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JOURNAL OF APPLIED ECONOMETRICS
J. Appl. Econ. 17: 549–564 (2002)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jae.688
GO-GARCH: A MULTIVARIATE GENERALIZED ORTHOGONAL
GARCH MODEL
ROY VAN DER WEIDE*
Department of Economics, CeNDEF, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
SUMMARY
Multivariate GARCH specifications are typically determined by means of practical considerations such as the
ease of estimation, which often results in a serious loss of generality. A new type of multivariate GARCH
model is proposed, in which potentially large covariance matrices can be parameterized with a fairly large
degree of freedom while estimation of the parameters remains feasible. The model can be seen as a natural
generalization of the O-GARCH model, while it is nested in the more general BEKK model. In order to avoid
convergence difficulties of estimation algorithms, we propose to exploit unconditional information first, so
that the number of parameters that need to be estimated by means of conditional information is more than
halved. Both artificial and empirical examples are included to illustrate the model. Copyright 2002 John
Wiley & Sons, Ltd.
1. INTRODUCTION
The ‘holy grail’ in multivariate GARCH modelling is without any doubt a parameterization of
the covariance matrix that is feasible in terms of estimation at a minimum loss of generality.
The general multivariate GARCH models available parameterize the covariance matrix by a very
large number of parameters that are hard to estimate, which often leads to convergence difficulties
of estimation algorithms. Therefore, the choice of the multivariate model is often determined by
means of practical considerations, i.e. the ease of estimation. The strong restrictions are often not
believed to reflect the ‘truth’, but they are imposed to guarantee feasibility.
Some of the best-known multivariate GARCH models available include the VECH model of
Bollerslev and Wooldridge (1988), the constant correlation model of Bollerslev (1990), the factor
ARCH model of Engle and Rothschild (1990), and the BEKK model studied by Engle and Kroner
(1995). For an overview of the multivariate GARCH models, as well as tests for misspecification,
see the paper by Kroner and Ng (1998). An extensive survey of empirical applications of time-
varying covariance models in finance can be found in Bollerslev, Chou and Kroner (1992). In
particular the model of Bollerslev (1990) has been a popular choice for modelling high-variate
time series. A test for its assumption of a constant correlation is introduced in a recent paper by
Tse (2000). Shortly after, both Engle (2002) and Tsui and Tse (2002) generalized the model to
allow for time-varying correlations.
A somewhat different approach is the Orthogonal GARCH (O-GARCH) or principal components
GARCH method. The principal components approach has first been applied in a GARCH-
type context by Ding (1994). Shortly after, Alexander and Chibumba (1996) introduced the
ŁCorrespondence to: Roy van der Weide, Department of Economics, CeNDEF, University of Amsterdam, Roetersstraat
11, 1018 WB Amsterdam, The Netherlands. E-mail: rvdweide@fee.uva.nl
Contract/grant sponsor: Netherlands Organization for Scientific Research (NWO).
Copyright 2002 John Wiley & Sons, Ltd. Received 21 November 2001
Revised 17 May 2002
550 R. VAN DER WEIDE
strongly related O-GARCH model. Thereafter, O-GARCH has been a popular choice to model
the conditional covariances of financial data (see e.g. Klaassen, 1999), mainly because the model
remains feasible for large covariances matrices (see e.g. Alexander, 2002). Recently, the model
has been elaborated along with applications by Alexander (1998, 2001).
The O-GARCH model implicitly assumes that the observed data can be linearly transformed into
a set of uncorrelated components by means of an orthogonal matrix. These unobserved components
can be interpreted as a set of uncorrelated factors that drive the particular economy or market,
similar to that in the Factor (G)ARCH approach of Engle and Rothschild (1990). The orthogonality
assumption, however, appears to be very restrictive. Indeed, if a linkage with a set of uncorrelated
economic components exists, why should the associated matrix be orthogonal? The O-GARCH
model is also known to suffer from identification problems, mainly because estimation of the matrix
is entirely based on unconditional information (the sample covariance matrix). For example, when
the data exhibits weak correlation, the model has substantial difficulties to identify a matrix that
is truly orthogonal (see e.g. Alexander, 2001).
The multivariate GARCH model proposed in this paper can best be seen as a natural
generalization of the O-GARCH model. Clearly, orthogonal matrices are very special, and they
reflect only a very small subset of all possible invertible linear maps. The generalized O-GARCH
model (GO-GARCH) allows the linkage to be given by any possible invertible matrix. Estimation
of the matrix requires the use of conditional information, which in turn solves possible identification
problems.1The parameters are relatively easy to estimate, so that a substantial increase in the
degrees of freedom is obtained at a very affordable price.
