# 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.

• 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 ? – user83346 Nov 19 '15 at 7:14

## 2 Answers

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.

• 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 $w_i = \sqrt{T_i}$ – Mayou Dec 5 '14 at 15:12
• 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. – Aksakal Dec 5 '14 at 15:16
• Great thank you. So in your opinion, what is the difference (advantage) of using $\sqrt{T_i}$ instead of $T_i$ as weights? – Mayou Dec 5 '14 at 15:28
• Stocks with larger samples will have less impact than in linear weight – Aksakal Nov 11 '16 at 14:34

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.