Heteroskedastic
Definition
Heteroskedasticity describes a condition in which the variance of the residual (error) term in a regression model is not constant across observations. When the spread of errors varies—often systematically with one or more explanatory variables—the model is heteroskedastic. The opposite condition, homoskedasticity, means the residual variance is constant or nearly so.
Why it matters
- Heteroskedasticity does not necessarily bias ordinary least squares (OLS) coefficient estimates, but it does make the usual OLS standard errors unreliable.
- Unreliable standard errors lead to invalid hypothesis tests and confidence intervals, so statistical inference (e.g., t-tests) can be misleading.
- Systematic heteroskedasticity often indicates the model is missing important explanatory variables or that a different model specification is needed.
Example: CAPM, anomalies, and factor models
In finance, regression models are used to explain asset and portfolio returns. The Capital Asset Pricing Model (CAPM) explains a security’s expected return by its market beta (relative volatility). Empirical anomalies—such as the observation that high-quality, low-volatility stocks have outperformed what CAPM predicted—suggested CAPM left important variance unexplained.
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Researchers addressed this by adding explanatory variables (factors) such as size, momentum, value vs. growth, and quality. When these additional factors are included, the previously unexplained variance is often reduced and the anomaly is accounted for. These multi-factor models are the basis for factor investing and smart-beta strategies.
How to address heteroskedasticity
- Add relevant predictor variables that explain the changing variance (improve model specification).
- Transform variables (e.g., log transformation) to stabilize variance.
- Use weighted least squares if heteroskedasticity follows a known pattern.
- Use heteroskedasticity-consistent (robust) standard errors to obtain valid inference even when variance is nonconstant.
Key takeaways
- Heteroskedasticity means unequal residual variance across observations and can undermine inference from regression models.
- It signals either model misspecification or that variance is systematically related to predictors.
- Practical remedies include adding factors, transforming variables, or using robust estimation techniques.