Stein-like 2SLS estimator

Maasoumi (1978) proposed a Stein-like estimator for simultaneous equations and showed that his Stein shrinkage estimator has bounded finite sample risk, unlike the three-stage least square estimator. We revisit his proposal by investigating Stein-like shrinkage in the context of two-stage least square (2SLS) estimation of a structural parameter. Our estimator follows Maasoumi (1978) in taking a weighted average of the 2SLS and ordinary least square estimators, with the weight depending inversely on the Hausman (1978) statistic for exogeneity. Using a local-to-exogenous asymptotic theory, we derive the asymptotic distribution of the Stein estimator and calculate its asymptotic risk. We find that if the number of endogenous variables exceeds 2, then the shrinkage estimator has strictly smaller risk than the 2SLS estimator, extending the classic result of James and Stein (1961). In a simple simulation experiment, we show that the shrinkage estimator has substantially reduced finite sample median squared error relative to the standard 2SLS estimator.