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. Copyright The Econometric Society 2007.
Year of publication: |
2007
|
---|---|
Authors: | Chamberlain, Gary |
Published in: |
Econometrica. - Econometric Society. - Vol. 75.2007, 3, p. 609-652
|
Publisher: |
Econometric Society |
Saved in:
Saved in favorites
Similar items by person
-
Heterogeneity, comitted variable bias and duration dependence
Chamberlain, Gary, (1979)
-
Econometrics and decision theory
Chamberlain, Gary, (2000)
-
Funds, factors, and diversification in arbitrage pricing models
Chamberlain, Gary, (1982)
- More ...