Palgrave Handbook of Econometrics: Applied Econometrics

Stephen G. Hall and James Mitchell 219

for density forecast evaluation, as well as comparison and combination, to which
we turn below.
The KLIC offers a measure of distance, or more accurately “divergence,” between
the “true” but unknown conditional densityf(yt|!t−h), defined with respect
to the total information set!t−h, and theith conditional density forecastg(yt|
it−h), defined with respect to forecasteri’s information setit−h:

KLICit|t−h=E

[

lnf(yt|!t−h)−lng(yt|it−h)

]

=

∫

f(yt|!t−h)ln

{

f(yt|!t−h)

g(yt|it−h)

}

dyt. (5.27)

KLICit|t−h=0 if and only ifg(yt|it−h)=f(yt|!t−h). But, as explained, since
f(yt|!t−h)is unknown evenex post, typically density forecasts are evaluated by

employing a goodness-of-fit test on thepit’szit|t−h=

∫yt

−∞

g(u|it−h)du. These

amount to a test for whetherKLICit|t−h=0.

Estimates of KLICit|t−h can be obtained when we follow Bao, Lee and
Saltoglu (2007), invoke Proposition 2 of Berkowitz (2001) and note the following
equivalence:

dti|t−h=lnf(yt|!t−h)−lng(yt|it−h)=lnq(zit∗|t−h)−lnφ(z∗it|t−h)=lnh(zit|t−h),
(5.28)
wherez∗it|t−h=#−^1 zit|t−h,q(.)is the unknown density ofzit∗|t−h, which needs to
be specified,φ(.)is the standard normal density and#is the c.d.f. of the standard
normal. Equation (5.28) offers a direct link between the Berkowitz test and the
KLIC. For the nonparametric uniformity tests we see that, whenh(zit|t−h)=1, as

it does under the null of correct conditional calibration,dit|t−h=0.

Under some regularity conditions,E

[

KLICit|t−h

]

can be consistently estimated

by sample (t=1,...,T) information:

KLIC

i

t−h=

1

T

∑T

t= 1 d

i

t|t−h, (5.29)

where, following Berkowitz (2001), forh=1, we could assume:

q(zit∗|t− 1 )=φ

[(

z∗it|t− 1 −μ−ρz∗it− 1 |t− 2

)

/σ

]

/σ. (5.30)

Noting that the LR test as traditionally written equals:

LRiB= 2

∑T

t= 1

[

lnq(z∗it|t− 1 )−lnφ(z∗it|t− 1 )

]

, (5.31)

reveals thatKLICit− 1 =LRiB/ 2 T.
More general specifications forq(.), allowingεtto follow a more general
distribution than the Gaussian, could also be specified. Whenh > 1we
should expect dependence, due to overlapping observations, and we might then