combining forecasts at different time horizons

External question — Quantitative Finance Stack Exchange Author: manav Original post: https://quant.stackexchange.com/questions/73669 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I define a prediction of return of an asset as the following: at time $t=0$ , I use my data and output that I expect the asset to make the following returns (in expected value) in the next n intervals $[r_1, r_2, \dots, r_n]$ . Here, I am doing a multi-horizon forecast, i.e. at time $t=0$ I have (expected value and variance) predictions for next $n$ steps. We ignore the variance predictions for this discussion. The way $r_i$ is defined is if the asset is held from time $t=i-1$ to $t=i$ , I expect it to make a return $r_i$ . Now, say I have another prediction on the same asset at time $t=0$ , as $[p_1, p_2, \dots, p_m]$ But now the frequency is different. For example $n=8$ , while $m=2$ but both predict return over next 48 hour period. The first predictor provides 8 predictions over the 6 hr periods, while the second provides two predictions over 24 hr periods, thus, both predicting the return profile over the next two days, but at different frequency. We call this this the prediction profile. We are also given scalars $\pi_r$ and $\pi_p$ giving relative importance of the two prediction profiles and they sum to one, so can be taken as probability of belief. How do we go about principally combining them into a single prediction. Please feel free to make any assumptions that make it practically suitable and closer to reality.

External answer — Quantitative Finance Stack Exchange Author: Russlan Ramdowar Original post: https://quant.stackexchange.com/a/85870 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I think I’d start by putting both forecasts on the same time intervals. The tricky bit is that a 24-hour prediction doesn’t tell you what happens during each of the four 6-hour periods, so you have to make an assumption there. The simplest one would be to spread the daily return evenly, allowing for compounding. So if the daily prediction is p, the equivalent 6-hour return would be (1 + p)^(1/4) - 1 . For example, a 4% daily prediction becomes roughly 0.985% per 6 hours. You can then take a weighted average for each interval: multiply the original 6-hour forecast by its weight, multiply the converted daily forecast by its weight, and add them together. Do that for all eight intervals and you have your combined profile. That seems like a reasonable starting point to me. But you are adding an assumption: the daily model never actually said the return would be spread evenly. If you know something about the timing, you could use that to distribute it differently. There’s also a small trap with expected returns. Compounding expected returns doesn’t necessarily give you the expected compounded return, because returns across intervals can be related. The conversion above works if we assume the four returns within each day are independent under that model and have the same mean. So I’d keep the original forecasts as well. If you want the combined daily forecast, combine the two models’ daily forecasts directly using the same weights. To get the daily forecast from the 6-hour model, you’d need either an independence assumption or some information about how those returns move together. Basically, align the intervals, make the missing timing assumption explicit, then average using your weights. Just keep in mind that the finer detail you added to the daily forecast is an assumption, rather than something that model predicted.