194 Forecast Combination and Encompassing
As∂μ∂β 0 e=−1, the optimal value forβ 0 ,β 0 ∗, solves
∂g(
μe,σe^2)∂μe =0.β∗
0 depends on
L(.), and is set to generate the optimal amount of bias (μ∗e) given the form ofL(.).
For squared-error loss,L
(
et)
=e^2 t,E[
L(
et)]
=μ^2 e+σe^2 , and∂g(
μe,σe^2)∂β 0 =−^2 μe,so
that the optimal amount of bias is, of course, zero (μ∗e=0).
Consider the first-order condition with respect toβ:
∂g(
μe,σe^2)∂β=∂g(
μe,σe^2)∂σe^2∂σe^2
∂β=0.Provided
∂g(
μe,σe^2)∂σe^2 =0,∂g(
μe,σe^2)∂β =0 implies that∂σe^2
∂β =0, so from (4.30),2 22 β∗= 2 σ 21 andβ∗=− 221 σ 21 irrespective of the form ofL(.), matching the
expression for squared-error loss.
As Elliott and Timmermann (2004) remark, if an element ofβ∗is zero under
squared-error loss, then the corresponding forecast will also receive zero weight
under any other loss function, assuming that the stated properties of the fore-
cast error distribution hold. In the general case, it will not be possible to set up a
general forecast encompassing test that does not depend on the form of the loss
function.
4.7 Conclusions
We have discussed the different types of standard linear forecast combination that
are commonly applied in the literature and the related tests of forecast encom-
passing. The tests of forecast encompassing depend upon whether the forecasts
are generated by models with unknown parameters and on whether the underly-
ing aim is to compare the forecasts themselves or the models on which they are
based. There is also an important distinction to be drawn between conditional and
unconditional tests.
More sophisticated forms of combination are reviewed, including nonlinear
forms of combination that might be useful when large numbers of forecasts are
available, and types of combination that might be preferable when the forecasts
are density or probability forecasts. For the most part, forecast accuracy is assessed
by the standard squared-error loss, although under certain conditions on the
data-generating process forecast encompassing is invariant to the form of the loss
function.
Notes
- For the specification FE(2′) one would usedt=(e 1 t− ̄e 1 )[(e 1 t− ̄e 1 )−(e 2 t− ̄e 2 )], and for
FE(3′),dt=(e 1 t−e ̄ 1 )(f 2 t− ̄f 2 ). - These include,inter alia, Granger (1969), Zellner (1986), Christoffersen and Diebold
(1997), Clements (1997), Elliott, Komunjer and Timmermann (2005), Patton and
Timmermann (2007) and Clements (2008).