Semiparametric partially linear varying coefficient higher-order spatial autoregressive model

In this paper, we propose a semiparametric higher-order spatial autoregressive model by allowing regression function to admit a partially linear varying coefficient structure. The proposed model makes a balance between interpretability of linear higher-order spatial autoregressive models and flexibility of varying coefficient higher-order spatial autoregressive models. We develop a computationally efficient estimation procedure for the proposed model and derive asymptotic properties of resulting estimators. We develop a Wald testing procedure to test linear constraint hypothesis on parameters in parametric component of the proposed model, and obtain asymptotic distributions of the resultant statistic under both null and alternative hypotheses. Moreover, a generalized likelihood ratio testing procedure is developed to test whether the coefficient functions have interesting parametric forms, in which a bootstrap procedure is suggested to appropriate the null distribution of the resulting statistic. Simulation studies and empirical analysis of Boston house price data and 1980 U.S. presidential election data are conducted to evaluate the usefulness of the proposed model and its estimation and testing procedures.