Latent Variable Models
Consider again the bivariate measurement error model (8.7), where the unobservable random variables En, є n, and vn are assumed to be mutually independent with expectation zero. The variables yn and xn are observable. Now let us assume that there is a third observable variable, zn, say, that is linearly related to E n in the same way as yn is:
Zn = lEn + Un, (8.18)
where un is independent of E n, є n, and vn, and has also mean zero. From this extended model, we obtain the following equations for the variances and covariances of the observable variables (the "covariance" equations):
о y = о EP2 + о2 cyx = о IP cyz = о EPy о 2x = о E + о v ох2 = о Ey
о Z = о Ey 2 + о2.
This system of six equations in six unknown parameters can be solved uniquely for the unknown parameters. Since the left-hand variables are consistently estimated by their sample counterparts consistent estimators for the parameters follow immediately. For example, the estimator of P is S = cyz/cxz = y'z/x'z, which is equivalent to the IV estimator with z as instrumental variable.