Owing to the generality of the random effects model (see for instance http://polmeth.wustl.edu/media/Paper/FixedvsRandomSept2013.pdf), you can fit such models using software such as gllamm (type: findit gllamm).
Gllamm is highly complex, and you really need to know what you're doing to undertake it - but there are some excellent examples of what it can do for a multinomial logit model here (http://www.gllamm.org/examples.html; see the Skrondal & Rabe-Hesketh (2003), Psychometrika paper on polytomous logit models). They include both the paper, data, and do-file to follow along.
Alternatively, a multilevel, multinomial probit model can be fit with cmp (http://www.stata-journal.com/article.html?article=st0224; type: findit cmp - see "Multinomial probit with heterogeneous preferences (random effects by individual)" example in the help file for cmp), which comes from a different perspective on estimation and is arguably equally complex in terms of model fitting.
an easy option is to download Sabre from the statistics dept, Lancaster University (it is free) - see chapter 5 (section 5.3 in Shahtahmasebi S, Berridge D: Conceptualising behaviour in health and social research: a practical guide to data analysis. New York, Nova Sci, 2010) or contact Damon Berridge. I am assured by colleagues that Sabre models can be fitted using Stata although I have not tried it myself. I am not too sure about R.
Thank you a lot for your answers. Yet, my main goal is to run a fixed effects model, meaning to control for unobservable heterogeneity by accounting only for the within variation in the dependent and independent variables. I still don't know if this would work. In the case of the linear and logistic regression I can use the Hausman test to determine whether the random effects model is consistent and I would use it only if it is. I have no idea how it works with multinominal logistic regression yet.
Sabre will also provide you with a parameter estimate (called Scale parameter) of unobserved heterogeneity which you can use to assess the statistical effect of unobserved heterogeneity in your data. hope this helps.
This article challenges Fixed Effects (FE) modelling as the ‘default’ for time-series-cross- sectional and panel data. Understanding differences between within- and between-effects is crucial when choosing modelling strategies. The downside of Random Effects (RE) modelling – correlated lower-level covariates and higher-level residuals – is omitted-variable bias, solvable with Mundlak’s (1978a) formulation. Consequently, RE can provide everything FE promises and more, and this is confirmed by Monte-Carlo simulations, which additionally show problems with another alternative, Plümper and Troeger’s Fixed Effects Vector Decomposition method, when data are unbalanced. As well as being able to model time-invariant variables, RE is readily extendable, with random coefficients, cross-level interactions, and complex variance functions. An empirical example shows that disregarding these extensions can produce misleading results. We argue not simply for technical solutions to endogeneity, but for the substantive importance of context and heterogeneity, modelled using RE. The implications extend beyond political science, to all multilevel datasets.
you can then use MLwin to fit multinomial panel models - mlwin can be called from within Stata and from within R - its advantage is that it is very fast and that you can switch to MCMC estimation (for often higher quality - less biased) estimates
see http://www.bristol.ac.uk/cmm/software/ specifically
R2MLwiN
An R command interface to the MLwiN multilevel modelling software package, allowing users to fit multilevel models using MLwiN from within the R environment.
runmlwin
A Stata command to fit multilevel models in MLwiN from within Stata.
Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...
I am also struggling with the same thing and gsem if not reaching convergence for me. I also tried accessing MLwiN, but it's for sale. I can't afford it. Has anyone found a breakthough apart from these approaches?