When you know the variance, weight instead of just robustifying

Question

Each observation in your regression is an average over a group of firms: row $i$ summarizes $n_i$ firms, and group sizes range from a handful to thousands. Averaging over more firms means less noise, so the error variance of row $i$ is $\sigma^2 / n_i$ , which you can compute because you know each $n_i$ .

Ordinary least squares with robust standard errors would give valid inference. Explain why weighted least squares is the better choice here, how you would set it up, and what you gain.

Accepted Answer

Why plain OLS is not wrong, just wasteful. OLS stays unbiased under heteroskedasticity, and White standard errors would make its inference valid. But OLS treats a row built from 5 firms and a row built from 5,000 firms as equally trustworthy, which they are not. It throws away information about which observations are reliable. You lose efficiency: wider confidence intervals than necessary. The weighted least squares fix. When you know the variance of each observation, the efficient estimator down-weights the noisy ones. Weight each observation by the inverse of its variance, wi = 1 / (\sigma^…

When you know the variance, weight instead of just robustifying

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