Errors-in-variables models

In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.

In the case when some regressors have been measured with errors, estimation based on the standard assumption leads to inconsistent estimates, meaning that the parameter estimates do not tend to the true values even in very large samples. For simple linear regression the effect is an underestimate of the coefficient, known as the attenuation bias. In non-linear models the direction of the bias is likely to be more complicated.

Consider a simple linear regression model of the form

where ${\displaystyle x_{t}^{*}}$ denotes the true but unobserved regressor. Instead we observe this value with an error:

where the measurement error ${\displaystyle \eta _{t}}$ is assumed to be independent from the true value ${\displaystyle x_{t}^{*}}$.

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