Spatial Weight Matrix Research Articles

Multiple environmental regulations and green technology innovation: a boundary theory perspective

Purpose Based on boundary theory, the authors try to clarify the boundaries of multiple environmental regulations. This paper aims to explore the strategic choices of multiple environmental regulations, analyze their spatial impact on green technology innovation and clarify the implementation priorities of multiple environmental regulations in various regions. Design/methodology/approach The authors construct an asymmetric response model based on the boundary theory perspective to analyze the strategic choices of different environmental regulations. In addition, dynamic spatial econometric methods are used to empirically analyze the differentiated regulatory boundaries of command-control type, market-incentive and voluntary-participation in environmental regulations and their spatial effects on green technology innovation. Findings Strict compulsory boundaries are formed in China when the command-control and market-incentive types are selected in a dynamic spatial and competitive game environment. When voluntary participation is selected, strict compulsory or loose basic boundaries are formed in each region. The command-control type is highly dependent on geographical distance, the market-incentive type is more dependent on an economic connection and the voluntary-participation type has obvious spatial spillover effects under the spatial weight matrix of geographical distance and economic connection. When Eastern China chooses voluntary participation in environmental regulation, the incentive effect on green technology innovation is the strongest. Middle China is the market-incentive type, and Western China is the command-control type. Command-control type and market-incentive type form strong complementary effects in dynamic spatial environments. Originality/value From the boundary theory perspective, this study analyzes the differentiated dynamic spatial boundaries of command-control, market-incentive and voluntary-participation in environmental regulations. Analyze the spatial impact of multiple environmental regulations on green technology innovation and clarify the implementation priorities of multiple environmental regulations in various regions.

Analysis of the degree of correlation of spatial distribution of electricity theft and exogenous variables: case study of Florianopolis, Brazil

Abstract This article presents a geospatial study case on electricity theft. The main objective is to identify the degree of correlation between exogenous variables and areas with a high density of irregular cases. Firstly, the geospatial study is carried out to asses the null hypothesis and check whether the data pattern presents clustering, for this the ANN method is applied, which ruled out the null hypothesis for the data set. Once the clustering pattern is confirmed, the spatial weight matrix is created to study spatial autocorrelation by applying Global Moran’s I and Local Moran’s I. Moran scatterplot is used to evaluate the degree of fitness, identify outliers, and local pockets of stationarity. The Local Moran index is used to determine the location of the clusters and the relationship between the points. In the data pre-processing step, spatial interpolation is implemented to the exogenous variables as a tool to better association of consumer units points and socioeconomic variables, the method utilized is IDW interpolation. The R-squared value of the spatial lag model after model tuning by feature selection was 87 % indicating that the model fit the observed data well.

A groundwater level spatiotemporal prediction model based on graph convolutional networks with a long short-term memory

ABSTRACT The performance of regional groundwater level (GWL) prediction model hinges on understanding intricate spatiotemporal correlations among monitoring wells. In this study, a graph convolutional network (GCN) with a long short-term memory (LSTM) (GCN–LSTM) model is introduced for GWL prediction utilizing data from 16 wells located in the northeastern Xiangtan City, China. This model is designed to account for both the hybrid temporal dependencies and spatial autocorrelations among wells. It consists of two parts: the spatial part employs GCNs to extract spatial characteristics from a spatial self-similarity weight matrix and an attribute self-similarity wight matrix among wells; the temporal part utilizes a LSTM module to capture the temporal patterns of GWL sequences, along with monthly precipitation and temperature data. This model dynamically predicts changes in groundwater levels, achieving higher accuracy on average compared to single-well predictions using LSTM. By incorporating both temporal dependencies and spatial autocorrelations, the GCN–LSTM model demonstrated an average improvement in goodness-of-fit of approximately 11.21% over the LSTM-based model for individual wells. Its application holds significant reference value for the sustainable utilization and development of groundwater resources in Xiangtan City.

The generalized STAR modelling with three-dimensional of spatial weight matrix in predicting the Indonesia peatland’s water level

The release rate of CO2 gas can be influenced by peatlands’ physical properties, such as water level and soil moisture, and rainfall. To anticipate the unstable condition which is when the peatland emit more carbon, we developed the Generalized Space Time Autoregressive (GSTAR) model in predicting these physical properties for the following weeks. As the innovation in modelling, the spatial weight matrix was based on three-dimensional coordinates with a modification on the height factor. The data we used are real-time data of water level on the peatlands in Pulang Pisau Regency, Central Kalimantan Province from 20 February 2021 to 18 March 2023. We then used Ordinary Kriging interpolation on the prediction results to create contour maps on different dates. There were empty data on several dates, especially from 24 March until 3 August 2022. To fill the empty data, we used linear interpolation and then we added white noise to the interpolation results. From the data, the water level has a downward trend pattern from around November to September and an upward trend pattern from October to November. Furthermore, we found that the best model for water level was GSTAR (2;0.1) with a modified matrix a=0.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a=0.1$$\\end{document} and b=1.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b=1.1$$\\end{document}. Based on the predicted water level, there is a risk of changes in the properties of the peatlands in several areas in Pulang Pisau Regency.

