-0.4 \ , , , , , , , , , , j

Jan-93 May-93Sep-93 Jan-94May-94Oct-94 Feb-95 Jun-95 Oct-95 Mar-96 Jul-96May-96

(a) - Diagonal Veen -Orthogonal GARCH

0 \-,-,-,-, ,-,-, ,-,-,-j

Jan-93 May-93 Sep-93 Jan-94 May-94 Oct-94 Feb-95 Jun-95 Oct-95 Mar-96 Jul-96 May-96

Jan-93 May-93 Sep-93 Jan-94 May-94 Oct-94 Feb-95 Jun-95 Oct-95 Mar-96 Jul-96 May-96

-BEKK - Diagonal VECH - Orthogonal GARCH

Figure 4.13 GARCH models correlation comparison: (a) US dollar-sterling and Japanese yen-sterling exchange rates; (b) FTSE 100 and DAX equity indices; (c) FTSE 100 and CAC equity indices.

Currently many practitioners prefer to use EWMA rather than GARCH correlations. They do have limitations, including the fact that their term structure forecasts do not mean-revert (§3.3), and GARCH is certainly the superior method for modelling volatility: one of the great advantages of univariate GARCH models is that they generate mean-reverting term structure forecasts for volatility in a very simple analytic form. But is there really a robust term structure of correlation that mean-reverts? Correlation is such a limited concept anyway, because it assumes that co-dependencies are linear and it ignores the co-dependency through common trends in prices.

Given the uncertainties surrounding correlation in financial markets, it may indeed be sufficient, from a modelling perspective, to use EWMA correlations. They can be particularly useful for constructing a time-varying commodity futures hedge because univariate GARCH models of commodity volatilities are often close to being integrated. An EWMA is an approximation to an I-GARCH model (§4.3.3) and they are very easy to compute. In short, the end results from a skilful parameterization and calibration of a bivariate GARCH model may just not be worth the effort.

Figure 4.14 compares two different estimates of beta for Danone stock in the CAC from July 1994 to February 1999. At certain times these estimates differ substantially, for example during most of 1996 and in the first few months of 1998. The 120-day equally weighted estimate is the beta that would be obtained when applying ordinary least squares to a CAPM with about 6 months of daily data. It ranges between about 0.6 and 1, but typically an even longer period would have been used for the betas supplied by data analytics firms, and these betas would have stayed roughly within the 0.7-0.8 range for the whole time. However, the exponentially weighted beta (with X - 0.94) ranges from less than 0.3 to over 1.7, and changes rapidly over the course of a few days. This EWMA beta estimate uses about 100 days of data (0.94100%0.002) so the statistical error of these estimates is certainly higher than the error on the 120-day equally weighted average (§5.2.1). However, it is not so much greater that all the differences between the two estimates in Figure 4.14 could be attributed to sampling error.

There has been some interesting research on the use of bivariate GARCH to calculate the optimal futures hedge (Cecchetti et al., 1988; Baillie and Myers, 1991; Kroner and Claessens, 1991; Lien and Luo, 1994; Park and Switzer, 1995). Recall from §4.2.3 that the daily WTI spot and prompt futures prices have integrated GARCH models for their volatilities. This is often the case for commodities. Table 4.11 shows the GARCH(1, 1) parameter estimates for the natural gas futures prices that were discussed in Chapter 3. Figure 4.15a shows the GARCH volatilities that are estimated from these models.

Since the GARCH models are near to being integrated, an EWMA model will be adequate for the calculation of an optimal time-varying proxy hedge ratio and there is no real need to use a bivariate GARCH. Figure 4.15b shows the

In short, the end results from a skilful parameterization and calibration of a bivariate GARCH model may just not be worth the effort

1.8 1.6 1.4 1.2 1

0.8 0.6 0.4 0.2

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-120-day --EWMA

Figure 4.14 Equally and exponentially weighted betas for Danone in CAC.

Table 4.11: GARCH(1, 1) models for natural gas futures

EWMA hedge ratio corresponding to X = 0.87. It is more variable than the hedge ratios that were based on the equally weighted average methods that were discussed in §3.1.3. Therefore, there may be a greater need for rebalancing during the course of the hedge than is normally assumed in practice.

Another important application of time-varying correlation is the derivation of optimal weights in a minimum-risk portfolio (§7.2). However, since the correlations used to derive these weights must be consistent with a positive definite covariance matrix, it is not possible to estimate them in a bivariate setting. Instead we must use a full multivariate GARCH model on all the assets in the portfolio. These models are described in the next subsection, and if there are very many assets in the portfolio the methods that are outlined in §4.5.3 will be necessary to overcome the difficulties of computing a large-dimensional GARCH covariance matrix.

4.5.2 Multivariate GARCH Parameterizations

In an -dimensional multivariate GARCH model there are n conditional mean equations, which can be anything, but for the sake of parsimony it is normal to assume the form