Variable selection of partially linear varying coefficient spatial autoregressive model

The assumption of equal variances is not always appropriate and different approaches for modelling variance heterogeneity have been widely studied in the literature. One of these approaches is joint location and scale model defined with the idea that both the location and the scale depend on explanatory variables through parametric linear models. Because the joint location and scale model includes two models, it does not deal well with a large number of irrelevant variables. Therefore, determining the variables that are important for the location and the scale is as important as estimating the parameters of these models. From this point of view, a combine robust estimation and variable selection method is proposed to simultaneously estimate the parameters and select the important variables. This is done using the least favorable distribution and least absolute shrinkage and selection operator method. Under appropriate conditions, we study the consistency, asymptotic distribution and the sparsity property of the proposed robust estimator. Simulation studies and a real data example are provided to demonstrate the advantages of the proposed method over existing methods in literature.

This paper introduces three Bayesian variable selection methods for spatial panel autoregressive quantile (SPARQ) model based on LASSO penalty, adaptive LASSO penalty, and spike-and-slab distribution. Meanwhile, the MCMC algorithm combining Gibbs and Metropolis-Hastings sampling is designed for each method by specifying different priors. The proposed methods enable simultaneous parameter estimation and variable selection. The numerical simulations suggest that the MCMC algorithm converges quickly and Bayesian variable selection methods are reliable. A real example is given to analyze the relationship between China’s regional innovation capability and 11 explanatory variables at different quantile levels. Both simulations and data example demonstrate that the proposed methods are feasible and reliable.

Spatial autoregressive (SAR) models with varying coefficients are useful for capturing heterogeneous effects of the impacts of covariates as well as spatial interaction in empirical studies, and a wide range of popular models can be seen as its special cases, such as linear SAR models. In this study, we will propose a unified model selection method for the SAR model with varying coefficients to achieve two targets simultaneously: (1) variable selection (eliminate irrelevant covariates), and (2) identification of the covariates with constant effect among the relevant covariates. To do so, we follow the idea of group LASSO to incorporate two penalty functions to simultaneously do model selection and estimation. Monte Carlo experiments show that the proposed method performs well in finite samples. Finally, we illustrate the method with an application to the housing data of Chinese cities.

The paper analyzes the determinants of apartments’ prices in Belo Horizonte, MG, Brazil, with the use of hierarchical models, spatial models and a hierarchical-spatial approach. Besides the apartments’ characteristics, such as area, age and building standard, prices were determined by local urban amenities. The hierarchical models indicated that local variables, such as urban violence, infrastructure and services, explained over 75% of prices’ remaining variability. The spatial models analyzed if, after controlling for price variability with the explanatory variables in the second level of the hierarchical models, spatial correlations still existed in price determination. The positive and significant spatial coefficient in spatial autoregressive models (SAR) indicated spatial dependency. The hierarchical-spatial approach showed that over 70% of apartment prices’ remaining variability could be explained by local variables and that the lagged urban services variable explained another 12% of this variability.

In this article, we consider variable selection of partially functional linear spatial autoregressive model with a diverging number of parameters, in which the explanatory variables include infinite dimensional predictor procedure, treated as functional data, and multiple real valued scalar variate. By combining series approximation method, two-stage least squares method and a class of non convex penalty function, we propose a variable selection method to simultaneously select significant explanatory variables in the parametric component and estimate the corresponding parameter related to spatial lag of the response variable. Under appropriate conditions, we derive the rate of convergence of the series estimator of the functional and parametric component, and show that the proposed variable selection method processes the oracle property. That is, it can estimate the zero components as exact zero with high probability, and estimate the non zero components as efficiently as if the true model was known beforehand. Simulation result show that our proposed variable selection method has better finite sample property.Notably, in the case where the correlation among the explanatory variables in the parametric component is low, the proposed variable selection method performs well. An application of the proposed variable selection method serves as a practical illustration.

In this paper, a Bayesian variable selection method for spatial autoregressive (SAR) quantile models is proposed on the basis of spike and slab prior for regression parameters. The SAR quantile models, which are more generalized than SAR models and quantile regression models, are specified by adopting the asymmetric Laplace distribution for the error term in the classical SAR models. The proposed approach could perform simultaneously robust parametric estimation and variable selection in the context of SAR quantile models. Bayesian statistical inferences are implemented by a detailed Markov chain Monte Carlo (MCMC) procedure that combines Gibbs samplers with a probability integral transformation (PIT) algorithm. In the end, empirical numerical examples including several simulation studies and a Boston housing price data analysis are employed to demonstrate the newly developed methodologies.

