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

Direct multi-step estimation and forecasting
the multi-step point forecasts are more accurate than those obtained by the other models. The
prediction interval does not necessarily increase with the horizon for the “adaptive” forecast and it
has the same amplitude as that of the AR(2) but, contrary to the latter, always (except at 12 steps
ahead) contains the true outcome. The ARIMA model does contain the realised value too, but the
forecast interval is much larger than that of the previous two models. The author concludes that
multi-step estimation does indeed yield a positive outcome.
In their article, Lin and Tsay (1996) study whether using cointegrating properties improve
long-term forecasting. In an empirical analysis, they compare several forecasting techniques to
an ‘adaptive’ procedure of multi-step forecasts, which they use as a benchmark since it does not
postulate the existence of a true model. They use a non-stationary VAR(p) model for the vector
of n variables:
x t = τ +
p∑
Υ i x t−i + ɛ t ,
i=1
where the ɛ t ∼ IN (0, Σ). It is assumed that, letting Υ (L) = ∑ p
i=1 Υ iL i , the series {Ψ i } 0 ∞
is such
that [Υ (L)] −1 = ∑ ∞
i=0 Ψ iL i . The h–step ahead forecast error is then given by
h−1
∑
e T +h,h = Ψ i ɛ T +h−i .
i=0
The DMS multi-step parameter estimates are given by minimizing the in-sample sum of the squared
e t,h . This implies a non-linear function of the elements of {Υ i } 1 p
. For simplicity, Lin and Tsay suggest
to simply use the least squares projection of x t onto the space spanned by (x t−h , ..., x t−h−p+1 )
and a constant, for t = h+p, ..., T . The computing time of this alternative estimator is much lower.
Lin and Tsay compare their DMS forecasts to those obtained by cointegrated VARs for seven
financial and macro-economic data sets. The vector processes are of dimension varying from 3 to
5 and are estimated over samples of 230 to 440 observations. The criterion used for analysis is
the square root of the average trace of the MSFE. Their results show that multi-step techniques
provide a greater forecast accuracy (up to a 60% gain), but for long horizon (beyond 50) only two
of the series still exhibit a gain from using DMS. The authors find it difficult to account for these
results.
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