An asymptotically optimal two-sample instrumental variables estimation using many instruments

Abstract We investigate the statistical estimation of the coefficients in a linear structural equation within a simultaneous equation system, utilizing two-sample data and a large number of instrumental variables. Specifically, we derive the asymptotic properties of the two-sample least variance ratio (2SLVR) estimator, an extension of the limited information maximum likelihood (LIML) estimator designed for one-sample contexts, in the two-sample setting with numerous instruments. It is well-documented that the one-sample two-stage least squares (TSLS) estimator and the generalized method of moments (GMM) suffer from non-negligible bias. Despite their widespread practical use, these methods often lose consistency when a large number of instruments are employed. In contrast, our findings demonstrate that the variance–covariance matrix of the limiting distribution of the 2SLVR estimator and its modifications frequently achieve the asymptotic lower bound, even when the disturbance terms deviate from a normal distribution. The results of this study hold significant potential for applications in econometrics and biometrics, particularly in contexts such as Mendelian Randomization (MR) analyses using DNA data.