Multivariate ARCH Models: Finite Sample to an LM-Type Test

Multivariate ARCH Models: Finite Sample
Properties of QML Estimators and an Application
to an LM-Type Test
E M. I∗† and G D.A. P
†
University of Alicante
Cardiff University
April 2, 2004
Abstract
This paper provides two main new results: the first shows theoretically
that large biases and variances can arise when the QML estimation method is
employed in a simple bivariate structure under the assumption of conditional
heteroscedasticity; and the second examines how these analytical theoretical
results can be used to improve the finite sample performance of a test for
multivariate ARCH effects providing an alternative to a traditional Bartlett
type correction. We analyse two models: one proposed in Wong and Li (1997)
and another proposed by Engle and Kroner (1995) and Liu and Polasek (1999,
2000). We prove theoretically that a relatively large difference between the
intercepts in the two conditional variance equations which leads to the two
series having correspondingly different volatilities in the restricted case, may
produce very large variances in some QML estimators in the first model and
very severe biases in some QML estimators in the second. Later we use our
bias expressions to propose an LM type test of multivariate ARCH effects and
show through simulations that small sample improvements are possible when
we bias correct the estimators and use the expected hessian version of the test.
JEL-code: C13, C32.
Keywords: Multivariate GARCH, QML, Finite Sample Properties.
Corresponding author. Dpt. Fundamentos del Análisis Económico. University of Alicante.
Campus de San Vicente, 03080, Alicante, Spain. E-mail address: [email protected].
† Both authors thank H. Wong for providing us with the Gauss program to simulate the Wong
and Li (1997) model. We also thank the comments received at seminars given in Cardiff University,
University of Exeter and Michigan State University. We acknowledge gratefully as well the financial
support from an ESRC grant (Award number: T026 27 1238).
∗
1
1 Introduction
The multivariate-ARCH (autoregressive conditional heteroscedastic) model was first
introduced by Kraft and Engle (1983), and Bollerslev, Engle and Wooldridge (1988).
Since then, new combinations of this specification in the variance equation with different structures in the mean equation have been proposed: for example Baba, Engle,
Kraft and Kroner (1991), Harmon (1988), and Engle and Kroner (1995) introduced
the theoretical framework of simultaneous equation models and Calzolari and Fiorentini (1994) have considered some cases of non-linear simultaneous equations, while
Polasek and Kozumi (1996) proposed the VAR-GARCH structure.
The multivariate model implies that the conditional variance - covariance matrix
(H ) of the disturbances (ε ) depends on the information set (I −1 ). If we assume
normality in the conditional distribution (following Engle and Kroner (1995)), the
multivariate-GARCH (Generalised-ARCH) model can be written as:
t
t
t
εt /It−1
∼ N (0, H )
t
The main problem to be faced in this specification is the relatively large number
of parameters that are involved. There are, however, many possible parameterisations for H in order to reduce the number of parameters to estimate. We begin by
considering the “vech” (vec-half) representation, which in the simple bivariate case
becomes:
t
⎛ h ⎞ ⎛ c ⎞ ⎛ a a a ⎞ ⎛ ε2
⎞
11
01
11
12
13
1 −1
= ⎝ h12 ⎠ = ⎝ c02 ⎠ + ⎝ a21 a22 a23 ⎠ ⎝ ε1 −1 ε2 −1 ⎠
h22
ε22 −1
⎛ g c03g g ⎞a31⎛ ha32 a33⎞
11
12
13
11 −1
+ ⎝ g21 g22 g23 ⎠ ⎝ h12 −1 ⎠
t
vechHt
,t
t
,t
t
,t
,t
,t
,t
g31 g32 g33
h22,t−1
For the estimation of this model, it is necessary to restrict the number of parameters still further. Another possible specification is the diagonal representation, where
each element of the covariance matrix h is a function, only, of past values of itself
and past values of ε ε . In the case of the bivariate model, it becomes:
jk,t
⎛ h ⎞ ⎛ c ⎞ ⎛ a 0 0 ⎞ ⎛ ε2
⎞
11
01
11
1 −1
= ⎝ h12 ⎠ = ⎝ c02 ⎠ + ⎝ 0
a22 0 ⎠ ⎝ ε1 −1 ε2 −1 ⎠
h22
ε22 −1
⎛ g c03 0 0 ⎞0⎛ h0 a33⎞
11
11 −1
+⎝ 0
g22 0 ⎠ ⎝ h12 −1 ⎠
j,t
k,t
t
vechHt
,t
t
,t
t
,t
,t
,t
,t
0
0
g33
h22,t−1
The drawback is that we must still ensure that H is a positive definite matrix
for all values of the ε , and it can be a difficult task to check this in the previous
t
t
2
specifications. This is why Engle and Kroner (1995) proposed a new parameterisation: the BEKK (Baba, Engle, Kraft and Kroner (1991)) representation, where “...it
includes all positive definite diagonal representations, and nearly all positive definite
vech representations...” (Engle and Kroner (1995)). In the simple bivariate case, it
becomes:
Ht
= C0∗
a∗11 a∗12
a∗21 a∗22
C0∗ +
∗
∗
g11
g12
∗
∗
g21
g22
+
ε1,t−1ε2,t−1
ε21,t−1
ε1,t−1 ε2,t−1
ε22,t−1
∗
∗
g11
g12
∗
∗
g21
g22
Ht−1
a∗11 a∗12
a∗21 a∗22
Hence, this solution is not as restrictive as the diagonal representation, but “...comparing this model to the vech form of the model, we see that this model economises
on parameters by imposing restrictions both across and within equations...” (Engle
and Kroner (1995)).
Nowadays there exists an extensive literature about multivariate-ARCH models
that have been applied to different varieties of data. Most of them use (Quasi)Maximum Likelihood (QML-ML) as the estimation procedure. However, there are
relatively few theoretical papers that examine the consequences of doing that.
The relevant part of the (conditional) log-likelihood function in these models is
denoted by:
T
L (y, θ) =
Lt (yt , θ) =
=1
t
−
1
2
T
log |Ht | −
=1
t
1
2
T
(yt
−
µt ) Ht−1 (yt − µt )
=1
(1.1)
t
Liu and Polasek (1999) gave the following representation of the conditional information matrix of the ML estimator (I (θ)) in a general multivariate heteroscedastic
model:
I (θ) =
1
2
T
∂vechHt
∂θ
=1
t
T
+
=1
t
Ht−1
D
∂µt
∂θ
Ht−1
Ht−1 D
∂vechHt
∂θ
∂µt
∂θ
(1.2)
(see Liu and Polasek (1999), page 103), where µ = E (y /I −1) is an M 1 conditional
mean vector, H = var (y /I −1) is an M Mconditional variance matrix, D is the
M 2 M (M + 1) /2 duplication matrix and
indicates Kronecker product. They
argue that this formula corrects the initial work by Wong and Li (1997) who omitted
t
t
t
t
×
×
3
t
t
×
the last expression of the equation. However, Liu and Polasek (1999) themselves introduced an error when they applied this expression to a VAR(1)-VARCH(1) (VectorAR(1)-Vector-ARCH(1)) model, because they neglected the influence of changes in
the parameters in the variance equation on the own disturbance (see Liu and Polasek
(1999, page 105)), which illustrates the difficulties of dealing with these models from
the theoretical point of view.
Regarding asymptotic theory, Tuncer (1994, 2000), Bauwens and Vandeuren (1995)
and Comte and Lieberman (2003) have established the strong consistency of the
Quasi-Maximum Likelihood estimator (QMLE) in a simple multivariate-ARCH model.
Asymptotic normality is proved provided that the initial state is either stationary or
fixed. More recently, Ling and McAleer (2003) have shown the asymptotic normality
in a VARMA-GARCH model requiring only the existence of the second-order moment of the unconditional errors, and a finite fourth-order moment of the conditional
errors, which represents an important advance. On the other hand, Hafner (2000),
analyses the fourth moment in this model. However, these papers do not consider the
issue of the sample size needed in order to have confidence in the asymptotic result,
and in this paper we provide results which go some way towards addressing this.
In relation to finite samples, in a more recent paper, Liu and Polasek (2000)
have compared through Monte Carlo simulation the biases that are generated using
the Splus+GARCH program package of MathSoft (1996), the BASEL package of
Polasek (1999) and the application of the method of scoring for MLE using the exact
information matrix (given previously). The generated biases are seen to be striking,
and the Bayesian method seems to be the best alternative (there are in fact some
recent papers that analyse different types of Bayesian bivariate-ARCH models applied
to economic data, such as Osiewalski and Pipien (2004)). For a sample size of 200
observations, these results show the existence of severe biases, and this is precisely
what has motivated the work in our present paper. On the other hand, Wong and
Li (1997) reported through Monte Carlo simulation that in their model, the biases
in the parameters were very small (see Wong and Li (1997) pages 119-122). In this
paper we are interested in a further analysis of these two models.
It is interesting to note as well that in a recent paper, Jensen and Rahbek (2004)
prove how in univariate ARCH processes the QMLE is always asymptotically normal
providing that the fourth moment of the innovation process exists, whether or not
the process is stationary. This gives support to the estimation of ARCH processes
without being subject to contraints, and in this paper we carry out the estimation
through unrestricted QML.
The plan of the paper is as follows. In the next section we will begin analysing a
bivariate model under two specifications that have been proposed in the literature so
far: the one given in Wong and Li (1997), where they allow the two disturbances to
be dependent but not correlated, and the one proposed in Engle and Kroner (1995)
and Liu and Polasek (1999, 2000), where linear dependence between the disturbances
is introduced. We provide theoretical results of the O (T −1) biases for the QML esti4
mators in each specification under the assumption of conditional heteroscedasticity.
