Errors-in-variables model

In statistics, an errors-in-variables model or a measurement error model is a regression model that accounts 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.^{[citation needed]}

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 of the true value ${\displaystyle x_{t}^{*}}$.
A practical application is the standard school science experiment for Hooke's law, in which one estimates the relationship between the weight added to a spring and the amount by which the spring stretches.
If the ${\displaystyle y_{t}}$′s are simply regressed on the ${\displaystyle x_{t}}$′s (see simple linear regression), then the estimator for the slope coefficient is

which converges as the sample size ${\displaystyle T}$ increases without bound:

This is in contrast to the "true" effect of ${\displaystyle \beta }$, estimated using the ${\displaystyle x_{t}^{*}}$,:

Variances are non-negative, so that in the limit the estimated ${\displaystyle {\hat {\beta }}_{x}}$ is smaller than ${\displaystyle {\hat {\beta }}}$, an effect which statisticians call attenuation or regression dilution. Thus the ‘naïve’ least squares estimator ${\displaystyle {\hat {\beta }}_{x}}$ is an inconsistent estimator for ${\displaystyle \beta }$. However, ${\displaystyle {\hat {\beta }}_{x}}$ is a consistent estimator of the parameter required for a best linear predictor of ${\displaystyle y}$ given the observed ${\displaystyle x_{t}}$: in some applications this may be what is required, rather than an estimate of the 'true' regression coefficient ${\displaystyle \beta }$, although that would assume that the variance of the errors in the estimation and prediction is identical. This follows directly from the result quoted immediately above, and the fact that the regression coefficient relating the ${\displaystyle y_{t}}$′s to the actually observed ${\displaystyle x_{t}}$′s, in a simple linear regression, is given by