14.3 Estimating a system

Limited-information methods estimate one equation at a time, using only the exclusions in that equation: 2SLS, and the limited-information maximum likelihood estimator (LIML) of Anderson and Rubin (1949), which has the same asymptotic distribution but is better behaved with many or weak instruments. Full-information methods estimate all equations together, using all the restrictions and the correlation between the structural disturbances. The most common is three-stage least squares (3SLS) of Zellner and Theil (1962): estimate each equation by 2SLS, use the residuals to estimate \(\Sigma \), and apply generalised least squares to the stacked system, with the instruments as in 2SLS. 3SLS is asymptotically efficient among IV estimators when all the equations are correctly specified, and equal to 2SLS when every equation is exactly identified or \(\Sigma \) is diagonal. Full-information maximum likelihood (FIML) has the same asymptotic distribution as 3SLS under normality. The price of system methods is robustness: a misspecification in one equation contaminates the estimates of all the others. Software 14.7.2 shows the same danger for the simpler system that follows.

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