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

Direct multi-step estimation and forecasting
They show that
( )
T (̂ρ 1 S ) h − 1
T
( )
(̂ρ IV ) h − 1
T (˜ρ DMSh − 1 )
T (˜ρ IV DMSh − 1 )
⇒
⇒
⇒
⇒
(∫ 1
−1
W (r) dr) 2 h
0
(∫ 1
0
[
1
( )
W (1) 2 − 1 +
2
) −1
W (r) 2 h
( )
dr W (1) 2 − 1 ,
2
(∫ 1
−1
W (r) dr) 2 h
0
(∫ 1
0
[
1
( )
W (1) 2 − 1 +
2
) −1
W (r) 2 h
( )
dr W (1) 2 − 1 ,
2
]
θ
(1 + θ) 2 ,
]
θ
h (1 + θ) 2 ,
and provide small sample approximations of the distributions. The leftward non-centrality of IMS
therefore increases with h, whereas that of DMS does not.
The instrumental estimators seem
better. Simulations illustrate these results.
This framework is also analysed in Chevillon and Hendry (2005) who now allow for a drift in the
random walk. This induces the presence of a deterministic trend which asymptotically dominates
estimation, yielding the same asymptotic accuracy for both methods. In finite sample though,
disparities appear: DMS is more accurate when the drift is ‘small’ compared to the variance of the
disturbances and when the latter exhibit negative serial correlation. Introducing the concept of
‘weak’ trend whereby the drift coefficient vanishes to zero asymptotically at the rate of O ( T −1/2) ,
Chevillon (2005b) derives asymptotic distributions where he allows for both the stochastic and
deterministic trends to have an impact on estimation. The model he uses is:
( h−1
) ∑
y t = ρ i τ T + ρ h y t−h + ε t , for h ≥ 1,
i=0
where τ T = √ ψ
[ ∑T
]
(
, Var[ε t ] = σ ε and σ 2 = lim T →∞ T −1 E
t=1 ε t . The resulting IMS,
T
and DMS, (˜τ )
h,T , ˜ρ h,T , estimators are such that
⎡
⎤ ⎡
√ √ ) ⎤
⎢ T (˜τ h,T − τ h,T ) ⎥ ⎢ T
(̂τ {h}
[∫
T
− τ 1 h,T ⎥
(h − 1) θ
⎣
T (˜ρ h,T − 1 ) ⎦−⎣
( ) ⎦ ⇒
0
T ̂ρ h ∫ ( 1 ∫ ) K ]
ψ,φ (r) dr
T − 1
0 [K ψ,φ (r)] 2 1 2
,
dr −
0 K −1
ψ,φ (r) dr
where K ψ,φ is a drifting Ornstein-Uhlenbeck process defined as
K ψ,φ (r) = ψf φ (r) + σ
∫ r
0
e φ(r−s) dW (s) ,
with W (r) a Wiener process on [0, 1] and:
̂τ {h}
T
, ̂ρh T
)
,
(18)
f φ (·) : r → eφr − 1
φ
if φ ≠ 0, and f 0 (r) = r. (19)
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