We impose the restriction that the variance parameters are zero; hence following an
approach that can be found in a number of other studies (see Engle, Hendry and
Trumble (1985) and Linton (1997)). In effect, we consider the case where conditional
heteroscedasticity is assumed when, in fact, it is absent. For easy of manipulation,
we assume as well that the intercept in the mean equation is known, although more
complicated structures could be analysed following the same methodology. In fact,
the results given in Iglesias and Phillips (2003) can be applied as well in this setting
so that we could allow for the inclusion of any number of exogenous variables in the
conditional mean equation. Although our theoretical results are obtained in a very
restricted model, we are able to prove how in the Wong and Li (1997) model the
variances for some estimators can be large when there is a relatively large difference
between the intercepts in the variance equations (they only showed results for cases
when the intercept parameters had very similar numerical values) and in the second
model which is examined in Liu and Polasek (1999, 2000) we show that a large difference in the intercepts under the null of no ARCH-effects (when the two series can
have very different volatitilies) can produce very large biases in some of the QMLEs.
We show as well theoretically how in the Wong and Li and Liu and Polasek models
some assumptions should be imposed for the QML estimator to be well-defined. We
provide evidence that the biases can be very different depending on both the structure we impose on the model, and on the combinations of the parameters we study.
We also analyse some invariance properties extending the Lumsdaine (1995) work
in univariate framework. Later, in Section 3, we show how the bias approximations
obtained in the null case can be used to improve the finite sample performance of a
test for multivariate ARCH effects. There are many papers that propose improving
the finite sample performance of LR tests by using Bartlett-type corrections (see for
example very recently Johansen (2002)); however, such a correction is not available
for the LM test. We show that, in the context of an LM test, the novel approach of
bias correcting the QML estimators may be a suitable alternative. Finally, Section 4
concludes.
2 A theoretical approach of a bivariate-ARCH model
2.1 Case 1: Allowing for dependent but uncorrelated disturbances
We begin by analysing the framework proposed in Wong and Li (1997) for the variance
equation, where the model is specified as:
yt = β + εt
5
(2.1)
where y
t = (y1t , y2t ) , εt = (ε1t , ε2t ) , E (εt ) = 0 , and we assume the intercept vector
β = (β 10β 20 ) to be known. We could allow for the estimation of the intercept and the
introduction of any number of exogenous variables in the mean equation by applying
directly the results of Iglesias and Phillips (2003). The conditional variance equation
follows the structure: Ht =
where:
h11t
0
0
h22t
2
h11t = E ε21t /It
1
h22t = E ε2t /It
1
−
−
2
= α0 + α1ε21
t−1
(2.2)
(2.3)
2
+ α2 ε2t−1
2
= γ 0 + γ 1 ε1t−1 + γ 2 ε2t−1
Expressions (2.2) and (2.3) can be re-written as:
ε21t = α0 + α1 ε21t
1+
α2ε22t
1+
η 1t
ε22t = γ 0 + γ 1 ε21t
1+
γ 2ε22t
1+
η 2t
−
−
−
−
where, due to the uncorrelatedness of the epsilons:
E (η 1t ) = E (η 2t ) = 0; E (η 1t η 2t ) = 0
E η 21t = E 2h211t ; E η22t = E 2h222t
After some algebra, we find:
E ε21t =
−
α0 (1 γ 2 ) + α2 γ 0
;
(1
γ 2 ) (1 α1 ) γ 1 α2
−
−
−
E ε22t =
−
γ 0 (1 α1 ) + γ 1 α0
(1
γ 2 ) (1 α1) γ 1 α2
−
−
−
From the above we may deduce the following restrictions on the variance equation
parameters:
γ 2 < 1; α1 < 1; (1
− γ 2) (1 − α1) − γ 1α2 > 0
Our objective is to analyse the QML biases of O (T 1) in this simple model. The
methodology we will use has been proposed by Cox and Snell (1968), where they
−
6
showed that for independent, butnot necessarily identically distributed observations,
the bias (b) of the MLE of β β reduces to:
bs = E
βs − β s
p
=
k k
si jl
i,j,l=1
kijl + kij,l
2
1
+
O T −2
(2.4)
, kijl = E
, kij,l = E
,
for s = 1, ..., p, where kij = E
for i, j,l = 1, ...,p (L denotes the Log-likelihood function). The total Fisher Information matrix and its inverse are defined by K = {−kij } and K −1 = {−kij } respectively.
The formula is valid, even for non-independent observations, provided that all k s are
of O (T ) (see Cordeiro and McCullagh (1991)), and this justifies the application of
the methodology in our case. Because under the null, QML is the same as ML, we
are, in fact, obtaining the biases of the two estimators in this case (under the null of
no-ARCH effects, the formula of McCullagh (1987) in terms of cumulants to capture
the bias of the QML estimator equals the one of Cox and Snell (1968)).
In order to proceed to obtain the expectations of the second and third order
derivatives, we can follow the matrix differential calculus techniques of Magnus and
Neudecker (1991). Liu and Polasek (1999) provided the expression of the conditional information matrix (I (θ)) of a general V AR (k) − V ARCH (q) model for
yt = (y1t, y2t, ..., yM t), by specialising (1.2):
∂2L
∂β i ∂βj
∂ 3L
∂βi ∂β j ∂β l
I (θ) =
I11 I12
I21 I22
∂2L
∂β i ∂βj
∂L
βl
with
I11 =
T
t=1
Wt Ht 1Wt, I21 = I12 = 0
−
and
I22 = 2
1
T
t=1
Vt D H t 1
−
Ht 1 DVt
−
where
Wt = (IM , Xt 1, ..., Xt k ), Vt = (IN , Zt 1,..., Zt q ),
Xt i = diag (y1t i, y2t i, ...,yM t i) , for
i = 1, ..., k ,
2
2
Zt j = diag ε1t j ,ε1t j ε2t j , ...εMt j , for j = 1, ..., q.
Note that IM and IN are an M × M and an N × N identity matrices respectively
and D is the duplication matrix defined in (1.2).
−
−
−
−
−
−
−
−
−
−
−
−
7
−
However, if we follow their analysis (see Liu and Polasek (1999, page 105)), the
previous expression is found to be in error because it neglects the dependence of the
disturbances in the variance equation with respect to the parameters in the mean
equation in the maximisation procedure. Their formula is valid only in the situation
where there are no parameters to estimate in the mean equation, which is precisely
our case. We extend the work by Liu and Polasek (1999) to include all the cumulants
we need for our analysis and Appendix 1 provides the expressions for the second
and third order derivatives of (1.1) in our model on applying the differential matrix
calculus.
Once we know the expressions of all the k components, we are in the position
to apply expression (2.4), obtaining the bias results and the variances (given by the
information matrix) as presented in the Theorem below.
Unfortunately, although Iglesias and Phillips(2003) were able to find a bias approximation for the variance parameter estimators in a univariate ARCH(1) model
without imposing restrictions, the additional complexity of the multivariate model
prevents similar derivations unless restrictions are imposed. To make progress under
the assumption that we specify the conditional variance structure given in (2.2) and
(2.3), we impose the restrictions that α1 = α2 = γ 1 = γ 2 = 0. This type of restriction, has been imposed in many of the theoretical analyses that have been carried
out in univariate ARCH models so far (eg. Engle, Hendry and Trumble (1985), Linton (1997)), and it facilitates, especially here, the analysis and the interpretation of
the results. It is to be noted that if, for example, it is demonstrated through the
bias approximations that severe biases and/or large variances are possible in the restricted model, then these characteristics will surely be found in unrestricted models.
Of course, if such problems do not arise in the restricted case the same may be true
in the unrestricted model but it need not be so. Hence care is needed in drawing
conclusions from the approximations.
T 2.1: If yt = εt where yt = (y1t , y2t ) , εt = (ε1t , ε2t ) is a vector of random
variables that has the structure given in (2.2) and (2.3), with α1 = α2 = γ 1 = γ 2 = 0,
then the biases and the variances of the QML estimators to order T
E (α 0 −
−
E (α1 −
E (γ 1 − γ 1 ) = o (T 1)
2
var (α0 ) = 4Tα0 + o (T 1 )
var (α1 ) = T1 + o (T 1 )
2
var (γ 1 ) = Tγα020 + o (T 1)
are given by:
1)
E (γ 0 −
1
E (α2 − α2 ) = o (T )
E (γ 2 − γ 2 ) = − T1 + o (T 1 )
2
var (γ 0 ) = 4Tγ 0 + o (T 1 )
2
var (α2 ) = Tαγ020 + o (T 1)
var (γ 2 ) = T1 + o (T 1 )
α0 ) = αT0 + o (T 1 )
α1 ) = T1 + o (T 1)
−
−
−
−
−
−
−
−
−
−
−
Proof.
1
−
γ 0 ) = γT0 + o (T
−
Given in Appendix 1.
Notice that the biases in the restricted model are relatively small suggesting that
estimation bias may not be a particular problem in this model. However, it is interesting to note how, when the intercept parameters α0 and γ 0 differ substantially,
8
the above model can generate severe and large variances in the QML estimators of
the α2 and γ 1 parameters (at least in one of them). In practical applications that
fit a model with this specification to real data, one should be mindful of this fact
when interpreting the estimation results. It is very easy to find an interpretation
in this situation: under the null of no-ARCH effects, the two intercepts reflect the
two unconditional volatilities of the two series. So our results show that the severe
variances result in this case when the two series have very different volatilities.
