I am trying to calculate the average correlation between two simple time series.
Say, $X_i$ (and $Y_i$) is an iid standard normal random variable.
Now, I generate the series $(X_i,Y_i)$ with some correlation $\rho_1$, say $0.5$, till the time $0.5T$. And for the time from $0.5T$ to $T$, I generate the time series with a correlation of $\rho_2$, say $0.7$.
When I tried to find the correlation between $X_i$ and $Y_i$ for the entire time $T$, by simulation it was found to be 0.6. (intuitively makes sense).
Now, instead of the correlation change (from 0.5 to 0.7) happening at $0.5T$, say the correlation change happen at $0.25T$, can I say something similar about the correlation between $X_i$ and $Y_i$ for the entire time $T$?
I want to derive a general expression for overall the correlation.
Here is my approach:
Say, $(X_i,Y_i)$ are plotted on a XY graph.
Overall correlation is the $tan$ (average angle of the vectors $(X_i,Y_i)$ with the x-axis)
So, the correlation is $\rho_1$ for time $t_1$ and $\rho_2$ for time $t_2$, then the overal correlation is $tan(\frac{t_1*tan^{-1}\rho_1 + t_2*tan^{-1}\rho_2}{t_1+t_2})$
Is my approach correct? Any other solutions are also welcome.
Thank you.
Why not to use a simple weighted average, i.e., $$ \rho_T = \alpha\rho_{t_1} + (1-\alpha) \rho_{t_2}? $$ Where $\alpha$ (assuming uniform distribution of the observations; i.e., constant time intervals of the time series sampling) is the proportion of time before the "regime shift".
EDIT:
Whether it is a good estimator depends on what exactly was changed. The correlation between two r.v.s depends on the variance of every one of them and their covariance. If the variances remained unchanged then it's a consistent estimator because $$ \operatorname{cov}(\alpha X_{t_1} + (1-\alpha)X_{t_2}, Y) = \alpha \operatorname{cov}( X_{t_1} , Y) + (1-\alpha)\operatorname{cov}(X_{t_2},Y). $$ However if the variance changed it becomes more complicated as $$ \operatorname{Var}(\alpha X_{t_1} + (1-\alpha)X_{t_2})=(\alpha \sigma_{X_{t_1}} + (1-\alpha)\sigma_{X_{t_2}})^2. $$