Michael P. Clements and David I. Harvey 181As in the previous section, assume that a record ofnpast forecasts and actuals
are available for evaluation, denoting the corresponding forecast errors byeˆ 1 tand
ˆe 2 t. Thenforecasts can be derived using model parameter estimates obtained in
one of three main ways. First, afixedestimation scheme involves a one-off esti-
mation ofθˆ 1 tandθˆ 2 tusing data from, say,t=1,...,R, and then using that same
set of estimates to producenforecasts fromt=R+htoR+n+h−1. Second, a
recursiveestimation scheme might be adopted, where the sample used for estima-
tion uses all available information at each point, increasing by one observation per
period, i.e., the models are first estimated overt=1,...,Rto produce forecasts for
t=R+h, then the model parameters are re-estimated overt=1,...,R+1 to give
forecasts fort=R+ 1 +h, etc. Finally, arollingscheme uses a moving window ofR
observations to estimate the models, so that recent data is included, but more dis-
tant observations discarded, i.e., the initial estimation sample is againt=1,...,R
for forecasts of the periodt=R+h, thent=2,...,R+1 for use in forecasts for
t=R+ 1 +h, etc.
The encompassing tests of the previous section must now be constructed using
dˆt, defined asdtfor the appropriate forecast encompassing specification FE(1),
FE(2) or FE(3), but based on the quantitiesfˆ 1 t,fˆ 2 t,eˆ 1 tandeˆ 2 t, which embody the
estimated parametersθˆ 1 tandθˆ 2 t. The results of West and McCracken (1998) and
West (2001) show that the additional uncertainty implicit indˆt affects the
asymptotic variance ofdˆ. We now have:
√
n[dˆ−E(dt)]⇒N(0,), (4.15)wheredˆ=n−^1
∑R+n+h− 1
t=R+h dˆtand:=S+δdg(DBS′dg+SdgB′D′)+δggDBSggB′D′, (4.16)withSdenoting the long run variance ofdtas before, and:
D=E[
∂dt/∂θ 1 ∂dt/∂θ 2]B=[
[E(X^21 t)]−^10
0 [E(X 22 t)]−^1]Sgg=∑∞
j=−∞E(gtg′
t−j), gt=[
e 1 tX 1 t
e 2 tX 2 t]Sdg=∑∞
j=−∞E{[dt−E(dt)]g′
t−j},