Instrumental variables interpretations of FIML and nonlinear FIML

FIML estimates of a simultaneous equation econometric model can be obtained by iterating to convergence an instrumental variables formula that is perfectly consistent with the intuitive textbook-type interpretation of efficient instruments: instruments for an equation must be uncorrelated with the error term of the equation, but at the same time must have the highest correlation with the explanatory variables. However, if our purpose is to obtain FIML from iterating to convergence some full information instrumental variables, the intuitive textbook-type interpretation of the efficient instruments is not necessarily helpful, and can be too restrictive. The purpose of this paper is to show that, in the full information framework, there is a much wider flexibility in the choice of the instruments. Against intuition, instruments may be not purged enough of correlation with the error term: for example, the instruments for the endogenous variables or functions of endogenous variables included in one equation do not need to be purged of the residuals of equations that are correlated with the given one. Viceversa, instruments can be purged too much: for example, if there are zero covariance restrictions, instruments may be purged also of the estimated residuals of equations uncorrelated with the given one.