2
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • $\begingroup$ If I understand it well, then you have $N$ stocks, and within each of these stocks you have observations $(x_{ij}, y_{ij})$, $i=1 \dots N, j=1 \dots n_i$ ? So I would suggest you to use the generalised least squares estimator. An R-implementation can be found in package nlme, function 'gls' where your grouping variable is the stock. You know R ? $\endgroup$ – user83346 Nov 19 '15 at 7:14
2
$\begingroup$

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.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Thanks for your answer. Two follow-up questions please: 1) Do weights need to be normalized in weighted-least squares? 2) What do you think about $w_i = \sqrt{T_i}$ $\endgroup$ – Mayou Dec 5 '14 at 15:12
  • $\begingroup$ No, the weights don't have to be normalized, but it's nice to have them to be equal to 1 when num of obs are equal because then SSE will be the same as in OLS. It's just easier to compare and track the results. Square of $T_i$ is good too, because it links to random walk properties of volatility/time. $\endgroup$ – Aksakal Dec 5 '14 at 15:16
  • $\begingroup$ Great thank you. So in your opinion, what is the difference (advantage) of using $\sqrt{T_i}$ instead of $T_i$ as weights? $\endgroup$ – Mayou Dec 5 '14 at 15:28
  • $\begingroup$ Stocks with larger samples will have less impact than in linear weight $\endgroup$ – Aksakal Nov 11 '16 at 14:34
0
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.