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  "body": "\n## Definition\n\nLet $\\varepsilon_1, \\dots, \\varepsilon_n$ be [[random variables|0PNRKMRC55]] with finite [[variances|0477R9PYAE]].\nThey are *homoscedastic* if $\\operatorname{Var}(\\varepsilon_1) = \\cdots = \\operatorname{Var}(\\varepsilon_n)$, and *heteroscedastic* otherwise.\n\nNow let $y_i = f(x_i) + \\varepsilon_i$ be a regression model with random regressors $x_i$.\nIts errors are *homoscedastic* if the [[conditional variance|0VGPX25NXQ]] $\\operatorname{Var}(\\varepsilon_i \\mid x_i)$ is a constant $\\sigma^2$, not depending on $x_i$.\nThey are *heteroscedastic* otherwise.\n\n## Remarks\n\nExample: 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 $\\sigma \u003e 0$.\nThen $\\operatorname{Var}(\\varepsilon_1) = \\sigma^2 \\ne 4 \\sigma^2 = \\operatorname{Var}(\\varepsilon_2)$, so the errors are heteroscedastic.\n\nNon-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 $x_i$.\n\nIn [[linear regression|41FVC4P5JB]] 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 $\\Omega = \\operatorname{diag}(\\operatorname{Var}(\\varepsilon_i))$.\nThe heteroscedasticity-consistent estimator of White replaces $\\Omega$ by $\\operatorname{diag}(e_i^2)$, where $e_i$ are the residuals.\nThe spelling *heteroskedastic* is also used.\n",
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