Profile Maximum Likelihood Estimation of Single-Index Spatial Dynamic Panel Data Model

In this paper, the spatial dynamic panel data (SDPD) model is extended to the single-index spatial dynamic panel data (Si-SDPD) model by introducing a nonlinear connection function to reflect the interaction between explanatory variables. The Si-SDPD model not only retains the advantages of the parametric SDPD model in dealing with spatial and temporal interaction effects and spatio-temporal dependencies, but also solves the limitations of the parametric SDPD model that may lead to missed bias. It reduces the data dimension of non-parametric models and enhances the practicability and explanatory power of parametric models. Since the parts of the model to be estimated contain unknown functions, we propose a new estimation method, a profile maximum likelihood (PML) method, to solve the problem of incidental parameters in the estimation. Under the assumption that the spatial coefficients are known, we preliminarily estimate the unknown function by carrying out local polynomial estimation, so as to transform the model into the parametric form for solving purposes. We then solve the dynamic panel parametric model via quasi-maximum likelihood (QML) estimation. We derive the asymptotic properties of profile maximum likelihood estimators (PMLEs) and find that, under certain regularity conditions, both parametric and non-parametric estimators are consistent. Monte Carlo results show that PMLEs have good finite sample performance.