Heteroscedasticity is often encountered in spatial-data analysis, so a new class of heterogeneous spatial autoregressive models is introduced in this paper, where the variance parameters are allowed to depend on some explanatory variables. Here, we are interested in the problem of parameter estimation and the variable selection for both the mean and variance models. Then, a unified procedure via double-penalized quasi-maximum likelihood is proposed, to simultaneously select important variables. Under certain regular conditions, the consistency and oracle property of the resulting estimators are established. Finally, both simulation studies and a real data analysis of the Boston housing data are carried to illustrate the developed methodology.

Testing for random effects and serial correlation in spatial autoregressive models

Previous studies neglected the implicit information in the spatial pattern of the aerosol optical depth (AOD), such as spatial configuration characteristics, spatial heterogeneity and spatial dependence. The understanding of AOD is helpful in accurately estimating the environmental and climatic effects from aerosols. In this study, a method of spatial autocorrelation statistics from geostatistics was applied to investigate the spatial pattern of AOD in Hubei province in central China during the period 2003–2008. Spatial autoregressive models were used to quantize the correlations between AOD and affecting factors, such as elevation, forest coverage and population density. In addition, the difference between the standard linear regression model and the spatial models was discussed. The results were as follows: the spatial pattern of AOD in Hubei province shows significant spatial autocorrelation, indicating that AODs are clustered such that higher AODs tend to be surrounded by higher AOD neighbors, while lower AODs are surrounded by lower AOD neighbors, and the spatial autocorrelation scale of the AOD over Hubei is approximately 400 km. The high–high zone is mainly distributed in the Wuhan city circle and the Jianghan plain areas, while the low–low zones are mainly located in the middle and high mountain areas of northwest Hubei province. The overall degree and pattern of spatial autocorrelation do not change largely from 2003 to 2008, which indicates a stable spatial configuration of AOD. A significant negative spatial autocorrelation exists between AOD and elevation, which may suggest that AOD and elevation have an inverse spatial distribution, and the same applies for forest coverage. Population density and the AOD show a significant positive spatial autocorrelation, which may imply that they have similar spatial distribution, while industrial production and AOD do not show an obvious positive spatial autocorrelation. Spatial autoregressive models show better predictive ability and stability than the standard linear regression model because the spatial autocorrelation of the AOD is taken into consideration.

Is It a Curse or a Blessing to Live Near Rich Neighbors? Spatial Analysis and Spillover Effects of House Prices in Beijing

A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stability α, α ∈ (0, 2]. It is shown that when the model is stationary, the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution, which avoids the difficulty of Roknossadati and Zarepour (2010) in deriving their limiting distribution for an M-estimate. On the contrary, we show that when the model is not stationary, the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour. Furthermore, a Wald test statistic is proposed to consider the test for a linear restriction on the parameter, and it is shown that under a local alternative, the Wald statistic has a non-central chisquared distribution. Simulations and a real data example are also reported to assess the performance of the proposed method.

To present and compare socioeconomic status (SES) rankings of households using consumption and an asset-based index as two alternative measures of SES; and to compare and evaluate the performance of these two measures in multivariate analyses of the socioeconomic gradient in malaria prevalence. Data for the study come from a survey of 557 households in 25 study villages in Tanzania in 2004. Household SES was determined using consumption and an asset-based index calculated using Principal Components Analysis on a set of household variables. In multivariate analyses of malaria prevalence, we also used two other measures of disease prevalence: parasitaemia and self-report of malaria or fever in the 2 weeks before interview. Household rankings based on the two measures of SES differ substantially. In multivariate analyses, there was a statistically significant negative association between both measures of SES and parasitaemia but not between either measure of SES and self-reported malaria. Age of individual, use of a mosquito net, and wall construction were negatively and significantly associated with parasitaemia, whilst roof construction was positively associated with parasitaemia. Only age remained significant when malaria self-report was used as the measure of disease prevalence. An asset index is an effective alternative to consumption in measuring the socioeconomic gradient in malaria parasitaemia, but self-report may be an unreliable measure of malaria prevalence for this purpose.

Variable selection in high-dimensional partially linear additive models for composite quantile regression

Capturing household-level spatial influence in agricultural management using random effects regression

Adjusted quasi-maximum likelihood estimator for mixed regressive, spatial autoregressive model and its small sample bias