Determine weights in weighted least squares regression Assume we have a cross-section of $N$ stocks. $Y_i$ is an sample variance estimate of stock returns for stock $i$. This sample variance is estimated using $T_i$ number of observations. All $T_i$ are not necessarily equal, i.e. the sample size for $Y$ estimation differ for i = 1,2,.., N.
Now I want to run a cross-sectional weighted least squares regression:
$Y_i = \beta X_i + \epsilon_i$
What is the best choice of weights here, such that the weights are based on $T_i$ for each $Y_i$. In other words, I want to assign a smaller weight to stock $i$ if $T_i$ is small.
 A: I don't think there's a single optimal weight scheme here. I'd try first $w_i=\frac{NT_i}{\sum_iT_i}$. This way $\sum_iw_i=N$ and if $T_i=T_j\to w_i=1$, nice qualities.
A: Yi (sample variance estimate of stock returns for stock i) is going to be too volatile. Replace it with a robust estimator like Median Absolute Deviation (M.A.D) in the weight function. I employed the latter successfully in a solvency model for insurance companies.
Also, if you regressed the sample variance estimate of stock returns against log (capitalization), a measure of a company's size, you should get an inverse smoothing effect as large companies have, on average, lower volatility in earnings and a lower stock beta. I would combine this with the M.A.D estimate.
A: If my description of what you are doing is wrong please please correct me:
We are supposed to have a set of valus {Xi} and the corresponding {Yi}.
From simple least squares Y = A.X + B.
Then we compute the total variance V = Σ(Yi - A.Xi - B)^2.
It's a kind of iteration.
Then we repeat minimization of the variance functional using the weights:
Wi = Vi / (Yi - A.Xi - B)^2
But then some of the Wi's may be infinite. I don't like this.
