Models for which the MLE and the conditional MLE coincide

The MLE, CMLE and MMLE coincide in a linear regression model with fixed individual effects. In this case, there is no incidental parameters problem and the MLE is consistent. The equivalence of these estimators is important because CMLE=MLE implies both the consistency of the MLE and the efficiency of the CMLE. In general, we cannot expect to find a CMLE or MMLE, since there may be no fixed-dimension sufficient statistic for the effects, nor an appropriate transformation of the data whose distribution does not depend on the effects. However, we show that the MLE, CMLE and MMLE do coincide in systems of seemingly unrelated regressions and in systems of simultaneous equations. We establish this result for systems in which (exogenous) variables in addition to (or other than) the intercept may have coefficients which vary over individuals, provided that the set of such variables is the same in every equation.The financial support of the National Science Foundation is gratefully acknowledged.