Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 185

the FE(3) regression with the regressors from Model 1, i.e., in the above example,
replacing the FE(3) regression with:

ˆe 1 t=λfˆ 2 t+γX 1 t+εt.

West and McCracken (1998) show that tests ofλ=0 from this augmented regres-
sion only require standard autocorrelation and heteroskedasticity robust variance
estimators, such as those of the previous section, and do not need estimators that
explicitly account for the model parameter estimation uncertainty. In terms of an
MDM-type testing approach, this augmented version of FE(3) can be executed by
computing (4.14) withdˆt=uˆ 1 tuˆ 2 tin place ofdt, whereuˆ 1 tanduˆ 2 tdenote the
residuals from regressions ofeˆ 1 tandfˆ 2 t, respectively, onX 1 t.

4.4 Nested model comparisons

The results of the previous section apply in the case where the rival forecasting
models are non-nested. However, it is also common in forecast evaluation exercises
for the forecasts under consideration to be obtained from models that are, in fact,
nested. The primary situation where this arises is when forecast encompassing
tests are employed to help determine whether a particular variable is useful for
prediction, by testing whether a forecast based on a model including that variable
as a regressor is encompassed by a forecast from the same model with that variable
excluded. In such situations, the forecasts are asymptotically equivalent under the
encompassing null hypothesis, and this affects the usual asymptotic results derived
under a non-nested model assumption.
Clark and McCracken (2001) examine the asymptotic properties of FE(2)-based
forecast encompassing tests for the special case of one-step-ahead forecasts (h=1),
when the models are nested, linear and estimated by OLS. Consider the following
nested models:

Model 1:yt=X 1 ′tθ 11 +e 1 t

Model 2:yt=X 1 ′tθ 21 +X 2 ′tθ 22 +e 2 t,

where the vectorsX 1 tandX 2 tcontaink 1 andk 2 regressors, respectively. The
corresponding forecasts are denoted by:

fˆ 1 t=X 1 ′tθˆ 11 t

fˆ 2 t=X 1 ′tθˆ 21 t+X 2 ′tθˆ 22 t,

where the parameter vectors are first estimated from the above models, using data
prior to timet. Under the null hypothesis thatf 1 tencompassesf 2 t, Model 2 con-
tainsk 2 redundant variables (those inX 2 t), and the population forecastsf 1 tand
f 2 tare identical. Under the conditions outlined by Clark and McCracken (2001),
which include conditionally homoskedastic forecast errors, the asymptotic null
distribution of the FE(2)MDMstatistic (4.14) (withdˆtreplacingdt), for the general