# Combining Mulitple Forecasts? Budged Constraints?

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:

http://sdrv.ms/RszqML

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?

• I don't understand why you wouldn't want to the weights to sum to 1. Say one forecasts 10%, another forecasts 15%, it makes more sense to average the two (e.g. as in the literature on Bayesian model averaging) than sum the two. – John Nov 13 '12 at 15:21
• Yes, I kind of agree tentatively. If you have time, do you mind taking a look at the spreadsheet I provided. In that example there are two predictions of a time series. The optimal combination it turns out is them added not averaged. Any ideas? – Stewart Charles Nov 13 '12 at 15:27
• This is more like a regression model than combining multiple forecasts. – John Nov 13 '12 at 15:54
• @John: In your example with 10% and 15% please consider this example and tell me where I am misunderstanding. Mum and Dad each pay me a weekly pocket money or charge me board. It varies from week to week and the two amounts are independent (they don't speak nor live together). All up the expected amount that Mum pays me (pocket money-board) each week is 0. Same with Dad. So, both income streams have no bias. Now, I ask Mum and Dad separately how much my pocket money less board is likely to be? Mum says £10, Dad says £15. Clearly the best estimate for my weekly income is £25 not £12.50, right? – Stewart Charles Nov 13 '12 at 16:02
• @John - I agree this is more like a regression model. In some ways combining multiple forecasts is usually, isn't it? However each of the two predictions time series is a valid prediction in it's own right, is it not? – Stewart Charles Nov 13 '12 at 16:08