Re: [Gretl-devel] OLS Forecasting
On Sat, 7 Feb 2009, Sven Schreiber wrote:
On 07.02.2009 20:35, Allin Cottrell wrote:
> On that basis, we calculate
>
> se(k) = sqrt(k*s2)
>
> where k = 1,2,...n is the forecast step and s2 is the square of
> the Standard Error of the Regression.
>
> If this is rubbish -- or if there is a clearly better way to
> proceed -- I hope Jack or Sven will tell me so!
>
Indeed it seems to me that this is not entirely correct in general; with
this formula the forecast uncertainty grows linearly (and thus w/o
bounds) irrespective of the property of the considered variable. But for
a mean-stationary variable the long-run forecast is just its mean, and
the associated confidence interval is finite. So in effect here it looks
as if you're assuming something like a random walk, and I'm not sure why
this assumption would be warranted in general. Just because the variable
was differenced by the user doesn't qualify IMHO.
The key is "mean-stationary" . Remember we're talking about the forecasts
on the level, while the statistical model is in differences. If you assume
that your model is correctly specified, then the k periods ahead forecast
is your current value of the level plus the sum of the k forecasts of the
difference.
Another equivalent way of saying this is that if the "right" model is in
differences, then the level is I(1), and possesses no unconditional mean.
However, you can make forecasts of y_{t+k} conditional on your information
set up to time t (call it F_{t}). However, as k grows, your confidence
band becomes wider and wider, as \lim_{k \to \infty} V(y_{t+k} | F_{t})
diverges.
Riccardo (Jack) Lucchetti
Dipartimento di Economia
Università Politecnica delle Marche
r.lucchetti(a)univpm.it
http://www.econ.univpm.it/lucchetti