  Methods for choosing a fixed set of knot locations in additive spline models
are fairly well established in the statistical literature. While most of these
methods are in principle directly extendable to non-additive surface models,
they are less likely to be successful in that setting because of the curse of
dimensionality, especially when there are more than a couple of covariates. We
propose a regression model for a multivariate Gaussian response that combines
both additive splines and interactive splines, and a highly efficient MCMC
algorithm that updates all the knot locations jointly. We use shrinkage priors
to avoid overfitting with different estimated shrinkage factors for the
additive and surface part of the model, and also different shrinkage parameters
for the different response variables. This makes it possible for the model to
adapt to varying degrees of nonlinearity in different parts of the data in a
parsimonious way. Simulated data and an application to firm leverage data show
that the approach is computationally efficient, and that allowing for freely
estimated knot locations can offer a substantial improvement in out-of-sample
predictive performance.
