Simultaneous inference in structured additive conditional copula regression models: a unifying Bayesian approach
- 704 Downloads
While most regression models focus on explaining distributional aspects of one single response variable alone, interest in modern statistical applications has recently shifted towards simultaneously studying multiple response variables as well as their dependence structure. A particularly useful tool for pursuing such an analysis are copula-based regression models since they enable the separation of the marginal response distributions and the dependence structure summarised in a specific copula model. However, so far copula-based regression models have mostly been relying on two-step approaches where the marginal distributions are determined first whereas the copula structure is studied in a second step after plugging in the estimated marginal distributions. Moreover, the parameters of the copula are mostly treated as a constant not related to covariates and most regression specifications for the marginals are restricted to purely linear predictors. We therefore propose simultaneous Bayesian inference for both the marginal distributions and the copula using computationally efficient Markov chain Monte Carlo simulation techniques. In addition, we replace the commonly used linear predictor by a generic structured additive predictor comprising for example nonlinear effects of continuous covariates, spatial effects or random effects and furthermore allow to make the copula parameters covariate-dependent. To facilitate Bayesian inference, we construct proposal densities for a Metropolis–Hastings algorithm relying on quadratic approximations to the full conditionals of regression coefficients avoiding manual tuning. The performance of the resulting Bayesian estimates is evaluated in simulations comparing our approach with penalised likelihood inference, studying the choice of a specific copula model based on the deviance information criterion, and comparing a simultaneous approach with a two-step procedure. Furthermore, the flexibility of Bayesian conditional copula regression models is illustrated in two applications on childhood undernutrition and macroecology.
KeywordsArchimedean copulas Generalised Metropolis–Hastings algorithm Gaussian copula Marginal distribution Penalised splines
We thank two reviewers for helpful comments on the submitted version. Financial support by the German research foundation via the research training group 1644 and the projects KN 922/4-1/2 is gratefully acknowledged. We would also like to thank Holger Kreft for providing the data on species richness patterns.
- Belitz, C., Hübner, J., Klasen, S., Lang, S.: Determinants of the socioeconomic and spatial pattern of undernutrition by sex in india: a geoadditive semi-parametric regression approach. In: Kneib, T., Tutz, G. (eds.) Statistical modelling and regression structures, pp. 155–179. Physica-Verlag, Heidelberg (2010)CrossRefGoogle Scholar
- Dagum, C.: A new model of personal income distribution: specification and estimation. Econ. Appl. 30, 413–437 (1977)Google Scholar
- Fermanian, J., Scaillet, O.: Nonparametric estimation of copulas for time series. J. Risk 5, 25–54 (2003)Google Scholar
- Klasen, S., Moradi, A.: The nutritional status of elites in India, Kenya, and Zambia: an appropriate guide for developing reference standards for undernutrition? Technical Report. Sonderforschungsbereich 386: Analyse Diskreter Strukturen. Discussion Paper No. 217. http://epub.ub.uni-muenchen.de/view/subjects/160101.html (2000)
- Klein, N., Kneib, T., Lang, S.: Bayesian structured additive distributional regression. Working papers in economics and statistics 2012-23. Faculty of Economics and Statistics, University of Innsbruck (2013). http://eeecon.uibk.ac.at/wopec2/repec/inn/wpaper/2013-23.pdf
- Klein, N., Kneib, T., Klasen, S., Lang, S.: Bayesian structured additive distributional regression for multivariate responses. J. R. Stat. Soc. Ser. C (2015a)Google Scholar
- Klein, N., Kneib, T., Lang, S.: Bayesian generalized additive models for location, scale and shape for zero-inflated and overdispersed count data. J. Am. Stat. Assoc. 110, 405–419 (2015b)Google Scholar
- Marra, G., Radice, R.: SemiParBIVProbit: Semiparametric Bivariate Probit Modelling. R package version 3.3 (2015)Google Scholar
- Radice, R., Marra, G., Wojtys, M.: Copula regression spline models for binary outcomes. Technical report (submitted) (n.d.)Google Scholar
- Wood, S.: mgcv: Mixed GAM Computation Vehicle with GCV/AIC/REML Smoothness Estimations. R package version 1.8-5 (2015)Google Scholar
- Yee, T.W.: VGAM: Vector Generalized Linear and Additive Models. R package version 0.9-7 (2015)Google Scholar