Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 191

under (i). Granger, White and Kamstra (1989, note c to Table 1, p. 91) suggest using
a contingency table approach to test the conditional aspect, based on whether the
number of occurrences of (say) zeros followed by zeros is consistent with there being
no association between the occurrence of a zero in one period and the occurrence
of a zero in the following period. Clements and Taylor (2003) suggest a regression-
based approach that facilitates the inclusion of variables besides lagged values of{
It

}
in the information set.
The tests of Christoffersen (1998) illustrate the distinction between conditional
and unconditional tests in the context of the evaluation of a single sequence of fore-
casts. From a conceptual point of view, the approach of Giacomini and Komunjer
(2005) can be viewed as replacing the single quantile forecastqˆtby a combination
of two (or more) quantile forecasts,θ′qˆt, whereθ=

(

θ 1 ,θ 2

)′

andqˆt=

(

qˆ 1 t,qˆ 2 t

)′
,
followed by the development of tests of conditional and unconditional forecast
encompassing based on the estimated weightsθˆ. Their treatment follows Giaco-
mini and White (2006) (although complications arise due to the discontinuous
nature of the moment conditions on which the generalized method of moments
(GMM) estimation ofθis based).
The conditionalα-quantile ofYt,Qt, is the optimal forecast for a “tick” or “check”
loss function:
Lα

(

et

)

=

(

α− 1

(

et< 0

))

et,

whereet=yt−qˆt, so thatL(.)is used as the basis for assessing whether combi-
nations of forecasts reduce loss. A straightforward application of the definition of
forecast encompassing to quantile forecasts gives the following definition of con-
ditional quantile forecast encompassing based on Giacomini and Komunjer (2005,
Definition 1, p. 418):qˆ 1 tencompassesqˆ 2 tat timetif:

Et− 1

[

Lα

(

Yt−qˆ 1 t

)]

=Et− 1

[

Lα

(

Yt−

(

θ 1 ∗tqˆ 1 t+θ∗ 2 tqˆ 2 t

))]

,

whereEt− 1 (.)≡E

(

.|Ft− 1

)

and whereθ∗=

(

θ 1 ∗,θ 2 ∗

)′
are the optimal weights in
that they minimize tick loss:
(
θ 1 ∗,θ 2 ∗

)

≡arg min

(θ 1 ,θ 2 )∈

Et− 1

[

Lα

(

Yt−

(

θ 1 tqˆ 1 t+θ 2 tˆq 2 t

))]

.

Thus the optimal weights are

(

θ∗ 1 ,θ 2 ∗

)
=(1, 0), so assigning a zero weight toˆq 2 t.
Giacomini and Komunjer (2005, Lemma 1, p. 419) show thatθ∗satisfies the
first-order condition:

Et− 1

[

α− 1

(

Yt−θ∗qˆt< 0

)]

=0, (4.25)

which is the correct conditional coverage condition of Christoffersen (1998). These
moment conditions are used to estimate the optimal weights by GMM and to
test for forecast encompassing (thatqˆ 1 tencompassesqˆ 2 t):

(

θ 1 ∗,θ 2 ∗

)

=(1, 0). The

conditional moment conditions (4.25) are replaced by:

E

[(

α− 1

(

Yt−θ∗qˆt< 0

))

Wt∗− 1

]

=0,