190 Forecast Combination and Encompassing
leads to those elements not being included inht− 1 :ht− 1 should include variables
that are thought likely to distinguish between the two forecasting methods: such
variables are likely to include indicators of past performance, as well as context-
specific variables, such as business cycle indicators if it is thought that the relative
performance of the two sets of forecasts may vary systematically with the busi-
ness cycle. Further details of conditional forecast encompassing tests are provided
in the discussion of the application of these methods to quantile forecasts in the
following section.
4.5.1 Quantile forecasts
Giacomini and Komunjer (2005) present an application of the general approach of
Giacomini and White (2006) to forecasting tests for quantile forecasts. The condi-
tional aspect of their approach can be brought to the fore by considering the tests of
correct conditional and unconditional coverage of Christoffersen (1998). Accord-
ing to Christoffersen (1998), a set of quantile forecasts is efficient with respect to
an information set (denotedt)ifE
(
α− 1(
Yt−qˆt< 0)
|t− 1)
=0, whereqˆtis
a forecast ofQt,α, theα-quantile of the distribution ofYtconditional onFt− 1 ,
namelyQt,α≡Ft−^1 (α), withFtthe conditional distribution function ofYt.1(.)
is the indicator function that takes the value one when the argument is true and
zero otherwise. If we defineIt≡ 1
(
Yt−qˆt< 0)
, then the condition for conditional
efficiency can be written more succinctly asE
(
α−It|t− 1)
=0. Testing whether
this holds is a test of correct conditional coverage, because it requires both that
(i) on average over the sample (t=1,...,n) the probability of an exceedence is
not significantly different fromα, and (ii) that there is no systematic relation-
ship between these exceedences and any variables in the agent’s information set
at the time the forecast was made. The first requirement is that of correctuncondi-
tionalcoverage, often termed a test for unbiasedness, as it is based on whether the
sample proportion of exceedences (say,πˆ=n−^1
∑n
t= 1 It) is significantly different
from the nominal proportionα. The null hypothesis is thatE
(
α−It)
=0 versus
E
(
α−It)
=0, and the standard likelihood ratio test is:LR=− 2[
n 0 ln(
1 −α
1 −ˆπ)
+n 1 ln
α
πˆ]asy
∼χ 12 ,wheren 1 =nπˆandn 0 =n−n 1. Tests for correct unconditional coverage, or bias,
can also be found in Granger, White and Kamstra (1989), Baillie and Bollerslev
(1992) and McNees (1995).
The second requirement can be tested by restricting the information set to past
values ofIt, namelyt− 1 =
{
It− 1 ,It− 2 ,...}. The suggestion of Christoffersen (1998)
is to test whetherE
(
α−It|t− 1)
=0 by testing whether{
It}
follows a binary
first-order Markov chain. If the transition probabilities are defined as:
πij=Pr(It=j|It− 1 =i),wherei,j={0, 1}, a lack of a systematic relationship betweenItandt− 1 (here
t− 1 =It− 1 ) requires thatπ 0 j =π 1 j,j={0, 1}, which gives rise to a simple
likelihood ratio test. Note that this test does not consider unbiasedness mentioned