Statistical Inference Of Spatial Autoregressive Models For Complex Data

Under the classical regression model,it is usually assumed that the explained variables are independent mutually.However,there is often a certain spatial dependence between adjacent data collected in many fields,such as economics,biology,meteorology and geology.If this dependence is ignored in statistical inference,serious bias results may be generated.Therefore,in the case of spatial dependence,the classical regression model is no longer applicable.The spatial autoregressive(SAR)model can better describe the quantitative relationship between variables because it take into account the possible spatial dependence among data.Up to now,remarkable achievements have been made in the theoretical research on SAR models,but inadequacies are in the research on SAR models with missing data or measurement error,etc.With the rapid development of computer technology in recent years,the analysis of spatially dependent data has attracted more and more attention in the field of statistics and related application fields.In addition,since the classical regression model is only a special case of the SAR model,many theoretical methods established in the classical regression model may be invalid for SAR models.Therefore,it is of great theoretical significance and practical value to study the SAR model in the case of missing data,measurement error data,etc.To this end,this paper investigates the following aspects.(1)For the partially linear varying-coefficient SAR(PLVCSAR)model and the partially linear additive SAR(PLADSAR)model,in which the parametric part satisfies certain constraint conditions,the restricted estimators of parametric and nonparametric components are established by combining sieve method,two-stage least squares(2SLS)method and Lagrange multiplier method.Meanwhile,the asymptotic properties of the proposed estimators are derived.In addition,to test hypotheses on the parametric component,the Lagrange multiplier test statistic are constructed.Under the null hypothesis,it is shown that the Lagrange multiplier test statistic asymptotically follows the standard chi-squared distribution.The simulation results show that proposed estimators and tests perform well.An example is used for illustration.(2)The empirical likelihood(EL)inference for a semiparametric varyingcoefficient SAR model is investigated.The maximum empirical likelihood estimators(MELE)for the parametric component and the nonparametric component are established.Asymptotic normality of the proposed estimators are derived under certain mild conditions.Furthermore,it is proved that the empirical loglikelihood ratio for the parametric component and the residual-adjusted empirical log-likelihood ratio for nonparametric component asymptotically follow two standard chi-squared distributions,respectively.The corresponding confidence regions/bands for parametric component and nonparametric component are constructed.Simulation results indicate that the EL confidence regions/bands perform better than those derived from the normal approximation.(3)For the semiparametric varying-coefficient spatial autoregressive models with a diverging number of parameters,a variable selection procedure be proposed by combining sieve method and 2SLS method with the SCAD(smoothly clipped absolute deviation)penalty.Under some mild conditions,the oracle property of the resulting estimators are established.Some simulation studies are conducted to assess the finite sample performance of the proposed variable selection procedure,and the developed method is applied to a real data set.(4)For SAR models with missing data,we consider the estimation for two SAR models:the linear SAR model and the PLVCSAR model,respectively.The IPW-based robust estimation is developed by combining the inverse probability weighted method and the 2SLS method with imputation for the linear SAR model.Under some conditions,the asymptotic normality and the robustness of the proposed estimator are proved.Furthermore,by combing sieve method,the proposed method and the corresponding theoretical results are extended to the PLVCSAR model.The simulation results and the real data show that the proposed estimator is robust.(5)For the SAR model with covariate measurement error,a three-stage least squares(3SLS)method is proposed by introducing'Berkson instrumental variables'-type and 'classical instrumental variables'-type.Under some mild conditions,the asymptotic normality of the proposed estimators are derived with two different types of instrumental variables,respectively.The simulation results show that the proposed 3SLS estimation performs well whether the instrumental variables are 'Berkson instrumental variables'-type or 'classical instrumental variables'-type.