Palgrave Handbook of Econometrics: Applied Econometrics

Michael P. Clements and David I. Harvey 189

become apparent by contrasting their null hypotheses for a general test of equal
forecast accuracy for a general loss functionL(.). The DMW null is that:

H 0 : E

[

L

(

Yt,f 1 t

(

β 1 ∗

))

−L

(

Yt,f 2 t

(

β∗ 2

))]

=0,

so thatLis defined over the random variableYtand the one-step forecast based on
information up tot−1 from Modeliwith population parameter vectorβi∗, denoted
fit

(

βi∗

)

. The expectation of the loss differential is unconditional. By contrast, the

Giacomini and White (2006) null is:

H 0 : E

[

L

(

Yt,f 1 t

(

βˆ 1

))

−L

(

Yt,f 2 t

(

βˆ 2

))

|t− 1

]

=0, (4.24)

so that loss depends on the estimates, and the expectation is with respect to an
information sett− 1.
In terms of testing for forecast encompassing with standard squared-error loss

(i.e.,L

(

Yt,fit

(

β∗i

))

=eit^2 , whereeit=Yt−fit∗), the DMW null becomes:

H 0 : E(dt)=0,

wheredt=e 1 t

(

e 1 t−e 2 t

)
for the null that Model 1 encompasses Model 2 using
FE(2). Assumingβ 1 ∗andβ∗ 2 are known, under standard conditions (see Diebold
and Mariano, 1995; Harvey, Leybourne and Newbold, 1998; West and McCracken,
1998) we obtain (4.12), namely:
√
n[d ̄−E(dt)]⇒N(0,S).

Whenβ 1 ∗andβ 2 ∗are not known and the forecasts are based onβˆ 1 andβˆ 2 , the
variance of the limiting normal distribution will in general include an additional
term for the effect of estimation uncertainty, as described in section 4.3.
To test for forecast encompassing using (4.24), letdˆt=ˆe 1 t

(

ˆe 1 t−eˆ 2 t

)

, whereeˆit=

yt−fit(βˆi)to make explicit the use of parameter estimates. ThenE(dˆt|t− 1 )= 0
is equivalent toE(ht− 1 dˆt)=0 whent− 1 =Ft− 1 (whereFtis the information
available at timet) andht− 1 is aFt− 1 -measurable function of dimensionq. Stan-
dard asymptotic normality arguments then give rise to the (one-step) Conditional
Forecast Encompassing Test (see Giacomini and White, 2006, Theorem 1, p. 1553):

Thn=nZ ̄n′ˆ−n^1 Z ̄n,

whereZ ̄n=n−^1
∑n
t= 1 Zt,Zt=ht− 1 dˆt, andˆnis the standard variance estimator,
ˆn=n−^1 ∑nt= 1 ZtZ′t. Under the null:

Tnh⇒χq^2 ,

asn→∞. The sequences of forecasts are based on rolling estimation windows of
fixed size to ensure non-vanishing parameter estimation uncertainty as the sam-
ple of forecasts (n) goes to infinity. (This aspect is suppressed in the notation for
convenience.) The choice ofht− 1 is crucial, in that the test will have no power
ifE(dˆt|t− 1 )=0 for some elements oft− 1 , but an injudicious choice ofht− 1