n "1 for k"j, n "0 otherwise.
(A.8a)
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M. Karanasos / Journal of Econometrics 90 (1999) 63–76
The autocovariance of the GARCH(p, q) process is equal to the sum of p terms.
The last term is the product of (a*)H and a coefficient which depends on
a*, 2, a*, b , 2, b . Each of the first p!1 terms is the product of (a*)H, where
N O
G
i"2, 2, p, and a coefficient which depends on a*, 2, a*, b , 2, b [e ) z ].
N O G GO
Given the symmetry of the autocovariance function in the inverse of the roots
(i-roots) of the autoregressive polynomial, if we interchange the i-root a* with the
G
first i-root a in the coefficient e z we will get the coefficient of (a*) H, i.e. the
G GO
coefficient f. Hence, f is given by
f"e z .
H
(A.9)
Finally, substituting Eq. (A.9) into Eq. (A.8) we get Eq. (2.2). 䊐
Case (ii): j*q. Since x is a GARCH(p!1, q) process its autocovariance is given
R
by
N eL z var(v )"N eL (z #l ) var(v ), 0)j)q!1,
R
G GH GO
GH
R
cov (x )" G GH GH
H R
N eL z var(v ),
j*q,
G GH GO
R
(A.10)
.where eL , l are given by Eqs. (A.2a), (A.2b) and z is given by Eq. (2.2b).
GH GH
GO
In Eq. (A.10) we expressed the z coefficients as functions of the z coefficients.
GH
GO
The covariance of x and e is given by
R
R\H
cov (x ,e )" (a*)Gcov(x , x
).
R R\H
G
R R\ H>G
G
(A.11)
Substituting Eq. (A.10) into the above equation we get
N
N
eL z
GH GO #n eL f var(v )
cov(x , e )" R R\H
GH GH
R
1!a*a*
G
G
G
O\
where f " l (a*a*)T\H
GH
GT G
TH
(A.12)
and n"0 for j*q, n"1 for 1)j)q!1.
The jth autocovariance of e is given by
R
H\
cov (e)" cov(x , e )(a*)J#(a*)Hvar(e)
H R
R\J R\H R
J
H
" cov(x , e )(a*) H\T#(a*) Hvar(e).
R R\T R
T
(A.13)
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
73
Substituting Eq. (A.12) into the above equation, and after some algebra, we get
N
a* H
cov (e)" "e z (a*)H 1! H R
G GO G
a*
G
G
O\ N
# (a*)H\TeL f var(v )#(a*) Hvar(e)
GT GT
R
R
T G
N
" e z var(v )#(a*)Hf, where
GH GO
R
G
(A.14)
O\ N
N
f"var(e)# (a*)\TeL f ! e z var(v ).
R
R
GT GT
G GO
T G
G
(A.14a)
We employ the same reasoning with the one used for the j)q!1 case to get
f"e z var(v ).
O
R
(A.15)
Finally, substituting the above equation into Eq. (A.14) we get Eq. (2.2). 䊐
Appendix B. Proof of Theorem 3.1
Using Eq. (3.1), the law of iterated expectations, and the following equations:
j "E(h e )"E(ee ), c "E(h h )"E(eh ),
G
R R\G
R R\G
G
R R\G
R R\G
E(eh )"E(h)"f /3
R R
R
(B.1)
we get
N
O
f
"a f # a j # b c ,
G G
GG
3
G
G
(B.2)
where
f
H\
c "a f #(a #b ) # (b #a )c
H
H
H 3
H\G
H\G G
G
O\H
N\H
# b c# a j,
H>G G
H>G G
G
G
(B.2a)
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M. Karanasos / Journal of Econometrics 90 (1999) 63–76
f
H\
j "a f #(3a #b ) # (b #a )j
H
H
H 3
H\G
H\G G
G
O\H
N\H
# b c# a j,
H>G G
H>G G
G
G
(B.2b)
where b "0 for k'q, and a "0 for 'k'p.
I
I
Note that in Eqs. (B.2a) and (B.2b) when j"1, the first summation terms become
zero, when j*q the second summation terms become zero, and when j*p the
third summation terms become zero, since the lower limit exceeds the upper limit of
the summations.
Substituting c and j into Eq. (B.2) we get
O
N
f
f
"a f [1#a #b ]# [3a#b#a b #a b ]
N
O
O
N N
O O
3
3 N
N\
O\
# [a #a (a #b )]j # [b #b (a #b )]c
G
N N\G
N\G G
G
O O\G
O\G G
G
G
N\O
O\N
#b a j #a b c .
O
O>G G
N
N>G G
G
G
(B.3)
The system of Eqs. (B.2a) and (B.2b) can be written in matrix form as A ) cN "B ) cN is
a (p#q!2);1 column vector given by cN "[c 2c j 2j ]. B is
O\ N\
a (p#q!2);1 column vector. Its i1th element is given by a f #f /3(a #b ) for
G
G
i)q!1, and its q!1#i, 1th element is given by a f #(3a #b )f /3 for
G
G i)p!1.
Solving this system of equations we can express the c ’s and the j ’s as functions of
G
G
f.
N>O\
f N>O\
c "a f C # (c a #b )C ,
HG 3
H
G G\BG
G\BG HG
G
G
(B.4a)
N>O\
f N>O\
j "a f " C
# (c a #b )!C
,
H
H>O\G 3
G G\BG
G\BG
H>O\G
G
G
(B.4b)
where d "q!1 for i*q, zero otherwise, and c "3 for i*q, one otherwise.
G
G
Substituting Eqs. (B.4a) and (B.4b) into Eq. (B.3) and solving for f we get
f "3a f /(1!d ). 䊐
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
75
Appendix C. Proof of Lemmas 3.1 and 3.2
Substitution of the h ’s into the h of Eq. (3.3) yields
GR
R
L
L
h " w a e # w b h
#w a .
R
G G R\
G G GR\
G
G
(C.1)
In the above equation, when we add and subtract sequentially the following
terms:
L
I
L\I\H
I
“
“ (b )w h
, 1)k)n!1, i "0,
GH G G R\I
G H GHGH\> GH$G H
(C.2)
we get Eq. (3.4). 䊐
The sum of the coefficients of the e ’s terms and of the h ’s terms
R\G
R\G
(i"1,2, 2, n) are
L
L
I
1! “
(1!b ) “ (a )w (1!a !b )(1
H
J G
G
G
GI> HI>H$G
J
(C.3)
We substitute Eq. (3.5a) into Eq. (3.5) and we get (for simplicity we will
assume that a "0)
L
L
L
h " (w a #w a )e #w b h
#w b h
.
R
G
G R\G
G R\G
G R\G
G
G
G
(C.4)
In the above equation, when we add and subtract sequencially the following
terms:
w b h
#w b h
, 1)k)n,
I R\I
I R\I
(C.5)
we get Eq. (3.6). 䊐
The sum of the coefficients of the e ’s and of the h ’s terms (i"1, 2, 2, 2n)
R\G
R\G
are
w
L
1! b
G
G
#w
L
L
L
a # b # b
H
H
G
H
H
G
L
L
L
L
1! b
a # b # b (1
G
H
H
G
G
H
H
G
(C.6)
76
M. Karanasos / Journal of Econometrics 90 (1999) 63–76
If L a #L b "1 (i.e. the first component is integrated of order one),
G G
G G
then the sum of the coefficients of e and h is
R\G
R\G
L
L
L
1!w a 1! a ! b (1 䊐
H
H
G
G
H
H
(C.7)
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