Re: [Gretl-devel] OLS Forecasting
On Mon, 9 Feb 2009, Riccardo (Jack) Lucchetti wrote:
On Sun, 8 Feb 2009, Sven Schreiber wrote:
> Well I'm kind of shooting in the dark here because I'm not familiar with
> what model setups we're talking about *exactly*. But it kind of sounded
> that it was all about whether the dependent variable was in differences
> or not. That, however, would not be enough to warrant an I(1) assumption
> for the levels. For example, it's perfectly legitimate to formulate a
> single-equation error-correction model for I(0) variables, and in that
> case the left-hand side variable would appear in differences, but the
> levels would be I(0), and still the model would *not* be mis-specified.
You're right. However, in that case you'd have the lagged level as an
explanatory variable. In order to compute proper k-steps-ahead forecasts, it
seems to me (but I need to think about this a little more) that you have to
unravel the autoregressive structure somehow. Hmm.
After some more reflection: we ought to distinguish two cases. If the
model does not include lags of the dependent variable or of its level,
we're ok. If it does, things are more complicated: assume the model can be
written as
A(L) y_t = x_t \beta + \epsilon_t
(note that this includes the "difference" case as A(L) could well be 1-L).
In general, we need to reconstruct the A(L) polynomial, which can be far
from trivial, since variables can be renamed etc; but if this can be done,
then the variance of the k-step ahead predicition error can be
reconstructed quite easily via inversion of the A(L) polynomial (details
omitted here for brevity).
This would be quite nice to have because it would enable us to do all
sorts of fancy tricks, like for instance automatic conversion of an ADL
model to ECM form and vice versa, computation of dynamic multipliers and
so on. However, it may be tricky to implement "right". We do have, at
present, a mechanism for keeping track of a variable being a
lag/difference of another one, but it may be a little fragile for this
purpose. Allin?
Riccardo (Jack) Lucchetti
Dipartimento di Economia
Università Politecnica delle Marche
r.lucchetti(a)univpm.it
http://www.econ.univpm.it/lucchetti