Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 183

+ 2 R^1 /^2 (θˆ 1 −θ 1 )R^1 /^2 (θˆ 2 −θ 2 )(n−^1

∑R+n+h− 1

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

− 1 ∑R+n+h− 1

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

}

+op( 1 )

⇒S+π{Vθ,11[E(e 2 tX 1 t)]^2 +Vθ,22[E(e 1 tX 2 t)]^2 + 2 Vθ,12E(e 2 tX 1 t)E(e 1 tX 2 t)}

=S+πDVθD′,

whereVθ,ijdenotes the(i,j)element ofVθ.
If forecast encompassing tests are conducted without taking account of the addi-
tional terms present in, i.e., by simply usingdˆtin place ofdtin the tests in
the previous section, and using the usual implicit long run variance estimator
that only estimatesS, then asymptotic size distortions are generally obtained. For
example, simulations by West (2001), using the example considered above with
θ 1 =1,θ 2 =0 and(e 1 t,X 1 t,X 2 t)′i.i.d. normal with variance-covariance matrix
diag(1, 1, 2), show that the FE(2)MDMtest of the previous section, run at the
nominal 5% significance level against a two-sided alternative, has empirical size
around 25% for largenwhenn/R=2.
In order to obtain asymptotically correctly-sized tests in general, the variance
estimators implicit in the forecast encompassing tests must be modified so as to
consistently estimate. Consistent estimation ofcan be obtained by estimating
the constituent quantitiesS,Sdg,Sgg,BandDusing their natural sample coun-
terparts, and usingn/Rin place ofπfor determiningδdgandδgg. To illustrate,
consider again the FE(2)MDMtest for the simple example above, assuming the
forecasts have been obtained via the fixed estimation scheme. A further simplifi-
cation ofis possible, since under the encompassing null hypothesis,e 1 tcannot
be predicted by model 2, soE(e 1 tX 2 t)=0, yieldingD=[E(e 2 tX 1 t),0]and:

=S+πVθ,11[E(e 2 tX 1 t)]^2.

A consistent estimator is:

ˆ=Sˆ+(n/R)Vˆθ,11

[

n−^1

∑R+n+h− 1

t=R+h

ˆe 2 tX 1 t

] 2

,

where:

Sˆ=[n+ 1 − 2 h+n−^1 h(h− 1 )]−^1

∑h− 1

j=−(h− 1 )

∑R+n+h− 1

t=|j|+R+h

(dˆt−dˆ)(dˆt−|j|−dˆ),

andVˆθ,11is the heteroskedasticity consistent estimator of the asymptotic variance
ofθˆ 1 :

Vˆθ,11=R

∑R

t= 1 eˆ

2

1 tX

2

1 t

(∑

R

t= 1 X

2

1 t

) 2.

The resultingMDMstatistic is then given by:

MDM=

n^1 /^2 dˆ

√

ˆ

.