Re: [Gretl-devel] ARIMA news
Allin Cottrell schrieb:
Model C:
\phi(L) ((1-L) y_t - X_t\beta) = \theta(L) \epsilon_t
Model D:
\phi(L) (1-L) (y_t - X_t\beta) = \theta(L) \epsilon_t
In practical terms the issue between C and D is, given that we're
differencing y_t, do we also difference X_t? (No = C, Yes = D.)
Up till now we have been doing C, and there's nothing inherently
wrong with that, but for the sake of compatibility with most other
ARIMA software we should be doing D. And in current CVS we do D,
unless the new option -y (or --y-diff-only) is given in
conjunction with the arima command, in which case C is restored.
I haven't used this setup myself, so my opinion isn't strong on this.
Having said that, in principle I like C better. The reason is that if
you have two (or more) I(1) variables, IMHO you should start to think
about the possibility of cointegration, otherwise you risk
mis-specification by ignoring the levels relationship. Differencing
everything right away looks very 1960-ish to me. This historical legacy
presumably is also the reason why other software is doing D. (But I
admit that comparability with other software is important, too, and
model C doesn't help you either if the variables are cointegrated.)
Also, if y is I(1) and X is I(0), then with D you are over-differencing
X. So the only special case where D makes sense is that all variables
are I(1) but no cointegration is present. I think a more sensible
default for an ARIMAX model would be to assume a situation where y is
I(1) and X is I(0).
But of course, as long as everything is possible with options, this is
maybe not too important.
Fwiw,
Sven