Grilli-Rampichini (2006) A Multilevel Multinomial Logit Model for the Analysis of Graduates' Skills

Summary

Grilli and Rampichini (2006) develop a multilevel multinomial logit model for analysing where university graduates acquired their professional and technical skills (university, workplace, or other), correcting for non-ignorable missing data via a joint selection-and-principal (S&P) model. The selection equation is a binary logit for whether the skill is currently used; the principal equation is a multinomial logit for the acquisition source. Random effects at the degree-programme level are introduced in both equations and linked via cluster-level correlations. Estimation uses adaptive Gaussian quadrature in Stata's gllamm. Applied to 2,540 employed University of Florence graduates across 56 programmes (year 2000 survey), selection bias turns out to be negligible (likelihood-ratio test $\mathrm{LR} = 5.52$, $p = 0.063$), but degree-programme effects are substantial (intraclass correlation, ICC 4.45%–11.16%).

Key Claims

• The model couples a binary selection equation (skill currently used?) with a $M=3$ multinomial principal equation (skill source: university / workplace / otherwise) through shared cluster-level random effects $\xi_j^{(m)}$.
• Linear predictor: $\eta_{ij}^{(m)} = \alpha^{(m)} + \beta^{(m)'} x_{ij} + \xi_j^{(m)} + \delta_{ij}^{(m)}$; subject-level selection-principal covariance $\text{Cov}(\epsilon_{ij}^S, \delta_{ij}^{(m)})$ is empirically unidentified (confounded with the fixed intercept) and omitted.
• ICC for workplace vs. university: 4.45%; ICC for otherwise vs. university: 11.16%; correlation between the two random effects $\hat\rho = 0.848$.
• Selection bias test: $\mathrm{LR} = 5.52$ on 2 df, $p = 0.063$ — borderline non-significant; ignoring selection is justified in this application.
• Degree programme is a significant source of heterogeneity ($\mathrm{LR} = 142.5$, 5 df, $p < 0.001$); short-degree students more likely to cite university as source; Business Economics and Engineering highest $P(\text{university})$; Foreign Languages and Political Science lowest.

Concepts Introduced or Extended

• Multilevel Model
• Bayesian Hierarchical Model
• Selection Model
• Discrete Choice

Entities Mentioned

• Leonardo Grilli
• Carla Rampichini

Quotes

"The multilevel multinomial logit model is a natural tool for the analysis of categorical outcomes nested within groups, but it is rarely extended to handle non-ignorable missing data."

"The selection and principal equations share cluster-level random effects, allowing the degree programme to affect both the probability of skill use and the source of skill acquisition."

My Take

The paper's main contribution is methodological: it provides a practical template for joint selection + multilevel multinomial modelling via gllamm, with clean identification coming from individual-level exclusion restrictions (proxies for skill applicability). The finding that selection bias is negligible here is substantively reassuring but does not diminish the method's value for applications where it is not. The ICC estimates reveal that degree programme explains a modest but non-trivial share of variance in skill-source attribution, suggesting curriculum design matters. The paper predates the widespread use of Bayesian Markov chain Monte Carlo (MCMC) for such models; a Bayesian treatment (e.g., via Stan) would allow full propagation of uncertainty in the random-effects covariance structure.