A pair of polychotomous random variables (Y 1 , Y 2 ) =: Y , where each Y j has a totally ordered support, is studied within a penalized Generalized Linear Model framework. We deal with a triangular generating process for Y , a structure that has been employed in the literature to control for the presence of residual confounding. Differently from previous works, however, the proposed model allows for a semi-parametric estimation of the covariateresponse relationships. In this way, the risk of model mis-specification stemming from the imposition of fixed-order polynomial functional forms is also reduced. The proposed estimation methods and related inferential results are finally applied to study the effect of education on alcohol consumption among young adults in the UK.

Summary A bivariate system of equations is developed to model ordinal polychotomous dependent variables within a simultaneous additive regression framework. The functional form of the covariate effects is assumed fairly flexible with appropriate smoothers used to account for non‐linearities and spatial variability in the data. Non‐Gaussian error dependence structures are dealt with by means of copulas whose association parameter is also specified in terms of a generic additive predictor. The framework is employed to study the effects of several risk factors on the levels of injury sustained by individuals in two‐vehicle accidents in France. The use of the methodology proposed is motivated by the presence of common unobservables that may affect the interrelationships between the parties involved in the same crash and by the possible heterogeneity in individuals’ characteristics and accident dynamics. Better calibrated estimates are obtained and misspecification reduced via an enhanced model specification.

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A conceptual framework for the analysis of dichotomous and ordinal polychotomous responses within a penalized multivariate Generalized Linear Model is introduced. The proposed structure allows for a rather flexible predictor specification through the inclusion of non-parametric and spatial covariate effects, and the characterisation of the distribution of the stochastic model components with copulae of univariate marginals. Analytic derivations for the particular case of Gaussian marginals within a bivariate system of dichotomous outcomes are also provided, and the framework is subsequently illustrated through the estimation of the HIV prevalence in Zambia using the 2007 DHS dataset.

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