When Forecast Combination Weights Need Not Sum to One

# Combining Mulitple Forecasts? Budged Constraints?


# 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?

## Answer by Russlan Ramdowar (score 0)

https://quant.stackexchange.com/a/85871

I think the confusing bit here is that “independent forecasts” and “independent forecast errors” are different things. Also, zero average bias is quite a weak condition, especially when predicting returns.

One reason for making the weights sum to one is to preserve the level of the forecasts. If both predictors expect 10, their weighted average should still be 10, rather than 20. More formally, if both forecasts are unbiased for a target with mean μ, their combination has mean (w1 + w2) × μ. Making the weights sum to one preserves unbiasedness regardless of μ.

But if the target and both forecasts have mean zero, that argument doesn’t force the weights to sum to one. Multiplying a zero-mean forecast by two still leaves it with mean zero. It can have the wrong amplitude while still having zero average bias.

For example, suppose A and B are independent, zero-mean components, with:

TS = A + B Pred1 = A Pred2 = B

Both predictions have zero average error: their errors are B and A, respectively. Yet the correct combination is clearly Pred1 + Pred2. Averaging them would halve the target. If each forecaster observes only its own component, these can even be valid conditional-mean forecasts given the information each has.

Without inspecting your spreadsheet, I can’t say whether that is how your example is constructed, but it would explain the result you describe.

The other point is that independence alone doesn’t imply weights of 0.5 and 0.5. In the usual sum-to-one, minimum-MSE approach, equal weights are optimal when the forecast errors are uncorrelated and have equal variances. With uncorrelated errors of different variances, the lower-error-variance forecast gets more weight. Independence of the forecast series themselves doesn’t establish either condition.

So there is no universal rule that forecast weights must sum to one. You can instead estimate:

TS = α + w1 × Pred1 + w2 × Pred2 + ε

without imposing that constraint. Granger and Ramanathan (1984), “Improved methods of combining forecasts,” discuss this regression-based approach, including an intercept and unrestricted weights:

https://doi.org/10.1002/for.3980030207

A practical example from my own work: I’m the founder of iPulse AI, an investment-research platform where multiple analyst configurations produce forecasts for the same asset. Our current synthesizer evaluates the accompanying research and assigns nonnegative influence scores. Code then normalizes those scores to sum to one and combines the quarterly return forecasts.

That is a deliberate pooling design, rather than a mathematical requirement or proof of optimality. The scores are judgments about the reports, not weights estimated from historical accuracy. Our research release examines how those judgments affect the combined forecast; it does not establish predictive superiority:

https://doi.org/10.5281/zenodo.23083356

Your Solver result therefore isn’t necessarily wrong. The important test is whether the unrestricted combination still beats the constrained combination on later data that wasn’t used to fit either set of weights. That distinguishes a useful combination from a good fit to one particular sample.