# 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.

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, $$Cov(X_t,Y_t)=E[(x_t-\mu_x)(y_t-\mu_y)+(x_{t-1}-\mu_x)(y_{t-1}-\mu_y)]+E[(x_t-\mu_x)(y_{t-1}-\mu_y)+(x_{t-1}-\mu_x)(y_{t}-\mu_y)]$$ $$Cov(X_t,Y_t)=2Cov(x_t,y_t) + Cov(x_{t-1},y_t) + Cov(x_t,y_{t-1})$$