-2mth direct EWMA volatility---2mth orthogonal EWMA volatility

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

-8mth direct EWMA volatility---8mth orthogonal EWMA volatility

- 10mth direct EWMA volatility---10mth orthogonal EWMA volatility

- 12mth direct EWMA volatility---12mth orthogonal EWMA volatility

Figure 7.9 Comparison of direct and orthogonal EWMA volatilities for crude oil futures.

illustrates some of the output of using the orthogonal EWMA method with three principal components of the crude oil futures data shown in Figure 6.4. The PCA of these data has already been discussed in §6.2.3. The great advantage in using the orthogonal EWMA method on term structure data is that all the volatilities and correlations in the system can be derived from just two or three EWMA variances. That is, instead of estimating 78 EWMA volatilities and correlations directly, using the same value of the smoothing constant throughout, only two or three EWMA variances of the trend, tilt and perhaps also the curvature principal components need to be generated. In some term structures, including the crude oil futures term structure used for Figure 7.9. a mere two components already explain over 99% of the variation, so adding a third component makes no discernible difference to the covariance results.

0.7 -I-1-1-1-1-1-r-

Feb-93 Feb-94 Feb-95 Feb-96 Feb-97 Feb-98 Feb-99

- 1 mth-3mth - - 3mth-6mth 6mth~9mth 9mth-12mth Figure 7.10 Some correlations from the orthogonal EWMA model.

From just two EWMAs, the entire 12 x 12 covariance matrix of the original system is recovered with negligible loss of precision. The orthogonal EWMA correlations shown in Figure 7.10 are very similar indeed to the correlations generated by direct EWMA. But there is a problem with using EWMAs at all in the crude oil futures market because there are some price decouplings between some of the near-maturity futures. This problem will be a point of discussion based on Figure 7.12 below, where the same correlations are measured by the orthogonal GARCH model.

It is extremely difficult to use multivariate GARCH to generate covariance matrices of more than a few dimensions

7.4.3 Orthogonal GARCH

It has been shown in §4.5 that large covariance matrices that are based on GARCH models would have many advantages. But it was also shown that it is extremely difficult to use multivariate GARCH to generate covariance matrices of more than a few dimensions. Some models that use only univariate GARCH to generate covariance matrices were discussed in §4.5.3, and there we referred to the orthogonal GARCH model that is now described.

A principal components representation is a multi-factor model, and the idea of using factor models with GARCH is not new. Engle et al. (1990) use the capital asset pricing model to show how the volatilities and correlations between individual equities can be generated from the univariate GARCH variance of the market risk factor. Their results have a straightforward extension to multi-factor models, but unless the factors are orthogonal a multivariate GARCH model on the risk factors will still be required.

The orthogonal GARCH model is a generalization of the factor GARCH model introduced by Engle et al. (1990) to a multi-factor model with orthogonal factors. The idea of using PCA for multivariate GARCH modelling goes back to Ding (1994). However, there Ding used the full number of

principal components in the representation and the strength of the orthogonal GARCH model rests, crucially, on using a reduced space of principal components, as explained by Alexander and Chibumba (1996) and more recently developed in Alexander (2000a, 2001b) and Klaassen (2000).24

The orthogonal GARCH model allows x GARCH covariance matrices to be generated from just m univariate GARCH models. Normally m, the number of principal components, will be much less than , the number of variables in the system. This is so that extraneous noise is excluded from the data and the volatilities and correlations produced become more stable. In the orthogonal GARCH model the m x m diagonal matrix of variances of the principal components is a time-varying matrix denoted D„ and the time-varying covariance matrix V, of the original system is approximated by

V, = AD, A,

(7.17)

where A is the x m matrix of rescaled factor weights. The model (7.17) is called orthogonal GARCH when the diagonal matrix D, of variances of principal components is estimated using a GARCH model. In the examples given here the standard vanilla GARCH(1,1) model (4.2) is used.

The representation (7.17) will give a positive semi-definite matrix at every point in time, even when the number m of principal components is much less than the number of variables in the system. Of course, the principal components are only unconditionally uncorrelated, but the assumption of zero conditional correlations has to be made, otherwise it misses the whole point of the model, which is to generate large GARCH covariance matrices from GARCH volatilities alone. The degree of accuracy that is lost by making this assumption is investigated by a thorough calibration of the model, comparing the variances and covariances produced with those from other models such as EWMAs or, for small systems, with full multivariate GARCH. Care needs to be taken with the model calibration, in terms of the number of components used and the time period used to estimate them, but once calibrated the orthogonal GARCH model may be run very quickly and efficiently on a daily basis.

The strength of the orthogonal GARCH model rests on using a reduced space of principal components

The remainder of this section examines how to calibrate orthogonal GARCH(1,1) models to term structures and to equity/foreign exchange systems. The first example of an orthogonal GARCH model is a straightforward extension of the crude oil term structure example presented in §7.4.2, using GARCH(1,1) variances of the first two principal components in place of EWMAs.

Table 7.4 presents the GARCH(1,1) parameter estimates of the first two principal components. The orthogonal GARCH volatilities that result are very close indeed to the direct GARCH volatilities. In fact, they are almost identical

24 Commercial software that generates large orthogonal GARCH matrices has been available since 1996 from S-Plus GARCH and since 1998 from www.algorithmics.com. Excel add-ins for the orthogonal GARCH model may be available from cleighfa- dial.pipex.com and MBRM (details on the CD).