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

Direct multi-step estimation and forecasting
it is possible, by iterated substitution, to find a set of non-linear function β h,i of the set of {α 1,i }
such that:
y T +h =
k∑
β h,i y T −i .
i=0
Denote by ̂β h,i the function of the estimated ˜α 1,i so that ŷ T +h,h = ∑ k
i=0 ̂β h,i y T −i . And then, the
IMS MSFE is defined as:
̂V IMS
h,k
[
= E (y T +h − ŷ T +h,h ) 2] .
Note that {y t } can expressed as an autoregressive process of order p according to Wiener–Kolmogorov’s
theorem, where p can be infinite. Then, if k ≥ p, the theoretical (for known parameters) MSFEs coincide
for DMS (V DMS
h,k
) and IMS (Vh,k
IMS ); but if k < p, the latter is larger than the former, which in
turn is larger than the ‘true’ MSFE from the correct—potentially infinitely parameterized—model
(see Bhansali, 1996). Define γ i as the ith autocorrelation of {y t } for i ≥ 1 and γ 0 its variance (in
stationary processes), then using an AR(1) as a forecasting model:
[γ 2 − (γ 1 ) 2] 2
V IMS
2,1 /V DMS
2,1 = 1 +
1 − (γ 2 ) 2 ≥ 1,
where the equality arises when the model is well specified. Similarly it can be shown that if the
( ) 4 θ
process follows an MA(1) with parameter θ, V2,1 IMS /V2,1 DMS = 1 +
1 + θ 2 > 1.
Bhansali recalls the main asymptotic findings. For a well specified model, the 1S estimation
procedure is asymptotically equivalent to maximum likelihood and in the case of Gaussian processes,
achieves the Cramér–Rao bound; yet this is not the case when k ≠ p. By contrast, DMS is
asymptotically inefficient for a well-specified model. However, if k (T ) → ∞, the distributions of
the DMS and IMS estimators coincide for T → ∞, under some regularity conditions (see Bhansali,
1993 when k (T ) = o ( T 1/3) ). In analyzing the ARMA(1, 1) model:
y t − ρy t−1 = ɛ t − θɛ t−1 ,
Bhansali notes that the two-step ahead forecast is given by:
y t+2 = −ρ (θ − ρ)
∞∑
θ i y t−i = τ (1 − θL) −1 y t , (10)
i=1
so that he recommends to minimize the in-sample sum of squared forecast errors:
∑ (
y t+2 − τ (1 − θL) −1 y t
) 2
,
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