184 Forecast Combination and Encompassing
There are, however, a number of special cases where the forecast encompass-
ing tests do not require correction for model parameter estimation uncertainty.
First, ifπ=0, then the parametersδdgandδggare zero regardless of the model
estimation scheme, ensuring thatin (4.16) reduces toS. Thus if the ration/Ris
very small, a case could be made for abstracting from the issue of model estimation
uncertainty and proceeding with the unadjusted tests of the previous section. Intu-
itively, this arises because, ifRis very large compared ton, the model parameter
estimation uncertainty becomes relatively insignificant compared to the uncer-
tainty that would be present in the testing problem even if the model parameters
were known.
A second case where model estimation corrections are not required is when the
two models are linear, estimated by least squares, with each involving just a single
regressor, as in the example above, and the forecast encompassing approach is
FE(1). This case is considered by Clements and Harvey (2006). As noted in the
previous section, for the FE(1)MDMtest,dt=η 1 tη 2 t, withη 1 tandη 2 tthe errors
from regressions ofytandf 2 t, respectively, on a constant andf 1 t. Hence, letting
C(., .)denote a covariance:
dt={
[yt−E(yt)]−C(yt,f 1 t)
V(f 1 t)
[f 1 t−E(f 1 t)]}×{
[f 2 t−E(f 2 t)]−C(f 1 t,f 2 t)
V(f 1 t)
[f 1 t−E(f 1 t)]}
.For forecasts from linear single regressor models, results such asE
(
fit)
=θiE(
Xit)
,
i=1, 2, are obtained, so thatdtcan be written as:
dt=θ 2{
[yt−E(yt)]−
C(yt,X 1 t)
V(X 1 t)
[X 1 t−E(X 1 t)]}×{
[X 2 t−E(X 2 t)]−
C(X 1 t,X 2 t)
V(X 1 t)
[X 1 t−E(X 1 t)]}
.Clearly,∂dt/∂θ 1 =0, and:
E(
∂dt
∂θ 2)
=C(yt,X 2 t)−C(X 1 t,X 2 t)C(yt,X 1 t)
V(X 1 t). (4.17)
The null hypothesis ofβ 2 =0 in the FE(1) regression implies that:
V(f 1 t)C(yt,f 2 t)−C(f 1 t,f 2 t)C(yt,f 1 t)=0.Substituting forf 1 tandf 2 tin this expression, and dividing both sides byθ 12 θ 2
(noting thatθ 1 =0,θ 2 =0), the right-hand side of (4.17) equals zero. SoD=(0, 0)
under the null, and estimation uncertainty is irrelevant asymptotically for FE(1),
contrasting with the findings using theMDMvariants of the FE(2) and FE(3) tests.
Finally, West and McCracken (1998) show that while tests based on the FE(3)
specification do require adjustment for estimation uncertainty, an alternative to
directly estimatingexists when the models are linear. Consider augmenting