Extending the standard formula for errors-in-variables bias32 in a multivariate regression to this setting, and under the assumption that ei is uncorrelated with zi and wi, the feasible-regression coefficients can be shown to be biased: (3)αg^=P(Vg+Ω)−1Vgαg≠αg, where is P≡(ρI1+|w|00I|z|),I|x| is the identity matrix with the dimensionality of x, Vg is the variance-covariance matrix of (gi, wi, zi)′, and Ω is the component of the variance-covariance matrix of g^i,w^i,zi′ that is due to error (see Supplementary Methods). In the special case of a univariate regression, in which the only covariate is a constant term, equation (3) implies that the regression slope coefficient βg^ converges to 1ρβg. This is a familiar form of attenuation bias, in which the degree of attenuation toward zero is greater the larger the amount of measurement error. In the multivariate case, however, the amount of attenuation bias for βg^ will also depend on the covariance matrix of gi with zi. Moreover, the other coefficients, ζg^ and δg^, will be biased as well, not necessarily toward zero. For example, a covariate whose coefficient in equation (1) is zero