Table 2.1 shows the standard errors of O (T −1) , and a comparison with the simulated errors for different combinations of the intercepts of the conditional variance
equation, confirming the results shown previously.
Table 2.1: Approximate Standard Errors when we overspecify the multivariate
ARCH effects. T= 400.
α0 = 0.81
γ 0 = 0.04
α0 = 0.04
γ 0 = 0.04
α0 0.081 (0.082) 0.004 (0.004)
α1 0.050 (0.051) 0.050 (0.050)
α2 1.012 (1.035) 0.050 (0.051)
γ 0 0.004 (0.004) 0.004 (0.004)
γ 1 0.002 (0.002) 0.050 (0.051)
γ 2 0.050 (0.050) 0.050 (0.050)
Simulated values are given in brackets for 20000 replications
On the other hand, the bias and variances of the QML estimators to O (T −1) in
a univariate ARCH(1) model, E (ε2/I −1) = α1 + α2ε2−1, when nothing is estimated
in the mean equation while α2 = 0, are given by (see Engle, Hendry and Trumble
(1985) and Iglesias and Phillips (2003)):
t
t
t
E (α1 − α1 ) = αT1 + o (T −1 )
2
var (α1 ) = 3Tα1 + o (T −1 )
E (α2 − α2 ) = − T1 + o (T −1 )
var (α2 ) = T1 + o (T −1)
Comparing these biases with those of Theorem 2.1, it is seen that, in the new
bivariate specification, the biases in the parameters that are common have the same
structure while, on the other hand, there is a loss of estimation efficiency to O (T −1)
in the intercept parameter estimator, and no gain or loss in efficiency for the estimator
of the ARCH parameter.
Extending the work in Lumsdaine (1995), the representation of the relevant part
of the log-likelihood involves:
Lt = −
1
2
ε21t
ε22t
log h11t + log h22t +
+
h11t h22t
Using the same argument as the one given in Lumsdaine (1995, page 10), we can
prove that if α0 and γ 0 change in the same proportion, the biases and t-statistics
9
in α1, α2, γ 1 and γ 2 will remain invariant. This result matches with the bias and
variance results obtained in Theorem 2.1. However, if α0 and γ 0 vary in different
proportions, the invariance property does not hold.
2.2 Case 2: Allowing for dependent and correlated disturbances
We analyse now the variance-specification proposed by Engle and Kroner (1995) and
Liu and Polasek (1999,2000), given by the bivariate model:
(2.5)
yt = β + εt
where y = (y1 , y2 ) , ε = (ε1 , ε2 ) , E (ε ) = 0, and we assume again the intercept
vector β = (β10, β20) to be known. We allow for possible misspecification of the
marginal distribution of the errors; thus we allow in for the QML estimator to be
used. The variance representation implies a diagonal structure for the disturbances
following an ARCH(1) process:
h
h
11
12
where var (ε /I 1) = H = h h
and:
t
t
t
t
t
t−
t
t
t
t
t
t
21t
22t
⎛
⎞ ⎛ ⎞ ⎛
h
α
α
0
⎝ h ⎠=⎝ α ⎠+⎝ 0 α
11t
10
12t
20
h22t
11
22
α30
0
0
0
0
α33
⎞⎛
⎞
ε ,t
⎠ ⎝ ε ,t ε ,t ⎠
1
2
1
−1
−1
2
ε22,t
−1
(2.6)
−1
Then, it follows that:
E (ε1t ε2s /It
−1
)=
0
−
−
otherwise
E ε21t /It
−1
E ε22t /It
(2.7)
α20 + α22ε1t 1 ε2s 1 , t = s
−1
=
α10 + α11 ε21t
=
α30 + α33 ε22t
(2.8)
−1
−1
The unconditional expectations become E (ε21t) = 1 α10α11 , E (ε22t) = 1 αα3033 , E (ε1tε2t) =
α20 , and the unconditional correlation coefficient between both disturbances is
1 α√
22
α20 (1 α11 )(1 α33 )
. This implies the restriction that in this model, in order to guaran(1 α22 ) α10 α30
tee that the correlation coefficient is absolutely smaller than 1:
−
−
−
−
−
√
10
−
α220
<
α10 α30
0
When α11
=
< α11, α33 < 1.
α22
=
α33
= 0
− α22)2
(1 − α11 ) (1 − α33 )
(1
, then
α220
α10 α30
< 1. In addition, α10 , α30 > 0, while
Our objective is again to analyse the biases of O (T −1) in this simple model when
we use the QML estimation procedure, under the assumption that we specify a diagonal structure in the conditional variance, when, in fact, the true model is the one
for which we have α11 = α22 = α33 = 0.
Using again the methodology proposed by Cox and Snell (1968) (Appendix 2 gives
the expressions for the second and third order derivatives), the bias results are given
in Theorem 2.2.
T 2.2: If yt = εt where εt = (ε1t, ε2t) is a vector of random variables that
has the structure given in (2.6) with α11 = α22 = α33 = 0, then the biases and the
1
+2
variances of the QML estimators to order T
E (α10 − α10
var (α10 ) =
E (α20 − α20
var (α20 ) =
E (α30 − α30
var (α30 ) =
E (α11 − α11
var (α11 ) =
E (α22 − α22
var (α22 ) =
E (α33 − α33
var (α33 ) =
Proof.
+
+4
+
−
are given by:
α210 α30 (α210 α230 α210 α220 α220 α230 α10 α220 α30 −α420 )
)=
+ ( − )
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
α210 ( α310 α330 α210 α220 α230 α10 α420 α30 α620 )
+ ( − )
T (α310 α330 α210 α220 α230 α10 α420 α30 α620 )
α20 (α410 α230 α210 α430 α210 α220 α230 α420 α210 α420 α230 − α620 )
)=
+ (
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
(α310 α330 α210 α220 α230 α10 α420 α30 α620 )
+ ( − )
T (α210 α230 α10 α420 α30 α420 )
α10 α230 (α210 α230 α210 α220 α220 α230 α10 α220 α30 −α420 )
)=
+ ( − )
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
α230 ( α310 α330 α210 α220 α230 α10 α420 α30 α620 )
+ ( − )
T (α310 α330 α210 α220 α230 α10 α420 α30 α620 )
α10 α30 (α210 α230 α210 α220 α220 α230 α10 α220 α30 −α420 )
)=
+ ( −
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
α210 α230 (α10 α30 α220 )
+ ( − )
T (α310 α330 α210 α220 α230 α10 α420 α30 α620 )
(α410 α230 α210 α430 α210 α220 α230 α210 α420 α230 α420 − α620 )
)=
+ (
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
(α210 α230 α420 )
−
)
2
T (α10 α230 α10 α420 α30 α420 ) + (
α10 α30 (α210 α230 α210 α220 α220 α230 α10 α220 α30 −α420 )
)=
+ ( −
T (α210 α230 α10 α220 α30 α420 )(α10 α30 −α220 )
α210 α230 (α10 α30 α220 )
+ ( − )
3
3
T (α10 α30 α210 α220 α230 α10 α420 α30 α620 )
3
+13
+5
+
+10
+2
+5
+
+
+6
+
2
+4
+
+6
+5
+2
+4
+
+
+
+2
+4
+
3
+13
+10
+2
+5
+5
+
+
+
+2
−
+4
+
+3
+5
+5
+
+
− 2 + +4+6
+
+
o T 1
+4
+
+
+
+2
−
+4
+
+3
+5
+5
+
Given in Appendix 2.
11
o T 1
o T 1
+
2
o T 1)
−
o T 1
o T 1
o T 1
o T 1)
o T 1
+
2
o T 1)
−
o T 1)
o T 1
In spite of the large and tedious expressions we get, it is important to highlight
the utility we can get from them, because they enable us to find approximations
to the biases for any combination of parameters, and to discover their evolution.
Notice that all the coefficient bias approximations contain in the denominator the
term (α10α30 − α220) which is the determinant of the unconditional covariance matrix
of the disturbances. Hence one obvious situation in which large estimator biases are
to be expected is when there is high correlation between the disturbances. However,
the biases can still be large even when this correlation is relatively modest as will be
shown below. Thus we can provide theoretical support for the large biases found by
Liu and Polasek (2000) even though our analytical results are obtained under strong
restrictions. An additional use for the approximations is for bias correction under the
assumption of overspecification of the conditional process; we can use the expressions
for bias correction, substituting estimates for the true values of the expressions. The
direct applicability of the results for testing will be shown in the next section of the
paper.
We have noted that our theoretical analyis supports the results in Liu and Polasek,
in the sense that the biases can be very large in these models -even though our setting
is different-, but our findings provide evidence that when the disturbances are not
highly correlated, the biases are only so large for some combinations of parameters.
Table 2.2 shows how the larger biases are those for the parameters of the ARCHcomponents, especially when there is a large difference between the intercepts of
the two conditional variance equations. For example, the approximate bias of the
estimator of α22 increases from around -0.005 to -0.249 when the constant terms α10
and α30 change from being the same and equal at 0.15 to α10 being kept constant at
0.15 and increasing α30 to 15.
Table 2.2: Biases and variances of O (T −1) for some different parameter
configurations, α10 = 0.15, α20 = 0.05 and T=200.
