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

Direct multi-step estimation and forecasting
9.2 A general forecast-error taxonomy
We now borrow from Clements and Hendry who suggest in Clements and Hendry (1998b) and
Hendry (2000) a general forecast error taxonomy which helps us in assessing the advantages of
multi-step estimation. We use the framework presented above but for ease of exposition modify it
slightly. Notice that in (20), f x (·), if it represents the true DGP, provides the—potentially time
dependent—joint density of x t at time t, conditional on X t−p
t−1 = (x t−1, ..., x t−p ), and z t . Assume,
without loss of generality, that {z t } contains only deterministic factors—such as intercepts, trends
and indicators—and that all stochastic variables are included in {x t }. As previously, it is desired to
forecast x T +h , or perhaps of function thereof (e.g. if z t originally contained stochastic variables),
over horizons h = 1, ..., H, from a forecast origin at T . Now, the dynamic model does not coincide
with the data generating process and it specifies the distribution of x t conditional on X t−r
t−1 , with lag
length r, deterministic components d t and implicit stochastic specification defined by its parameters
ψ t . This model is fitted over the sample t = 0, ..., T, so that parameter estimates are a function of
the observations, represented by:
̂ψ T = Ψ T
(
˜X0 T , D 0 T
)
, (23)
where ˜X denotes the measured data and, as before D 0 t = (d t , ..., d 0 ). A sequence of forecasts
{̂x T +h|T } is produced as a result. The subscript on ̂ψ in (23) denotes the influence of the sample
size. Let ψ e T = E T
[̂ψT
], where it exists. Because the underlying densities may be changing over
time, all expectation operators must be time dated. Future values of the stochastic variables are
unknown, but those of deterministic variables are known; there, therefore, exists a function g h (·)
such that
̂x T +h|T = g h
(
˜XT
−r+1
T
, D T +1
T +h , ̂ψ T
)
. (24)
The corresponding h–step ahead expected forecast error is, thus, the expected value of e T +h|T =
x T +h − ̂x T +h|T , and is given by
]
E T +h
[x T +h − ̂x T +h|T | X 0 T , {Z ∗ } 0 T +h
,
where the actual values of the deterministic factors over the forecast period (including any deterministic
shifts) are denoted by {Z ∗ } T +1
T +h and {Z∗ } 0 T +h = {Z ∗ } T +1
T +h , Z0 T ; and the
[
]
expectation
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