[Gretl-users] Re: ARIMA on log transformed data
Am 05.01.2025 um 02:13 schrieb Brian Revell:
Take the antilog of the log{y}=z transformed actual and fitted or
forecast values from the ARIMA model, which if in natural logs will
be EXP(z).
Brian, as explained (inter alia) by Dave Giles in the link that Allin
gave, this is not completely correct.
However, ... (see further down)
On Sat, 4 Jan 2025, 23:59 Cottrell, Allin, <cottrell(a)wfu.edu> wrote:
On Sat, Jan 4, 2025 at 2:55 PM <dbrilakis(a)yahoo.gr> wrote:
>
> Hi, I found that my data become stationary (after differentiate)
the log data with best ARIMA(p,d,q) How do I rebuilt the ARIMA
forecast to the origina scale before log?
There's a standard means of converting from a forecast or fitted value
of log(y) to that of y itself, if the error term is reckoned to be
normal, plus some variations on the theme. Dave Giles has quite a nice
discussion of the point: see
https://davegiles.blogspot.com/2014/12/s.html
One interesting point on that blog page is also the comment by user
"Daumantas", citing BÅRDSEN, G. & LÜTKEPOHL, H. 2011. Forecasting levels
of log variables in vector autoregressions. International Journal of
Forecasting, 27, 1108-1115: '...if specification and estimation
uncertainty are taken into account [...] in practice, using the
exponential of the log forecast is preferable to using the optimal
forecast." [...] (Log-normality is assumed...)'
In that sense Brian's simple recipe would not be misguided.
Apart from that, this natural question has been asked a couple of years
ago, see:
https://gretlml.univpm.it/hyperkitty/list/gretl-users@gretlml.univpm.it/t...
For convenience, I'm reproducing the script that Jack provided back then:
<hansl> open bjg.gdt --quiet series insample = t < "1960:3" series f =
NA smpl insample == 1 --restrict arima 0 1 1 ; 0 1 1 ; lg fcast
--out-of-sample matrix F = $fcast + 0.5 * $fcse.^2 smpl insample == 0
--restrict --replace f = exp(F) setinfo f --graph-name="forecast" smpl
full gnuplot g f --time-series --with-lines --output=display
</hansl>
Actually, I'm not sure whether in this script $fcse should be used as an
estimator of the theoretical sigma, or perhaps rather the model accessor
$sigma.
Again, all this assumes (log)normal errors, which might be completely
wrong for the given data.
Maybe we could offer something more automated if the dependent variable
of a model is recognized as being the log of something. As this is
becoming increasingly technical, I will post something to follow up on
this to the development list.
cheers
sven
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