In a regression of daily returns on a signal, residual variance is four times larger on high-volume days than on low-volume days. State what this does to the OLS coefficient and to its reported standard error, and give the correction you would use.

In a regression of daily returns on a signal, residual variance is four times larger on high-volume days than on low-volume days. State what this does to the OLS coefficient and to its reported standard error, and give the correction you would use.

Approach: Separate the assumptions needed for unbiasedness from those needed for the variance formula, then name the estimator that drops the constant-variance assumption.

OLS stays unbiased and consistent, its usual standard errors are wrong in an unknown direction, and you report White heteroskedasticity-robust standard errors. Unbiasedness needs only E[u|X] = 0, which a changing residual variance does not touch, so beta_hat is still centred on beta. The textbook variance sigma^2*(X'X)^{-1} assumes one constant residual variance; with variance four times higher on a subset it is too small when the high-variance days also carry extreme signal values and too large otherwise, so t-statistics built from it cannot be trusted either way. The sandwich estimator (X'X)^{-1} * (sum x_i*x_i'*u_hat_i^2) * (X'X)^{-1} puts the squared residuals where the single sigma^2 was and is consistent under any variance pattern. OLS also loses its minimum-variance property, so weighted least squares with weights proportional to the inverse variance is more efficient when the pattern is known.

Follow-up: Robust standard errors are consistent asymptotically. What goes wrong with them in a sample of 40 observations with one high-leverage point?

Key concepts: heteroskedasticity, robust standard errors, sandwich estimator, weighted least squares.