Heteroskedasticity

Heteroskedasticity is a property of the error terms in a regression model in which the variance of the errors is not constant across observations. In ordinary least squares regression, this violates the assumption of constant variance (homoscedasticity) and makes standard errors unreliable, so t statistics and confidence intervals may be invalid. The OLS estimator of the coefficients often remains unbiased and consistent under standard exogeneity assumptions but is no longer efficient when heteroskedasticity is present.

Causes and forms of heteroskedasticity include model misspecification, omitted variables, or data that exhibit changing dispersion across levels of the independent variable or groups of observations. Common patterns include increasing or decreasing spread of the residuals with fitted values, or different variances across subpopulations. Time series data can show conditional heteroskedasticity, where volatility changes over time.

Detection methods range from visual inspection of residual plots to formal tests. The Breusch-Pagan test and the White test assess whether the variance of the residuals is linked to the regressors or to missing nonlinearities. The Goldfeld-Quandt test evaluates variance changes across ordered samples. In time series, tests related to autoregressive conditional heteroskedasticity (ARCH) are used to detect changing variance driven by past errors.

Remedies aim to restore valid inference. Robust standard errors, also known as heteroskedasticity-consistent covariance estimators (HC0–HC3, White), provide valid standard errors without changing coefficient estimates. Transformations of the dependent variable (e.g., logarithmic, Box-Cox) or weighted least squares (when the form of heteroskedasticity is known) are alternative approaches. In practice, addressing heteroskedasticity often involves model refinement or specification of a more flexible error structure to achieve homoscedasticity.