combining subjective probability estimates and statistical estimates for forecasting At the end of the year forecasters usually struggle year to predict landing estimate for the financial year due to variety of reasons including volatility, unreliable demand projections, inventory control etc. . For example: a statistical method based on time series extrapolation would say we would end up at 100 - 110 Million Dollars. There is always going to be subjective/judgmental involved in forecasting. A judgmental forecast would end up in between 115 - 125 Millions. 
This year we plan to use a cool tool called Subjective Probability Interval Estimate (SPIES) which was recently published in Harvard Business Review that provides us the confidence interval of subjective estimates and we plan to collect year end estimates from multiple stake holders (Example: Marketing/Finance/Sales Force). 
As an example lets say we have 3 forecast, the first generated by a statistcal method, the second and third generated by SPIES tool from above:


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*Statsitcal Forecast: 100 - 110

*Judgmental Forecast 1: 115 - 125

*Judgmental forecast 2: 95 - 105


My Questions is two fold:


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*Is it possible to combine the interval estimates from the forecast above ? Would Averaging the low and upper intervals work ?

*If yes, how should I go about it ?


Many Thanks for your help.
 A: If you believe the intervals (and that's questionable, given the strong level of evidence that people are overconfident, even statisticians. See http://messymatters.com/calibration-results/ ) then a standard weighting would be to weight by the inverse of the variance.
If your judgmental forecast is supposed to indicate 95% confidence that the value is between 115 and 125, then you know at 10 units (125-115) is 4 standard deviations, so the standard deviation is 2.5 and the variance is 2.5 squared, or 6.25.
So, you could weight this forecast by 1/6.25 and do similar calculations for the other two, and then do a weighted average of the midpoints of the interval to get a combined forecast.
If you were willing to assume that the three forecasts were independent you could treat these as three points in a meta-analysis and get a combined confidence interval. But independence of the forecasts is so doubtful I wouldn't have faith in that approach. I think I'd be tempted to regard each of the endpoints as an estimate of the 2.5th percentile and 97.5th percentile (if these are supposed to reflect 95% confidence) and do a weighted average similar to the one above using the inverse of the variance. 
I look forward to seeing what others might suggest for this.
