Covariance of Overlapping Returns and Daily Covariance Estimates
Summary
The document explains why rolling multi-day returns are statistically dependent and how that dependence affects covariance. It expands the covariance of two overlapping two-day return series into same-day covariance terms and cross-time covariance terms. This illustrates that multi-day covariance is not generally just the single-period covariance multiplied by the window length.
The proposed simplification is to estimate daily covariance and scale it by the return-window length when the model assumes time-offset covariances are zero, as in Brownian-motion-style settings. That assumption may not hold in observed data, so the scaling is conditional on the model. The discussion does not provide a direct correction for regression significance or correlation estimates from a small sample; the example motivates checking serial dependence rather than treating overlapping observations as independent.
Key ideas
- Overlapping multi-day returns share daily observations and are therefore dependent.
- The covariance of aggregated returns includes cross-time covariance terms as well as same-period covariance.
- Scaling daily covariance by the window length assumes time-offset covariances vanish.
- Brownian-motion-style assumptions can justify the simplified covariance scaling, but empirical returns may violate them.
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# Error Metric For Regression of Overlapping Returns Series
# Error Metric For Regression of Overlapping Returns Series
I want to regress two returns series. I calculate 30 day returns and then use overlapping return windows for a regression over 360 days (regression uses 11 data points).
What is the right way to think about correlation? In this case, my 30 day returns are not independent, so I have a sense that the correlation will be drastically overestimated.
Ideally, I'd like to have a measure of correlation to know how good the fit is and in order to optimize the return window (30 days) and the time period (360 days) used in the regression.
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/40335
To avoid complicating the maths with vector notation, instead consider just a 2-day return as opposed to a 30-day.
Then your two, 2-day, series can be stated as ($X_t, Y_t$) where $X_t=x_t+x_{t-1}$ and $Y_t=y_t+y_{t-1}$, for little $x,y$ the daily, i.e. 1-day (log) returns.
Then, $$Cov(X_t,Y_t) = E[(X_t-\mu_X)(Y_t-\mu_Y)] = E[(x_t-\mu_x+x_{t-1}-\mu_x)(y_t-\mu_y+y_{t-1}-\mu_y)]$$, which after expanding out gives you four terms, $$Cov(X_t,Y_t)=E[(x_t-\mu_x)(y_t-\mu_y)+(x_{t-1}-\mu_x)(y_{t-1}-\mu_y)]+E[(x_t-\mu_x)(y_{t-1}-\mu_y)+(x_{t-1}-\mu_x)(y_{t}-\mu_y)]$$ $$Cov(X_t,Y_t)=2Cov(x_t,y_t) + Cov(x_{t-1},y_t) + Cov(x_t,y_{t-1})$$
In other words the covariance of the 2-day time frame is twice the covariance of the daily time frame plus the time offset covariances. In option theory and Brownian motion these time offset terms are set to zero so that the covariance of an n-long period is simply n times the covariance of a single period.
If you aspire to the same modelling characteristics in your data then you can use daily returns and multiply them by 30 as the estimator for your 30-day covariance.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.