Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 173

forecast errors bye 1 t,e 2 t,t=1,...,n, the obvious estimator is given by:

λˆopt=

∑n

t= 1 e

2

1 t−

∑n

∑ t=^1 e^1 te^2 t

n

t= 1 e

2

1 t+

∑n

t= 1 e

2

2 t−^2

∑n

t= 1 e^1 te^2 t

. (4.4)

This estimated weight can then be used in the future to produce out-of-sample
combined forecasts. The above estimator could equally be obtained from ordinary
least squares estimation of the regression:

e 1 t=λ(e 1 t−e 2 t)+εt, (4.5)

or, equivalently:
yt=( 1 −λ)f 1 t+λf 2 t+εt, (4.6)

which is essentially a rearrangement of (4.1) with the error of the combined
forecast,εt, now interpreted as a regression error.
The above analysis assumes that the individual forecasts are unbiased. However,
it may well be the case that the individual forecasts are biased, as has been observed
empirically for some macroeconomic forecasts (see, e.g., Zarnowitz and Braun,
1993; Stekler, 2002; Harvey and Newbold, 2003). In order to allow for bias in the
forecasts, an intercept can be added to the regression (4.5) or (4.6):

e 1 t=α+λ(e 1 t−e 2 t)+εt,

which ensures that the implied combination:

fct=α+( 1 −λ)f 1 t+λf 2 t, (4.7)

is unbiased.
In addition to the possibility of biased forecasts, forecasts may also be inefficient
in the sense of Mincer and Zarnowitz (1969). A generic forecastftis said to be
Mincer–Zarnowitz efficient ifα=0 andβ=1 in a regressionyt=α+βft+εt, which
implies that the forecast and forecast error are uncorrelated (see, e.g., Clements and
Hendry, 1998, Ch. 3, for a discussion). If the individual forecasts are inefficient, the
appropriate generalization of the combined forecast involves relaxing the implicit
assumption that the combination weights sum to one. This results in an efficient
combined forecast, with the more general formulation advocated by Granger and
Ramanathan (1984):
fct=α+β 1 f 1 t+β 2 f 2 t. (4.8)

The weights are now obtained from the corresponding regression:

yt=α+β 1 f 1 t+β 2 f 2 t+εt. (4.9)

Clearly (4.7) and (4.1) are special cases of (4.8), where the restrictionsβ 1 +β 2 =1,
andα=0,β 1 +β 2 =1 are imposed, respectively. Note that when the actuals and
forecasts are non-stationary integrated processes, (4.9) should be specified using
actual and predicted changes, rather than levels.