The next section will introduce the generalized Orthogonal GARCH model (GO-GARCH).
Estimation is discussed in Section 3. Sections 4 and 5 present some simulation results and an
empirical example, respectively. Section 6 concludes.
2. GENERALIZED ORTHOGONAL GARCH
2.1. Notation
In a multivariate GARCH setting, the conditional covariance matrix of the m-dimensional zero
mean random variable depends on elements of the information set up to time t1, denoted
by =t1. Assume that xtis normally distributed and that its conditional covariance matrix Vtis
measurable with respect to =t1, the multivariate GARCH model is then described by:
xtj=t1¾N0,V
t1
wherewehaveassumedthatxtis second-order stationary so that VDEVtexists. The information
set =tcontains both lagged values of the squares and cross-products of xtand elements of the
conditional covariance matrices up to time t, i.e. lagged values of Vt. The challenge in multivariate
GARCH modelling is to find a parameterization of Vtas a function of =t1that is fairly general
while feasible in terms of estimation.
In the following we will frequently use the terms conditional information and unconditional
information. We specify unconditional information as information that can be extracted from the
1For example, the data is not required to exhibit strong correlation for the method to work.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 551
unconditional covariance matrix. By conditional information we mean the information set =tas
introduced above.
2.2. Representation
The key assumption of the GO-GARCH model is the following:
Assumption 1 The observed economic process fxtgis governed by a linear combination of
uncorrelated economic components2fytg:
xtDZyt2
The linear map Z that links the unobserved components with the observed variables is assumed to
be constant over time, and invertible.
Without loss of generality,3we normalize the unobserved components to have unit variance,
so that:
VDExtxT
tDZZT3
An explicit example, which we will denote the GO-GARCH(1,1) model, would be:
xtDZytyt¾N0,H
t4
where each component is described by a GARCH(1,1) process:
HtDdiagh1,t,...,h
m,t5
hi,t D1˛iˇiC˛iy2
i,t1Cˇihi,t1iD1,...,m 6
where H0DIequals the unconditional covariance matrix of the components.4The conditional
covariances of fxtgare given by:
VtDZHtZT7
2.3. Identification
Let Pand denote the matrices with, respectively, the orthonormal eigenvectors and the
eigenvalues of the unconditional covariance matrix VDZZT.
2Note that there might be more components than the number of variables observed, so that exposing a set of reliable
components could be troublesome. However, if the components are assumed to be described by independent GARCH-type
models, a new set of uncorrelated components can be constructed by aggregating the ‘original’ components. Under certain
conditions, the (extracted) aggregated components are also described by GARCH-type processes. See, for example, Drost
and Nijman (1993) in which temporal aggregation of GARCH processes is considered. However, it is known that the
GARCH-type ‘features’ typically become weaker under aggregation. As a consequence, the accuracy with which the
components are described by GARCH-type models increases as more components can be extracted, which will result in
better fits.
3Note that the unconditional variances of the components and the matrix Zare directly related. Let fytgdenote the
components with original scaling, and let the normalized set of components be denoted by fQytg,sothatfQytgDfDytg,
where Drepresents the diagonal normalization matrix. The observed process is then given by fxtgDfZytgDfQ
ZQytg,where
Q
ZDZD1.
4Ling and McAleer (2002) provide a method for treating the initial value when it comes to asymptotic theory for
multivariate GARCH.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
552 R. VAN DER WEIDE
Let us assume that an orthogonal linear linkage Zindeed exists, so that xtDZyt.The
unconditional covariance matrix Vis then given by: VDZHZT,whereHis diagonal. Then
the orthogonal matrix P, the O-GARCH estimator for Z, is only guaranteed to coincide with Z,
when the diagonal elements of Hare all distinct. Identification problems thus arise when some
of the uncorrelated components have similar unconditional variance. To see this, suppose that all
components have unit variance, so that VDZIZTDI. Clearly, the matrix Zis no longer identified
by the eigenvector matrix of V, as for every orthogonal matrix Q,wehaveZQZQTDI.Note
that the eigenvalues of Vreflect the variances of the components when the model is well identified.
The estimations should therefore be interpreted with caution when some of the eigenvalues are
almost identical. Problems of this type are known to occur when, for example, the data exhibits
weak dependence.5The next lemma states that the linkage Zis well identified when conditional
information is taken into account.
