Nonparametric seemingly unrelated regression

A method is presented for simultaneously estimating a system of nonparametric regressions which may seem unrelated, but where the errors are potentially correlated between equations. We show that the advantage of estimating such a ‘seemingly unrelated’ system of nonparametric regressions is that less observations can be required to obtain reliable function estimates than if each of the regression equations is estimated separately and the correlation ignored. This increase in efficiency is investigated empirically using both simulated and real data. The method uses a Bayesian hierarchical framework where each regression function is represented as a linear combination of a large number of basis terms. All the regression coefficients, and the variance matrix of the errors, are estimated simultaneously by their posterior means. The computation is carried out using a Markov chain Monte Carlo sampling scheme that employs a ‘focused sampling’ step to combat the high-dimensional representation of the unknown regression functions. The methodology extends easily to other nonparametric multivariate regression models.