Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 175

where:
!

(

ztγi

)

=

(

1 +exp

[

−

(

γ 0 i+γ 1 iz 1 t+γ 2 iz 2 t

)])− 1

,

andzit=s−y^1

(

fit−y ̄

)
,i=1, 2, i.e., thefitare normalized by the in-sample mean
and standard deviation ofyt. Whenp=0 andk=2, the ANN specializes to
standard linear combination of the two sets of forecasts. Donaldson and Kamstra
(1996) allow values ofpup top=3, and find values of 1 or 2 are typically selected
by their cross-validation procedure. The combinations are estimated by choosing
theγias random drawings from aU(−1, 1)distribution, and then estimation ofα,
β,δcan be carried out by ordinary least squares (OLS).
Finally, for forecasts other than point forecasts, other forms of combination are
used. For example, experts’ subjective probability distributions are often combined
using the logarithmic opinion pool, or LoOP (see, e.g., Genest and Zidek, 1986;
Clemen and Winkler, 1999), which for discrete probability distributions is given by:

fj=

∏N

i= 1

(

fij

)βi

∑M

j= 1

∏N

i= 1

(

fij

)βi=

exp

(∑N

i= 1 βilogf

j

i

)

∑M

j= 1 exp

(∑

N

i= 1 βilogf

j

i

),

wheref
j
i is individuali’s probability of “classcj,” where there areMclasses. The
denominator is a scaling factor, and typically

∑N
i= 1 βi=1. Clements and Harvey
(2007) consider a form of LoOP for probability forecasts (M=2) and two rival
forecasts (N=2), and Kamstra and Kennedy (1998) (henceforth, KK) suggest a
form of combination for probability forecasts that involves the combining of log
odds ratios by logit regressions. The KK combination off 1 tandf 2 tis:

fct=

exp

[

α+β 1 ln

( f

1 t

1 −f 1 t

)

+β 2 ln

( f

2 t

1 −f 2 t

)]

1 +exp

[

α+β 1 ln

( f

1 t

1 −f 1 t

)

+β 2 ln

( f

2 t

1 −f 2 t

)]

=

exp(α)

( f

1 t

1 −f 1 t

)β 1 ( f

2 t

1 −f 2 t

)β 2

1 +exp(α)

( f

1 t

1 −f 1 t

)β 1 ( f

2 t

1 −f 2 t

)β 2 , (4.10)

whereβ 1 andβ 2 are the maximum likelihood estimates of the slope coefficients

from a logit regression ofyton a constant, ln

(

f 1 t

1 −f 1 t

)

and ln

(

f 2 t

1 −f 2 t

)

. Clements and

Harvey (2007) show that the KK combination is optimal when the data-generating
process has the form:

yt= 1

(

exp(δ 1 X 1 t+δ 2 X 2 t)

1 +exp(δ 1 X 1 t+δ 2 X 2 t)

>vt

)

(4.11)

f 1 t=

exp(θ 11 X 1 t)

1 +exp(θ 11 X 1 t)

f 2 t=

exp(θ 12 X 2 t)

1 +exp(θ 12 X 2 t)

,