# 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:

1. return of QMJ from Oct 1970 - Sep 1971 (12 month returns) regressed on MktLiquidity of Sep 1970 + FundingLiquidity of Sep 1970

2. return of QMJ from Nov 1970 - Oct 1971 (12 month returns) regressed on MktLiquidity of Oct 1970 + FundingLiquidity of Oct 1970

3. 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!

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.