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

puted,

dij,t = L(RV _mfi) - L(RV _t+∞,fi ),    i,J = l,∙∙∙,^o,    t = 1,∙∙∙,T

(H)

where LÇ) is chosen to be one of the loss functions described above. At each
step, the EPA hypothesis

Ho : E (d_ij,t) = 0,    ∀ i>J EM               (12)

is tested for a set of models M C Mo, with M = Mo at the initial step. If Ho
is rejected at the significance level a, the worst performing model is removed
and the process continued until non-rejection occurs with the set of surviving
models being the MGS, M*_a. If a fixed significance level a is used at each step,
MX contains the best model from Mo with (1 - a) confidence⁷.

At the core of the EPA statistic is the t-statistic
where d_ij = ɪ ∑J⅛₁ d_ijj,. tij provides scaled information on the average dif-
ference in the forecast quality of models i and J. var(dij) is an estimate of
var(dij) and is obtained from a bootstrap procedure described below. In order
to decide whether, at any stage, the MGS must be further reduced, the null
hypothesis in (12) is to be evaluated. The difficulty being that for each set M
the information from (m — 1) m/2 unique t-statistics needs to be distilled into
one test statistic. Hansen, et al. (2003, 2005) propose the following the range
statistic,

⁷See Hansen et al. (2005) for a discussion of this interpretation.

13

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