Michael P. Clements and David I. Harvey 179who presented such an approach for testing the null of equal forecast accuracy.
Under standard assumptions:
√
n[d ̄−E(dt)]⇒N(0,S), (4.12)
whereSdenotes the long-run variance ofdt, giving rise to the statistic:
DM=
nd ̄
√
∑h− 1
j=−(h− 1 )∑n
t=|j|+ 1 (dt−
d ̄)(dt−|j|−d ̄), (4.13)whered ̄=n−^1
∑n
t= 1 dtand the implied estimator ofSuses a rectangular lag win-
dow as above. This statistic has an asymptotic standard normal distribution under
the null of forecast encompassing, and is robust to the aforementioned forecast
error properties of autocorrelation and non-normality. Harvey, Leybourne and
Newbold (1998) propose a small modification of this test which has improved finite
sample properties, drawing on work by Harvey, Leybourne and Newbold (1997).
The modified statistic is:
MDM=n−^1 /^2 [n+ 1 − 2 h+n−^1 h(h− 1 )]^1 /^2 DM, (4.14)and the recommendation is to use critical values from thetn− 1 distribution rather
than those from the limiting standard normal. Simulation results in Harvey,
Leybourne and Newbold (1998) show thatMDMhas better finite-sample size
properties than the regression-based variantsR 1 andR 2 , although some loss in
size-adjusted power relative toR 1 is observed for small samples.
In addition to forecast errors being autocorrelated (forh> 1 )and possibly non-
normally distributed, it may also be the case that the errors exhibit autoregressive
conditional heteroskedasticity (ARCH – see, e.g., Engle, 1982), with the squared
forecast errors following a dependent sequence. Intuitively, this describes the sit-
uation where, if a variable proves difficult to forecast in one period, it is likely to
prove difficult to forecast in the next period as well. In such circumstances, and
again in the context of FE(2), Harvey, Leybourne and Newbold (1999) show that the
forecast encompassing tests suffer asymptotic size distortions, rejecting the forecast
encompassing null too frequently. They also propose a simple modification which
largely overcomes the size problem; this involves computingMDMas above, but
replacinghin (4.13) and (4.14) with0.5n^1 /^3 +h, where.denotes integer part.
This modification should be employed whenever ARCH in the forecast errors is
suspected, or is detected through prior testing.
Although Harvey, Leybourne and Newbold (1998) propose theMDMtest (4.14)
in the context of FE(2), the test can also be applied using the forecast encompassing
specifications FE(1) and FE(3), the only difference being the specification ofdt. For
FE(1), application of the Frisch–Waugh theorem shows thatβ 2 is identical to that in
the regressionη 1 t=β 2 η 2 t+νt, whereη 1 tandη 2 tdenote the errors from regressions
ofytandf 2 t, respectively, on a constant andf 1 t. This allows us to write the null
hypothesis as:
E(η 1 tη 2 t)=0,