The name is absent

10

where superscript j denotes the candidate forecast model (with 0 indicating the
benchmark), e_t^j₊^h_h is the forecast error made by candidate forecast model j at time t
for forecast horizon h, and T denotes the number of forecasts made. Subsequently,
we report the relative RMSFE (for each horizon h) by dividing the respective
RMSFE of each of our alternative specifications by the corresponding RMSFE of
the benchmark. If the relative RMSFE is below 1, the alternative specification
displays a better forecast performance than the benchmark. To test whether the
differences are statistically significant, we employ a DM test (Diebold and
Mariano 1995). This test is based on the null hypothesis that two non-nested series
of forecasts { f_t^0h}_t^T₌₁ and { f_t^jh}^T_t=1 are of equal accuracy,

E(d^j ) = EΓ(e^jh )2_-(_e0^h )21 = 0                     (3 4)

E (^ut+h ) E l(et+h ) (et+h ) I ʊ ,                             l-’.4J

In this test, the loss function is the difference of the squared forecast errors of the
candidate forecasts. Because the sample mean loss differential is asymptotically
normally distributed, the large-sample DM test statistic is

d^jh

DMjh = -t=7 ,                     (3'5)

J/

T rτ

where d^jh is the sample mean loss differential and γj^h is the cumulative sample
autocovariance up to order h-1.

Encompassing test

Even if a forecast { f_t^jh}^r_t=1 does not outperform the benchmark { f_t^0h}_t^r₌₁ , a combi-
nation of these two forecasts could nevertheless help to improve forecast accuracy.
Therefore, we consider here the combined forecast { f_t^ch }^r_t=1 estimating λ_jh as the
corresponding “best” weight.

ft^ch=(1-λjh)ft^0h+λjhft^jh.                           (3.6)

If the null hypothesis λ_jh = 0 is true, { f_t^0h }_t^r₌₁ is conditionally efficient with re-
spect to { f_t^jh}_t^r₌₁ (Granger and Newbold 1973; 1986) or encompassing { f_t^jh}_t^r₌₁
(Hendry 1993). In this case, the DM-statistic can be calculated for each period t as

d ^jh = (e^0h -e^jh ) e^0h                                (3 7)

d_t+h = (e_t+h -e_t+h ) e_t+h .                                (3.7)

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