E (α10 − α10)
var (α10)
E (α20 − α20)
var (α20)
E (α30 − α30)
var (α30)
E (α11 − α11)
var (α11)
E (α22 − α22)
var (α22)
E (α33 − α33)
var (α33)
α30 = 0.15 α30 = 15
0.00083
0.00032
0.00026
0.00013
0.00083
0.00032
-0.00553
0.00041
-0.00519
0.00347
-0.00553
0.00041
0.00083
0.00034
0.01246
0.01127
0.08322
3.37250
-0.00554
0.00498
-0.24921
0.00497
-0.00554
0.00498
Once we have found the bias expressions of O (T −1), we can again extend the
12
work by Lumsdaine (1995) to our model. In this case we need to change α10, α20 and
α30 in the same proportion to get invariance in the bias and t-statistics of α11 , α22
and α33. Otherwise, the invariance property becomes invalid (again, this is consistent
with the results in Theorem 2.2).
2.2.1
Special case when the correlation of the disturbances is overspecified
In this case, if we set α20 = 0, Theorem 2.2 now becomes:
C 2.1: If yt = εt where εt = (ε1t , ε2t ) is a vector of random variables
that has the structure given in (2.6) under overspecification of the conditional correlation (α20 = 0), then the biases and the variances of the QML estimators to order
T 1 are given by :
−
E (α10 − α10) = αT10 + o (T 1)
E (α20 − α20) = o (T 1)
E (α30 − α30) = αT30 + o (T 1)
2
var (α10 ) = 3αT10 + o (T 1 )
var (α20 ) = α10Tα30 + o (T 1)
2
var (α30 ) = 3αT30 + o (T 1 )
E (α11 − α11) = − T1 + o (T 1)
2
2
E (α22 − α22) = − 2αT10α+10αα3030 + o (T 1)
E (α33 − α33) = − T1 + o (T 1)
var (α11 ) = T1 + o (T 1 )
var (α22 ) = T1 + o (T 1 )
var (α33 ) = T1 + o (T 1 )
In the results given in Theorem 2.2, we set α20 = 0.
−
−
−
−
−
−
−
−
−
−
−
Proof.
−
The expression for the bias of α22 is now especially easy to interpret and it is easy
too to analyse the effect of a large distance between the two intercepts. On the other
hand, the bias and variances of the QML estimators in a univariate ARCH(1) model,
when nothing is estimated in the mean equation, were given at the end of Section
2.1. So we see that the effect of imposing a correlation between the disturbances,
when in fact it does not exist, again does not affect the bias structure, although on
the other hand, this time there is neither gain nor loss in efficiency to the order of
the approximation.
3 An LM type test allowing for bias correction in
the estimators
In this section, we examine how the biases of O (T 1) can be used to improve the
finite sample performance of a test for multivariate ARCH effects. We propose that
instead of improving the finite sample behaviour of the test by applying a Bartletttype correction (Bartlett (1937)) which, in any case is not available, we proceed by
bias correcting the estimates themselves. The justification for this is the following.
Let’s consider the LM test which takes the form (see Harvey (1989), page 169):
−
13
LM
= ( log
D
Ψ 0 )´ Ψ−1 log
L
I
0
D
L
Ψ 0
(3.1)
where log Ψ 0 is the vector of first order derivatives of the log likelihood
function evaluated under the null hypothesis, Ψ 0 is the vector of restricted estimates,
and Ψ0 is the estimated information matrix.
Harvey (1989) notes that log Ψ 0 ´≈ − Ψ∗ − Ψ 0 ´ 2 log (Ψ∗) where Ψ∗ is
the vector of unrestricted estimates; using this approximation we may write that:
D
L
I
D
LM
L
= ( log
D
D
Ψ 0 )´ Ψ−1 log
L
I
≈ Ψ∗ − Ψ 0 ´ 2 log (Ψ∗)
D
L
−1
Ψ0
I
0
D
D
L
Ψ 0
2 log L (Ψ∗)
L
Ψ∗ − Ψ 0
(3.2)
which has an asymptotically equivalent form given by:
Ψ
∗
− Ψ 0
−1
´Ψ Ψ
I
∗
0
− Ψ 0
.
(3.3)
In the case where there are no nuisance parameters and the null hypothesis is that
0 = Ψ∗, in which case the above statistic reduces to
0 : Ψ0 = 0 we see that Ψ∗ − Ψ
H
,
(Ψ∗)´ Ψ−1 (Ψ∗)
I
0
.
(3.4)
Our proposal is to use a bias-corrected estimate of Ψ∗, denoted by Ψ∗BC , in place
of Ψ∗ in the LM statistic. Since Ψ∗BC is second order efficient it is anticipated that the
statistics will converge to its limiting distribution faster so the size of the test in small
samples will be closer to its nominal level. If the bias correction is non-stochastic, then
the information matrix will be unchanged; this is the case for the situation we shall
consider below.Thus, if in (3.3) Ψ 0 is replaced by Ψ 0, the bias approximation for the
unrestricted estimate obtained when assuming the null is true, then Ψ∗ − Ψ 0 = Ψ∗BC
so that the above statistic is modified to:
(Ψ∗BC )´ Ψ−1 (Ψ∗BC )
I
0
(3.5)
Under the alternative, the bias correction that is used is incorrect so that as well as
improving the size some improvement in power seems likely. The statistic we actually
use is
14
−1
(D log L Ψ 0 )´IΨ0 D log L Ψ 0
(3.6)
which is asymptotically
equivalent to (3.5). To see this note that
0
D log L Ψ
≈ − Ψ − Ψ 0
∗
D2 log L (Ψ
∗
) = − (ΨBC )´D2 log L (Ψ )
∗
∗
On substituting from this approximation into (3.6) we may deduce that required
result.
So far we have assumed the absence of parameters not subject to test; however
the basic argument is unchanged when such parameters are present. We can choose
to ignore them and use the form of the test which tests only a subset of the complete
parameter vector, or we can include them in which case they too can be evaluated
at their bias corrected values. The argument for including them turns mainly on the
possibility that the power may be increased since the bias correction is invalid under
the alternative.
We show now in more detail through simulation how the bias-correction procedure works. For ease of application we use the Wong and Li model (Case 1 in the
previous section) as an example. In particular, since in the LM procedure estimation
is conducted only under the null, the bias approximations (which for the conditional
variance parameters are found only in the null case) can be employed directly since
they are non-stochastic and known.. As was seen in Theorem 2.1 bias approximations
were found for the constant terms in the variance equations (2.2) and (2.3); these are
nuisance parameters for the LM test on the variance parameters, since they are not
subject to the test. Bias corrected estimates for them are easily found. These bias
corrected estimates will be employed in the LM test. However, as has been noted
above, an additional use of the bias approximations for the conditional variance parameters in the null case can also be found. Rather than evaluate these parameters
as zero under the null, we may set them at the O (T 1) biases since the expected
values of the QML estimators are not zero but are close to the bias approximation.
This yields the test statistic given in (3.6). To analyse the effect of this use of the
bias corrections, we shall first conduct simulations with the bias corrected constant
terms in the LM while setting the parameters under test to zero. Then in further
simulations we both use bias corrected estimates for the constant terms and set the
parameters under test to their asymptotic bias values. Hence in this case we are
effectively testing a null under which the conditional variance parameters are equal
to the expected value of the QML estimator rather than zero.
In this case Ψ = (α0, α1, α2, γ 0, γ 1, γ 2) is the 6×1 vector of unknown parameters
whereas the null hypothesis we wish to test is
−
H0 = α1 , α2 , γ 1, γ 2 = 0
In what follows we shall consider three versions of the LM test statistic:
15
Model 1: The nuisance parameters are replaced with uncorrected QML estimates
and the parameters under test are set to zero (M1). This is the standard test statistic
given in (3.1).
Model 2: The nuisance parameters are replaced with bias corrected QML estimates with the parameters under test set to zero (M2). This case is considered in
this paper for comparison purposes.
Model 3: The nuisance parameters are replaced with bias corrected QML estimates and the parameters under test are set to their asymptotic bias values (M3).
This is the statistic given in (3.6).
There are several variants of the LM test and generally they differ only in the
estimator of the information matrix; see for example Amemiya (1985) and Dagenais
and Dufour (1991) for some related literature. We may distinguish three types of such
estimators; the Outer Product (OP) matrix of the score vector, the Hessian (HES)
matrix and the expectation of the Hessian (ExpHES) matrix. A non-operational
procedure which we shall examine for comparative purposes, uses the true Hessian
(TrueHES) where the actual values of unknown parameters are employed rather than
estimates. Each of these four variants of the LM test will be examined in the simulations in the contexts of Models 1-3.
