Semiparametric spatial regression: theory and practice

Nonparametric methods have been very popular in the last couple of decades in time series and regression, but no such development has taken place for spatial models. A rather obvious reason for this is the curse of dimensionality. For spatial data on a grid evaluating the conditional mean given its closest neighbors requires a four dimensional nonparametric regression. In this paper, a semi-parametric spatial regression approach is proposed to avoid this problem. An estimation procedure based on combining the so-called marginal integration technique with local linear kernel estimation is developed in the semi-parametric spatial regression setting. Asymptotic distributions are established under some mild conditions. The same convergence rates as in the one-dimensional regression case are established. An application of the methodology to the classical Mercer wheat data set is given and indicates that one directional component appears to be nonlinear, which has gone unnoticed in earlier analyses.