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.