The linear GMM model with singular covariance matrix due to the elimination of a nuisance parameter

When in a linear GMM model nuisance parameters are eliminated by
multiplying the moment conditions by a projection matrix, the
covariance matrix of the model, the inverse of which is typically
used to construct an efficient GMM estimator, turns out to be
singular and thus cannot be inverted. However, one can show that
the generalized inverse can be used instead to produce an efficient
estimator. Various other matrices in place of the projection matrix
do the same job, i.e., they eliminate the nuisance parameters. The
relations between those matrices with respect to the efficiency of
the resulting estimators are investigated.