Error Metric For Regression of Overlapping Returns Series

Question

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 1

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.