Lemma 2 Let Z be the map that links the uncorrelated components fytgwith the observed process
fxtg. Then there exists an orthogonal matrix U0such that:
P
1
2U0DZ8
Proof. The result follows directly from Singular Value Decomposition, see e.g. Horn and
Johnson (1999).
Let the estimator for U0be denoted by U. Without loss of generality, we restrict the determinant
of Uto be 1.6
It can be verified that the orthogonal matrices Pand have [mm 1]/2andmdegrees
of freedom, respectively. Together with the [mm 1]/2 degrees of freedom for U,wehave
mCmm 1Dm2degrees of freedom for the invertible matrix Z. The matrices Pand will
be estimated by means of unconditional information, as they will be extracted from the sample
covariance matrix V. Conditional information is required to estimate U0.
Note that there is a continuum of matrices Qfor which a set of linearly independent components
utDQxtcan be obtained. For every choice of orthogonal matrix U, the linear transformation
QDUT1
2PTinduces an uncorrelated series with unit variance: EutuT
tDQVQTDUTUDI.
Clearly, these components often still exhibit a form of non-linear correlation. Therefore, linear
independence can be very deceiving, as it might give the impression that the linkage between the
observed variables and the uncorrelated components is uncovered, when more often it is not. The
original components can only7be restored by means of the inverse of Z.
According to Lemma 2, the model is well identified as there exists a U0that is associated with
the original Z. Indeed, the additional [mm 1]/2 degrees of freedom induced by the extra term
Uextends the representation to full generality, in the sense that any invertible linkage Zcan in
principle be estimated from the data, instead of orthogonal matrices only.
5Given that the observed data is normalized to have unit variance, which is common practice.
6More precisely, Uis considered an element of SO (m), which denotes the set of all m-dimensional orthogonal matrices
with positive determinant.
7Equivalent matrices, in the sense that they only exchange variables, for example, are included.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 553
One way to parameterize the estimator for the orthogonal matrix U0would be by means of
rotation matrices.8
Lemma 3 Every m-dimensional orthogonal matrix U with detU D1can be represented as a
product of m
2D[mm 1]/2rotation matrices:
UD
i<j
Rijij ij 9
where Rijijperforms a rotation in the plane spanned by eiand ejover an angle ij.
Proof. See Vilenkin (1968).
The rotation angles9fijgare commonly referred to as the Euler angles, which can be estimated
by means of maximum likelihood.
We have noted earlier already that the O-GARCH model suffers from identification problems,
for example when the data exhibits weak correlation. These problems should not arise when
conditional information is exploited, as proposed in the GO-GARCH model. For example, when
the uncorrelated components appear to be observed directly, we expect the estimator for U0to
be close to PT,since O
ZDP
1
2PTDV
1
2is approximately diagonal when the data is virtually
uncorrelated.
2.4. Time-varying Correlations
The implied conditional correlations fRtgof the observed process fxtgcan be computed as:
RtDD1
tVtD1
tDtDVt°I
1
210
where fVtgDfZHtZTgdenotes the conditional covariances of fxtgDfZytg,andwhere°denotes
the Hadamard product.
This theoretical example illustrates how possible lower and upper bounds for the correlation
depend on the type of linear map Z.LetZbe the following two-dimensional map:
ZD10
cos sin 11
where measures the extent to which the uncorrelated components are mapped in the same
direction. For D0 the map is not invertible yielding perfect correlation between the observed
variables, whereas for D1
2we have the identity map, so that the observed variables are
completely uncorrelated. Let the conditional variances of the uncorrelated components be denoted
8An alternative parameterization of the orthogonal matrix can be found in van der Weide (2002).
9Note that the values for the angles will depend on the ordering of the rotation matrices. The ordering should not affect
the estimation results.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
554 R. VAN DER WEIDE
by h1t,h
2t. It can be verified that the conditional correlation between the observed variables,
denoted by t, is given by:
tDh1tcos
h1th1tcos2Ch2tsin2
12
If we assume that hit >0, we can define ztDh2t/h1t,sothattcan be expressed as:
tD1
1Czttan2
13
For finite samples, the variable ztwill have finite lower and upper bounds. As a consequence, the
conditional correlation tis also bounded.
