Additive, Dynamic and Multiplicative Regression

We survey and compare model-based approaches to regression for cross-sectional and longitudinal data which extend the classical parametric linear model for Gaussian responses in several aspects and for a variety of settings. Additive models replace the sum of linear functions of regressors by a sum of smooth functions. In dynamic or state space models, still linear in the regressors, coeecients are allowed to vary smoothly with time according to a Bayesian smoothness prior. We show that this is equivalent to imposing a roughness penalty on time-varying coeecients. Admitting the coeecients to vary with the values of other covariates, one obtains a class of varying-coeecient models (Hastie and Tibshirani, 1993), or in another interpretation, multiplicative models. The roughness penalty approach to non-and semiparametric modelling, together with Bayesian justiications, is used as a unifying and general framework for estimation. The methodological discussion is illustrated by some real data applications.