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-
tions of i and j,
{<■}
for b = 1, ..., B. Values with a bar, e.g. dij = T 1 ɪ) dijj,
represent averages over all T observations. First we will establish how to esti-
mate the variance estimates rdτ(dij) and 7τr(di.), which are required for the
calculation of the EPA test statistics in (13), (14) and (15):
B , 4 9
υτ(dij )
τ(di,)
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
Tib or Tib,
tr or ' sq
b
pτ = B-1 £ 1 f∕'i"i >7'-) for τ = R, SQ
b=1
1(A) = I
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 ∣dij∙ jand (dij)2 inequations (13) and (14) with ∣dib — 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 .VLl 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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