Heteroskedasticity and robust errors

When the noise is not the same everywhere

Ordinary least squares fits $y_i=\beta_0+\beta_1x_i+\varepsilon_i$ and its textbook standard errors assume homoskedasticity: the error variance $\sigma^2$ is the same for every observation. In finance that is rarely true. Volatility clusters (calm periods, then turbulence), and the noise around a relationship often grows with the size of the variable. When the spread of the errors changes, the data are heteroskedastic.

The consequences are specific. The coefficient estimates $\hat\beta$ stay unbiased and consistent, so the fitted line is not wrong. But the usual standard errors, and so the $t$-statistics, confidence intervals and $p$-values built on them, can be too small or too large, and conclusions drawn from them mislead. Two standard remedies: heteroskedasticity-robust (White) standard errors, which replace the constant $\sigma^2$ by each observation's squared residual, and weighted least squares, which gives noisier observations less weight, $w_i\propto1/\sigma_i^2$, when you know how the variance behaves.

• Heteroskedasticity: the error variance changes across observations.
• Effect: coefficients stay unbiased, standard errors and tests become unreliable.
• Fixes: robust (White) standard errors, or weighted least squares with $w_i\propto1/\sigma_i^2$.

Lab · what's graded

• What heteroskedasticity means15 pts
• A robust t-statistic15 pts
• What it does to a regression15 pts
• Diagnose and repair15 pts
• A least squares weight10 pts
• A robust interval15 pts
• Return regressions need robust errors15 pts