...not sure, if this helps, but I had a very similar problem with a multinomial logit MLM using 2nd order PQL in Mlwin. Individuals were nested within small geographical areas, my response variable had 5 categories (I chose the largest as the refcat). It turned out to be exactly the issue you describe here: "I am wondering if the problem is related to the fact that for many countries I only have '0s' and no '1s' ?". In some of my areas (my level-2-units) no respondents had ticked one of my five categories, and my model just never converged. What worked for me (after numerous attempts), was simply to use fewer categories on my response variable (I collapsed two that made sense) and re-ran it. But every model is different. What I found really useful is this forum, Lemma, Univ of Britsol (just in case you haven't already seen it): http://www.bristol.ac.uk/cmm/software/support/support-faqs/errors.html . A few people have posted problems similar to yours in the past... Good luck!

Dear Stefanie, I am following your advice to reduce the categories, by running 3 binomial regressions for belonging to cluster A yes/no, B yes/no and C yes/no, instead of a multinomial. In this way the program does have enough 0s and 1s for each country and cluster, and works. Eventually, in case I have evidence the 2 variables don't matter, I will turn back to the multinomial. thanks a lot for your response!

The problem with the IGLS estimator is lack of information to estimate the model you have specified; then when you then take it over  as starting values for MCMC it cannot proceed as the starting values are inadmissible.

Two suggestions

1) choose a different reference base - I usually choose to drop the outcome that has the most frequent occurrence - this often works.

2) reduce the complexity of the model in the random part - In IGLS initially remove all the covariances at the higher level - you can click on each one to remove them one at a time  or  click on the Omega in the equations window and choose set diagonal matrix ; if its level 3 - the command is SMAT 3 0. You will see the covariances are no longer there but the variances on the diagonal are. Start in IGLS - hopefully to converge . Now put the covariances back in while in IGLS - they will have the values of zero (click on the Omega and request set full matrix - ie SMAT 3 1) . Do not press start in IGLS but switch to MCMC than Start - this should produce a positive definite matrix ). You will need a long run and a long burinn to get way from these initial values. Sometimes in IGLS even with no covariances  you get extreme results for the variance estimates - by that I mean an estimate of zero or greater than say 10. If you right  click on these problematic terms - you can set your own initial estimate - say something like 0.01 - in the IGLS mode and then switch to MCMC using these chosen estimates as your starting position. Again ensure by looking at the  trajectories of each parameter that the estimates have got away from the estimates you have imposed.

Apologies that this is rather involved but you are trying to fit a complex model with rather limited data and the algorithms sometimes need help - the art of modeling!

 See page 282  of the following for a bit more explanation and example

https://www.researchgate.net/publication/268331676_Vol2Traingmanual2014withminorcorrections

It is good to see that you are using CMM's resources - this site is a guide to all our material

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html

and this is the section on the multinomial

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#discrete

And yes if all this fails your only option is to combine categories so that the algorithms have something to work on (the covariances at the higher level represent the tendency for a country to favour  A to also favor B - and if you do not have A and B in a country - there is not a lot to go on!)

Data Vol2Traingmanual2014withminorcorrections

Thanks for all this, Kelvyn! It is really useful for me, too (especially the last example on ethnic segregation by Rebecca Pillinger is nice).

Dear Kelvin, many thanks! I have done both points 1 and 2 and it now works with one of the problematic variables. Yet, when I introduce the other variables, one of the variances converges to zero in the IGLS estimate, so that when I switch to MCMC it signals error. Nevertheless, I apparently found a way to run the model anyway. That is, after having estimated the empty model in MCMC, i switch back to IGLS to introduce a new variable, then switch to MCMC and run the model - I receive a warning because the starting value is zero -  yet with lots of iterations 25000 it seems to arrive to sound results - trajectories being fine. I hope it is an appropriate solution. Marco