Stephen G. Hall and James Mitchell 221then constructed based on
{
dt|t−h}T
t= 1
, where:dt|t−h=[
lnf(yt|!t−h)−lng(yt| 1 t−h)]−[
lnf(yt|!t−h)−lng(yt| 2 t−h)]
, (5.33)
=lng(yt| 2 t−h)−lng(yt| 1 t−h), (5.34)=[
lnq(z 1 ∗t|t−h)−lnφ(z∗ 1 t|t−h)]
−[
lnq(z∗ 2 t|t−h)−lnφ(z∗ 2 t|t−h)]. (5.35)
The null hypothesis of equal accuracy is then:H 0 :E(dt|t−h)= 0 ⇒KLIC
1
t−h−KLIC2
t−h=0. (5.36)The sample meandis defined as:d=
1
T∑T
t= 1[[
lnq(z 1 ∗t|t−h)−lnφ(z∗ 1 t|t−h)]
−[
lnq(z∗ 2 t|t−h)−lnφ(z∗ 2 t|t−h)]]
.
(5.37)A test can be constructed since we know thatd, under appropriate assumptions,
has the limiting distribution:
√
T(d−E(dt|t−h))
d
→N(0,̂ν), (5.38)where Bao, Lee and Saltoglu (2007), following West (1996), discuss estimatorŝν
for the long-run asymptotic variance ofdt|t−hallowing for parameter uncertainty,
e.g., when the forecasts are model-based and the models are estimated using an
expanding, not rolling (fixed length), window (see Giacomini and White, 2006).
In the absence of parameter uncertainty, the test (5.38) reduces to a DM-type test:
d/
√
Sd
Td
→N(0,1), whereSd=γ 0 + 2∑∞
j= 1 γjandγj=E(dt|t−hdt−j|t−j−h). As sug-
gested by White (2000), the test of equal predictive accuracy (5.36) can readily be
extended to multiple (greater than two) models.
To avoid having to postulate an unknown densityq(.), it is more convenient to
couch the test in terms of (5.34) rather than (5.35).^6 In this case we see clearly that
the test is equivalent to that proposed by Amisano and Giacomini (2007).
Amisano and Giacomini (2007), independently of Bao, Lee and Saltoglu (2007),
proposed tests that can be used to compare the accuracy of density forecasts where
evaluation is based on logarithmic scores, e.g., lng(yt| 1 t−h), rather than the
pit’s. These tests can be superficially related to the traditional Bayesian approach
to comparing models using Bayes factors (e.g., see Koop, 2003). When there are no
parameters to be estimated, the logarithmic Bayes factor is equal to the difference
of the two models’ logarithmic scores seen in (5.34) (see Gneiting and Raftery,
2007).
Related approaches of comparing density forecasts statistically have been pro-
posed by Sarno and Valente (2004) and Corradi and Swanson (2006b). Rather than
using the KLIC measure of “distance,” these rely on the integrated squared differ-
ence between the forecast density and the true density (Sarno and Valente) and