why it is advisable to use a long averaging period in historical volatility estimates. On the other hand, the longer the averaging period, the longer-lasting the ghost effects from exceptional returns (see Figure 3.1).

5.2.2 GARCH Models

The covariance matrix of the parameter estimates in a GARCH model can be used to generate GARCH confidence intervals for conditional variance. For example, the variance of the one-step-ahead variance forecast in a GARCH(1, 1) model is

V,($+i) = V,(S>) + K,(6t)e? + ,( ) ? + 2cov,(to, &)E2

+ 2 cov,(co, J3)cr2 + 2 cov,(a, #

The estimated covariance matrix of parameter estimates is part of the standard output in GARCH procedures. It will depend on the sample size used to estimate the model: as with the error bounds for moving average estimates given above, the smaller the sample size the bigger these quantities. However, in the GARCH forecast by far the largest source of uncertainty is the unexpected return at time t. If there is a large and unexpected movement in the market, in either direction, then e2 will be large and the effect in (5.4) will be to widen the confidence interval for the GARCH variance considerably.

Consider the GARCH(1, 1) models discussed in §4.3.2. Table 5.2 reports upper and lower bounds for 95% confidence intervals for the 1-day-ahead GARCH variance forecasts that are generated using (4.16) with a number of different values for unexpected daily return. Note that these returns will be squared in the symmetric GARCH(1, 1), so it does not matter whether they are positive or negative. The confidence interval for the variance is quoted in annualized terms assuming 250 days a year.

Note that the GARCH confidence intervals are much wider following a large unexpected return - rather than uncertainty in parameter estimates, it is market behaviour that is the main source of uncertainty in GARCH forecasts. These confidence intervals are for the variance, not the volatility, but may be translated into confidence intervals for volatility.17 For example, the confidence intervals for an unexpected return of 0.005 translate into a confidence interval in volatility terms of (30.8%, 33.1%) for the S&P 500, (44.6%, 46%) for the CAC and (36.5%, 37.9%) for the Nikkei 225.

5.2.3 Confidence Intervals for Combined Forecasts

Suppose a process volatility a is being forecast and that there are m different

Percentiles are invariant under monotonic differentiable transformations, so the confidence limits for volatility are the square root of the limits for variance.

The GARCH confidence intervals are much wider following a large unexpected return - rather than uncertainty in parameter estimates, it is market behaviour that is the main source of uncertainty in GARCH forecasts

Unexpected return

S&P 500

Nikkei 225

Lower

Upper

Lower Upper

Lower Upper

0.001 0.002 0.003 0.004 0.005

0.10080

0.099647

0.098144

0.096568

0.094967

0.10345 0.10460 0.10610 0.10768 0.10928

0.20367 0.20273 0.20144 0.20006 0.19865

0.20649 0.20742 0.20871 0.21009 0.21151

0.13829 0.13722 0.13606 0.13489 0.13371

0.14022 0.14130 0.14245 0.14362 0.14480

forecasting models available. Denote the forecasts from these models by ct,, 2, . . ., am and suppose that each of these forecasts has been made over a data period from t - 0 to t=T. Now suppose we have observed, ex post, a realization of the process volatility over the same period. The combined forecasting produce of Granger and Ramanathan (1984) applied in this context requires a least squares regression of the realized volatility a on a constant and ,, 2, . . ., . The fitted value from this regression is a combined volatility forecast that will fit at least as well as any of the component forecasts: the R2 between realized volatility and the combined volatility forecast will be at least as big as the R2 between realized volatility and any of the individual forecasts. The estimated coefficients in this regression, after normalization so that they sum to one, will be the optimal weights to use in the combined forecast.

Figure 5.3 shows several different forecasts of US dollar exchange rate volatilities for sterling, the Deutschmark and the Japanese yen. Some general observations on these figures are as follows:

> The GARCH forecasts seem closer to the implied volatilities.

> The historic 30-day and the EWMA forecasts with X = 0.94 are similar

(because the half-life of an EWMA with X - 0.94 is about 30 days). »- The EWMA forecasts are out of line with the other 90-day forecasts

(because the EWMA model assumes a constant volatility, the 90-day

forecasts are the same as the 30-day forecasts). > There is more agreement between the different 30-day forecasts than there is

between the different 90-day forecasts (because uncertainties increase with

the risk horizon).

*- The volatility forecasts differ most at times of great uncertainty in the markets, for example during 1994 in sterling and during 1990 in the Deutschmark. On the other hand, they can be very similar when nothing unusual is expected to happen in the market, for example during 1990 in the yen.

Now suppose that a combination of these forecasts is to be used as a single forecast for realized volatility. Table 5.3 summarizes the results of the ordinary

For the Granger-Ramanathan procedure these forecasts should also be independent.

Table 5.2: GARCH variance 95% confidence interval bounds on 11 September 1998