Note that a constant linkage Zgives rise to time-varying correlations between the observed
variables. These correlations rise on average when the components are mapped more in the same
direction. We can not exclude the possibility that the ‘economic mechanism’ Zevolves over
time. If so, endogenizing Zand making it time-varying might improve the fit of the time-varying
correlations. Extending the GO-GARCH model to allow for a non-constant Z, however, is left for
further research. A first step would be to test for a constant linkage, for example by means of test
on structural change such as the Chow test.
3. ESTIMATION
The parameters that need to be estimated by means of conditional information, include the vector
of rotation coefficients that will identify the invertible matrix Z(see Lemma 2-3), and the
parameters ˛, ˇ for the munivariate GARCH(1,1) specifications. The log likelihood L,˛,ˇ for
the GO-GARCH model can be represented as:
L,˛,ˇ D1
2
t
mlog2 Clog jVtjCxT
tV1
txt14
D1
2
t
mlog2 Clog jZHtZT
jCyT
tZT
ZHtZT
1Zyt15
D1
2
t
mlog2 Clog jZZT
jClog jHtjCyT
tH1
tyt16
where ZZT
DPPTis independent of . For the initial value of Htwe take the identity matrix,
which equals the implied unconditional covariance of fytg. Even in high-variate cases, when the
covariance matrices are very large, it should not be a problem to maximize the log likelihood
over the [mm 1]/2C2mparameters. Note that in order to avoid convergence difficulties
of estimation algorithms, we propose a kind two-step estimation. We exploit unconditional
information first, so that the number of parameters for Zthat are estimated through maximum
likelihood is [mm 1]/2 instead of m2(see lemma 2).
3.1. Consistency
Conditions for strong consistency of the maximum likelihood estimator for general multivariate
GARCH are derived by Jeantheau (1998). These conditions are verified by Comte and Lieberman
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 555
(2001) for the general BEKK model, in which a result of Boussama (1998), concerning the
existence of a stationary and ergodic solution to the multivariate GARCH(p, q) process, is used.
It can be verified that the more general BEKK model has the GO-GARCH model nested as a
special case (see van der Weide, 2002). Strong consistency of the quasi MLE for GO-GARCH can
therefore be established by appealing to Jeantheau’s conditions, following Comte and Lieberman.
To keep it simple, we focus on the GO-GARCH(1,1) model as in (4), but it can be verified that the
results also hold for the more general GO-GARCH(p,q) model. To apply the results of Jeantheau
(1998), we assume that the starting value of the process is drawn from its stationary distribution
P0, although Comte and Lieberman (2001) indicate that consistency holds for an arbitrary starting
value. We refer to Ling and McAleer (2002) for a more extensive discussion of the treatment of
the initial value and its implication for asymptotic properties.
Proposition 4 Consider the GO-GARCH(1,1) model, where ˛iand ˇidenote the GARCH(1,1)
parameters of the independent components. If the components are stationary, i.e.
˛iCˇi<1foriD1,...,m 17
then the MLE is consistent.
Proof. The result follows directly from the derivation of Comte and Lieberman (2001).
How to conduct inference is beyond the scope of this paper. However, as Comte and Lieberman
(2001) have proven asymptotic normality of the quasi-MLE for the BEKK model, having GO-
GARCH nested as a special case, we conjecture that this property is also inherited by GO-GARCH.
Some caution will be in place though, since we proposed a kind of two-step estimation which will
affect the distribution of the estimator. For example, the standard errors might be underestimated
by the Fisher Information matrix. We leave a precise study of the asymptotic distribution for
further work. For tests on possible misspecification of the multivariate GARCH model see Kroner
and Ng (1998), and the more recent paper by Tse (2002).
4. SIMULATION RESULTS
This section aims to illustrate the behaviour of the GO-GARCH model by experimenting with
artificial data.
4.1. Orthogonal Linkage
We constructed the independent components by generating from four univariate GARCH(1,1)
models to build a four-variate time series. The conditional variance of each component is
described by:
hi,t DciC˛iy2
i,t1Cˇihi,t1iD1,...418
The values that are assigned to the parameters c, ˛, ˇ aresummarizedinTableI.
The parameters are chosen so that variances are nearly integrated, which is commonly observed
in financial data. Also note that the parameters are chosen in such a way that some of the
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
556 R. VAN DER WEIDE
Table I. GARCH parameters
Component c˛ˇ
1 0.08 0.10 0.88
2 0.03 0.08 0.90
3 0.05 0.15 0.80
4 0.10 0.20 0.70
unconditional variances are identical. The length of the artificial data set is 3000 observations,
which is equivalent to approximately 12 years of daily data.
