We analyze daily quotes of two European currencies measured
against the USD, namely the DEM and
the GBP. The sample period is December 31, 1979 to
April 1, 1994, covering observations. Note that a
subperiod of our sample has already been investigated by
Bollerslev and Engle (1993) discussing common features of volatility
processes.
The data is provided in
fx
. The first column contains DEM/USD and the second
GBP/USD. In
XploRe
a preliminary statistical analysis is
easily done by the
summarize
command. Before inspecting the summary statistics,
we load the data,
, and take log differences,
.
XFGmvol01.xpl
produces the following table:
[2,] " Minimum Maximum Mean Median Std.Error" [3,] "-----------------------------------------------------------" [4,] "DEM/USD -0.040125 0.031874 -4.7184e-06 0 0.0070936" [5,] "GBP/USD -0.046682 0.038665 0.00011003 0 0.0069721"
Evidently, the empirical means of both processes are very close to zero (-4.72e-06 and 1.10e-04, respectively). Also minimum, maximum and standard errors are of similar size. First differences of the respective log exchange rates are shown in Figure 10.1.
As is apparent from Figure 10.1, variations of exchange rate returns exhibit an autoregressive pattern: Large returns in foreign exchange markets are followed by large returns of either sign. This is most obvious in periods of excessive returns. Note that these volatility clusters tend to coincide in both series. It is precisely this observation that justifies a multivariate GARCH specification.
The quantlet
bigarch
provides a fast algorithm to estimate the BEKK
representation of a bivariate GARCH(1,1) model. QML-estimation is implemented by means of
the BHHH-algorithm which minimizes the negative Gaussian log-likelihood function. The
algorithm employs analytical first order derivatives of the log-likelihood function
(Lütkepohl; 1996) with respect to the 11-dimensional vector of parameters containing the
elements of
and
as given in (10.6).
The standard call is
{coeff, likest}=bigarch(theta, et),
where as input parameters we have initial values theta for the iteration
algorithm and the data set, e.g. financial returns, stored in et. The
estimation output is the vector coeff containing the stacked elements of the
parameter matrices ,
and
in (10.6) after numerical
optimization of the Gaussian log-likelihood function. Being an iterative procedure the
algorithm requires to determine suitable initial parameters theta. For the
diagonal elements of the matrices
and
values around 0.3 and 0.9 appear
reasonable, since in univariate GARCH(1,1) models parameter estimates for
and
in (10.3) often take values around
and
. There is no clear
guidance how to determine initial values for off diagonal elements of
or
. Therefore it might be reasonable to try alternative initializations of these
parameters. Given an initialization of
and
the starting values for the
elements in
are immediately determined by the algorithm assuming the unconditional
covariance of
to exist, Engle and Kroner (1995).
Given our example under investigation the bivariate GARCH estimation yields as output:
Contents of coeff [ 1,] 0.0011516 [ 2,] 0.00031009 [ 3,] 0.00075685 [ 4,] 0.28185 [ 5,] -0.057194 [ 6,] -0.050449 [ 7,] 0.29344 [ 8,] 0.93878 [ 9,] 0.025117 [10,] 0.027503 [11,] 0.9391 Contents of likest [1,] -28599
The last number is the obtained minimum of the negative log-likelihood function. The
vector coeff given first contains as first three elements the parameters of the
upper triangular matrix , the following four belong to the ARCH (
) and the
last four to the GARCH parameters (
), i.e. for our model
The (co)variance is obtained by sequentially calculating the difference equation
(10.7) where we use the estimator for the unconditional covariance matrix as
initial value (
). Here, the
vector
contains
log-differences of our foreign exchange rate data. Estimating the covariance process is
also accomplished in the quantlet
XFGmvol02.xpl
and additionally provided in
sigmaprocess
.
We display the estimated variance and covariance processes in Figure 10.2. The upper and the lower panel of Figure 10.2 show the variances of the DEM/USD and GBP/USD returns respectively, whereas in the middle panel we see the covariance process. Except for a very short period in the beginning of our sample the covariance is positive and of non-negligible size throughout. This is evidence for cross sectional dependencies in currency markets which we mentioned earlier to motivate multivariate GARCH models.
Instead of estimating the realized path of variances as shown above, we could also use
the estimated parameters to simulate volatility paths (
XFGmvol03.xpl
).
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For this at each point in time an observation
is drawn from a multivariate normal
distribution with variance
. Given these observations,
is updated
according to (10.7). Then, a new residual is drawn with covariance
. We apply this procedure for
. The results, displayed in the upper
three panels of Figure 10.3, show a similar pattern as the original process
given in Figure 10.2. For the lower two panels we generate two variance processes
from the same residuals
. In this case, however, we set off-diagonal
parameters in
and
to zero to illustrate how the unrestricted BEKK model
incorporates cross equation dynamics. As can be seen, both approaches are convenient to
capture volatility clustering. Depending on the particular state of the system, spillover
effects operating through conditional covariances, however, have a considerable impact on
the magnitude of conditional volatility.