# Drawing inferences from multiple modal estimates of a random variable

I'm working to elicit a probability distribution from the Survey of Private Dealers, a ~monthly survey concerning the future direction of certain financial random variables. Latest example here.

There are two tables regarding a single random variable (future Federal Funds rate targets) that I'd like to turn into a prior probability of that random variable.

1. "Provide your estimate of the most likely outcome (i.e., the mode) for the target federal funds rate or range, as applicable, immediately following the FOMC meetings and at the end of each of the following quarters and years below. For the time periods at which you expect a target range, please indicate the midpoint of that range in providing your response."
Statistic Modal Estimate
25th percentile 0.63%
Median 0.63%
75th percentile 1.13%
# of respondents - 24

In other words, 24 respondents give 1 modal estimate, and 3 summary statistics are provided from those 24 data points.

1. "Please indicate the percent chance that you attach to the target federal funds rate or range falling in each of the following ranges at the end of 2023."
Bucket Average Probability
< 0.00% 1%
0.00 - 0.25% 15%
0.26 - 0.50% 17%
0.51 - 0.75% 24%
0.76 - 1.00% 17%
1.01 - 1.25% 13%
1.26 - 1.50% 7%
1.51 - 1.75% 3%
1.76 - 2.00% 2%
≥ 2.01% 2%
# of respondents - 21

In other words, 21 respondents gave a probability distribution of the same random variable, and the distributions were averaged.

Question: What is the right way for me to combine these two tables into a prior? The mode is easy in this case (clearly the 0.63% from the first table and the 0.51%-0.75% in the second table tell the same story), but unsure of how it would work in the general case.

The obvious prior is the stepwise distribution from the second table, assuming that all values in each bucket are equally likely, and that the buckets at the ends have the same widths as the other buckets.

Those assumptions give the following graph:

You could also smooth out the steps and the ends, which would be worthwhile if the stepped version gives unreasonable results.

The 0.63% and 1.13% in the first table look like rounded versions of the midpoints for the 0.50% - 0.75% and 1.00% - 1.25% buckets. So the first table probably tells you that:

• at least 50% of respondents had a mode of at most 0.75%
• at least 25% of respondents had a mode between 0.50% and 0.75%
• at least 25% of respondents had a mode of at least 1.00%.

This could help you model the dispersion of views among economists, each with their own probability distribution, or help you choose between this data source and other one. But again, unless the above prior or a smoothed version of it gives an unreasonable result, I wouldn't use the first table in creating a prior about the interest rate.

• The other advantage of the first table is that it is estimated for a larger number of time-periods: after each Fed meeting for the current year (~monthly), quarterly for the next 2 years, annual for 4 years after that. The second table, while more directly useful as you showed, is only available at annual resolution for the next 4 years. Ultimately I'd like to use the "overlapping" observations (i.e. at the end of each of the next 4 years), but somehow interpolate the dispersion to the other data points available in the first table but not the second. Commented Oct 29, 2021 at 0:39
• Interpolating is more complicated: to get curves that vary smoothly in time and reflect all the data, it probably helps to use a model whose parameters you can vary more easily. E.g., the Bank of England models data like this with split-normal distributions. Commented Oct 29, 2021 at 12:28