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

Let {dijj} be the sequence of T observed differences in loss functions for

models i and j. B block bootstrap counterparts are generated for all combina-

represent averages over all T observations. First we will establish how to esti-
mate the variance estimates rdτ(d_ij) and 7τr(d_i.), which are required for the
calculation of the EPA test statistics in (13), (14) and (15):

^B ,            4 9

B^-1 £ (3g> - 3ij.)

й=1

b            ₉

B^-1 £ ∙   - di.)

й=1

for all i, j ∈ M. In order to evaluate the significance of the EPA test a p-value is
required. That is obtained by comparing Tr or Tsq with bootstrap realisations

T^ib _or T^ib,
tr ^or ' sq

b

p_τ = B^-1 £ 1 f∕'i"ⁱ >7'-)  for τ = R, SQ

b=1

1 if A is true

0 if A is false .

The B bootstrap versions of the test statistics Tr or Tsq are calculated by re-
placing ∣di_j∙ jand (dij)² inequations (13) and (14) with ∣d^ib — d{j∖ and (d^ — dij^
respectively. The denominator in the test statistics remains the bootstrap esti-
mate discussed above.

This model elimination process can be used to produce model specific p-
values. A model is only accepted into .VL_l if its p-value exceeds a. Due to the
definition of .VL, this implies that a model which is not accepted into .VL, is
unlikely to belong to the set of best forecast models. The model specific p-values
are obtained from the p-values for the EPA tests described above. As the fcth

15

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