Constructing Overlapping Multi-Month Returns for Factor Regressions
Summary
The document asks whether monthly factor returns should be aggregated into overlapping quarterly, semiannual, or annual returns before regressing them on lagged market- and funding-liquidity variables. The proposed motivation is that monthly observations may be noisy, but the question receives no substantive answer on whether longer horizons improve inference or how overlapping observations affect statistical tests.
The reply explains return compounding: consecutive monthly simple returns are combined by multiplying their gross returns and subtracting one. Rolling the window forward by one month yields a monthly series of overlapping multi-month returns. Because adjacent windows share most of their months, the resulting observations are dependent; the response flags that overlap has statistical issues but does not explain corrections. Thus the construction is described, while the regression design and inference limitations remain unresolved.
Key ideas
- A multi-month simple return is calculated by compounding the monthly gross returns within the window.
- Overlapping returns arise when the horizon window advances one month at a time.
- Adjacent overlapping observations share monthly returns and are therefore not independent.
- The document leaves the effects on regression inference and the merits of longer horizons unanswered.
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Full text
# Difference between returns
# Difference between returns
I have a monthly time series of monthly returns for a specific factor that I'm investigating, the Quality factor QMJ as proposed by Asness et al. (https://www.aqr.com/Insights/Datasets/Quality-Minus-Junk-Factors-Monthly). I have monthly data on these monthly returns from July 1963 - December 2019, so 678 months.
I will run a regression of QMJ monthly returns on 2 variables that proxy for market and funding liquidity and try to interpret the results (i.e. how market and funding liquidity can affect QMJ returns).
So I will run a time-series OLS regression of $$QMJ (t) = MktLiquidity(t-1) + FundingLiquidity(t-1) + \text{some control variables}.$$
Now it was brought to my attention that rather than using the monthly returns that I have, I should use either 3, 6 or 12 month overlapping returns to run this regression. The reason being that the monthly returns contain too much noise and would thus be easy to get statistically insignificant results. So my 1st question is, is this true? Can someone explain this?
Also, how do I transform my monthly return data to either get 3 months, 6 months, or 12 months overlapping returns. If I would have 12 month overlapping returns, the regressions would be for example:
- return of QMJ from Oct 1970 - Sep 1971 (12 month returns) regressed on MktLiquidity of Sep 1970 + FundingLiquidity of Sep 1970
- return of QMJ from Nov 1970 - Oct 1971 (12 month returns) regressed on MktLiquidity of Oct 1970 + FundingLiquidity of Oct 1970
- return of QMJ from Dec 1970 - Nov 1971 (12 month returns) regressed on MktLiquidity of Nov 1970 + FundingLiquidity of Nov 1970
and so on...
Now the 2nd question is how do I get these 3/6/12 month overlapping returns from my monthly return data? So the end result should be that I get monthly observations (678 months) of yearly(or semi-annually or quarterly) returns.
Thanks!
## Answer by user42108 (score 1)
https://quant.stackexchange.com/a/59503
how do I get these 3/6/12 month overlapping returns from my monthly return data?
Assume that you start with a bankroll of `X`. After % return `r1` in month 1 that becomes `(1+r1)*X`, after % return `r2` in month 2 that becomes `(1+r2)*(1+r1)*X`, etc. You'll then be able to calculate your 3, 6 and 12mo returns.
There are issues with using overlapping returns which are addressed in other questions.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.