Heteroscedasticity: OLS Inference, Tests, and Remedies

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What is homoscedasticity?

It is the condition in which random disturbances have the same finite variance across observations, often written as $\operatorname{Var}(\varepsilon_i\mid x_i)=\sigma^2$.

What is heteroscedasticity?

It is the condition in which disturbance variance is not constant and may change across observations, over time, or with predictor values.

What does the diagonal of an error covariance matrix show?

Its diagonal entries are the variances of the individual errors. Comparing them reveals whether the error variances are constant.

How can income and meal spending illustrate heteroscedasticity?

People with higher incomes may show a wider range of meal spending, so the variance of spending can increase with income.

How is heteroscedasticity different from serial correlation?

Heteroscedasticity means unequal error variances. Serial correlation means errors at different observations have nonzero covariances; it can occur even when diagonal variances are equal.

Quiz questions

1. Under exogeneity, what does heteroscedasticity do to OLS coefficient estimates?Answer: It does not by itself bias them, though OLS may be inefficient.
2. Which estimator provides a consistent estimate of the coefficient covariance matrix under heteroscedasticity, given suitable conditions?Answer: White's heteroscedasticity-consistent covariance estimator.
3. Why can conventional OLS standard errors be unreliable under heteroscedasticity?Answer: Their variance formula assumes constant error variance.
4. What may happen to maximum-likelihood estimates in a misspecified Logit or Probit model with heteroscedasticity?Answer: They may be biased and inconsistent.
5. Which test can compare variation across groups or portions of the data?Answer: The Goldfeld–Quandt test.