The LM test based upon the expected Hessian is not always available since finding
the closed form solution for the expected Hessian may not be possible. In this case,
however, it is straightforward. Besides, to find the expected hessian for any higher
order specification of the Wong and Li (1997) model would be straightforward. From
Wong and Li (1997) we find on using (2.1), (2.2) and (2.3), that we may write:
log (Ψ) =
−
1−
−
1−
0
(Ψ) =
0
where:
− 1 = 1 2
=− = 1
On taking expectations through
(Ψ) we have:
⎛− − − 0 0 0
⎜⎜⎜ − − − 0 0 0
⎜
(Ψ)=⎜
⎜⎜⎜ −0 −0 − 0 −0 − 0 −0
⎜⎝ 0 0 0 − − −
0
0
0 −
−
−
D
T
L
hessian1
1
hessiani
dh
2h11t
t=1
Hessian
T
, ε21t
−
ε 21t
h11t
1
T
t=1
1
2
ε22t
h22t
1
dh,
2h22t
dh
hessian2
2ε2it
1
h2iit
hiit
2
1, ε2t−1
dhdh , i
,
Hessian
T
2α20
T
2α0
ExpHES
T γ0
2α20
T
2α0
3T
2
T γ0
2α0
T γ0
2α20
T γ0
2α0
3T γ 20
2α20
T
2γ 20
T α0
2γ 20
T
2γ 0
with inverse:
16
T α0
2γ 20
3T α20
2γ 20
T α0
2γ 0
T
2γ 0
T α0
2γ 0
3T
2
⎞
⎟⎟⎟
⎟⎟⎟
⎟⎟⎟
⎠
⎛ 4α
⎜⎜ −α T
⎜⎜ αT
(ExpHES (Ψ))−1 = ⎜
⎜⎜ T0γ
⎜⎜
⎝ 0
2
0
0
2
0
0
0
α0
T
1
T
α20
T γ0
0
0
0
0
0
0
⎞
⎟⎟
⎟⎟
⎟
γ ⎟
⎟
T ⎟
0 ⎟
⎠
0
0
0
−
0
0 − Tαγ
0 0 − 4Tγ Tγα
0 0 Tγα − Tγα
γ
0 − T1
0 0
T
2
0
2
0
2
0
0
0
2
0
2
0
0
2
0
2
0
0
We thus have four variants of the LM test. Their size and power are examined in
a set of 60000 simulation experiments. First the test sizes are examined for sample
sizes T=50, 100, 200 and 500 where the nuisance parameters are set to α0 = 0.81 and
γ 0 = 0.04. This choice of parameter values and sample sizes was made to ensure that
the small sample biases and variances were not trivial. In the simulations, to examine
the power of the tests we considered two sets of values for the variance parameters:
(i) α1 = α2 = γ 1 = γ 2 = 0.16, and (ii) α1 = α2 = γ 1 = γ 2 = 0.49. The first of this
represents a moderate departure from the null whereas the second lies close to the
stationarity bound and so is a relatively extreme departure.
The results on the test size are given in Table 3.1 and for size-adjusted power in
Table 3.2. The first clear result we find is that of the bad size properties in small
samples for the HES LM test (see Table 3.1), because it is clearly over-sized, even
at T=500, in marked contrast to the other tests. The OP and the ExpHES have
better size properties, although when we check the size-adjusted power of the tests
(Table 3.2) it reveals the lack of power of the OP test for finite samples. At the more
extreme alternative the ExpHES and the TrueHES tests have power close to unity at
all sample sizes. From the results, the first recommendation in practical applications
is to use the ExpHES to test for multivariate ARCH effects. Once we have selected
the ExpHES, we can concentrate on the selection among Model 1, Model 2 or Model
3. Model 3 seems to have much better size properties than Model 1 or Model 2.
Comparing Models 2 and 3 it is interesting to see the marginal effect of introducing
the QML biases in place of zeros in specifying the null hypothesis. As it was suggested
by our earlier theoretical analysis, the size of the test is clearly improved. Analysing
the TrueHES, the test presents the best size properties is again Model 3. Thus,
the use of bias-correction to improve the size of the test, as an alternative to the
traditional Bartlett (1937) type correction, is supported in our study. If we consider
the size-adjusted power, we observe how the test power in Model 2 and 3 improves
on that of Model 1 with Model 3 being slightly superior. So the overall conclusion
from the simulations is that, of the operational tests, only ExpHES performs well. Its
size is approximately correct even at T=50 while it has high power against both the
moderate and extreme alternatives at all sample sizes considered. It even dominates
the non-operational TrueHES test for the moderate alternative and has comparable
but slightly less power for the extreme alternative. Hence, our simulations support
the use of the ExpHES test while bias correcting all the QML estimates.
17
Table 3.1: Size Results based on 5% critical values
OP
HES
ExpHES
TrueHES
M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3
T=500 0.038 0.038 0.039 0.086 0.081 0.086 0.058 0.056 0.052 0.055 0.059 0.052
T=200 0.047 0.048 0.051 0.146 0.141 0.145 0.057 0.057 0.048 0.062 0.065 0.052
T=100 0.054 0.048 0.054 0.148 0.134 0.146 0.058 0.061 0.044 0.063 0.072 0.053
T=50 0.043 0.040 0.048 0.101 0.095 0.098 0.063 0.062 0.042 0.066 0.080 0.054
The results are based on 60000 Monte Carlo replications under the null of no-ARCH
effects. α0 = 0.81 and γ 0 = 0.04.
Table 3.2: Power Results based on 5% critical values size-adjusted
OP
HES
ExpHES
TrueHES
When the alternative hypothesis is α1 = α2 = γ 1 = γ 2 = 0.16
M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3
T=500 0.940 0.942 0.932 0.996 0.996 0.996 1.000 1.000 1.000 1.000 1.000 1.000
T=200 0.244 0.250 0.218 0.229 0.232 0.229 1.000 1.000 1.000 0.961 0.966 0.962
T=100 0.062 0.079 0.063 0.018 0.020 0.020 0.979 0.979 0.979 0.852 0.857 0.853
T=50 0.040 0.047 0.042 0.025 0.027 0.027 0.815 0.820 0.832 0.708 0.708 0.709
When the alternative hypothesis is α1 = α2 = γ 1 = γ 2 = 0.49
T=500 0.837 0.844 0.846 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
T=200 0.537 0.541 0.526 0.719 0.754 0.689 1.000 1.000 1.000 1.000 1.000 1.000
T=100 0.101 0.143 0.087 0.030 0.075 0.015 0.999 1.000 0.999 0.999 0.999 0.999
T=50 0.013 0.018 0.009 0.009 0.009 0.007 0.966 0.969 0.966 0.981 0.982 0.982
The results are based on 60000 Monte Carlo replications. α0 = 0.81 and γ 0 = 0.04.
4 Conclusions
In this paper we have provided theoretical evidence of the severe biases and large
variances that result from unconstrained-QML estimation of a simple bivariate-ARCH
model under overspecification of the conditional heteroscedasticity processes. When
we analyse the model in Wong and Li (1997), we find that some of the estimators can
have large variances if the difference between the intercepts in the model is relatively
large. In the case of the Engle and Kroner (1995) and Liu and Polasek (1999, 2000)
specification, we find that strongly contemporaneously correlated disturbances and/or
a large difference between the intercepts can produce large biases in the estimators of
the ARCH-terms for some combinations of parameters. Under the null of no-ARCH
effects, the intercepts capture the volatility of the series, and then the results of this
paper warn about the testing of multivariate ARCH effects among series that may
have very different degree of volatilities. One rule of thumb in practical applications,
would be to standardize always the volatilities of the series before they are used in a
multivariate model, although the best recommendation is to use the bias expressions
that are provided in this paper. We believe that the possibility of extreme biases and
18
variances should be taken into account in practical applications when QML is used as
the estimation procedure in this model, and this paper provides an analysis of what
happens in a simple bivariate process. In the last section of the paper we use our bias
results to improve the finite sample performance of an LM type test for multivariate
ARCH effects by bias-correcting the estimators of the parameters. We show that this
can be considered as an alternative way to improve the finite sample behaviour in
testing instead of applying a Bartlett-type correction. The general recommendation
from this paper is that when testing for multivariate ARCH effects by performing the
LM test, the expected Hessian form should be used and all QML estimators should
be bias corrected.
19
Appendix 1
The proof of Theorem 2.1, implies the use of expression (2.4) to find the kij ,
the
the
kij,l components. Using differential matrix calculus, defining Ht−1 =
¶
µ k11tijl and
h
h12t
, and assuming the parameter vector to be (α0 , γ 0 , α1 , α2 , γ 1 , γ 2 ) , we
h21t h22t
obtain:
SECOND ORDER DERIVATIVES
evaluation
2
T (h11t )
− 2
k11
k15
0
k24
k34
−
evaluation
k12
k16
0
2
T (h11t ) ε21t−1 ε22t−1
2
k25
k35
−
0
k13
0
k22
2
h22t ε21t−1
T(
)
2
0
k26
evaluation
2
T ε2
(h11t )
− 1t−12
2
T (h22t )
− 2
2
T (h22t ) ε22t−1
−
2
k36
0
k55
T (h22t ) ε41t−1
−
2
evaluation
2
T (h11t ) ε22t−1
−
2
k14
k23
0
−
k33
k44
2
k45
0
k66
T (h22t ) ε42t−1
−
2
k46
2
0
k56
−
2
h11t ε41t−1
T(
)
2
2
T (h11t ) ε42t−1
−
2
2
T (h22t ) ε21t−1 ε22t−1
2
THIRD ORDER DERIVATIVES
k111
k114
k212
k215
k223
k226
k135
k234
k144
k244
k155
k255
k333
k336
k436
k446
k455
k466
k656
evaluation
3
2T (h11t )
3
2T (h11t ) ε22t−1
0
0
0
22t 3 2
2T (h ) ε2t−1
0
0
11t 3 4
2T (h ) ε2t−1
0
0
22t 3 4
2T (h ) ε1t−1
3
2T (h11t ) ε61t−1
0
0
0
0
0
22t 3 2
2T (h ) ε1t−1 ε42t−1
k112
k115
k213
k216
k224
k133
k136
k235
k145
k245
k156
k256
k334
k434
k444
k335
k456
k555
k666
evaluation
0
0
0
0
0
11t 3 4
2T (h ) ε1t−1
0
0
0
0
0
22t 3 2
2T (h ) ε1t−1 ε22t−1
3
2T (h11t ) ε41t−1 ε22t−1
3
2T (h11t ) ε21t−1 ε42t−1
3
2T (h11t ) ε62t−1
0
0
22t 3 6
2T (h ) ε1t−1
3
2T (h22t ) ε62t−1
k113
k116
k214
k222
k225
k134
k233
k236
k146
k246
k166
k266
k335
k435
k445
k356
k366
k556
evaluation
3
2T (h11t ) ε21t−1
0
0
3
2T (h22t )
3
2T (h22t ) ε21t−1
3
2T (h11t ) ε21t−1 ε22t−1
0
0
0
0
0
22t 3 4
2T (h ) ε2t−1
0
0
0
0
0
22t 3 4
2T (h ) ε1t−1 ε22t−1
The Cox and Snell (1968) expressions that are required (apart from the second
order derivatives, and the third order derivatives previously given), once we evaluate
them when α1 = α2 = γ 1 = γ 2 = 0, are:
1
k
2 111
1
k
2 116
1
k
2 215
1
k
2 224
1
k
2 133
1
k
2 232
1
k
2 141
1
k
2 146
1
k
2 245
1
k
2 154
+ k11,1
+ k11,6
+ k21,5
+ k22,4
1
k
2 253
1
k
2 162
1
k
2 261
1
k
2 266
1
k
2 335
1
k
2 434
1
k
2 443
1
k
2 352
1
k
2 451
+ k25,3
+ k16,2
+ k26,1
+ k13,3
+ k23,2
+ k14,1
+ k14,6
+ k24,5
+ k15,4
+ k26,6
+ k33,5
+ k43,4
+ k44,3
+ k35,2
+ k45,1
eval.
