Obtaining the mean forecast instead of the median forecast

[newsagent]

Suppose we have a time series variable y_t, where t=1,2...,T.

We want to forecast the variable in periods T+1, T+2,...

Suppose also we want to model the log of y_t, and that an ARIMA(1,1,2) is selected for this transformation.
So, an ARMA(1,2) is used for x=diff(log(y_t))

How can I obtain the mean forecasts (without bias) for y_t?

Based on this site (https://otexts.com/fpp3/ftransformations.html), I think that in this case the

mean forecasts of y_t =median forecasts*(1+0.5*sigma^2_h)
               
                          =exp[log(y_T)+cumsum(x)]*(1+0.5*sigma^2_h),

where sigma^2_h is the forecast variance at horizonn h that we obtain from the ARMA(1,2) for x.

I tried this "manuall approach" but I have not managed to get the same results with the automated process of the fable package.

So, I suspect that my formula above for the mean forecasts of y_t is not correct but I can not find it. Some guidance is greatly appreciated

For the case where x=log(y_t) the formula (given also in the above link)

mean forecasrs =exp[log(y_T)]*(1+0.5*sigma^2_h),
`
works fine and gives me the same result with the fable package.

[robjhyndman]

Yes, for a log transformation, the mean forecast is approximately equal to the median forecast times $(1+0.5\sigma^2_h)$. This is what is returned by the fable package, provided the log transformation is included in the model statement.

However, your expression for the median forecast is incorrect for an ARIMA(1,1,2) model. You have not taken account of either the AR or MA parts of the model.

[newsagent]

thank you. To finf the correct median I have to use the taylor approximation, I guess

[robjhyndman]

No, the median is simply the back-transformed forecast. You don't need a Taylor approximation. See https://otexts.com/fpp3/arima-forecasting.html for computing forecasts from an ARIMA model