Errors-in-variables model


Errors-in-variables model

In statistics, an error-in-variables model is a statistical model which is similar to a regression model but where the independent variables (or explanatory variables) are observed with error. A full statistical model includes components describing these observation errors.

The terms "functional relationship" and "structural relationship" are also used in connection with errors-in-variables models.

Error-in-variables models can be estimated in several different ways. Besides those outlined here, see::*total least squares for a method of fitting which does not arise from a statistical model;:*instrumental variables for a method that makes use of an additional set of observations.

pecification of model

The pairs of variables {"Xi","Yi"} that would arise in a linear regression model are assumed to be related to pairs of unobserved variables {"ξi","ηi"} which themselves follow a straight line relationship: [Draper N.R., Smith, H. (1998) Section 3.4]

:eta_i=eta_0+eta_1 xi_i , ,

where "β"0 and "β"1 are the parameters of the "true relationship" which is to be estimated. The observed variables are then modelled by

:Y_i=eta_i + epsilon_i , ,:X_i=xi_i + delta_i , ,

where "εi" and "δi" represent observation errors. The observation errors are assumed to have an expected value of zero and to be uncorrelated across the pairs but not necessarily uncorrelated within the pairs. The variances of the observation errors are assumed to be constant across the pairs but the variances are taken to be unknown: however, in some instances it is assumed that the ratio between the two variances is known.

Notes

References

Draper N.R., Smith, H. (1998) "Applied Regression Analysis" (3rd Edition), Wiley.

Torsten Söderström. Errors-in-variables methods in system identification, "Automatica", Volume 43, Issue 6, June 2007, Pages 939-958.


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