172 Forecast Combination and Encompassing
encompassing), we take the forecasts as given, and do not consider issues related
to the method employed in obtaining the predictions, for example those of model
estimation uncertainty and nesting considered in later sections. Assuming the fore-
casts to be unbiased, i.e., that the forecast errorseit=yt−fit(i=1, 2)have zero
mean, Bates and Granger (1969) suggest the use of a combined forecast,fct, of the
following form:
fct=( 1 −λ)f 1 t+λf 2 t. (4.1)When 0≤λ≤1,fctcomprises a simple weighted average of the two individual
forecasts.
The optimal choice for the weighting parameterλdepends on the relative accu-
racy of the individual forecastsf 1 tandf 2 t, and can easily be obtained for a given
loss, or cost of error, function. By far the most commonly assumed cost of error
function in the literature is that of squared error loss, with forecast accuracy deter-
mined by the mean squared forecast error (MSFE) measure. Denoting the forecast
error associated withfctbyεt=yt−fct, we obtain the following expression for the
MSFE of the combined forecast:
E(εt^2 )=( 1 −λ)^2 σ 12 +λ^2 σ 22 + 2 λ( 1 −λ)ρσ 1 σ 2 , (4.2)whereσ 12 andσ 22 denote, respectively, the MSFEs off 1 tandf 2 t, andρdenotes
the correlation between the forecast errorse 1 tande 2 t. The optimal combination
weight associated with a squared error loss function is then derived by choosingλ
to minimize (4.2), i.e.:
λopt=
σ 12 −ρσ 1 σ 2
σ 12 +σ 22 − 2 ρσ 1 σ 2. (4.3)
The expected squared error associated with the optimal combination weightλopt
is given by:
E(
ε^2 t(
λopt))
=
σ 12 σ 22(
1 −ρ^2)σ 12 +σ 22 − 2 ρσ 1 σ 2,where, of necessity,E
(
ε^2 t(
λopt))
≤min{
σ 12 ,σ 22}. Suppose thatf 1 andf 2 are equally
accurate, i.e.,σ 12 =σ 22. Then:
E(
ε^2 t(
λopt))
=1
2
σ 12(
1 +ρ)
.Given that|ρ|≤1,^12 σ 12 ( 1 +ρ)<σ 12 for all values ofρother thanρ=1. So there
are diversification gains when the forecasts are equally accurate unless the forecasts
are perfectly correlated.
In practice the optimal weight parameter, and its constituent parametersρ,σ 12
andσ 22 , will not be known. However, given time series data on past forecasts and
actuals, these quantities can be estimated, resulting in a sample analogue of the
population weight parameter (4.3). Denoting the time series of pasth-steps-ahead