Bayesian Reduced Rank Regression in SEMs with Weak Identification

This paper is concerned with Bayesian reduced rank regression when instruments are weak. There have been a number of studies on weak identification problem with the application of reduced rank regression combined with singular value decomposition (SVD) method in the Bayesian framework, see Anderson (1951), Forsythe and Moler (1967), Geweke (1996), Kleibergen and van Dijk (1998), Gao and Lahiri (1999), van Dijk (2002), and Kleibergen and Papp (2003). Kleibergen and van Dijk (1998) used SVD method, and showed that the issue can be addressed well when at least some of the instruments are strong. Gao and Lahiri (1998) find, using Monte Carlo simulation, that the performance of the estimator is asymmetric depending on the values of the canonical form parameters. In some situations it performs better than most competing Bayesian and Classical estimators, but in other times it is worse than even simple OLS. In this paper we derive analytical results justifying the findings in Gao and Lahiri (1999). We find that SVD and the posterior distribution of the structural parameters by reduced rank method cannot handle the weak instruments problem properly; the bias is an asymmetric function of the nuisance parameters. Thus, the structural parameters may not be properly recovered from the reduced form parameter using SVD under the weak instrument problem