Forecasting Path Averages

Consider two standard types of \(h\)-step forecast:

(a).  \(h\)-step forecast, \(y_{t+h,t}\), of \(y_{t+h}\)

(b).  \(h\)-step path forecast, \(p_{t+h,t}\), of \(p_{t+h} =  \{ y_{t+1}, y_{t+2}, ..., y_{t+h} \}\).

Clive Granger used to emphasize the distinction between (a) and (b).

As regards path forecasts, lately there's been some focus not on forecasting the entire path \(p_{t+h}\), but rather on forecasting the path average:

(c).  \(h\)-step path average forecast, \(a_{t+h,t}\), of \(a_{t+h} =  1/h [y_{t+1} +  y_{t+2} + ... +   y_{t+h}]\)

The leading case is forecasting "average growth", as in Mueller and Waston (2016).

Forecasting path averages (c) never resonated thoroughly with me.  After all, (b) is sufficient for (c), but not conversely -- the average is just one aspect of the path, and additional aspects (overall shape, etc.) might be of interest. 

Then, listening to Ken West's FRB SL talk, my eyes opened.  Of course the path average is insufficient for the whole path, but it's surely the most important aspect of the path -- if you could know just one thing about the path, you'd almost surely ask for the average.  Moreover -- and this is important -- it might be much easier to provide credible point, interval, and density forecasts of \(a_{t+h}\) than of \(p_{t+h}\).

So I still prefer full path forecasts when feasible/credible, but I'm now much more appreciative of path averages.