Palgrave Handbook of Econometrics: Applied Econometrics

182 Forecast Combination and Encompassing

and where the parametersδdgandδggare given in the following table:

Estimation scheme δdg δgg

Fixed 0 π

Recursive 1 −π−^1 ln( 1 +π) 2

[

1 −π−^1 ln( 1 +π)

]

Rolling,π≤ 1 π/ 2 π−π^2 / 3

Rolling,π> 11 −( 2 π)−^11 −( 3 π)−^1

withπ=limR,n→∞(n/R),0≤π<∞. Note thatBSggB′in the last term of (4.16)
defines the asymptotic(R→∞)variance-covariance matrix of the estimator of the
parameter vectorθ=[θ 1 ,θ 2 ]′, denotedVθ. The above results can also be general-
ized beyond the example considered of scalar linear regression models estimated by
least squares, provided the models continue to be non-nested. For results pertain-
ing to multiple regressors in a linear framework, and also more general models and
estimation techniques, see West and McCracken (1998) and West (2001). Essen-
tially, (4.15) and (4.16) continue to hold, but involve more general representations
for the constituent components of.
Comparing (4.12) and (4.15)–(4.16), it can be seen that the uncertainty involved
through estimation of the model parameters gives rise to additional terms in the

asymptotic variance ofdˆ. To see how this arises, consider the FE(2)MDMtest for the
simple example above, assuming the forecasts have been obtained via the fixed esti-

mation scheme, so thatθˆit=θˆi=

∑R

t= 1 ytXit/

∑R

t= 1 X

2
it,t=R+h,...,R+n+h−1,
i=1, 2. In this case, the expression (4.16) simplifies to=S+πDVθD′with
D=[E(e 2 tX 1 t),E(e 1 tX 2 t)]. Now the forecast errors can be written aseˆit =eit
−(θˆi−θi)Xit,i=1, 2, resulting in the decomposition:

dˆ=d ̄+n−^1 ∑R+n+h−^1

t=R+h

[

(θˆ 1 −θ 1 )e 2 tX 1 t+(θˆ 2 −θ 2 )e 1 tX 2 t

+(θˆ 1 −θ 1 )^2 X 12 t−(θˆ 1 −θ 1 )(θˆ 2 −θ 2 )X 1 tX 2 t− 2 (θˆ 1 −θ 1 )e 1 tX 1 t

]

.

It then follows that:

V

[√

n[dˆ−E(dt)]

]

=V

[√

n[d ̄−E(dt)]

]

+nE

{

n−^2

[

(θˆ 1 −θ 1 )^2 (

∑R+n+h− 1

t=R+h e^2 tX^1 t)

2

+(θˆ 2 −θ 2 )^2 (

∑R+n+h− 1

t=R+h e^1 tX^2 t)

2

+ 2 (θˆ 1 −θ 1 )(θˆ 2 −θ 2 )

∑R+n+h− 1

t=R+h e^2 tX^1 t

∑R+n+h− 1

t=R+h e^1 tX^2 t

]}

+op( 1 )

=V

[√

n[d ̄−E(dt)]

]

+(n/R)E

{

[R^1 /^2 (θˆ 1 −θ 1 )]^2 (n−^1

∑R+n+h− 1

t=R+h e^2 tX^1 t)

2

+[R^1 /^2 (θˆ 2 −θ 2 )]^2 (n−^1

∑R+n+h− 1

t=R+h e^1 tX^2 t)

2