Are combination forecasts of S&P 500 volatility statistically superior?

model is eliminated from M, save the (bootstrapped) р-value of the EPA test
in (13) or (14) as p (к). For instance, if model Mi was eliminated in the third
iteration, i.e. к = 3. The р-value for this zth model is then p) = ιιιax∕,<₃p(k).
This ensures that the model eliminated first is associated with the smallest
р-value indicating that it is the least likely to belong into the MCS⁸.

4 Empirical results

Results pertaining to individual forecasts, including the VIX and MBF will
be discussed first, followed by those including the combination forecasts. The
exact composition of the combinations will be outlined once the individual
forecasts are compared as their composition is motivated by the performance
of the individual forecasts. Rankings based simply on the loss functions, MSE
and QLIKE will be discussed first followed by an examination of the MCS.

4.1 Individual forecasts

Table 1 reports the ranking based on both MSE and QLIKE for all of the
individual forecasts. The rankings given the two different loss functions differ
slightly, as they penalise forecast errors differently. A number of interesting
patterns emerge. The ARMA and ARFIMA time series forecasts based on RV
produce the most accurate forecasts of RV¢+22, confirming similar results (e.g.
Anderson et al., 2003). Another obvious result is that VIX⁹ is not amongst the
most accurate forecasts under either loss function, although it does better under
the asymmetric loss function. Further it is apparent that the GARCH models
⁸See Hansen et al. (2005) for a detailed interpretation for the MCS p-values.

⁹Poon and Granger (2003) suggest to divide the VIX by /∕365∕252 to account for the
difference between calender month and trading days.

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