# Forecasting a transformed time series

I have fitted a seasonal ARIMA model using R to a log transformed times series which I called lnseries.

I can forecast fine for the transformed time series (lnseries) storing the ARIMA model (which I called fit) then using the command:

$\texttt{plot(forecast(fit))},$

this shows me the forecast and 95% confidence interval. But I'm stuck on how to get the actual original time series forecast plot using this model.

Help anyone!?

• Use the lambda=0 argument when fitting the ARIMA model. Jan 3, 2015 at 13:08
• Careful with the use of logs. Like drugs, they can have side effects. They are an overused tool that can be handled in other ways in a less risky fashion. Take a look at this post stats.stackexchange.com/questions/121592/… Jan 6, 2015 at 20:56

If your sample sizes are quite large and you have approximate normality on the log scale you could treat your variance estimate as "known" and produce a reasonable approximation of a mean forecast by taking $\exp(\hat{y}_t+\frac{1}{2}\sigma^2_{t})$ where $\sigma^2_{t}$ is the conditional variance of the predicted observation.
• @Gleb_b is adding $.5\sigma^2$ how you transform the median to the mean under normality assumption? Also, OP is modelling ARIMA, you may need to drop the subscript $t$ in the variance. Jan 3, 2015 at 12:37
• @mugen $\exp(\mu)$ and $\exp(\mu+\frac{1}{2}\sigma^2)$ are the median and mean of a lognormal distribution. [However, if we account fully for the estimation error in $\sigma$ as well, then you'd get a distribution which doesn't have a mean; but the prediction limits for an individual observation still apply just fine.] Jan 3, 2015 at 14:32
• For individual observations the events correspond under monotonic increasing transformation (including logs and exponentiation), so $P(l\leq Y_t\leq u)=P(\log(l)\leq \log(Y_t)\leq \log(u))$. But for a parameter, the log or exp of a mean is not itself the mean on the transformed scale. Nov 24, 2021 at 23:14