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

Direct multi-step estimation and forecasting
interest, not the one-step estimated by a multi-step criterion or (iv) the gains are to be found in
small samples. These assumptions are studied by other authors as we see below and will lead to
the preferred approach to direct multi-step estimation which no longer aims to estimate the 1S
model via multi-step techniques.
4 Parametric DMS estimation under misspecification.
The first contributions to the parametric approach to multi-step forecasting suggest some forms of
model misspecification which could provide a sufficient rationale for the use of DMS. By parametric,
it is meant that the one-step ahead parameters are the object of interest but that they are estimated
by minimizing functions of the multi-step errors.
Stoica and Nehorai (1989) extend the concept of direct multi-step estimation which was suggested
by Findley to ARMA models:
A (L) y t = C (L) ɛ t ,
where A (L) = ∑ p
i=0 a iL i and C (L) = ∑ p
i=0 c iL i . The forecasts ŷ T +h,h are computed as the
conditional expectation of y T +h given y T for the model with parameter θ = (a 0 , ..., a p , c 0 , ..., c p ).
Define B h (L) = ∑ h−1
i=0 b iL i and D h (L) = ∑ p
i=0 d iL i , such that:
C (L) = A (L) B h (L) + L h D h (L) ,
so that
y t =
(
B h (L) + D h (L)
A (L) Lh )
ɛ t .
The h–step ahead forecast error is thus given by
e T +h,h = y T +h − ŷ T +h,h = B h (L) ɛ T +h .
The authors define the multi-step parameter estimator as that which minimizes a function of the
in-sample squared multi-step forecast errors,
̂θh = argmin F (V 1,T , ..., V h,T ) ,
˜θ h ∈Θ
12