0
0
0
0
− 2αT 0
0
− T2αγ30
−
0
T γ0
2α20
0
0
− T2γα20
0
0
− 2αT0 γ
0
− 2γT
0
− 3Tγ α0
0
T γ2
− 2α20
0
3T γ 20
− α2
0
0
0
1
k
2 112
1
k
2 211
1
k
2 216
1
k
2 225
1
k
2 134
1
k
2 233
1
k
2 142
1
k
2 241
1
k
2 246
1
k
2 155
+ k11,2
+ k21,1
+ k21,6
+ k22,5
1
k
2 254
1
k
2 163
1
k
2 262
1
k
2 331
1
k
2 336
1
k
2 435
1
k
2 444
1
k
2 353
1
k
2 452
+ k25,4
+ k16,3
+ k26,2
+ k13,4
+ k23,3
+ k14,2
+ k24,1
+ k24,6
+ k15,5
+ k33,1
+ k33,6
+ k43,5
+ k44,4
+ k35,3
+ k45,2
eval.
0
0
0
0
− T2αγ20
0
0
− 2αT 2
0
0
0
0
− 2γT
0
0
− 2γT 2
0
− 3T
α0
−3T
− T2
3T γ 3
− α3 0
0
0
0
1
k
2 113
1
k
2 212
1
k
2 221
1
k
2 226
1
k
2 135
1
k
2 234
1
k
2 143
1
k
2 242
1
k
2 151
1
k
2 156
+ k11,3
+ k21,2
+ k22,1
+ k22,6
1
k
2 255
1
k
2 164
1
k
2 263
1
k
2 332
1
k
2 431
1
k
2 436
1
k
2 445
1
k
2 354
1
k
2 453
+ k25,5
+ k16,4
+ k26,3
+ k13,5
+ k23,4
+ k14,3
+ k24,2
+ k15,1
+ k15,6
+ k33,2
+ k43,1
+ k43,6
+ k44,5
+ k35,4
+ k45,3
eval.
0
0
0
0
− 2γT
0
0
− T2αγ20
0
0
0
0
T α2
− 2γ 30
0
0
− 2γT
0
− 3T
γ0
T γ0
− 2α2
0
− T2αγ00
− 3Tαγ0 0
0
0
1
k
2 114
1
k
2 213
1
k
2 222
1
k
2 131
1
k
2 136
1
k
2 235
1
k
2 144
1
k
2 243
1
k
2 152
1
k
2 251
+ k11,4
+ k21,3
+ k22,2
+ k13,1
1
k
2 256
1
k
2 165
1
k
2 264
1
k
2 333
1
k
2 432
1
k
2 441
1
k
2 446
1
k
2 355
1
k
2 454
+ k25,6
+ k16,5
+ k26,4
+ k13,6
+ k23,5
+ k14,4
+ k24,3
+ k15,2
+ k25,1
+ k33,3
+ k43,2
+ k44,1
+ k44,6
+ k35,5
+ k45,4
eval.
0
0
0
− 2αT 2
0
− 2αT 0
0
T γ2
− 2α30
0
0
0
− 2γT 2
0
− T2γα20
0
0
− 2αT 0
−3T
− 2αT 0
3T γ 20
α30
3T γ 20
− α2
0
−
0
0
1
k
2 115
1
k
2 214
1
k
2 223
1
k
2 132
1
k
2 231
1
k
2 236
1
k
2 145
1
k
2 244
1
k
2 153
1
k
2 252
+ k11,5
+ k21,4
+ k22,3
+ k13,2
1
k
2 161
1
k
2 166
1
k
2 265
1
k
2 334
1
k
2 433
1
k
2 442
1
k
2 351
1
k
2 356
1
k
2 455
+ k16,1
+ k16,6
+ k26,5
+ k23,1
+ k23,6
+ k14,5
+ k24,4
+ k15,3
+ k25,2
+ k33,4
+ k43,3
+ k44,2
+ k35,1
+ k35,6
+ k45,5
0
0
0
− 2αT0 γ
0
0
0
− 2αT 0
0
0
− T2γα30
0
0
0
− T2γα20
0
− 3Tα0γ 0
− T2αγ00
− 3Tα2γ 0
0
0
0
0
1
k
2 456
1
k
2 365
1
k
2 464
+ k45,6
+ k36,5
+ k46,4
1
k
2 553
1
k
2 652
1
k
2 661
1
k
2 666
+ k55,3
+ k65,2
+ k66,1
+ k66,6
eval.
0
0
0
3T α20
γ 20
T α0
− 2γ 2
0
− 3T
α0
−
−3T
1
k
2 361
1
k
2 366
1
k
2 465
+ k36,1
+ k36,6
+ k46,5
1
k
2 554
1
k
2 653
1
k
2 662
+ k55,4
+ k65,3
+ k66,2
eval.
0
0
0
− 3Tγ 2α0
0
− T2γα0
0
− 3T
γ
0
1
k
2 362
1
k
2 461
1
k
2 466
+ k36,2
+ k46,1
+ k46,6
1
k
2 555
1
k
2 654
1
k
2 663
+ k55,5
+ k65,4
+ k66,3
eval.
0
0
0
3T α30
γ 30
− T2
−
−3T
1
k
2 363
1
k
2 462
1
k
2 551
+ k36,3
+ k46,2
+ k55,1
1
k
2 556
1
k
2 655
1
k
2 664
+ k55,6
+ k65,5
+ k66,4
eval.
0
0
3T α0
− γ2
0
3T α20
γ 20
T α20
− 2γ 2
0
− 3Tα0γ 0
−
1
k
2 364
1
k
2 463
1
k
2 552
+ k36,4
+ k46,3
+ k55,2
1
k
2 651
1
k
2 656
1
k
2 665
+ k65,1
+ k65,6
+ k66,5
0
0
3T α20
γ 30
− 2γT
0
− T2γα0
0
− 3Tγ α0
0
−
Appendix 2
The proof of Theorem 2.2, implies the use of expression (2.4) to find the
kij , kij,l µand the kijl components.
