Comparing the asymptotic and empirical (un)conditional distributions of OLS and IV in a linear static simultaneous equation

In designing Monte Carlo simulation studies for analyzing finite sample properties of econometric inference methods, one can use either IID drawings in each replication for any series of exogenous explanatory variables or condition on just one realization of these. The results will usually differ, as do their interpretations. Conditional and unconditional limiting distributions are often equivalent, thus yielding similar asymptotic approximations. However, when an estimator is inconsistent, its limiting distribution may change under conditioning. These phenomena are analyzed and numerically illustrated for OLS (ordinary least-squares) and IV (instrumental variables) estimators in single static linear simultaneous equations. The results obtained supplement–and occasionally correct–earlier results. The findings demonstrate in particular that the asymptotic approximations to the unconditional and a conditional distribution of OLS are very accurate even in small samples, and that the actual absolute estimation errors of inconsistent OLS in finite samples are often much smaller than those of consistent IV, even when the instruments are not extremely weak. It is also shown that conditioning reduces the estimation errors of OLS, whereas it deranges the distribution of IV when instruments are weak. Finally it is indicated how OLS could be modified to produce accurate inference under assumptions regarding the degree of simultaneity.