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

and a semi-quadratic statistic,

as test statistics to establish EPA. Both test statistics indicate a rejection of the

EPA hypothesis for large values. The actual distribution of the test statistic
is complicated and depends on the structure between the forecasts included
in M. Therefore p-values for each of these test statistics have to be obtained
from a bootstrap distribution (see below). When the null hypothesis of EPA
is rejected, the worst performing model is removed from M. The latter is
identified as Mi where

■                          d_{i                                               z}

г = arg max—                           (15)

ⁱ∈^ ∖jυad(di,'}

and di, = ^~γ ^_j∙∈^ dij. The tests for EPA are then conducted on the reduced
set of models and one continues to iterate until the null hypothesis of EPA is
not rejected.

Bootstrap distributions are required for the test statistics Tr and Tsq.
These distributions will be used to estimate p-values for Tr and Tsq tests and
hence calculate model specific p-values. At the core of the bootstrap procedure
is the generation of bootstrap replications of di_j∙y. In doing so, the tempo-
ral dependence in dʊ^ must be accounted for. This is achieved by the block
bootstrap, which is conditioned on the assumption that the {di_j∙y} sequence is
stationary and follows a strong geometric mixing assumption. The basic steps
of the bootstrap procedure are now described.

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