A factor model can have a stable expected beta while its residual risk explodes in a sell-off. If a manager reports the same OLS standard error for an equity beta estimated across quiet and crisis days, the point estimate may still be useful, but its confidence interval is built from the wrong covariance model. This note turns the A4 warning in the classical linear model into an operational workflow: detect non-constant variance with White's general test, preserve OLS coefficients but replace their standard errors with a heteroscedasticity-consistent sandwich, and use weighted least squares only when a variance model is economically credible.
The distinction matters in practice. Robust standard errors repair inference without claiming to know why volatility changed; GLS and WLS seek efficiency by making that claim. Conditional volatility is then the Stage 4 problem addressed by ARCH and GARCH volatility models, not something a single robust covariance estimate can model dynamically.
Learning objectives. By the end of this note you should be able to:
- distinguish the validity of OLS coefficients from the validity of classical OLS standard errors under heteroscedasticity;
- construct and interpret White's test and state its main limitation;
- write the heteroscedasticity-consistent sandwich covariance estimator and explain HC0, HC1, and HC3;
- derive WLS as GLS for known diagonal and calculate a weighted factor regression;
- outline feasible WLS when the variance function is estimated rather than known.
In this lesson
- What changes when variance is not constant
- White's general test and the targeted Breusch–Pagan alternative
- Robust standard errors: repair the inference first
- GLS and WLS when the variance model is known
- Feasible WLS: useful, but model-dependent
Before you start
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