j - 12mth direct EWMA volatility -- 12mth orthogonal GARCH volatility J

MM *° M g i & i\ &

- 4yr direct EWMA volatility -- 4yr orthogonal GARCH volatility }

« 5 <s 2 « i hi sa

- 6mth direct EWMA volatility -- 6mth orthogonal GARCH volatility

-2yr direct EWMA volatility -- 2yr orthogonal GARCH volatility

50 i---------------

0 I- ->-1-P-,--T-,- ,- ,- - - - -,-1-rJ

Cl;i3i>csii>3Q.icii>Tliii:i:

j - 7yr direct EWMA volatility - - 7yr orthogonal GARCH volatility

Figure 7.13 Calibration of an orthogonal GARCH model to UK zero-coupon yields. Table 7.5a: Eigenvalue analysis

Component Eigenvalue Cumulative R2

Pi Pi

5.9284117 1.9899323 0.97903180

0.53894652 0.71984946 0.80885235

Table 7.5b: Factor weights

model is good, but could be improved further if the 10-year yield were excluded from the system.

All elements of the covariance matrix are obtained from the variances of just two principal components that represent the most important sources of information - other variations are ascribed to noise and are not included in the model

The GARCH(1,1) model estimates for the first two principal components are given in Table 7.6. This time the second principal component has a better-conditioned GARCH model: the tilts in the UK yield curve are less temporary and more important than they are in the crude oil term structure discussed above. As a consequence the orthogonal GARCH correlations will be more stable than the correlations in Figure 7.12.

Figure 7.14 shows some of the orthogonal GARCH correlations for the UK zero-coupon yields. Not only does the orthogonal method provide a way of estimating GARCH volatilities and volatility term structures that may be impossible to obtain by direct univariate GARCH estimation, they also give very sensible GARCH correlations, which would be very difficult indeed to estimate using direct multivariate GARCH. All elements of the covariance matrix are obtained from the variances of just two principal components that represent the most important sources of information - other variations are ascribed to noise and are not included in the model.

Table 7.6: GARCH(1,1) models of the first and second principal component

Jan-92 Jul-92 Jan-93 Jul-93 Jan-94 Jul-94 Jan-95

- 1yr-2yr -- 3mth-2yr - 6mth-12mth

Figure 7.14 Orthogonal GARCH correlations of UK zero-coupon yields.

A useful technique for parameterizing multivariate GARCH models is to compare the GARCH volatility estimates from the multivariate GARCH with those obtained from direct univariate GARCH estimation. Similarly, when calibrating an orthogonal GARCH model one could compare the volatility and correlation estimates with those obtained from other models, such as an EWMA correlation model or other multivariate GARCH models. There are, of course, problems with this. What choice of smoothing constants should be made when the comparison is with EWMA volatilities? If the system is large, convergence problems may very well be encountered, so how sure can one be about the validity of the diagonal vech or BEKK multivariate GARCH parameter estimates?

We now show some results from calibrating orthogonal GARCH models to multivariate GARCH models. Since multivariate GARCH is not easy to use for large systems, a relatively small system of four European equity indices is used: France (CAC), Germany (DAX), the Netherlands (AEX) and the UK (FTSE 100).27 A PCA on daily return data from Morgan Stanley Index prices, from 1 April 1993 to 31 December 1996 gives the results in Table 7.7.

Table 7.7a: Factor weights

The weights on the first principal component are comparable and quite high. Since this is the trend component the indices are, on the whole, moving together, and the eigenvalue analysis in Table 7.7b indicates that common movements in the trend explain 67% of the total variation over the 4-year period.

The third and fourth principal components are often more important in equity systems than in term structures, and this case is no exception. Thus all four principal components have been used in the orthogonal GARCH model, that

27 Many thanks to Dr Aubrey Chibumba for producing these results as part of his MPhil thesis at Sussex University.