1058 The Econometrics of Exchange Rates
Kilian extends the sample period from 1991:4 to 1997:4 and applies both Mark’s
bootstrap methodology and the VECM bootstrap. The former methodology indi-
cates that for the extended data set thep-values of the various statistics are stable
or increasing and there is overall predictability only for Switzerland.^65 Kilian also
shows that the effect of small sample bias, together with the fact that Mark’s boot-
strap is inconsistent, has a substantial impact on inference. When allowing for
a drift in the forecasts of the RW, both bootstrap methodologies detect overall
predictability for Switzerland and Canada.
22.6.1.2 Forecast evaluation measures
In the late 1990s, the scientific consensus was that the Meese and Rogoff (1983a,
1983b) results still stood (Rogoff, 1999). However, the findings of Clark and
McCracken (2001, 2003, 2005) raise concerns regarding the power of commonly
usedt-type tests. This motivates McCracken and Sapp (2005) to investigate the
out-of-sample performance of exchange rate determination models using new test
statistics regarding the comparison of nested models.^66
Letyt+k=st+k−stdenote the variable to be predicted andx2,t=(x′1,t,x′22,t)′be a
vector of predictors. The number of in-sample and out-of-sample observations isR
andP, respectively, so thatT=R+P. Forecasts are generated by estimating two
linear models of the formx1,tβ 1 andx′22,tβ 2 recursively by OLS. The forecast errors
areuˆ1,t+k=yt+k−x′1,tβˆ1,tanduˆ2,t+k=yt+k−x′22,tβˆ2,t. Under the null hypothesis
the first model is nested within the second and the two forecast errors are identical.
In the present context, the first model is the RW with a drift and the second the
long-horizon regression model, which implies thatx1,t=1 andx22,t=zt.
The first two test statistics examined are used to test for forecast accuracy. The
former is at-type test and the latter is itsF-type counterpart. The null hypothesis
is that the two mean squared errors (MSEs) are equal against the alternative that
the MSE for the second model is smaller. Let̂dt+k=̂u^2 1,t+k−̂u^2 2,t+k,d ̄=(P−
k+ 1 )−^1
∑T−k
t=R̂dt+k=MSE 1 −MSE 2 ,̂dd(j)=(P−k+^1 )
− 1 ∑T−k
t=R+ĵdt+k̂dt+k−jforj 0 and̂dd(j)=̂dd(−j), and let̂Sdd =
∑ ̄j
j=− ̄jK(j/M)̂dd(j)be the long-run covariance ofdt+kestimated using a kernel-based estimator with function
K(·), bandwith parameterMand maximum number of lags ̄j. The tests for forecast
accuracy are:
MSE−t=(P−k+ 1 )^1 /^2
d ̄
̂S^1 /^2
dd, (22.86)MSE−F=(P−k+ 1 )^1 /^2
d ̄
MSE 2. (22.87)
The next two tests concern forecast encompassing and are built upon the statis-
tic used by Harveyet al.(1998) for non-nested models. In this case, the null
hypothesis is that the forecast of the RW model encompasses that of the struc-
tural exchange rate model and, therefore, the covariance between the forecast