I'm hoping that someone can lend a hand. I have been reading various papers on how to combine multiple forecast time series. The main paper is Granger and Bates 1969. The suggestion here is that there is a closed form solution for combining independent forecast time series (eg returns on FTSE).
I'm hoping someone can shed some light on a query I have. Most of these papers suggest a budget constraint of 1, meaning that if I have two forecast time series and wish to combine them to make a superior forecast time series then it will have the form CombinedForecast = k * Forecast1 + (1-k) * Forecast2. In other words the combined forecast is a linear combination of individual forecasts such that the coefficients (ie (k) and (1-k)) sum to 1. Intuitively this doesn't make sense to me, despite being common across many papers on combining forecasts.
I have prepared an Excel example which will hopefully highlight the problem:
You will notice I have two prediction time series being Pred1 and Pred2. Each of these has zero bias and the two are independent of each other. The TS time series is the time series we wish to predict. You can ignore the Err column.
So, given Pred1 and Pred2 are predictions of TS, literature would expect that the optimal weightings should be 0.5 and 0.5. However if we use solver to find W1 and W2 to minimise MSE we find that the optimal weightings sum to 1+1=2.
I'm sure there is something obvious that I am overlooking here. Why should the sum of all prediction weightings be 1?