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,
coefficients 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 coefficients.
Admitting the coefficients to vary with the values of other
covariates, one obtains a class of varying-coefficient models
(Hastie and Tibshirani, 1993), or in another interpretation,
multiplicative models. The roughness penalty approach to non- and
semiparametric modelling, together with Bayesian justifications, is
used as a unifying and general framework for estimation. The
methodological discussion is illustrated by some real data
applications.