Palgrave Handbook of Econometrics: Applied Econometrics

Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1059

errors of the RW and the difference of the forecasts errors of the two models will

be equal to or smaller than zero. Under the alternative, the deviations from fun-

damentals contain valuable information, implying that the covariance is positive.

Let̂ct+k=̂u1,t+k(̂u1,t+k−̂u2,t+k). The forecast-encompassing test statistics are:

ENC−t=(P−k+ 1 )^1 /^2

c ̄

Sˆ^1 cc/^2

, (22.88)

ENC−F=(P−k+ 1 )^1 /^2

̄c

MSE 2

. (22.89)

The limiting distributions of the test statistics considered so far are non-standard

and depend upon nuisance parameters. Thus, critical values for the test statistics

should be generated by bootstrap procedures, such as the VECM bootstrap of Kilian

(1999).

The final statistic, derived by Chaoet al.(2001), is also used to test for fore-

cast encompassing and follows aχ^2 distribution. Let̂ht+k=̂u1,t+kx1,t,̂bt+k=

̂u1,t+k+x22,t,b ̄=(P−k+ 1 )−^1
∑T−k
t=R̂bt+k,̂F=−(P−k+^1 )

− 1 ∑T−k

t=Rx22,tx

′

1,t,

and̂B =[(P−k+ 1 )−^1

∑T−k

t=Rx1,tx

′

1,t]

− (^1). Furthermore, letˆ
bb(j) = (P−k+
1 )−^1
∑T−k
t=R+jbˆt+kbˆ
′
t+k−j,ˆhh(j)=(P−k+^1 )
− 1 ∑T−k
t=R+jhˆt+khˆ
′
t+k−j,ˆbh(j)=(P−
k+ 1 )−^1
∑T−k
t=R+j
ˆbt+khˆ′t+k−j, forj0, withˆbb(j)=ˆbb(−j),ˆhh(j)=ˆhh(−j),
ˆbh(j)=ˆbh(−j). Finally, letSˆbb=∑ ̄j
j=− ̄j
K(j/M)ˆbb(j),ˆShh=
∑ ̄j
j=− ̄j
K(j/M)ˆhh(j),
ˆSbh=∑ ̄j
j=− ̄j
K(j/M)ˆbh(j). The test statistic is written as:
CCS=(P−k+ 1 )b ̄′b ̄, (22.90)
where=Sˆbb+λˆbh(FˆBˆSˆ′bh+Sˆ′bhBˆ′Fˆ′)+λˆbbFˆBˆSˆ′hhBˆ′Fˆ′,πˆ=(P−k+ 1 )R−^1 ,λˆbh=
1 −ˆπ−^1 ln( 1 +ˆπ),λˆbh= 2 [ 1 −ˆπ−^1 ln( 1 +ˆπ)]. The null hypothesis of no predictability
based on macroeconomic fundamentals requires the covariance betweenu1,t+k
andx22,tto be zero. If fundamentals contain predictive power then the covariance
should deviate from zero.
McCracken and Sapp (2005) employ the long-horizon regression (22.80), but
follow Meese and Rogoff (1983a, 1983b) and determine the fundamental value of
the exchange rate according to various structural models. This approach results in
a vast number of tests, which raises concerns about the reliability of inference.^67
In order to mitigate the multiple testing problem, McCracken and Sapp (2005) fol-
low recent developments in the statistical genetics literature and calculateq-values
along with thep-values.^68 Using bothp-values andq-values, they find evidence of
predictability for many cases. The encouraging results can be attributed to the fact
that the newF-type tests are more powerful than thet-type tests. Despite the fact
that RMSEs are similar to those reported by Kilian (1999), theF-type tests are able
to detect the superiority of the structural models over the RW with a drift. As far as
the monetary model is concerned,F-type tests of equal forecast accuracy indicate