Heteroskedasticity and Robust Standard Errors in Finance

Changing return variance leaves regression coefficients unbiased but corrupts their standard errors, and the fix has a specific name.

Heteroskedasticity means the variance of a regression's error term is not constant across observations, and in finance it is closer to the default condition than the exception, since the variance of returns rises and falls with the market's own volatility regime. Ignoring it does not bias a regression's coefficients, it corrupts the standard errors used to decide whether those coefficients mean anything.

Under heteroskedasticity, ordinary least squares coefficient estimates remain unbiased and consistent, meaning they are still centered on the true value and converge to it as the sample grows. What breaks is the standard formula used to compute the standard error of each coefficient, which assumes the error variance is the same for every observation. When that assumption fails, the reported standard errors are typically too small during volatile subperiods and too large during calm ones, which distorts every t-statistic and every confidence interval built from the regression, even though the coefficients themselves are still pointing in the right direction on average.

In a regression of one asset's returns on a market factor, the residual variance tends to expand during periods of market stress and contract during calm stretches, a pattern connected to the volatility clustering fact covered in GARCH models. A regression run across a sample that spans both regimes is treating two different variance environments as one, and the classical standard error formula has no way to know that.

The Breusch-Pagan test regresses the squared residuals from the original regression on the same set of regressors and tests whether that relationship is statistically significant, which would indicate the error variance depends on the regressors rather than staying constant. The White test generalizes this idea without assuming a specific functional form for how the variance changes, testing a broader pattern of dependence between the squared residuals and the regressors and their cross products, at the cost of lower power against any one specific form of heteroskedasticity compared with a test built for that exact form.

Both tests operate on the same basic idea, checking whether the squared residuals relate systematically to the regressors, and a significant result from either one is the signal to move from classical standard errors to a heteroskedasticity-consistent version before reporting any t-statistic from the regression as final.