Using differential matrix calculus, defining
¶
h11t h12t
−1
Ht =
, and ordering the parameters as: α10 , α20 , α30 , α11 , α22 , α33 ,
h21t h22t
we obtain:
SECOND ORDER DERIVATIVES
evaluation
2
T (h11t )
−
2
k11
k15
k24
11t 22t
−T h
h
ε1t−1 ε2t−1
−T h11t h12t ε21t−1
evaluation
k12
k16
k25
2
k34
k45
−
T (h12t ) ε21t−1
2
−T ε31t−1 ε2t−1 h11t h12t
2
k66
−
T (h22t ) ε42t−1
2
k35
k46
−T h11t h12t
2
T (h12t ) ε22t−1
−
2
³
¡
¢2 ´
−T ε1t−1 ε2t−1 h11t h22t + h12t
−T ε1t−1 ε2t−1 h12t h22t
2
T (h12 ) ε21t−1 ε22t−1
−
2
k13
k22
k26
evaluation
2
T (h12t )
−
2
³
¡
¢2 ´
−T h11t h22t + h12t
−T h22t h12t ε21t−1
k14
k23
−
k55
−
T (h22t ) ε22t−1
³ 2
k33
¡
¢2 ´
−T ε21t−1 ε22t−1 h11t h22t + h12t
k44
k56
2
−T h12t h22t
−
2
k36
evaluation
2
T (h11t ) ε21t−1
−
T (h22t )
2
2
2
T (h11t ) ε41t−1
2
−T ε32t−1 ε1t−1 h22t h12t
THIRD ORDER DERIVATIVES
k111
k114
k212
k215
k223
k226
k135
k234
k144
k244
k155
k255
k333
k336
k436
k446
k455
k466
k656
evaluation
¡
¢3
2T h11t
¡ 11t ¢3 2
2T
ε1t−1
³ h
¡
¢2 ´
11t
11t 22t
2T h
h h + 3 h12t
³
¡
¢2 ´
2T h11t ε1t−1 ε2t−1 h11t h22t + 3 h12t
³
¡
¢2 ´
2T h22t h11t h22t + 3 h12t
³
¡
¢2 ´
2T h22t ε22t−1 h11t h22t + 3 h12t
³
¡
¢2 ´
2T h12t ε1t−1 ε2t−1 h11t h22t + h12t
³
¡
¢2 ´
2T h12t ε21t−1 h11t h22t + h12t
¡
¢3
2T h11t ε41t−1
¡
¢2
4T h11t h12t ε41t−1
³
¡
¢2 ´
2T h11t ε21t−1 ε22t−1 h11t h22t + 3 h12t
³
¡
¢2 ´
4T h12t ε21t−1 ε22t−1 3h11t h22t + h12t
¡
¢3
2T h22t
¡
¢3
2T h22t ε22t−1
¡
¢2
2T h12t h22t ε21t−1 ε22t−1
¡ 12t ¢2 11t 4
2T h
h ε1t−1 ε22t−1
³
¡
¢2 ´
2T h11t ε41t−1 ε22t−1 h11t h22t + 3 h12t
¡
¢2
2T h12t h22t ε21t−1 ε42t−1
¡
¢2
4T h22t h12t ε1t−1 ε52t−1
k112
k115
k213
k216
k224
k133
k136
k235
k145
k245
k156
k256
k334
k434
k444
k335
k456
k555
k666
evaluation
¡
¢2
4T h11t h12t
¡
¢2 12t
4T h11t
h ε1t−1 ε2t−1 ´
³
¡
¢2
12t
11t 22t
2T h
h h + h12t
³
¡
¢2 ´
2T h12t ε22t−1 h11t h22t + h12t
³
¡
¢2 ´
2T h11t ε21t−1 h11t h22t + 3 h12t
¡
¢2
2T h12t h22t
¡
¢2
2T h22t h12t ε22t−1
³
¡
¢2 ´
2T h22t ε1t−1 ε2t−1 h11t h22t + 3 h12t
¡
¢2
4T h11t ³h12t ε31t−1 ε2t−1
¡
¢2 ´
2T h11t ε21t−1 ε2t−1 h11t h22t + 3 h12t
³
¡
¢2 ´
2T h12t ε1t−1 ε32t−1 h11t h22t + h12t
³
¡
¢2 ´
2T h22t ε1t−1 ε32t−1 h11t h22t + 3 h12t
¡
¢2
2T h22t h12t ε21t−1
¡
¢2
4T h12t h11t ε41t−1
¡
¢3 6
2T h11t
ε1t−1
³
¡
¢2 ´
2T h22t ε21t−1 ε22t−1 h11t h22t + 3 h12t
³
¡
¢2 ´
2T h12t ε31t−1 ε32t−1 h11t h22t + h12t
³
¡
¢2 ´
4T h12t ε31t−1 ε32t−1 3h11t h22t + h12t
¡
¢3
2T h22t ε62t−1
k113
k116
k214
k222
k225
k134
k233
k236
k146
k246
k166
k266
k335
k435
k445
k356
k366
k556
evaluation
¡
¢2
2T h12t h11t
¡
¢2
2T h11t h12t ε22t−1
¡
¢2
4T h11t h12t ε21t−1
³
¡
¢2 ´
4T h12t 3h11t h22t + h12t
³
¡
¢2 ´
4T h12t ε1t−1 ε2t−1 3h11t h22t + h12t
¡
¢2
2T h11t h12t ε21t−1
¡
¢2
4T h22t h12t
¡
¢2
4T h12t h22t ε22t−1
¡
¢2
2T h12t h³11t ε21t−1 ε22t−1
¡
¢2 ´
2T h12t ε21t−1 ε22t−1 h11t h22t + h12t
¡
¢2
2T h12t h22t ε42t−1
¡
¢2
4T h22t h12t ε42t−1
¡
¢2
4T h22t h³12t ε1t−1 ε2t−1
¡
¢2 ´
2T h12t ε31t−1 ε2t−1 h11t h22t + h12t
¡
¢2
4T h11t h12t ε51t−1 ε2t−1
¡ 22t ¢2 12t
4T h
h ε1t−1 ε32t−1
¡ 22t ¢3 4
2T h
ε2t−1
³
¡
¢2 ´
2T h22t ε21t−1 ε42t−1 h11t h22t + 3 h12t
The Cox and Snell (1968) expressions that are required (apart from the second order derivatives,
and the third order derivatives previously given), once we evaluate them when α11 = α22 = α33 = 0, are:
1
2 k111
1
2 k113
1
2 k115
1
2 k211
1
2 k213
1
2 k215
1
2 k221
1
2 k223
1
2 k225
1
2 k131
1
2 k133
1
2 k135
1
2 k231
1
2 k233
1
2 k235
1
2 k141
+ k11,1
+ k11,3
+ k11,5
+ k21,1
+ k21,3
+ k21,5
+ k22,1
+ k22,3
+ k22,5
+ k13,1
+ k13,3
+ k13,5
+ k23,1
+ k23,3
+ k23,5
1
2 k143
+ k14,3
1
2 k145
+ k14,5
1
2 k241
+ k24,1
−T α10 α330
+ k14,1
1
2 k243
+ k24,3
1
2 k245
+ k24,5
1
2 k151
+ k15,1
1
2 k153
+ k15,3
1
2 k155
evaluation
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+ k15,5
1
2 k251
+ k25,1
1
2 k253
+ k25,3
1
2 k255
+ k25,5
1
2 k161
+ k16,1
1
2 k112
1
2 k114
1
2 k116
1
2 k212
1
2 k214
1
2 k216
1
2 k222
1
2 k224
1
2 k226
1
2 k132
1
2 k134
1
2 k136
1
2 k232
1
2 k234
1
2 k236
1
2 k142
+ k11,2
+ k11,4
+ k11,6
+ k21,2
+ k21,4
+ k21,6
+ k22,2
+ k22,4
+ k22,6
+ k13,2
+ k13,4
+ k13,6
+ k23,2
+ k23,4
+ k23,6
evaluation
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+ k14,2
T α20 α230 (α10 −α30 )
3
1
2 k144
+ k14,4
3
1
2 k146
+ k14,6
1
2 k242
+ k24,2
1
2 k244
+ k24,4
1
2 k246
+ k24,6
1
2 k152
+ k15,2
1
2 k154
+ k15,4
3
2(α10 α30 −α220 )
T α220 α230
2(α10 α30 −α220 )
T α220 α230 (α10 −α30 )
2(α10 α30 −α220 )
T α10 α20 α230
(α10 α30 −α220 )3
−T α320 α30
3
α10 α30 −α220
3
T α20 α30 (α30 −α10 )
3
α10 α30 −α220
2
T α20 α30 (α30 −α10 )
3
2 α10 α30 −α220
2
T α20 α30 (α10 −α30 )
3
2 α10 α30 −α220
2
2
2
T α20 α30 α30 +α10 −2α220
3
2 α10 α30 −α220
T α20 (α10 −α30 ) α220 +α10 α30
3
2 α10 α30 −α220
2
T α20 (α30 −α10 ) α20 +α10 α30
3
2 α10 α30 −α220
2
2
2
2
T α20 2α20 −α10 −α30 α20 +α10 α30
3
2 α10 α30 −α220
T α420
3
2 α10 α30 −α220
(
)
(
)
(
)
(
)
(
)
(
(
)
(
)
(
)
(
(
1
2 k156
)
)
)(
(
)
(
)
)
+ k15,6
1
2 k252
+ k25,2
1
2 k254
+ k25,4
1
2 k256
+ k25,6
1
2 k162
+ k16,2
2(α10 α30 −α220 )
−T α210 α330
2(α10 α30 −α220 )
T α220 α330
3
3
3
2(α10 α30 −α220 )
T α220 α30 (α30 −α10 )
(α10 α30 −α220 )3
T α210 α20 α230
(α10 α30 −α220 )
−T α320 α230
3
3
(α10 α30 −α220 )
T α20 α30 (α230 +α210 −2α220 )
3
2(α10 α30 −α220 )
T α10 α220 α30 (α30 −α10 )
2(α10 α30 −α220 )
3
T α220 α230 (α10 −α30 )
3
2 α10 α30 −α220
T 2α220 −α210 −α230 α220 +α10 α30
3
2 α10 α30 −α220
2
T α10 α20 (α10 −α30 ) α20 +α10 α30
3
2 α10 α30 −α220
2
T α20 α30 (α30 −α10 ) α20 +α10 α30
3
2 α10 α30 −α220
T α320 (α30 −α10 )
3
2 α10 α30 −α220
(
(
)
)(
(
)
)
(
(
)
)
(
(
)
(
)
)
evaluation
1
2 k163
+ k16,3
1
2 k165
+ k16,5
1
2 k261
+ k26,1
1
2 k263
+ k26,3
1
2 k265
1
