186 Forecast Combination and Encompassing
case 0<π<∞ (withπ=limR,n→∞(n/R)as before), is given by:
MDM⇒
1
√
2, (4.18)where the terms 1 and 2 depend on the estimation scheme as follows:
Estimation scheme 1 2Fixed λ−^1 [W( 1 )−W(λ)]′W(λ) πλ−^1 W(λ)′W(λ)
Recursive∫ 1
λr− (^1) W(r)′dW(r) ∫^1
λr
− (^2) W(r)′W(r)dr
Rolling λ−^1
∫ 1
λ[W(r)−W(r−λ)]
′dW(r)λ− 2 ∫^1
λ[W(r)−W(r−λ)]
′
[W(r)−W(r−λ)]dr
withλ=( 1 +π)−^1 andW(r)a(k 2 × 1 )vector standard Brownian motion. In
the case of the fixed estimation scheme,( 2 )−^1 /^2 1 ∼N(0, 1), so that standard
critical values can be employed. However, for the recursive and rolling estimation
schemes, the forecast encompassing statistic no longer has a standard limit dis-
tribution under the null; critical values for these non-standard distributions are
provided for a range of values ofk 2 andπby Clark and McCracken (2000, 2001).
The above results assume the presence of model estimation uncertainty, with
0 <π<∞. If, on the other hand,π=0, Clark and McCracken (2001) show that
theMDMstatistic is again standard normally distributed in the limit under the
null hypothesis. Thus, if the ration/Ris very small, standard normal critical values
may be employed.
In the more general case whereh>1 and conditionally heteroskedastic fore-
cast errors are permitted, the above results no longer hold in general. Clark and
McCracken (2005) analyze this situation for predictions from nested linear mod-
els, where the forecasts are obtained using direct multi-step methods (see, e.g.,
Bhansali, 2002; Marcellino, Stock and Watson, 2006), as opposed to forecasts
obtained by iterated one-step methods. They find that, for 0<π<∞, FE(2)
MDM-type test statistics do not have pivotal asymptotic null distributions, instead
depending on nuisance parameters that vary with the second moments of the fore-
cast errors, the model regressors, and the orthogonality conditions implicit in the
OLS model estimations.
Two exceptions exist where FE(2)MDM-type forecast encompassing tests do
possess pivotal limit distributions under the null forh>1. First, ifk 2 =1, then the
nuisance parameters vanish and the limit distribution of the test statistics reduces
to that forh=1, as given by (4.18) above. Second, ifπ =0, the test statis-
tics are standard normally distributed. Aside from these exceptions, however, no
nuisance parameter-free asymptotic distributions exist from which critical values
can be obtained. In such cases, critical values must instead be generated by sim-
ulation or bootstrap methods. Clark and McCracken (2005) outline a method for
simulating the asymptotic critical values using estimates of the nuisance parame-
ters, and also an algorithm for obtaining critical values via a parametric bootstrap;