188 Forecast Combination and Encompassing
An implication of evaluating methods rather than models is that it may be opti-
mal to combine forecasts from the data-generating process with those from other
models. This situation is ruled out when models are compared. Clements and
Hendry (1998, Ch. 10) provide the following illustration based on an AR(1) pro-
cess,yt=ψyt− 1 +vt, wherevt∼i.i.d.N(0,σv^2 )and|ψ|<1. Then theh-step-ahead
conditional MSFE, assuming an in-sample size ofRobservations, is:
E[eˆ^2 R+h|yR]=σv^2(
1 −ψ^2 h)
(
1 −ψ^2) +E[(
ψh−ψˆh) 2
]y^2 R, (4.19)whereψˆis the OLS estimator of the unknown parameterψ. The first term is the con-
tribution of future disturbances, and the second is due to parameter uncertainty.
Using the asymptotic formula in Baillie (1979, equation 1.6, p. 676):
E[
ˆeR^2 +h|yR]
=σv^2(
1 −ψ^2 h)(
1 −ψ^2) +h^2 ψ^2 (h−^1 )R−^1(
1 −ψ^2)
y^2 R. (4.20)Chong and Hendry (1986) note thath^2 ψ^2 (h−^1 )in the second term of (4.20) has a
maximum ath=− 1 /lnψ, and so is not monotonic. It is straightforward to show
that (4.20) will exceed the unconditional variance of the process (σy^2 =E
(
y^2 t)
=σv^2
(
1 −ψ^2)− 1
) when:
yR^2
σy^2>Rψ^2
h^2(
1 −ψ^2),where the unconditional variance can be viewed as the expected squared forecast
error of a forecast of zero (the unconditional mean), ̄y=0. This establishes that
the data-generating forecast can be beaten in terms of (squared-error loss) accuracy
when there is estimation uncertainty. Moreover, consider the combined forecast:
y ̃R+h=βy ̄+( 1 −β)yˆR+h=( 1 −β)yˆR+h, (4.21)where 0≤β≤1, withh-step-ahead forecast error:
e ̃R+h=yR+h−y ̃R+h=βe ̄R+h+( 1 −β)eˆR+h, (4.22)whereeˆR+h≡yR+h−yˆR+hande ̄R+h≡yR+h−y ̄=yR+h. MinimizingE
[
̃eR^2 +h|yR]with respect toβyields:
βh∗=(
1 +
Rψ^2
h^2(
1 −ψ^2))− 1. (4.23)
Clements and Hendry (1998) compare the performance ofˆyR+h,y ̄andy ̃R+hin
terms of the unconditional MSFE, and establish that there are gains to forecast
combination.
The key differences between Giacomini and White (2006) and the approach of
Diebold and Mariano (1995) and West (1996) (DMW) to testing predictive ability