I ran into problems of non-convergence in conducting a GMM with free variances. Although the best loglikelihood has been replicated, the systems says that a number of perturbed starting value run(s) did not converge. Even if a gradually increase random starts (from 20 4, to 100 20, to 200 40, to 400 80 and so on), I still get this warning that a number of perturbed starting values did not converge. Is this really something to be worried about? Can I take any other measures to overcome non-convergence?
GMM with free variances is a very flexible model for which there may be relatively little information in the data to estimate its parameters. This is reflected in the many non-convergencies. But if you have a substantial number of best logL convergencies you need not worry too much about the solution - although with a different sample in the future, it may may be hard to replicate your findings.