Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 193

so that the combination weights are selected to minimize
∑
te

2
t, the sum of squares
of the forecast error for the combined forecast. Similarly, the “variance-covariance
approach” of Bates and Granger (1969) selects the weights to minimize the variance
of the forecast error of the combination of forecasts. Nevertheless, a number of
papers have allowed for asymmetric loss, and have considered how the properties
of optimal forecasts change once we dispense with the assumption of symmetric
loss, as well as providing tests of rationality once we allow forecasters to have
asymmetric loss functions.^2
A key paper that investigates forecast combination in the context of asymmetric
loss is Elliott and Timmermann (2004). They show that, for general loss func-
tions and forecast error distributions, the optimal combination weights depend on
higher-order moments of the forecast error distribution, such as the skew. How-
ever, under certain restrictions on the form of the forecast error distribution, they
establish aninvarianceresult, whereby the optimal combination weights on the
individual forecasts are identical to the squared-error loss weights for almost all
loss functions and that only the value of the constant term in the combination
will differ. The value of the constant is chosen to generate the optimal amount
of bias in the combination given the degree of asymmetry of the loss function.
Their invariance result holds when the marginal distribution of the forecast errors
depends only on the first two moments of the forecast errors, which holds when the
joint distribution of the actual and forecasts(yt,ft)′is elliptically symmetric (which
includes the multivariate normal andt-distributions: see Elliott and Timmermann,
2004, Proposition 2, p. 53).
Suppose:

E

(

yt

ft

)

=

(

μy

μ

)

, C

(

yt

ft

)

=

(

σy^2 σ 21 ′

σ 21  22

)

,

then the forecast combination error is:

et=yt−β 0 −β′ft,

with moments:
μe=μy−β 0 −β′μ (4.29)

σe^2 =σy^2 +β′ 22 β− 2 β′σ 21. (4.30)

The decision maker selects

(

β 0 ,β

)

according to:

min

β 0 ,β

∫

L

(

et

)

dF

(

et

)

,

i.e., to minimizeE

[

L

(

et

)]

. Under elliptical symmetry we can writeE

[

L

(

et

)]

=

g

(

μe,σe^2

)

. From (4.29) and (4.30), onlyμedepends onβ 0. Thus the first-order

condition for minimizingE

[

L

(

et

)]

with respect toβ 0 is:

∂g

(

μe,σe^2

)

∂β 0

=

∂g

(

μe,σe^2

)

∂μe

∂μe

∂β 0

=0.