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Bodnar

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									       Multivariate GARCH Process: an Elliptical Approach
                                     Taras Bodnar∗
   Department of Statistics, European University Viadrina, PO Box 1786, 15207
                                Frankfurt (Oder), Germany

   Modeling the distributional properties of asset returns turns out to be an impor-
tant point for the both researches and practitioners of the financial sector. Different
distributional assumptions imposed on the returns distribution may influence the
portfolio allocation and the decision about the reconstruction of the holding portfolio
or keeping the weight for the individual assets at the same level as in the previous
time period. In the following we distinguish between the two existence in the recent
literature approaches (modeling the unconditional distributions of the asset returns
or using the models for the conditional characteristics of the returns like conditional
mean vector and covariance matrix) while make an attempt to give a link between
them.
   The first distribution applied in modeling financial data appears to be a normal
distribution. Fama (1976) found that the normal assumptions provide a rather good
fit in case of data taken with the monthly or smaller frequency. However, for the more
frequent observations the heavy-tailed distributions should be maintained. Fama
(1965) proposed to use the mixture of normality, while Blattberg and Gonedes (1974)
compared the ability of the multivariate t- and symmetric stable distributions in
explaining the financial data.
   The possible generalization of these both models is based on the elliptical con-
toured distributions. This is the more general class of distributions that includes as
a partial case all the mixtures of normality (see e.g., Fang et. al (1990, p. 48)).
They have been already applied in modeling financial data. For instance, Owen and
Rabinovitch (1983) extended several well-known in finance theorems, like, Tobin’s
separation theorem, Bawa’s rules of ordering certain prospects to elliptical contoured
distributions while Chamberlain (1983) showed that this family of distributions im-
plies the mean-variance utility functions. More recently, Berk (1997) proved that
the one of the necessary condition of the validity of the capital asset pricing model
(CAPM) is an elliptical distribution of asset returns. Zhou (1993) and Hodgson
et. al (2002) derived the tests for the CAPM under the assumption of the elliptical
   ∗
       E-mail address: bodnar@euv-frankfurt-o.de

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symmetry.
   The situation becomes more difficult when the uncertainty about the mean vector
µ and the covariance matrix V of the asset returns are taken into account. In this
case the estimator for the weights of the optimal portfolio is obtained by plugging
               ˆ     ˆ
the estimators µ and V instead of the unknown parameters µ and V. Because, in
                              ˆ     ˆ
general, the distributions of µ and V do depend on the selected model of asset returns
we would loss the nice properties derived under the assumption of normality (see e.g.,
Britten-Jones (1999), Gibbons et. al (1989), Bodnar and Schmid (2005)). To avoid
the point the matrix elliptical distributions were suggested in the literature among
others by Eftekhari and Satchell (1996) and Bodnar (2004). The other possibility is
to forecast the pseudo-generating variable of the vector of asset returns and then to
estimate their dispersion matrix. This is done in the present paper.
   The second approach applied in the financial literature is based on modeling
the conditional characteristics of asset returns. Since the seminal work of Engle
(1982) several multivariate generalizations of the GARCH(p,q) process were sug-
gested. While the described above models deal with modeling the unconditional
distribution of asset returns, the multivariate GARCH processes were designed based
on the conditional covariance structure. In general, it is assumed that the vector of
asset returns Xt follows the following stochastic process
                                         1
     Xt = µt + η t    with η t = Ht2 εt .                                           (1)

µt is conditional mean vector given the sigma field Ft−1 generated by the all informa-
tion till time t − 1, i.e. µt = E(Xt |Ft−1 ), and Ht = Cov(Xt |Ft−1 ) is the conditional
covariance matrix of Xt . To avoid the problem with modeling the stochastic behavior
of µt , which can be done, for example, by fitting the VARMA process, usually, the
process {Xt − µt } is considered as the new data for Ht .
   In the paper we introduce a new class of multivariate models, called multivariate
elliptical GARCH (MElGARCH) process. The presented model puts together the
conditional time varying properties of the multivariate GARCH processes and the
elliptical symmetry of the mixture of normal distributions given by

     Xt = Rt εt ,                                                                   (2)
                      p                       q
      2                              2                 2
     Rt = α0 +             αi Xt−i   Σ   +         βj Rt−j ,                        (3)
                     i=1                     j=1


                                                    2
where α0 > 0, αi ≥ 0, βj ≥ 0, and y      Σ   is the norm of the vector y with respect to
the positive definite matrix Σ equal to       y Σ−1 y. εt are assumed to be independently
identically normally distributed with the mean vector 0 and the covariance matrix
Σ, i.e. εt ∼ N (0, Σ). The assertion we denote by {Xt } ∼ M ElGARCHk (Σ, p, q).
   The idea behind the MElGARCH process is to replace the unobservable pseudo-
generating variable R by its forecast Rt given the information available at time point
t. From the equation (2) and the independency of Rt and εt it follows that Xt is
elliptically contoured distributed. More precisely, Xt follows the mixture of normal
distribution with the unknown shape of elliptical symmetry. Its density function does
always exists and is completely specified by the density function of Rt (Fang et. al
(1990)).
   The designed process possesses the generality of the GARCH process and the
similarity of the elliptical distributions. While it is suggested that the conditional
covariance matrix of asset returns varies with time, the dispersion matrix Σ is kept to
be a constant. The last assumption can be violated by considering the time-varying
covariance matrix of the White noise process εt . This generalization is not treated in
the paper, but present a possible extension of the obtained results and is left for future
researches. The other appealing properties of the MElGARCH process is that the
linear combinations of its elements follow the MElGARCH with the same coefficients
α0 > 0, αi ≥ 0, βj ≥ 0. In the partial case we obtained that the components of the
vector Xt follow the univariate GARCH processes. This result is explained by the
elliptical nature of the considered process.
   The one of the most important applications of the MElGARCH process leads
to the portfolio theory. Because of its elliptical structure the MElGARCH process
satisfies the necessary conditions of the validity of the CAPM model derived by Berk
(1997). Fitting the MElGARCH model to asset returns we suggests the non-varying
efficient frontier in the mean-variance space. From the other side it is allowed the
variability of the optimal portfolios within the efficient frontier. The changes are
explained by the time-varying investor’s coefficient of risk aversion. These results are
not unexpected and possess the intuitive economical interpretation. When the gut
news arrive the market an investor expects more from the market profitability, which
drives the coefficient of risk aversion down. From the other side the bad news are
accompanied with the higher volatility of the market and lead the larger investor’s


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risk aversion in order to avoid the possible loses.
   In the empirical study the daily data of EUR/USD, JAP/USD, and GBP/USD
exchange rates are used for estimating the dispersion matrix of the process. The two
stage quasi maximum likelihood estimation is used for estimating the parameters of
the MElGARCH process. The obtained results do not support the volatile behavior of
the dispersion matrix even during the Asian and Russian crisis. The time variability
of the covariance matrix is explained by the time varying behavior of the generating
variable that influences the coefficient of the investor’s risk aversion. Our results are
in line with the findings of Bollerslev et. al (2004), who argued that the coefficient
of risk aversion is not constant through time.




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