Errors in Variables

Abstract

This essay surveys the history and recent developments on economic models with errors in variables. These errors may arise from the use of substantive unobservables, such as permanent income, or from ordinary measurement problems in data collection and processing. The point of departure is the classical regression equation with random errors in variables:

$$y = {X^*}\beta + u,$$

where y is a n × 1 vector of observations on the dependent variable, X* is a n × k matrix of unobserved (latent) values on the k independent variables, ß is a k × 1 vector of unknown coefficients, and u is a n × 1 vector of random disturbances. The matrix of observed values on X* is

$$X = {X^*} + V,$$

where V is the n × k matrix of measurement errors. If some variables are measured without error, the appropriate columns of V are zero vectors. In the conventional case the errors are uncorrelated in the limit with the latent values X* and the disturbances u; and the errors have zero means, constant variances, and zero autocorrelation. In observed variables the model becomes

$$y = X\beta + (u - V\beta ).$$