2 k331
1
2 k333
1
2 k335
1
2 k431
+ k26,5
1
2 k433
+ k43,3
1
2 k435
+ k43,5
1
2 k441
+ k44,1
1
2 k445
+ k44,5
1
2 k351
+ k35,1
1
2 k353
+ k35,3
1
2 k355
+ k35,5
1
2 k451
+ k45,1
1
2 k453
+ k45,3
1
2 k455
+ k45,5
1
2 k361
+ k36,1
1
2 k363
1
2 k365
+ k36,3
+ k36,5
3
1
2 k166
+ k16,6
1
2 k262
+ k26,2
2(α10 α30 −α220 )
T α10 α220 (α10 −α30 )
1
2 k264
+ k26,4
−T α210 α320
+ k26,6
3
1
2 k266
1
2 k332
1
2 k334
1
2 k336
1
2 k432
3
1
2 k434
+ k43,4
3
1
2 k436
+ k43,6
1
2 k442
+ k44,2
(α10 α30 −α220 )3
T α210 α20 α30
3
α10 α30 −α220
3
T α10 α20 (α10 −α30 )
3
α10 α30 −α220
)
(
)
0
0
0
−T α10 α220 α30
+ k43,1
+ k44,3
+ k16,4
2(α10 α30 −α220 )
−T α10 α320
(
2(α10 α30 −α220 )
T α420
2(α10 α30 −α220 )
T α420 (α10 −α30 )
2(α10 α30 −α220 )
−3T α210 α330
3
(α10 α30 −α220 )
3T α10 α220 α230
3
α10 α30 −α220
3T α10 α220 α230 (α10 −α30 )
3
α10 α30 −α220
2
T α10 α20 (α30 −α10 )
3
2 α10 α30 −α220
2
T α10 α20 (α10 −α30 )
3
2 α10 α30 −α220
2
2
2
T α10 α20 α30 +α10 −2α220
3
2 α10 α30 −α220
2
3T α10 α20 α30 (3α30 −α10 )
3
2 α10 α30 −α220
2
2
2
3T α20 α30 α10 α10 α30 −α20 −α10 α30 α10 α30 +α220 +2α420
4
2 α10 α30 −α220
2
2
2
2
3T α20 α30 α10 α10 −4α20 α10 α30 −α20 +α230 α10 α30 +α220 −2α420 α30
4
2 α10 α30 −α220
T α210 α220
3
2 α10 α30 −α220
−T α310 α30
3
2 α10 α30 −α220
T α210 α220 (α30 −α10 )
3
2 α10 α30 −α220
(
)
(
)
(
)
(
)
(
)
(
)
(
[
(
)
)
(
(
[
((
)(
(
(
)
(
)
(
)
]
)
)
(
T α10 α420
3
1
2 k164
2(α10 α30 −α220 )
T α420 (α30 −α10 )
+ k33,1
+ k33,3
+ k33,5
1
2 k443
evaluation
−T α10 α220 α30
)
)
))
]
3
(α10 α30 −α220 )3
(α10 α30 −α220 )3
T α210 α20 α230
(α10 α30 −α220 )3
0
0
0
+ k33,2
+ k33,4
+ k33,6
T α320 (α10 −α30 )
+ k43,2
1
2 k444
+ k44,4
1
2 k446
+ k44,6
1
2 k352
+ k35,2
1
2 k354
+ k35,4
1
2 k356
+ k35,6
1
2 k452
+ k45,2
1
2 k454
+ k45,4
1
2 k456
+ k45,6
1
2 k362
+ k36,2
1
2 k364
+ k36,4
1
2 k366
+ k36,6
3
2(α10 α30 −α220 )
−T α10 α220 α230
3
2(α10 α30 −α220 )
−T α210 α220 α30
3
2(α10 α30 −α220 )
T α420 α30
3
2(α10 α30 −α220 )
3T α10 α20 α230 (α10 −α30 )
3
(α10 α30 −α220 )
−3T α310 α330
3
α10 α30 −α220
3T α10 α220 α330
3
α10 α30 −α220
T α10 α20 α230 +α210 −2α220
3
2 α10 α30 −α220
2
2
T α10 α20 (α30 −α10 )
3
2 α10 α30 −α220
T α10 α220 α30 (α10 −α30 )
3
2 α10 α30 −α220
2
2
2
3T α20 α30 α10 α10 −4α20 α10 α30 −α20 +α230 α10 α30 +α220 −2α420 α30
4
2 α10 α30 −α220
3T α210 α220 α30 (3α30 −α10 )
3
2 α10 α30 −α220
2
2
2
2
3T α20 α30 α10 α10 α30 −α20 −α10 α30 α10 α30 +α220 +2α420
4
2 α10 α30 −α220
T α210 α20 (α30 −α10 )
3
2 α10 α30 −α220
T α310 α220
3
2 α10 α30 −α220
−T α310 α230
3
2 α10 α30 −α220
(
)
(
)
(
)
(
)
(
)
(
[
((
)
)(
)
(
(
[
(
(
))
)
)
)
(
(
)
(
)
(
)
(
)
)
]
]
evaluation
1
2 k461
1
2 k462
T α220 [(2α220 α30 +α10 α230 +6α10 α220 )(α10 α30 −α220 )+6α420 α30 −3α10 α230 (α10 α30 +α220 )]
+ k46,1
4
T α320
+ k46,2
1
2 k463
+ k46,3
1
2 k464
+ k46,4
1
2 k465
+ k46,5
1
2 k466
+ k46,6
1
2 k551
+ k55,1
1
2 k552
+ k55,2
1
2 k553
+ k55,3
1
2 k554
+ k55,4
1
2 k555
+ k55,5
1
2 k556
+ k55,6
1
2 k651
+ k65,1
1
2 k652
+ k65,2
1
2 k654
+ k65,4
1
2 k655
+ k65,5
1
2 k656
+ k65,6
1
2 k661
+ k66,1
1
2 k662
+ k66,2
1
2 k663
+ k66,3
1
2 k664
+ k66,4
1
2 k665
+ k66,5
1
2 k666
+ k66,6
T α220
4(α10 α30 −α220 )
)(α10 α30 −α220 )+3α10 α220 α30 −6α420 +3α210 α230 ]
2α220 +α10 α30 +3α210 +3α230
[−(
4
2α220 α10 +α210 α30 +6α30 α220
[(
2(α10 α30 −α220 )
)(α10 α30 −α220 )+6α420 α10 −3α30 α210 (α10 α30 +α220 )]
4
4(α10 α30 −α220 )
T α10 α220 [(2α220 α30 +α10 α230 +6α10 α220 )(α10 α30 −α220 )+6α420 α30 −3α10 α230 (α10 α30 +α220 )]
4
T α420
4(α10 α30 −α220 )
)(α10 α30 −α220 )+3α10 α220 α30 −6α420 +3α210 α230 ]
2α220 +α10 α30 +3α210 +3α230
[−(
4
2(α10 α30 −α220 )
T α220 α30 [(2α220 α10 +α210 α30 +6α30 α220 )(α10 α30 −α220 )+6α420 α10 −3α30 α210 (α10 α30 +α220 )]
4
4 α10 α30 −α220
2
2
2
2α20 α30 +α30 α10 +6α10 α20 α10 α30 −α220 +3α30 2α420 −α10 α220 α30 −α210 α230
T
4
2 α10 α30 −α220
2
3
2
2
5
T α20 +α10 α30 3 α10 α20 α30 +α10 α20 α30 −α20 −α20 2α220 +α10 α30 +3α210 +3α230 α10 α30 −α220
4
α10 α30 −α220
2
2
2
2
2α20 α10 +α10 α30 +6α30 α20 α10 α30 −α220 +3α10 2α420 −α10 α220 α30 −α210 α230
T α20 +α10 α30
4
2 α10 α30 −α220
2
2
2
2
2α20 α30 +α30 α10 +6α10 α20 α10 α30 −α220 +3α30 2α420 −α10 α220 α30 −α210 α230
T α10 α20 +α10 α30
4
2 α10 α30 −α220
T α20 α220 +α10 α30 3 α10 α320 α30 +α210 α20 α230 −α520 −α20 2α220 +α10 α30 +3α210 +3α230 α10 α30 −α220
4
α10 α30 −α220
2
2
2
2
T α30 α20 +α10 α30
2α20 α10 +α10 α30 +6α30 α20 α10 α30 −α220 +3α10 2α420 −α10 α220 α30 −α210 α230
4
2 α10 α30 −α220
3T α220 α10 α230 α10 α30 −α220 −α10 α30 α10 α30 +α220 +2α420
4
2 α10 α30 −α220
2
2
2
3T α20 α10 α30 α30 −4α20 α10 α30 −α20 +α210 α10 α30 +α220 −2α420 α10
4
2 α10 α30 −α220
2
2
2
2
3T α20 α10 α30 α10 α30 −α20 −α10 α30 α10 α30 +α220 +2α420
4
2 α10 α30 −α220
2
2
2
2
3T α20 α10 α30 α30 −4α20 α10 α30 −α20 +α210 α10 α30 +α220 −2α420 α10
4
2 α10 α30 −α220
2
2
3T α10 α20 α30 (3α10 −α30 )
3
2 α10 α30 −α220
2
2
3T α10 α20 α30
3
α10 α30 −α220
2
3T α10 α20 α30 (α30 −α10 )
3
α10 α30 −α220
3
2
−3T α10 α30
3
α10 α30 −α220
3T α310 α220 α30
3
α10 α30 −α220
3T α210 α220 α30 (α30 −α10 )
3
α10 α30 −α220
−3T α310 α330
3
α10 α30 −α220
(
(
α220 +α10 α30
)[(
)
)(
)
(
(
)[ (
)
)[(
)
)[(
)(
)[ (
)
)
)[(
(
((
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(
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))
]
)
]
)
)
(
(
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)]
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)(
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)
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]
References
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30
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31