Errors-in-variables model

Remarks

Assume that the $\xi_i$ are independent and identically distributed with variance $\sigma_\xi^2 > 0$, and that the $\eta_i$ have variance $\sigma_\eta^2$. Assume 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. Then $\operatorname{Cov}(x_i, y_i) = \beta \sigma_\xi^2$ and $\operatorname{Var}(x_i) = \sigma_\xi^2 + \sigma_\eta^2$. So the ordinary least squares slope of $y$ on $x$ converges in probability to $$\beta \, \frac{\sigma_\xi^2}{\sigma_\xi^2 + \sigma_\eta^2},$$ which differs from $\beta$ when $\beta \ne 0$ and $\sigma_\eta^2 > 0$; the slope is not consistent.

Example: $\beta = 2$ and $\sigma_\xi^2 = \sigma_\eta^2 = 1$ give the limit $1$.

Special case: if $\eta_i = 0$ for all $i$, the regressor is observed exactly and the model is ordinary linear regression. Error in the response alone is absorbed into $\varepsilon_i$ and does not bias the slope.

With 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 $\sigma_\varepsilon^2 / \sigma_\eta^2$.