I'm an undergraduate student of applied mathematics and while I was studying time series models I encountered this proposition in my text book that I don't understand.
The particular time model is this one: $F_{t,h}= \hat B_t\cdot\hat S_{t+h-s} $
Where $F_{t,h}$ is the forecast for time bucket t+h made at time t, $\hat B_t$ is the stationary demand computed at time bucket t, $\hat S_{t+h-s}$ is the multiplicative seasonality factor.
The textbook (Quantitative Methods - P. Brandimarte, page 558) states that for initialization we have to set s+1 parameters, $\hat B_0$ and $\hat S_{s-j}$ for $j=1,..,s$, but to do this we need just s observations because the average value of multiplicative seasonality factors must be 1.
$\hat S_0 + \hat S_1 + ... + \hat S_{s-1} = s $
I don't understand the reason of this last sentence, why should the average value of the seasonality factors be 1?