How are the confidence intervals calculated in the hw() function the the forecast package in R? I have been using the hw() function from the forecast package to predict some timeseries objects. I've noticed for some of my timeseries predictions, the prediction interval sizes are almost the exact same size, no matter how far into the future I'm predicting. I would expect the size to increase as you predict further into the future as you become more uncertain of your predictions, so I'm wondering does anyone know where I could find the actual code for the hw() function, as I can't seem to find it myself, or can anyone explain why my timeseries would get a confidence interval of the same size?
Below is a screenshot of my timeseries, when I make predictions for any number of months, the difference between the upper and lower bounds is always between 3226 and 3227. One explanation I can think of is that my timeseries is only 2 years so there is not a lot of data to make predictions with, but it still seems strange to me that the size of the prediction interval does not change much.


The code is visible if you type the name of the function in the console. Or you could go to the github repository for the package and find it there: https://github.com/robjhyndman/forecast/blob/master/R/HoltWintersNew.R#L516
In any case, the formula for the forecast variance (for additive Holt-Winters) is given by (see https://otexts.com/fpp3/ets-forecasting.html)
$$\sigma_h^2 = \sigma^2\Big[1 + (h-1)\big\{\alpha^2 + \alpha\beta h + \frac16\beta^2h(2h-1)\big\}+ \gamma k \big\{\alpha+\gamma + \beta m (k+1)\big\}\Big]$$
where $h$ is the forecast horizon. So if $\alpha$, $\beta$ and $\gamma$ are all close to zero, then the forecast variance is approximately a constant, and the prediction intervals will be almost equal at all horizons.
