two types of forecasts

[newsagent]

In section 5.6 of the book it is mentioned that

"For example, you may wish to add up sales forecasts from various regions to form a forecast for the whole country.
But medians do not add up, whereas means do."

So, suppose that we have this case

library(fpp3)

us_retail_employment <- us_employment |>
  filter(year(Month) >= 1990, Title == "Retail Trade") |>
  select(-Series_ID)
 

fc1 <-  us_retail_employment |>   model(m1=ARIMA(log(Employed)~pdq(0,2,1))) |>
  forecast(h = 7) %>% mutate(.median=median(Employed))

# A fable: 7 x 5 [1M]
# Key:     .model [1]
  .model    Month           Employed  .mean .median
  <chr>     <mth>             <dist>  <dbl>   <dbl>
1 m1     2019 Oct t(N(9.7, 7.1e-05)) 15665.  15664.
2 m1     2019 Nov t(N(9.7, 0.00017)) 15963.  15962.
3 m1     2019 Dec   t(N(9.7, 3e-04)) 15973.  15971.
4 m1     2020 Jan t(N(9.7, 0.00046)) 15621.  15617.
5 m1     2020 Feb t(N(9.6, 0.00066)) 15471.  15466.
6 m1     2020 Mar t(N(9.6, 0.00091)) 15458.  15451.
7 m1     2020 Apr  t(N(9.6, 0.0012)) 15471.  15462.

To calcuate the annual forecasts we have

fc1  %>% as.data.frame() %>%
 group_by(year(Month)) %>%  summarize(mean_annual=mean(.mean),
 median_annual=mean(.median))

 A tibble: 2 × 3
  `year(Month)` mean_annual median_annual
          <dbl>       <dbl>         <dbl>
1          2019      15867.        15962.
2          2020      15505.        15464.

Hence,

A) it is wrong to take the average of the monthly median forecasts in order to obtain the median_annual, as these types of forecasts do not add up/can not be averaged

B) The correct choice is the mean_annual.

[robjhyndman]

Note that this is a different situation from the one you reference in the book. You are averaging over time rather than aggregating over geographical regions.

Employed here is the number of employed people in each month for retail trade in the US. So aggregating over time would make no sense (mostly the same people are employed each month).

Your fc1 object contains means and medians of the forecast distribution for each month. Then you compute the mean of the means, and the median of the medians. It is not clear why, or how you wish to interpret these. It is not "wrong", but it is not clear what is intended, so it is not possible to say what is the "correct choice".

[newsagent]

It should have been mean(.median). That was a mistake. My bad. I made the correction above

The general question that i wanted to ask is that if I have a macroeconomic (say montly of quarterly) variable like inflation, unemployment, GDP etc, does it make sense to take the mean of the (monthly or quarterly) median forecasts in order to obtain the annual forecasts or should we take the mean of the (monthly or quarterly) mean forecasts?

[robjhyndman]

The mean of median forecasts is a little hard to interpret (what is it estimating?), whereas the mean of means can be simply presented as the annual mean. Because of small differences in month or quarter length, it is not quite the same as estimating the variable annually, but close enough.

[newsagent]

Thanks!