180 Forecast Combination and Encompassing
with the corresponding specification fordtbeingdt=η 1 tη 2 t; in practice,η 1 tand
η 2 tcan be replaced with their residual counterpartsηˆ 1 tandηˆ 2 t, respectively. For
FE(3), testing can proceed by settingdt=e 1 tf 2 t.^1 Variants ofR 1 andR 2 can also
be constructed for FE(1) and FE(3) in a straightforward manner; see, for example,
Newbold and Harvey (2002) for FE(1).
Tests for forecast encompassing can also be devised for the ANN combina-
tion. This would require the computationally more burdensome estimation of
θ=
(
α,β 1 ,...βk,γ 1 ,...γp,δ 1 ,...δp)
by NLS, rather than choosing theγias randomdraws from aU(−1, 1)distribution: i.e., minimizingQn(θ)=
∑n
t= 1
[
yt−ft(θ)] 2
to
giveθˆn. We can then use the results that, under general conditions,θˆnconverges
toθ∗, where:
θ∗=arg min
θ
E[
yt−ft(θ)] 2
,and that
√
n(
θˆ−θ∗)
⇒N(
0,θ)
, whereθcan be consistently estimated, to con-duct inference. The null hypothesis thatf 1 tencompassesf 2 tbased on the ANN
combination can be constructed as a test of the joint significance of all the param-
eters related tof 2 t, i.e.,β 2 =γ 21 =...=γ 2 p=0: see White (1989), Kuan and
White (1994) and the discussion by Franses and van Dijk (2000, pp. 230–2) for
details.
4.3 Model-based forecasts
The analysis and forecast encompassing tests considered in the previous section
treat the forecasts as given. However, in many practical applications forecasts are
obtained using estimated regression models, and the impact of estimation uncer-
tainty on the encompassing tests then needs to be examined if we wish to assess the
predictive ability of the underlying models. West and McCracken (1998) and West
(2001) study the impact of estimation uncertainty for the forecast encompassing
specifications FE(3) and FE(2) respectively, drawing on the work of West (1996),
although the general results are equally applicable to FE(1). Suppose, by way of a
simple example, that the forecastsf 1 tandf 2 tare generated using the non-nested
regression models:
Model 1: yt=θ 1 X 1 t+e 1 t
Model 2: yt=θ 2 X 2 t+e 2 t,where the scalar regressorsX 1 tandX 2 tare assumed to be stationary and well
behaved, and whereE(e 1 tX 1 t)=E(e 2 tX 2 t)=0. Given estimates of the model
parameters (θˆ 1 tandθˆ 2 t) using data prior to timet, the corresponding forecasts can
then be constructed as:
fˆ 1 t=θˆ 1 tX 1 t
ˆf 2 t=θˆ 2 tX 2 t.