Vasicek(1973), referenced in this paper(See bottom of page 16) explains a method of shrinking individual betas $\beta^{TS}$ toward a cross-sectional mean $\beta^{XS}$ as follows: for each time $t$, the shrinkage $\beta$ for stock $i$ is defined as;
$\beta_i^{shrink} = w_i * \beta_i^{TS} + (1 - w_i) * \beta^{XS}$
The shrinkage weight is defined as:
$w_i = 1 - \frac{\sigma_{i,TS}^2}{\sigma_{i,TS}^2 + \sigma_{XS}^2}$
where $\sigma_{i,TS}^2$ is the variance of the individual beta, and where $\sigma^2_{XS}$ is the variance of the cross-sectional variance of betas (dispersion of betas in the cross-section).
I would like to extend this methodology as follows:
Imagine now I have standard deviation estimates, instead of beta estimates, that I would like to shrink. I have individual standard deviation values $s_i^{TS}$ and a cross-sectional standard deviation value $s^{XS}$ at each time $t$. I would like to do the following:
$s^2_{i,shrink} = w_i * s^2_{i, TS} + (1 - w_i) * s^2_{XS}$
How can best define the shrinkage weight $w_i$ in this case?
Thanks!