Figure 5.3 Volatility forecasts: (a) 30-day US dollar-sterling; (b) 90-day US dollar-sterling; (c) 30-day German mark-US dollar; (d) 90-day German mark-US dollar: (e) 30-day Japanese yen-US dollar; (f) 90-day Japanese yen-US dollar.

Market Models Table 5.3: Combined forecasts of realized volatility

GBP DEM JPY

least squares regressions of realized volatility on the GARCH, historic, EWMA and implied volatility forecasts shown in Figure 5.3. Some general conclusions may be drawn from the estimated coefficients and the ?-statistics (shown in parentheses). Firstly, the GARCH forecasts take the largest weight in the combined forecast, although they are not always the most significant. In fact implied volatilities are often very significant, particularly for the 30-day realized volatilities. Historic and EWMA forecasts have low weights, in fact they are often less than zero, so they appear to be negatively correlated with realized volatility.19 In each case the fitted series gives the optimal combined forecast for realized volatility. Note that the intercept is quite large and negative in most cases, which indicates that the forecasts have a tendency to overestimate realized volatility.

Figure 5.4a shows two combined forecasts of 30-day realized volatility of the GBP-USD rate. Forecast 1 uses all four forecasts as in Table 5.3, and forecast 2 excludes the EWMA forecast because of the high multicollinearity with the historic 30-day forecast (§A.4.1). In each case the model was fitted up to April 1995 and the last 6 months of data are used to compare the model predictions with the realized volatility out-of-sample in Figure 5.4b. Since the 30-day realized volatility jumps up 30 days before a large market movement, it is a very difficult thing to predict, particularly when markets are jumpy. There is, however, a reasonable agreement between the forecast and the realized volatility during less volatile times, and the out-of-sample period marked with a dotted line on the figures happened to be relatively uneventful.

19The low and insignificant coefficients on the historic 30-day forecast and the EWMA are a result of a high degree of multicollinearity of these variables (§A.4.1). In fact their unconditional correlation estimate during the sample is over 0.95 in all three models. When there is a high level of correlation between some explanatory variables the standard errors on coefficient estimators will be depressed, and the danger is that the model will be incorrectly specified.

May-88 May-89 May-90 May-91 May-92 May-93 May-94 May-95

(a) -30-day Realised----Combined Forecast 1 - - Combined Forecast 2

Figure 5.4 (a) Thirty-day realized volatility and combined forecasts of the US dollar-sterling rate; (b) out-of-sample performance of the US dollar-sterling combined forecast; (c) 90-day realized volatility and combined forecasts of the US dollar-sterling rate.