176 Forecast Combination and Encompassing
withX 1 tandX 2 tscalar explanatory variables underpinning the two forecasts,
although KK combination is a computationally attractive method of combining
forecasts whilst ensuringft∈(0, 1)more generally.
4.2.2 Forecast encompassing
The concept of forecast encompassing relates to whether or not one forecast
encapsulates all the useful predictive information contained in a second forecast.
Formally, using a squared error loss function as above,f 1 tis said to encompassf 2 t
if, in a linear combination of the two forecasts,f 2 toptimally receives zero weight,
so that combiningf 1 twithf 2 tdoes not lead to a reduction in the MSFE. Thus, using
the simplest Bates and Granger (1969) form of linear combination,f 1 tencompasses
f 2 tif the optimal value ofλin (4.1) is zero. This concept was originally proposed
by Nelson (1972) and Granger and Newbold (1973), withf 1 treferred to as being
conditionally efficient with respect tof 2 t; the terminology has subsequently been
modified by Chong and Hendry (1986) to that of forecast encompassing, adopting
the language of the model encompassing literature (see,inter alia, Mizon, 1984;
Mizon and Richard, 1986).
Chong and Hendry (1986) focus on the fact that, iff 1 tencompassesf 2 t, the
forecast errors of the encompassing forecast,e 1 t, should be uncorrelated with the
encompassed forecastf 2 t. This obtains sincee 1 tshould be uncorrelated with infor-
mation available at the time of the forecast, while, on the other hand, correlation
betweene 1 tandf 2 twould imply that the accuracy off 1 tcould be improved by
linear combination withf 2 t. This approach results in an alternative definition of
forecast encompassing, namely thatf 1 tencompassesf 2 tif the optimal value ofλ
is zero in a regression:
e 1 t=λf 2 t+εt.
The two definitions of forecast encompassing presented thus far implicitly
assume that the forecasts are unbiased and efficient. To account for potential fore-
cast bias and inefficiency, alternative forecast encompassing specifications can also
be derived by defining encompassing asf 2 treceiving zero optimal weight in a more
general forecast combination such as (4.7) or (4.8). Both approaches have been pro-
posed in the literature, with Fair and Shiller (1989, 1990) using (4.8), allowing for
the forecasts to be both biased and inefficient, and Andrews, Minford and Riley
(1996) advocating use of (4.7), an in-between case allowing for forecast bias while
retaining the assumption that the combination weights sum to one.
The alternative forecast encompassing definitions can be summarized and
given a regression interpretation as follows. Beginning with the most general
formulation, the Fair and Shiller (1989) definition off 1 tencompassingf 2 tequates
toβ 2 =0 in the regression:
FE(1): yt=α+β 1 f 1 t+β 2 f 2 t+εt.Relative to this specification, the Nelson (1972) and Granger and Newbold (1973)
approach imposes the restrictionsα=0 andβ 1 +β 2 =1, with encompassing
defined byλ=0 in the regression:
FE(2): e 1 t=λ(e 1 t−e 2 t)+εt.