Decision theory applied to an instrumental variables model

This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (rho, phi, omega), where phi is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter rho and a parameter vector omega, which is normalized to be a point on the (k-1)-dimensional unit sphere; rho measures the strength of the association between the endogenous variables and the instrumental variables, and omega is a measure of direction. A prior distribution is introduced for the IV model. The parameters phi, rho, and omega are treated as independent random variables. The distribution for phi is uniform on the unit circle; the distribution for omega is uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for rho is left general. It turns out that given any positive value for rho, the Bayes estimator of phi does not depend on rho; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter omega in the IV model. This gives an integrated likelihood function with two scalar parameters, phi and rho. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for phi. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on rho and not on phi or omega. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for phi and omega, but a prior is also needed for the scalar rho, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on rho and not on phi or omega. The optimality result combines average risk and maximum risk. The confidence set minimizes the average-with respect to the prior distribution for rho-of the maximum risk, where the maximization is with respect to phi and omega.