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  "body": "\n## Definition\n\nThe *simple linear errors-in-variables model* for observed pairs $(x_i, y_i)$, $i = 1, \\dots, n$, states that\n$$y_i = \\alpha + \\beta \\xi_i + \\varepsilon_i, \\qquad x_i = \\xi_i + \\eta_i,$$\nwhere:\n\n1. $\\alpha, \\beta \\in \\R$ are unknown parameters;\n2. $\\xi_i$ is the true value of the regressor, which is not observed;\n3. $\\varepsilon_i$ and $\\eta_i$ are random errors with mean $0$, and $\\eta_i$ is the *measurement error* of the regressor.\n\nThe multivariate version replaces $\\beta \\xi_i$ by $\\beta^T \\xi_i$ with $\\xi_i, \\eta_i \\in \\R^p$.\n\n## Remarks\n\nAssume that the $\\xi_i$ are independent and identically distributed with [[variance|0477R9PYAE]] $\\sigma_\\xi^2 \u003e 0$, and that the $\\eta_i$ have variance $\\sigma_\\eta^2$.\nAssume also that all the $\\xi_i$, $\\eta_i$, $\\varepsilon_i$ are independent, with finite variances, that the $\\eta_i$ are identically distributed, and that the $\\varepsilon_i$ are identically distributed.\nThen $\\operatorname{Cov}(x_i, y_i) = \\beta \\sigma_\\xi^2$ and $\\operatorname{Var}(x_i) = \\sigma_\\xi^2 + \\sigma_\\eta^2$.\nSo the [[ordinary least squares|YM708KHV2W]] slope of $y$ on $x$ converges in probability to\n$$\\beta \\, \\frac{\\sigma_\\xi^2}{\\sigma_\\xi^2 + \\sigma_\\eta^2},$$\nwhich differs from $\\beta$ when $\\beta \\ne 0$ and $\\sigma_\\eta^2 \u003e 0$; the slope is not [[consistent|0AS800VAYX]].\n\nExample: $\\beta = 2$ and $\\sigma_\\xi^2 = \\sigma_\\eta^2 = 1$ give the limit $1$.\n\nSpecial case: if $\\eta_i = 0$ for all $i$, the regressor is observed exactly and the model is ordinary [[linear regression|41FVC4P5JB]].\nError in the response alone is absorbed into $\\varepsilon_i$ and does not bias the slope.\n\nWith normal $\\xi_i$, $\\eta_i$, $\\varepsilon_i$, the parameters are not identified from the joint distribution of $(x_i, y_i)$ unless extra information is given, such as the ratio $\\sigma_\\varepsilon^2 / \\sigma_\\eta^2$.\n",
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