The first orthogonal matrix considered is the identity matrix. As it preserves independence, the
components will be observed directly. In the second part, we simulate with an orthogonal matrix
that induces dependence among the observed variables.
Independent multivariate GARCH
In this part, we test whether the models are able to detect the independent nature of the observed
data. It is known, and demonstrated by a theoretical example in subsection 2.2, that the O-GARCH
model cannot deal properly with virtually independent data. In contrast, GO-GARCH should be
able to estimate a linear representation that induces weak dependence or even independence. The
results are presented in Table II.
As expected, O-GARCH was not able to detect the independence of the process. The estimated
matrix is far from being diagonal, so that conditional dependence is ‘brought into’ the residuals.
The substantial errors in the GARCH parameters estimates also indicate that O-GARCH did not
extract the independent components, but some dependent variables instead. The GO-GARCH
model, however, performs very well in this example. The estimated linkage correctly reflects the
independent nature of the data. Also the GARCH parameters are estimated properly.10
Dependent multivariate GARCH
The independent components are described by exactly the same process as in the first part. The
key difference is that in this example they are not observed directly. The observed process will
be an orthogonal representation of the components that exhibits strong dependence. In principle,
the O-GARCH model could also perform well in this example, as the observed variables are no
longer independent, while the associated matrix is orthogonal. However, note that some of the
Table II. Estimates for the linkage and GARCH parameters
O-GARCH GO-GARCH
Z10.39 0.25 0.64 0.58 1.00 0.02 0.01 0.01
0.27 0.83 0.06 0.47 0.01 1.00 0.01 0.01
0.88 0.36 0.32 0.08 0.00 0.02 1.00 0.00
0.06 0.34 0.70 0.66 0.00 0.00 0.04 1.00
c0.09 0.04 0.03 0.09 0.03 0.02 0.05 0.10
˛0.09 0.06 0.08 0.10 0.11 0.06 0.15 0.20
ˇ0.81 0.90 0.89 0.80 0.86 0.92 0.80 0.70
10 Note that some components might have been switched.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 557
components have a similar scaling (unconditional variance). As a consequence, O-GARCH might
still suffer from identification problems, see subsection 2.2.
The orthogonal matrix, denoted by Z, is constructed as a product of four rotation matrices, and
is shown in Table III. Table IV summarizes the results.
In the case of O-GARCH, the estimates for the GARCH parameters are clearly different
from the true parameters suggesting that the model was not able to identify the independent
components. In the previous subsection we have seen that the trivial orthogonal matrix, namely
identity, could also not be identified by O-GARCH. Thus even when the linkage is truly
orthogonal, there is no guarantee that O-GARCH is able to identify it. The model additionally
requires that all the components have a different scaling, which might often not be the case (see
subsection 2.2).
When we look at the estimates of the GO-GARCH model, we find that the GARCH parameters
of the components are estimated with reasonable accuracy.11 From this we conclude that the
linkage estimated by GO-GARCH cannot be far from the ‘truth’ as we build it.
4.2. Non-orthogonal Invertible Linkage
In this subsection, non-orthogonal invertible matrices are chosen to link the independent com-
ponents with the observed process. This will be an important example, as we generalized the
O-GARCH model to be able to expose linkages that are not orthogonal. Note that in section 3.3
it was illustrated with a theoretical example that lower bounds for the conditional correlations can
be observed when the matrices approach singularity.
In order to have a more controlled experiment, we confine ourselves to the two-dimensional
case. Similar to the first examples, we construct two independent components in order to build up
Table III. Orthogonal linkage
Map Z
Matrix
p3
2p21
21
21
2p2
p3
2p2
1
21
21
2p2
p3
41
2p2
1
2p23
4
1
4
p3
2p2
p3
2p2
p3
4
Table IV. Estimates for the GARCH parameters
O-GARCH GO-GARCH
c0.05 0.08 0.05 0.07 0.10 0.02 0.05 0.03
˛0.06 0.07 0.07 0.06 0.20 0.06 0.15 0.11
ˇ0.89 0.84 0.89 0.87 0.70 0.92 0.80 0.86
11 Note that some components have been switched.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
558 R. VAN DER WEIDE
Table V. The GARCH parameters
Component c˛ˇ
1 0.05 0.15 0.80
2 0.05 0.25 0.70
Table VI. The invertible linkages
Map Z1Z2Z3Z4
Matrix 11
01
1
21
02
22
11
12
21
Table VII. The unconditional covariances
Map Z1Z2Z3Z4
Vi21
11
5
42
24
21
15
54
45
Table VIII. The true linear representations
Map Z1Z2Z3Z4
Wi1.41 1
01
2.24 2
01
0.47 0.75
0.94 0.75
0.75 1.49
1.49 0.75
a bivariate time series. The conditional variances of both components are specified by means of
the same GARCH model, as in (18). Also the sample size is chosen to be identical, namely 3000
observations. Table V lists the values at which the GARCH parameters were set to simulate the
independent components. It can easily be verified that both components have unit unconditional
variance, so that their unconditional covariance matrix equals the identity matrix. We will consider
four different invertible linear maps for the linkage. The associated matrices, denoted by Z1to
Z4, are shown in Table VI. The unconditional covariance matrix of the observed process is simply
given by: ViDZiZT
i. The covariances V1to V4are listed in Table VII.
