Maximum likelihood estimator and confidence intervals for a simple errors in variables model
Recently Learner obtained the maximum likelihood estimator for β, the slope coefficient in an affine relation between normal unobservables when an instrumental normal variable is available and the errors of observation are normal and independent. The present paper establishes the conditions under which this result holds and finds estimators under other conditions. When the instrumental variable is error-free equals the socalled instrumental variable estimator, even when the errors of observation in the other variables are correlated. It is also shown that the 1 - α confidence set for β proposed by Durbin is an interval and finite if and only if the correlation between the observed values of the explanatory variable and those of the instrumental variable is significantly different from 0 at level α. This leads to some new proposed confidence intervals for β.