Re: [Gretl-users] Standard error for confidence intervals
On Viernes, 16 de Octubre de 2009 11:03:40 Francisco Sosa escribió:
Hi
I have sent an e-mail to the mailing list, but I haven't got so much
success, I am going to rewrite the question to make it more understandable
(I am not a native English speaker)
Basically, I have a dynamic prediction and I do not know why the standard
error grows up (see below), does anybody have the formula or any paper
where I can look into, thanks
You can read about this in any book about ARIMA models. In spanish:
"Introducción al Análisis de series Temporales" de Ezequiel Urial y Amado
Peiró. Ed: AC (Capítulo 7: Predicción) o
"Introducción al Análisis Univariante de Series temporales" de J.J Cáceres, G.
Martín y F.J. Martín. Ed: Delta publicaciones. (Capítulo 4 sección 4)
In short, if we write the model as: (without constant, but the problem is the
same if you have the constant)
Y_t=\phi Y_{t-1}+eps_t
The prediction error for period T+1 in a AR(1) model is
Y_{T+1}- E(Y_{T+1}/Y_1..Y_T)=\phi Y_T+eps_{T+1}-\phi Y_T=eps_{T+1}
and the variance for the prediction error in T+1 is =Var(eps).
The prediction error for period T+2 is:
Y_{T+2} - E(Y_{T+2}/Y_1..Y_T)=\phi Y_{T+1}+eps_{T+2}
- \phi E(Y_T+1/Y_1..Y_T) =
\phi^2 Y_T + \phi eps_{T+1} + eps_{T+2} - \phi^2 Y_T =
\phi eps_{T+1} + eps_{T+2}
and the variance for the prediction error in T+2 is =
(\phi^2 + 1)*Var(eps).
--
Ignacio Diaz-Emparanza
DEPARTAMENTO DE ECONOMÍA APLICADA III (ECONOMETRÍA Y ESTADÍSTICA)
UPV/EHU
Avda. Lehendakari Aguirre, 83 | 48015 BILBAO
T.: +34 946013732 | F.: +34 946013754
www.et.bs.ehu.es