The observed data is commonly normalized to have unit variance by a diagonal matrix
D, so that the covariance matrices of the normalized series Q
ViDDiZiZT
iDT
ihas 1’s along
the diagonal. In our example, the diagonal elements of D1to D4are easily seen to be
f1/p2,1g,f2/p5,1/2g,f1/p2,1/p5g,andf1/p5,1/p5g, respectively. It follows that the true
matrix that links the normalized observed variables with its independent components, is given by
DiZi1. These matrices, denoted by Wi, are shown in Table VIII.
Note that in all cases the orthonormal eigenvectors of the unconditional covariance matrix are
given by
PD1
p211
11
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 559
Table IX. The estimates for the linkages and the GARCH parameters
Map O-GARCH
Z1Z2Z3Z4
O
Wi0.54 0.54 0.51 0.51 0.61 0.61 0.53 0.53
1.31 1.31 2.17 2.17 0.86 0.86 1.59 1.59
Oci0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.06
O˛i0.18 0.12 0.21 0.13 0.14 0.23 0.11 0.12
O
ˇi0.76 0.83 0.73 0.82 0.82 0.72 0.84 0.81
Table X. The estimates for the linkages and the GARCH parameters
Map GO-GARCH
Z1Z2Z3Z4
O
Wi0.01 0.99 0.02 0.98 0.48 0.73 1.49 0.74
1.41 1.01 2.23 2.00 0.94 0.76 0.77 1.51
Oci0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
O˛i0.25 0.14 0.25 0.14 0.14 0.25 0.25 0.14
O
ˇi0.71 0.81 0.71 0.81 0.81 0.71 0.71 0.81
so that O-GARCH is expected to estimate scaled versions of Pfor the linkages. The results for
both the O-GARCH and the GO-GARCH model are presented in Table IX and X, respectively.
This example illustrates that the GO-GARCH model is able to deal with decompositions that
are not of the orthogonal type. The estimated linkages are in all cases very close12 to the ‘truth’,
the matrices from Table VIII. The estimates for the GARCH parameters are also accurate.
Apriori we know that the O-GARCH model cannot uncover the non-orthogonal linkages, as
it restricts the matrix to be orthogonal. As a consequence, it extracts components that are not
independent, which is reflected by the biased estimates for the GARCH parameters. Particularly
in example 4, the O-GARCH estimates for the GARCH parameters show substantial error. The
difference between the estimated correlations is therefore most notable in Example 4, which can
be seen in Figure 1.
In example 4, the GO-GARCH estimates for the correlations never fall below 0.8, say,
whereas the correlations estimated by O-GARCH show much stronger declines and some-
times even drop to below 0.4. The reason for this effect is that the matrix from example 4
shows the strongest ‘deviation’ from an orthogonal matrix. The linkage from example 4 maps
both independent components in almost the same direction which induces a strong correlation
between the observed variables. Exactly the same feature is observed in the empirical exam-
ple described in the next section. Indeed, it seems reasonable that observed variables that are
strongly related exhibit high correlation at all times. As the linkage with the components that
induces the high correlation is assumed constant over time, it will be surprising to observe
periods in time where the variables suddenly appear almost uncorrelated. This feature is illus-
trated by a theoretical bivariate example, where the lower bound and upper bounds of the
12 Neglect signs, as they do not yield a different representation. Also note that some components might have been switched.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
560 R. VAN DER WEIDE
1.0
0.0
500 1000 1500 2000 2500 3000
500 1000 1500 2000 2500 3000
500 1000 1500 2000 2500 3000
500 1000 1500 2000 2500 3000
0.2
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-1.0
-0.5
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RHO1_GOGARCH
RHO2_GOGARCH
RHO3_GOGARCH
RHO4_GOGARCH
1.0
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500 1000 1500 2000 2500 3000
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0.5
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RHO3_OGARCH
RHO1_OGARCH
RHO2_OGARCH
RHO4_OGARCH
Figure 1. The estimated time evolutions of the (conditional) correlations for all four examples. The figures
on the left correspond with the GO-GARCH model and those on the right with O-GARCH
correlation are derived as a function of a characteristic parameter of the linkage Z,in
subsection 2.4.
