Statistical Inference For Dynamic Spatial Partially Linear Single Index Models With Time-Varying Spatial Weight Matrix

Spatial econometrics,as a branch of econometric research,effectively reflects the dependence between units of study by taking into account the location and state factors of the units of study.The first law of geography explains that the closer the spatial units are,the stronger the dependence tends to be,and conversely,the weaker the dependence.As a measure of inter-unit connectivity,the spatial weight matrix has a direct impact on the estimation of spatial correlation coefficients.0-1 weight matrices,k-order weight matrices,geographically weighted weight matrices and other exogenous weight matrix forms have been increasingly recognised and applied by scholars,but such spatial weight settings do not use information from the sample data,nor do they take into account the inter-unit weights that change over time.So the setting of weights between spatial units is a research issue worth investigating.This paper is based on the Spatial Autoregressive Model(SAR),which assumes that the dependent variable in this unit is not only influenced by the independent variable,but also by the spatial spillover effect of the dependent variable in neighbouring units.As the exogenous weight matrix disregard the sample information,it does not reflect the degree to which the spatial spillover effect varies with the distance between units,nor does it reflect the change in the weight between units at a given moment due to external shocks at that moment.On the other hand,the effect of the independent variable on the dependent variable should not be portrayed only in the form of parametric regression;in general,a non-parametric regression setting is more conducive to reflecting the characteristics of a data-driven model.However,nonparametric regression models can lead to ’dimensionality disaster’ in multivariate situations.Semi-parametric regression models can not only reduce the dimensionality of the model,but also maintain the flexibility of non-parametric models.For these reasons,this paper extends the spatial autoregressive model to Dynamic Spatial Partially Linear Single-Index Models with Time-Varying Spatial Weight Matrix(DSPLS).This paper studied from two aspects of statistical theory and empirical application,mainly studies the following three points:First,this paper improves the SAR model in two ways and proposes a dynamic spatial partially linear single index model with a time-varying spatial weight matrix.On the one hand,the spatial weight matrix is set in a time-varying form,and the timevarying spatial weight matrix is constructed through the form of a function of negative exponential distance.By setting the time-varying parameters in the negative exponential function in the form of a Generalized Autoregressive Scoring Model(GAS),it is not only possible to determine the degree of variation of the weight matrix with distance,but also to reflect the impact of external shocks on the weight matrix at certain times.On the other hand,this paper extends the linear part of the SAR model into a partially linear single indicator form to increase its applicability.Second,the DSPLS model is estimated in this paper using a two-step estimation method.In the first step,Generalized Estimating Equations(GEE)is used to estimate the single index,and in the second step,Maximum Likelihood Estimation(MLE)is used to estimate the parameters other than the single index.This paper demonstrates the asymptotic theory of the single index estimation and parameter estimation,gives the parameter estimation results for different sample sizes and different time lengths and simulates the real connection function shape and time-varying parameter sequences.Thirdly,this paper selects the empirical problems under the background of medicine and economy,and discusses them using the DSPLS model.The first demonstration is based on the enhanced Nathan Kline Institute Rockland Sample of the United States,which compares the spatial correlation characteristics of healthy people and drug users in the brain regions of interest.The results showed that the spatial correlation of drug addicts’ brains was weaker than that of healthy people,and the correlation fluctuated more obviously with time.The second empirical investigation explores the factors influencing the economies of OECD member countries.The results show that the single index consisting of three variables,namely the share of foreign direct investment,the share of human capital and the population growth rate,shows a slower growth relationship on the growth rate of GDP per capita,while the coefficient of increase of the share of fixed capital and the urbanisation rate on the growth rate of GDP per capita is 2.31% and 0.22% respectively.The spatial weights have shown an overall upward trend over the past 40 years,indicating that OECD member countries are becoming more cooperative.