direct multi-step estimation and forecasting - OFCE - Sciences Po

Direct multi-step estimation and forecasting
˜Υ h has, therefore, some potential when ̂Υ is badly biased for Υ, or when E[x T +1 | x T ] = Ψx T but
E[x T +h | x T ] ≠ Ψ h x T . However, they remark also that in stationary processes, misspecification
of the DGP is not sufficient to advocate the use of DMS, since ̂Υ is the OLS and ˜Υ h converges
towards the unconditional expectation with Υ h tending to zero as h increases. Hence, increasing
divergence between ̂Υ and ˜Υ h is unlikely. Moreover, DMS is inefficient in small samples, so
that if ɛ t ∼ IN ( 0, σ 2 ɛI n
)
, biases are unlikely to be enough for a gain to appear. Thus, Clements
and Hendry note that if ɛ t follows a negative moving average, there may be some potential for
DMS. They derive a taxonomy of forecast errors and show that the only terms in common for
both methods are those of error accumulation, namely ∑ h−1
i=0 Υi ɛ T +h−i in the framework above.
Simulating the small sample estimation biases, they show, for several stationary values—0, 0.4
and 0.8—of the autoregressive coefficient in a univariate AR(1) process without intercept, that for
sample sizes ranging from 10 to 100, the two step ahead DMS does not yield better estimates of
the powered coefficient than the squared IMS.
This result is specific to finite samples as Chevillon and Hendry (2005) show: when estimating
(16) with an additional drift by OLS and (17), with a drift also, by GMM, with a HAC covariance
matrix estimator, DMS is asymptotically more efficient than IMS in the case of stationary processes
with positive slope. Indeed, in the univariate case, denoting by ê h and ẽ h the IMS and DMS forecast
errors at horizon h using GMM estimation, and ρ the slope coefficient, these authors show that:
h ( E [ ê 2 [ẽ2 ] ) 2ρ
h]
/E h − 1ρ →
h→∞ (1 − ρ 2 ) (2 − ρ) ,
the latter being of the same sign as ρ, as long as |ρ| < 1. In the case of integrated processes, this
result collapses and IMS always dominates DMS.
A Monte Carlo analysis by Clements and Hendry, of the forecasts from the ‘nonseasonal Holt–
Winters Model’ illustrates the relative behaviours of the IMS and DMS techniques in their framework.
The data is generated by the sum of unobserved components for the trend, intercept and
irregular elements:
y t = µ t + ε t ,
µ t = µ t−1 + β t + δ 1t ,
β t = β t−1 + δ 2t .
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