A note on the application of EC2SLS and EC3SLS estimators in panel data models

Baltagi and Li [Baltagi, B.H., Li, Q., 1992. A note on the estimation of simultaneous equations with error components. Econometric Theory 8, 113-119] showed that for estimating a single equation in a simultaneous panel data model, EC2SLS has more instruments than G2SLS. Although these extra instruments are redundant in White's [White, H., 1986. Instrumental variables analogs of generalized least squares estimators. In: Mariano, R.S. (Ed.), Advances in Statistical Analysis and Statistical Computing, vol. 1. JAI Press, New York, pp. 173-277] terminology, they may yield different estimates and standard errors in empirical studies with finite N and T. We illustrate this using the crime data of Cornwell and Trumbull [Cornwell, C., Trumbull, W.N., 1994. Estimating the economic model of crime with panel data. Review of Economics and Statistics 76, 360-366]. We show that the standard errors of EC2SLS are smaller than those of G2SLS for this example. In general, we prove that the asymptotic variance of G2SLS differs from that of EC2SLS by a positive semi-definite matrix. Although this difference tends to zero as the sample size tends to infinity, in small samples, this difference may be different from zero and can lead to gains in small sample efficiency. This proof is extended to the system equations' 3SLS counterparts.