I face the following problem: I have a time series $x_t,\;t=1,...,T$ where each value $x_t$ is a weighted average for various groups, that is $x_t = \sum_{i=1}^Nf_{it}*x_{it}$ with $\sum_{i=1}^Nf_{it}=1$. Now I want to know if I can decompose the variance of the overall time series $x_t$ in a sense that a part of the variance is due to variance in the weights $f_{it}$ and another part is due to variance in the group values $x_{it}$ (and a potential remainder).
I thought about something along the lines of classical variance decomposition into intra and inter group variance. But the situation should be different because when $x_t = \bar{x}\,\forall\,t$ then there is no overall variance, regardless of any variance in weights or group values. Any thoughs on that?
Best Daniel
