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.

Motivating example

Consider a simple linear regression model of the form where x_{t}^{} denotes the true but unobserved regressor. Instead we observe this value with an error: where the measurement error \eta_{t} is assumed to be independent of the true value 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 y_{t}′s are simply regressed on the x_{t}′s (see simple linear regression), then the estimator for the slope coefficient is which converges as the sample size T increases without bound: This is in contrast to the "true" effect of \beta, estimated using the x_{t}^{*},: Variances are non-negative, so that in the limit the estimated is smaller than \hat{\beta}, an effect which statisticians call attenuation or regression dilution. Thus the ‘naïve’ least squares estimator is an inconsistent estimator for \beta. However, is a consistent estimator of the parameter required for a best linear predictor of y given the observed x_t: in some applications this may be what is required, rather than an estimate of the ‘true’ regression coefficient \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 y_{t}′s to the actually observed x_{t}′s, in a simple linear regression, is given by It is this coefficient, rather than \beta, that would be required for constructing a predictor of y based on an observed x which is subject to noise. It can be argued that almost all existing data sets contain errors of different nature and magnitude, so that attenuation bias is extremely frequent (although in multivariate regression the direction of bias is ambiguous ). Jerry Hausman sees this as an iron law of econometrics: "The magnitude of the estimate is usually smaller than expected."

Specification

Usually measurement error models are described using the latent variables approach. If y is the response variable and x are observed values of the regressors, then it is assumed there exist some latent variables y^{} and x^{} which follow the model's “true” functional relationship g(\cdot), and such that the observed quantities are their noisy observations: where \theta is the model's parameter and w are those regressors which are assumed to be error-free (for example when linear regression contains an intercept, the regressor which corresponds to the constant certainly has no "measurement errors"). Depending on the specification these error-free regressors may or may not be treated separately; in the latter case it is simply assumed that corresponding entries in the variance matrix of \eta's are zero. The variables y, x, w are all observed, meaning that the statistician possesses a data set of n statistical units which follow the data generating process described above; the latent variables x^, y^, \varepsilon, and \eta are not observed however. This specification does not encompass all the existing errors-in-variables models. For example in some of them function g(\cdot) may be non-parametric or semi-parametric. Other approaches model the relationship between y^* and x^* as distributional instead of functional, that is they assume that y^* conditionally on x^* follows a certain (usually parametric) distribution.

Terminology and assumptions

Linear model

Linear errors-in-variables models were studied first, probably because linear models were so widely used and they are easier than non-linear ones. Unlike standard least squares regression (OLS), extending errors in variables regression (EiV) from the simple to the multivariable case is not straightforward, unless one treats all variables in the same way i.e. assume equal reliability.

Simple linear model

The simple linear errors-in-variables model was already presented in the "motivation" section: where all variables are scalar. Here α and β are the parameters of interest, whereas σε and ση—standard deviations of the error terms—are the nuisance parameters. The "true" regressor x* is treated as a random variable (structural model), independent of the measurement error η (classic assumption). This model is identifiable in two cases: (1) either the latent regressor x* is not normally distributed, (2) or x* has normal distribution, but neither εt nor ηt are divisible by a normal distribution. That is, the parameters α, β can be consistently estimated from the data set without any additional information, provided the latent regressor is not Gaussian. Before this identifiability result was established, statisticians attempted to apply the maximum likelihood technique by assuming that all variables are normal, and then concluded that the model is not identified. The suggested remedy was to assume that some of the parameters of the model are known or can be estimated from the outside source. Such estimation methods include Estimation methods that do not assume knowledge of some of the parameters of the model, include

Multivariable linear model

The multivariable model looks exactly like the simple linear model, only this time β, ηt, xt and x*<sub style="position:relative;left:-.4em">t are k×1 vectors. In the case when (εt,ηt) is jointly normal, the parameter β is not identified if and only if there is a non-singular k×k block matrix [a A], where a is a k×1 vector such that a′x* is distributed normally and independently of A′x*. In the case when εt, ηt1,..., ηtk are mutually independent, the parameter β is not identified if and only if in addition to the conditions above some of the errors can be written as the sum of two independent variables one of which is normal. Some of the estimation methods for multivariable linear models are

Non-linear models

A generic non-linear measurement error model takes form Here function g can be either parametric or non-parametric. When function g is parametric it will be written as g(x*, β). For a general vector-valued regressor x* the conditions for model identifiability are not known. However in the case of scalar x* the model is identified unless the function g is of the "log-exponential" form and the latent regressor x* has density where constants A,B,C,D,E,F may depend on a,b,c,d. Despite this optimistic result, as of now no methods exist for estimating non-linear errors-in-variables models without any extraneous information. However there are several techniques which make use of some additional data: either the instrumental variables, or repeated observations.

Instrumental variables methods

Repeated observations

In this approach two (or maybe more) repeated observations of the regressor x* are available. Both observations contain their own measurement errors, however those errors are required to be independent: where x* ⊥ η1 ⊥ η2. Variables η1, η2 need not be identically distributed (although if they are efficiency of the estimator can be slightly improved). With only these two observations it is possible to consistently estimate the density function of x* using Kotlarski's deconvolution technique.