192 Forecast Combination and Encompassing
whereWt∗is anFt-measurable function.Wt− 1 plays the same role asht− 1 in the
Conditional Forecast Encompassing Test of Giacomini and White (2006) and the
same issues relate to its selection. The sample moment function is given by:
gn(θ)=
1
n∑nt= 1[
α− 1(
yt−θ′qˆt< 0)]
w∗t− 1 , (4.26)and the GMM estimator ofθ∗, denoted byθˆn, is the solution of:
min
(θ 1 ,θ 2 )∈[
gn(θ)]′ˆ
S−n^1[
gn(θ)]
, (4.27)where:
Sˆn=^1
n∑nt= 1[
α− 1(
yt−θt′qˆt< 0)] 2
w∗t− 1 w∗′t− 1 , (4.28)andθˆnis obtained by solving (4.27) and (4.28) iteratively, starting with (4.27) and
Sˆn=I. This givesθˆn
(
Sˆn=I)
, after which we obtain an updated estimate ofSˆnfrom
(4.28), etc.
Giacomini and Komunjer (2005, Propositions 1 and 2, p. 420) establish the
consistency and asymptotic normality ofθˆn. Specifically, under some conditions:
(
γ′S−^1 γ)− 1 / 2 √
n(
θˆn−θ∗)
⇒N(0, 1),whereγ≡−E[ft
(
θ∗′qt)
Wt∗− 1 q′t],S=E[g(
θ∗;Yt,W∗t− 1)
g(
θ∗;Yt,Wt∗− 1)′
], andftis
the conditional density ofYt. Given the asymptotic distribution ofθˆn, the CQFE
(Conditional Quantile Forecast Encompassing) test thatqˆ 1 tencompassesqˆ 2 tis
given by:
ENCn=n(
θˆn′−(1, 0))
ˆ−n^1(
θˆn′−(1, 0))′
,whereˆ−n^1 is a consistent estimate of=
(
γ′S−^1 γ)− 1. Under the null,ENCn⇒χ 22
asn→∞. The estimate forSis given by (4.28), and that forγis given by Giacomini
and Komunjer (2005).
4.6 Loss functions and forecast combination
Ever since the early work on forecast combination of Bates and Granger (1969)
and Granger and Ramanathan (1984), combination weights have generally been
chosen to minimize a symmetric, squared-error loss function, and the empir-
ical forecast performance of the combination has typically been assessed by
squared-error loss. This reflects the widespread use of squared-error loss in the
forecast evaluation literature. For example, the “regression method” of Granger
and Ramanathan (1984) estimates by OLS an equation such as:
yt=β 1 f 1 t+β 2 f 2 t+et,