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

where f_t^c is the combination forecast. In this context, the combination parame-
ters have been estimated based on a rolling window of forecasts. A fixed set of
weights was deemed to be inappropriate in this context as the level of volatility
was substantially higher during the latter part of the sample period, see Figure
1. Therefore, to form a combination forecast at time t, fξ, combination weights
were obtained by estimating equation 7 on forecasts from t — 500 to t — 1, and
then combining the various individual forecasts ft formed at time t using these
weights. Allowing for the initial period of 500 forecasts to be used for estima-
tion, the final I960 forecasts are used for comparative purposes. The specific
composition of the combination forecasts will be discussed in Section 4 as they
are motivated by results based on the individual forecasts.

3.3 Evaluating forecasts

As argued above, it is the objective of this paper to determine whether com-
bination forecasts are superior to individual MBF and IV. At the heart of the
model confidence set (MCS) methodology (Hansen, Lunde and Nason, 2003)
as it is applied here, is a forecast loss measure. Such measures have frequently
been used to rank different forecasts and the two loss functions utilised here are
the, MSE and QLIKE,

MSEⁱ = (RV _i+22 — ft)²,                      (9)

QLIKEⁱ = log(ft) ₊ ^RV'"" ,                  (10)

^ft

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