178 Forecast Combination and Encompassing
FE(2) and FE(3), the null and alternative hypotheses take the same form as above
but with the parameterλreplacingβ 2.
The most straightforward approach to testing would be to simply estimate the
relevant regression, FE(1), FE(2) or FE(3), by OLS, and then perform a standard
t-test of the null thatβ 2 =0orλ=0. However, as Harvey, Leybourne and New-
bold (1998) show in the context of the FE(2) regression, such an approach is not
robust to properties of the forecast errors that one might expect to encounter
in practice. First, an implicit assumption of the regression errorεtbeing identi-
cally and independently distributed (i.i.d.) is not plausible for forecasts at horizons
greater than one, since even optimal forecasts in this setting would be expected to
have errors that follow a moving-average process of orderh−1. Second, in some
applications it is likely that the forecast errors are non-normally distributed (for
example, Harvey and Newbold, 2003, find substantial evidence of non-normality
in US macroeconomic forecast errors). Non-normality in the errors induces con-
ditional heteroskedasticity in the regression FE(2), resulting in oversized tests if
conventionalt-tests are applied.
Harvey, Leybourne and Newbold (1998) consider two ways of obtaining asymp-
totically correctly-sized tests in these situations. One is to continue with a
regression-basedt-test, but using standard errors that are robust to heteroskedas-
ticity and autocorrelation. Specifically, assuming the forecast errors are at most lag
(h− 1 )dependent (in line with forecast optimality), they propose the use of a rect-
angular lag window for long-run variance estimation, as in Diebold and Mariano
(1995). This approach yields the test statistic:
R 1 =
λˆ
√√
√
√√∑h− 1
j=−(h− 1 )∑n
t=|j|+ 1 (e 1 t−e 2 t)εˆt(e1,t−|j|−e2,t−|j|)εˆt−|j|
[∑n
t= 1 (e^1 t−e^2 t)
2 ]^2,whereλˆis the least squares estimate ofλin the FE(2) regression. Alternatively, one
could impose the information thatλ=0 under the null, replacingˆεtwithe 1 tin
the variance estimator:
R 2 =
λˆ
√√
√√
√∑h− 1
j=−(h− 1 )∑n
t=|j|+ 1 (e^1 t−e^2 t)e^1 t(e1,t−|j|−e2,t−|j|)e1,t−|j|
[∑n
t= 1 (e 1 t−e 2 t)
2 ]^2.Under standard assumptions about the forecast errors, both R 1 and R 2 are
asymptotically standard normally distributed.
The second approach proposed by Harvey, Leybourne and Newbold (1998)
observes that the null hypothesis under the FE(2) specification requires:
E[e 1 t(e 1 t−e 2 t)]=0.This motivates testing for forecast encompassing via a test for whether the series
dt=e 1 t(e 1 t−e 2 t)has zero mean, along the lines of Diebold and Mariano (1995),