5. EMPIRICAL EXAMPLE
We include an example from real life, as an attempt to gain insight in the relation between
observed economic and financial variables and the uncorrelated factors that are assumed to drive
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 561
the market. Our example considers the Dow Jones Industrial Average (DJIA) versus the NASDAQ
composite. The sample contains more than ten years of daily observations, starting on 1 January
1990 and ending in October 2001. First, we estimate a first-order13 vector autoregressive (VAR)
model to account for the linear structure present in the data. Subsequently, we use the residuals to
estimate the conditional covariances from which the (conditional) correlations between the DJIA
and the NASDAQ can be computed. Questions that arise naturally include: (i) are non-orthogonal
linkages common in real-life examples? (ii) will allowing for a more general linkage (all invertible
matrices) typically induce a better description of the time-varying correlations between economic
and financial variables?
We estimate both the O-GARCH and GO-GARCH model, and compare their results. The
estimates are summarized in Table XI. To address the question whether non-orthogonal linkages
can be found in financial data, we first verify whether the estimated unrestricted matrix is
approximately orthogonal. Let
Z1
go denote the unrestricted representation, then
Z1
go
Z1
go
TD1.49 1.09
1.09 1.95
should be close to the identity matrix, which it is not. This confirms our conjecture that the
orthogonality assumption of O-GARCH is probably too restrictive, in that it might exclude many
of the linkages ‘observed’ in financial markets. In order to quantify the impact of the restrictions
imposed by O-GARCH, we compute the Likelihood Ratio statistic (LR) to test the O-GARCH
specification against GO-GARCH for several lengths of the time series. The results are listed in
Table XII. For all lengths of the time-series considered, the hypothesis of an orthogonal linkage
is rejected at a 1% level.
Even though the GO-GARCH model provides a better fit when compared with O-GARCH, it
might be that our more general model is still seriously misspecified. For this reason, we include
Table XI. The estimates for the linkages and GARCH parameters
Model O-GARCH GO-GARCH
Z10.55 0.55 1.18 0.32
1.19 1.19 0.58 1.27
c0.009 0.003 0.009 0.003
˛0.070 0.039 0.054 0.079
ˇ0.922 0.957 0.939 0.915
Table XII. Likelihood-Ratio Test of O-GARCH
against GO-GARCH
Length 250 500 1000 3082
LR 23.7 13.8 42.6 731.4
Note: The critical value of 2
1at a 1% level is 6.63.
13 Higher-order specifications do not significantly contribute to a better linear fit.
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
562 R. VAN DER WEIDE
Table XIII. Misspecification test: VAR model on the squares and the product of the
two residuals
Model Variable O-GARCH GO-GARCH
ε2
1,t ε2
2,t ε1,tε2,t ε2
1,t ε2
2,t ε1,tε2,t
c4.11 4.29 4.22 3.55 4.39 3.89
ε2
1,t11.40ŁŁ 0.08 0.27ŁŁ 1.09Ł0.52 0.70
ε2
2,t11.06ŁŁ 0.36 0.09 0.92 0.90 0.81
ε1,tε2,t 2.44ŁŁ 0.44 0.36 1.99Ł1.46 1.52
adj.R20.14 0.00 0.01 0.00 0.00 0.00
Note:Ł,ŁŁ significant at the 5% and 1% level.
a simple test for misspecification. Since we are interested in heteroskedasticity in particular, we
estimate a VAR model on the squares and the product of the two standardized residuals to verify
whether the conditional covariance has been modelled correctly. The results, which are shown
in Table XIII, suggest that GO-GARCH is not seriously misspecified. The remaining structure
found in the standardized GO-GARCH residuals is fairly weak. In contrast, the residuals from the
O-GARCH model still seem to exhibit significant persistence in volatility.
