Estimation of single-index model with spatial interaction

• Our model can describe possible nonlinear relationships besides spatial dependence.
• The two-step estimation procedures is proposed to estimate the unknowns.
• We establish the asymptotic normality of the estimators.
• We establish the simultaneous confidence bands of the nonparametric function.
• Monte Carlo simulations support the proposed method.

This article is concerned with the single-index model in spatial dependence data, where the spatial lag effect enters the model linearly and the relationship between variables is a nonparametric function of a linear combination of multivariate regressors. This setup avoids the so-called curse of dimensionality while still capturing important nonlinear features in high dimensional data. It also provides a convenient framework in which to model interactions between the regressors. We propose a two stage estimation strategy where the nonparametric component is established by a local linear approach and the estimation of the parametric part by GMM method, which can be seen as a direct nonlinear least squares method. We derive the asymptotic distributions of the unknowns in our model, and the procedures for constructing simultaneous confidence bands of the nonparametric function are also established. In addition, a simulation study is performed.