Least-Squares versus Instrumental Variables Estimation in a Simple Errors in Variables Model - Leamer, Edward E. (Econometrica, 1978)
IF ONE OF THE EXPLANATORY VARIABLES in a linear regression model is measured with error, the ordinary least squares estimator is known to be biased and inconsistent. Given suitable assumptions, an instrumental variables estimator is known to be consistent. In a large sample, the instrumental variables estimator is thus unambiguously preferred, but the choice of an estimator in a small sample remains a puzzle. The method of maximum likelihood sheds light on this puzzle. It will be shown below that the instrumental variables estimate is the maximum likelihood estimate if, and only if, it lies between the ordinary least squares estimate and the reverse least squares estimate, that is, if and only if it satisfies the bounds implied by the simple errors in variables model. The letters Y, x, and z will indicate, respectively, the vector of observations of the dependent variable, the vector of error-ridden measurements of the explanatory variable and the vector of observations of an instrumental variable, each measured around its mean. The ordinary least squares estimate is then