10.2 An empirical illustration


10.2.1 Data description

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 $ T = 3720$ 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 20606 summarize command. Before inspecting the summary statistics, we load the data, $ R_t$, and take log differences, $ \varepsilon_t=\ln (R_t) - \ln
(R_{t-1})$. 20609 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"

20613 XFGmvol01.xpl

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.

Figure 10.1: Foreign exchange rate data: returns.

\includegraphics[width=1.5\defpicwidth]{mvolmfxrate.ps}

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.


10.2.2 Estimating bivariate GARCH

The quantlet 20719 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 $ C_0, \, A_{11}$ and $ G_{11}$ 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 $ C_0$, $ A_{11}$ and $ G_{11}$ 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 $ A_{11}$ and $ G_{11}$ values around 0.3 and 0.9 appear reasonable, since in univariate GARCH(1,1) models parameter estimates for $ a_1$ and $ g_1$ in (10.3) often take values around $ 0.3^2=0.09$ and $ 0.81=0.9^2$. There is no clear guidance how to determine initial values for off diagonal elements of $ A_{11}$ or $ G_{11}$. Therefore it might be reasonable to try alternative initializations of these parameters. Given an initialization of $ A_{11}$ and $ G_{11}$ the starting values for the elements in $ C_0$ are immediately determined by the algorithm assuming the unconditional covariance of $ \varepsilon _t$ 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

20723 XFGmvol02.xpl

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 $ C_0$, the following four belong to the ARCH ($ A_{11}$) and the last four to the GARCH parameters ($ G_{11}$), i.e. for our model

$\displaystyle \Sigma_t = C_0^\top C_0 + A_{11}^\top \varepsilon _{t-1} \varepsilon _{t-1}^\top A_{11} + G_{11}^\top \Sigma_{t-1} G_{11}$ (10.7)

stated again for convenience, we find the matrices $ C_0$, $ A$, $ G$ to be:

$\displaystyle C_0 = 10^{-3} \begin{pmatrix}1.15&.31\\ 0&.76\end{pmatrix},$    
$\displaystyle A_{11}= \begin{pmatrix}{\hspace*{0.2cm}.282}&-.050\\ -.057&{\hspa...
...293}\end{pmatrix},\, G_{11}= \begin{pmatrix}.939&.028\\ .025&.939\end{pmatrix}.$ (10.8)


10.2.3 Estimating the (co)variance processes

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 ( $ \Sigma_0=\frac{E^\top E}{T}$). Here, the $ T \times 2$ vector $ E$ contains log-differences of our foreign exchange rate data. Estimating the covariance process is also accomplished in the quantlet 20863 XFGmvol02.xpl and additionally provided in sigmaprocess .

Figure: Estimated variance and covariance processes, $ 10^5\hat{\Sigma_t}$.

\includegraphics[width=1.5\defpicwidth]{mvolmcovar.ps}

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 ( 20875 XFGmvol03.xpl ).

Figure: Simulated variance and covariance processes, both bivariate (blue) and univariate case (green), $ 10^5\hat{\Sigma_t}$.

\includegraphics[width=1.6\defpicwidth]{mvolmcovarsimul.ps}

For this at each point in time an observation $ \varepsilon _t$ is drawn from a multivariate normal distribution with variance $ \Sigma_t$. Given these observations, $ \Sigma_t$ is updated according to (10.7). Then, a new residual is drawn with covariance $ \Sigma_{t+1}$. We apply this procedure for $ T=3000$. 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 $ \xi_t$. In this case, however, we set off-diagonal parameters in $ A_{11}$ and $ G_{11}$ 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.