Michael P. Clements and David I. Harvey 177The Chong and Hendry (1986) definition can be obtained by assumingf 1 tin FE(1)
to be efficient, i.e., imposingα=0 andβ 1 =1, with encompassing then defined
byλ=0 in the regression:
FE(3): e 1 t=λf 2 t+εt.The FE(2) and FE(3) cases can also be modified to allow for forecast bias by
relaxing theα=0 assumption, yielding the Andrews, Minford and Riley (1996)
regression:
FE(2′): e 1 t=α+λ(e 1 t−e 2 t)+εt,
and a modified Chong–Hendry regression:
FE(3′): e 1 t=α+λf 2 t+εt.In the remainder of this chapter, we focus on the more commonly used definitions
FE(1), FE(2) and FE(3).
It is clear that if the restrictions imposed by FE(2) and FE(3) are not satisfied,
the three definitions of encompassing are not equivalent, and the forecastf 1 tmay
encompassf 2 taccording to one definition, but not another. FE(1) is the most
general approach of the three, and allows an analysis of the forecast encompassing
hypothesis without the additional requirements that the individual forecasts are
unbiased and efficient. Note that if the optimal value ofβ 2 in FE(1) is zero, it
does not follow that the optimal forecast is simplyf 1 t: the correct inference is
that alinear function of f 1 t, i.e.,α+β 1 f 1 t, cannot be improved (in terms of MSFE)
through combination withf 2 t. If the restrictions underlying FE(2) and FE(3) do
hold, tests based on these approaches should be more powerful.
When the actuals and forecasts are integrated time series processes, FE(1) should
be implemented using actual and predicted changes rather than the levels of the
series, as otherwise the test statistics will not have their standard distributions.
When the data (and forecasts) are integrated, the FE(3) approach can be problem-
atic, as noted by Ericsson (1992). Because a forecastf 1 twould be expected to be
cointegrated withyt, the resulting forecast errore 1 tis integrated of order zero. The
regression FE(3) is then unbalanced in the sense that the dependent and explana-
tory variables have differing orders of integration, and Phillips (1995) shows that
the resulting forecast encompassing tests have an asymptotic size of one.
Tests of the null hypothesis of forecast encompassing can be conducted using
any of the above definitions. In terms of the more general FE(1) definition, the
null and alternative can be expressed as:
H 0 : β 2 = 0 (f 1 tencompassesf 2 t)
H 1 : β 2 > 0 (f 1 tdoes not encompassf 2 t).The alternative hypothesis is often chosen to be one-sided, to rule out the possi-
bility of negative combination weights. Note, however, that negative weights can
arise: from (4.3) it is apparent that the weight onf 2 tin the Bates–Granger combi-
nation will be negative ifσ 2 ρ>σ 1 (and the weight onf 1 twill exceed unity). For