To examine to what extent the restrictions on the linkage affect the (conditional) correlations, we
compare the implied correlations of both models. The time evolution of the correlations is shown
in Figure 2, which reveals several interesting features. Perhaps most striking is that the correlations
estimated by GO-GARCH are much less volatile. Furthermore, the GO-GARCH correlations never
seem to fall below 0.6, say, whereas the more volatile correlations estimated by O-GARCH show
several substantial drops. In the beginning of 2000, the correlation according to O-GARCH is
even just below zero where the GO-GARCH correlations show no decline at all. Since the matrix
estimated by GO-GARCH is explicitly not orthogonal, we have reason to believe that O-GARCH
often underestimates the correlations (see subsection 2.4). Indeed, it seems plausible that the DJIA
and the NASDAQ, which are strongly related, exhibit high correlation at all times. Assuming that
the linkage does not change over time, it will be surprising to observe periods where the DJIA and
the NASDAQ appear to be almost unrelated. The differences observed for the year 2000, however,
Figure 2. The estimated correlations between the DJIA and the NASDAQ
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
GO-GARCH: A MULTIVARIATE GARCH MODEL 563
are kind of extreme. This might suggest that the linkage is not constant over time, or stronger,
that it was subject to a structural change. Indeed, at the beginning of 2000 we experienced a
technological boom which could explain our findings. To test the assumption of a constant linkage
will be left for further research.
6. CONCLUDING REMARKS
A new type of multivariate GARCH model is proposed that can best be seen as a generalization
of the O-GARCH model. It supports the assumption that the observed variables are driven by
some unobserved uncorrelated components, linked by a linear map. In order to identify these
components, we only need invertibility of the associated matrix. Under the null of O-GARCH,
however, the matrix is assumed orthogonal which only covers a very small subset of all possible
invertible matrices. Moreover, even when the matrix is truly orthogonal, the estimator proposed
by O-GARCH is not always able to identify it. The GO-GARCH model considers every invertible
matrix as a possible linkage, which will be parameterized in such a way that it is not expected to
complicate estimation while excluding any identification problems.
The model is tested on both artificial and financial data. The simulation results show that the
model correctly estimates both orthogonal and non-orthogonal invertible linkages. The results are
not affected by the scaling of the uncorrelated factors or a possible weak dependence among the
observed variables. The latter is known to be responsible for the identification problems of O-
GARCH, which is confirmed by our experimental results. The nature of the linkage, for example
whether it is orthogonal or not, is strongly related with the implied correlations between the
observed variables. This relation is made explicit by a theoretical example, and illustrated by
some of the simulation results. The effect of the linkage on the correlations is also observed
in the empirical example, the Dow Jones Industrial Average versus the NASDAQ. A likelihood
ratio test rejects the hypothesis that the associated matrix is orthogonal. In addition a simple test
for misspecification suggests that the GO-GARCH model provides a better description of the
process, and the conditional correlations in particular. We argue that by restricting the matrix to
be orthogonal, O-GARCH will often underestimate the correlations. The differences are kind of
extreme during the year 2000, which coincides with the technology boom that initiated early that
year. It could be that the linkage is not constant over time and that it experienced a structural
change during the technological bust in 2000. A test for such a structural break, and perhaps even
extending the model to allow for a time-varying linkage, is left for further research. Probably
the most important question that remains is to what extent GO-GARCH is able to improve the
modelling of very large covariance matrices. Indeed, it would be very interesting to compare
the model with other recently developed multivariate GARCH models, such as the Dynamic
Conditional Correlation model of Engle (2002). The misspecification tests recently proposed by
Tse (2002) can be used to measure performance.
ACKNOWLEDGEMENTS
An earlier version of this paper has been presented at the International Conference on Modelling
and Forecasting Financial Volatility, Perth, Western Australia, 7–9 September 2001. Stimulating
discussions with participants, in particular Robert Engle and the organizers Philip Hans Franses
and Michael McAleer, are gratefully acknowledged. I am especially grateful to Peter Boswijk,
Copyright 2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 549–564 (2002)
564 R. VAN DER WEIDE
Cees Diks, Roald Ramer and two anonymous referees as they provided helpful comments and
suggestions. I also thank Cars Hommes and Carol Alexander for reading preliminary versions
of the paper. This research has been supported by the Netherlands Organization for Scientific
Research (NWO) under a NWO-MaG Pionier grant. None of the above are responsible for errors
in this paper.
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