Heteroscedasticity

Remarks

Example: let $x_i = i$ and $\varepsilon_i \sim N(0, \sigma^2 x_i^2)$ for $i = 1, \dots, n$ with $n \ge 2$ and $\sigma > 0$. Then $\operatorname{Var}(\varepsilon_1) = \sigma^2 \ne 4 \sigma^2 = \operatorname{Var}(\varepsilon_2)$, so the errors are heteroscedastic.

Non-example: if $\varepsilon_i = x_i u_i$ with $u_i \sim N(0, 1)$ independent of $x_i$ and $x_i \in \set{-1, 1}$, then $\operatorname{Var}(\varepsilon_i \mid x_i) = x_i^2 = 1$ for all $i$, so the errors are homoscedastic, although they depend on $x_i$.

In linear regression with heteroscedastic, uncorrelated errors, the covariance of the least squares estimator is $(X^T X)^{-1} X^T \Omega X (X^T X)^{-1}$ with $\Omega = \operatorname{diag}(\operatorname{Var}(\varepsilon_i))$. The heteroscedasticity-consistent estimator of White replaces $\Omega$ by $\operatorname{diag}(e_i^2)$, where $e_i$ are the residuals. The spelling heteroskedastic is also used.