Table 7.4: GARCH(1,1) models of the first and second principal components

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Figure 7.11 Orthogonal GARCH term structure volatility forecasts for 1-month crude oil futures.

to the EWMA volatilities illustrated in Figure 7.10. Why use GARCH, then? There are two important reasons. The first is that EWMA volatility term structure forecasts do not converge to the long-term average, but GARCH forecasts do. provided a + p < 1. Although univariate GARCH analysis of the crude oil data indicated IGARCH behaviour and therefore no convergence in term structures (§4.2.3), Table 7.4 shows that the principal components do have convergent volatility term structures.25 It is therefore a simple matter to extend the orthogonal GARCH parameter estimates to provide forecasts of the average volatility over the next n days, for any n (using the formulae given in §4.4.1). Orthogonal GARCH volatility terms structures for the 1-month future are shown in Figure 7.11, for every day during the six year period.

Another good reason to use orthogonal GARCH rather than orthogonal EWMA is that the orthogonal GARCH correlations will more realistically reflect what is happening in the market. As already mentioned, the correlations

25 Recall from §4.4.1 that the size of u + p determines the speed of convergence of the volatility term structure. So the second (tilt) component in Table 7.4 has a more rapidly convergent volatility term structure than the first (trend) component. This reflects the fact that the prices of short-maturity crude oil futures are more variable than the prices of long-maturit\ futures.

0.85-

0.8-

0.75- j

0.7--1-1-J-1-1-

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j-1mth-3mth 3mth-6mth 6mth-9mth 9mth-12mth Figure 7.12 Some of the correlations from the orthogonal GARCH model.

shown in Figure 7.10 that were generated by the orthogonal EWMA are a little worrying. One would expect correlations between commodity futures to be more or less perfect most of the time, but the EWMA correlations between the 1-month futures and other futures, and between other pairs at short maturities, appear to be considerably below 1 for long periods of time. For example, during long periods of 1996 and 1998 the EWMA correlations are nearer to 0.8 than 1.

The reason for this is that the smoothing constant of 0.95 - which from Table "".4 is an appropriate choice for the EWMA volatility of the first and most important principal component - is clearly too large for the correlations. Unfortunately, if one were to reduce the values of the smoothing constants used in the orthogonal EWMA model so that the correlations were less persistent, the volatilities would also be less persistent.

In the crude oil futures market, price decoupling only occurs over very short time spans so correlations may deviate below 1, but only for a short time. Now, if the orthogonal model were to be used with just one principal component (the results from §6.2.3 indicate that this trend component explains over 95% of the variation) the correlations would of course be unity. So all the variation in the orthogonal GARCH correlations is coming from the movements in the second principal component. This second principal component is the tilt component, and it only explains about 3% of the movement (see Table 6.4a).

The GARCH(1,1) models of the first two principal components of this term structure, given in Table 7.4, indicate that the second principal component has a lot of reaction (a is about 0.22) but little persistence (P is about 0.66). In other words these tilt movements in the term structure of futures prices are intense but short-lived. So one would expect the correlations given by the orthogonal GARCH model in Figure 7.12 to be more accurately reflecting real market conditions than the orthogonal EWMA correlations in Figure 7.10.

This example has shown how 78 different volatilities and correlations of the term structure of crude oil futures between 1 month and 12 months can be generated, very simply and very accurately, from just two univariate GARCH models of the first two principal components. It has also shown how volatility forecasts of different maturities can be generated as simple transformations of these two basic GARCH variances.

The orthogonal GARCH model is particularly useful for term structures that have some illiquid maturities

The orthogonal GARCH model is particularly useful for term structures that have some illiquid maturities. When market trading is rather thin, there may be little autoregressive conditional heteroscedasticity in the data and what is there may be rather unreliable, so the direct estimation of GARCH volatilities is very problematic. The orthogonal GARCH model has the advantage that the volatilities of such assets, and their correlations with other assets in the system, are derived from the principal component volatilities that are common to all assets and the factor weights that are specific to that particular asset.

To illustrate this point let us step up the complexity of the data a little. Still a term structure, but a rather difficult one: the daily zero-coupon yield data in the UK with 11 different maturities between 1 month and 10 years from 1 January 1992 to 24 March 1995, shown in Figure 6.3b. It is not an easy task to estimate univariate GARCH models on these data directly because yields may remain relatively fixed for a number of days. Particularly on the more illiquid maturities, there may be insufficient conditional heteroscedasticity for GARCH models to converge well and the direct estimation of GARCH models on these data is rather problematic. Since it has not always been possible to obtain convergence for even univariate GARCH models on these data, the orthogonal GARCH volatilities in Figure 7.13 have been compared instead with EWMA volatilities.26

The orthogonal GARCH volatilities are not as closely aligned with the EWMA volatilities as they were in the previous example, but there is sufficient agreement between them to place a fairly high degree of confidence in the orthogonal GARCH model. Again two principal components were used in the orthogonal GARCH, but the PCA of Table 7.5 shows that these two components only account for 72% of the total variation (as opposed to over 99% in the crude oil term structure).

Clearly the lower degree of accuracy from a representation with two principal components is one reason for the observed differences between the orthogonal GARCH volatilities and the EWMA volatilities. Another is that the 10-year yield has a very low correlation with the rest of the system, as reflected by its low factor weight on the first principal component, which is quite out of line with the rest of the factor weights on this component. The fit of the orthogonal

Interest rates have less persistence in volatility than equities or commodities, so a smoothing constant of 0.9 has been used.