Statistical Inference Of Partially Linear Spatial Autoregressive Model Under Constraint Conditions

In many application fields of regression analysis,in addition to sample information,some prior knowledge or information about regression coefficients can be obtained and this prior information can be expressed in linear form.Ignoring the prior knowledge or information on regression coefficients often lead to incorrect estimate of parameters of interest.Thus,it is of great theoretical meaning and application value to study statistical inference of spatial autoregressive models under constraint conditions.Partially linear spatial autoregressive model has received wide attention from theoretical and applied researchers due to it is of the explanatory power of linear spatial autoregressive model and flexibility of nonparametric spatial autoregressive model.However,the current study about partially linear spatial autoregressive model all ignores the constraint that the regression coefficients may satisfy.Therefore,this paper considers statistical inference of partially linear spatial autoregressive model under constraint conditions.In part 3 of this paper,we consider the estimation and hypothesis testing of partially linear spatial autoregressive model under constraint conditions and independent and identically distributed error terms.First,by combining series approximation method,twostage least squares method and Lagrange multiplier method,we obtain constrained estimators of the parameters and function in the partially linear spatial autoregressive model and investigate their asymptotic properties.Furthermore,we propose a Wald test statistic to check whether the regression coefficients in the parametric component of the partially linear spatial autoregressive model satisfy linear constraint conditions,and derive asymptotic distributions of the resulting test statistic under both null and alternative hypotheses.Finally,simulation studies and real data analysis are conducted to demonstrate finite sample performance and practical application effect of the proposed estimation and testing methods.In many applications of regression analysis,regression coefficients usually satisfy constraint conditions and error terms are heteroscedastic.Therefore,in part 4 of this paper,we consider statistical inference of partially linear spatial autoregressive model with constraint conditions and heteroscedasticity.Firstly,the constrained estimators of unknown parameters and function are established by combining local polynomial smoothing method,generalized method of moments and Lagrange multiplier method,and their asymptotic properties are investigated under appropriate regularity conditions.Furthermore,a Wald test statistic is constructed to test appropriateness of a linear constraint condition on the regression coefficients.It is shown that the proposed test statistic asymptotically follows Chi-squared distribution under both null and alternative hypotheses.Finally,simulation studies and real data analysis are conducted to demonstrate finite sample performance of the proposed estimation and testing methods.