STAFGCN:A spatial-temporal attention-based fusion graph convolution network for pedestrian trajectory prediction

Pedestrian trajectory prediction provides crucial data support for the development of smart cities. Existing pedestrian trajectory prediction methods often overlook the different types of pedestrian interactions and the micro-level spatial-temporal relationships when handling the interaction information in spatial dimension and temporal dimension. The model employs a spatial-temporal attention-based fusion graph convolutional framework to predict future pedestrian trajectories. For the different types of local and global relationships between pedestrians, it first employs spatial-temporal attention mechanisms to capture dependencies in pedestrian sequence data, obtaining the social interactions of pedestrians in spatial contexts and the movement trends of pedestrians over time. Subsequently, a fusion graph convolutional module merges the temporal weight matrix and the spatial weight matrix into a spatial-temporal fusion feature map. Finally, a decoder section utilizes TimeStacked Convolutional Neural Networks to predict future trajectories. The final validation on the ETH and UCY datasets yielded experimental results with an Average Displacement Error(ADE) of 0.34 and an Final Displacement Error(FDE) of 0.55. The visualization results further demonstrated the rationality of the model.

Unravelling habituation for COVID-19-related information: A panel data study in Japan.

This study examines people's habituation to COVID-19-related information over almost three years. Using publicly available data from 47 Japanese prefectures, I analyse how human mobility responded to COVID-19-related information, such as the number of COVID-19-infected cases, the declaration of a state of emergency (DSE), and several doses of vaccine using an interactive effects model, which is a type of panel data regression. The results show that Japanese citizens were generally fearful and cautious during the first wave of the unknown infection. As such, a 1% week-on-week increase in the number of infected cases results in a decrease in human mobility by 1.09-percentage-point (pp) week-on-week. However, they gradually became habituated to similar infection information during the subsequent waves, which is reflected in 0.71 pp and 0.29 pp decreases in human mobility in the second and third waves. Nevertheless, the level of habituation decreased in response to the different types of the infection, such as new variants in the fourth wave, with 0.50 pp decrease. By contrast, regarding the DSE, it is more plausible to consider that human mobility responds to varying requests rather than habituate them. Whereas a rapid vaccination program could alleviate people's concerns. I also find spatial spillovers of infection information on human mobility using a spatial weight matrix included in the regression model. However, there is no evidence of DSE or vaccination spatial spillovers, likely because both are valid only in one's own prefecture. The implementation of flexible human mobility control policies by closely monitoring human mobility can prevent excessive or insufficient mobility control requests. Such a flexible policy can efficiently suppress infection spread and prevent economic activity reduction more than necessary. These implications are useful for evidence-based policymaking during future pandemics.

Bayesian Estimation of the Semiparametric Spatial Lag Model

This paper proposes a semiparametric spatial lag model and develops a Bayesian estimation method for this model. In the estimation of the model, the paper combines Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm, random walk Metropolis sampler, and Gibbs sampling techniques to sample all the parameters. The paper conducts numerical simulations to validate the proposed Bayesian estimation theory using a numerical example. The simulation results demonstrate satisfactory estimation performance of the parameter part and the fitting performance of the nonparametric function under different spatial weight matrix settings. Furthermore, the paper applies the constructed model and its estimation method to an empirical study on the relationship between economic growth and carbon emissions in China, illustrating the practical application value of the theoretical results.

Local Walsh-average regression for spatial autoregression single index varying coefficient models

. In real life, a large number of variables are spatially correlated in adjacent regions in various fields, such as finance, sociology, ecology, and geographic information systems. And the factors affecting these variables often have both linear and non linear relationships with them, how to accurately depict the relationship between them with a spatial regression model is a significant work. In order to further accurately describe the relationship between explanatory variables and response variables, we establish a spatial autoregressive single index varying coefficient model. The least squares method is the main method of model estimation but it is not robust enough when the parameter distribution is not normal. We present a new estimation method based on local Walsh average-regression. Under some general assumptions, we establish the asymptotic properties of the parametric and non parametric estimators. We further verify the accuracy and efficiency of the estimators under the condition of non normal error distribution through numerical experiments. In addition, the robustness of the proposed method is verified when the spatial weight matrix is interfered and when the bandwidth is changed within a certain range. Finally, we demonstrate the accuracy of forecasting by our proposed model compared with traditional spatial autoregressive model through real data analysis.

Spatial Nonlinear Effects of Street Vitality Constrained by Construction Intensity and Functional Diversity—A Case Study from the Streets of Shenzhen

As an important part of urban vitality, street vitality is an external manifestation of street economic prosperity and is affected by the built environment and the surrounding street vitality. However, existing research on the formation mechanism of street vitality focuses only on the built environment itself, ignoring the spatial spillover effect on street vitality. This study uses 5290 street segments in Shenzhen as examples. Utilizing geospatial and other multisource big data, this study creates spatial weight matrices at varying distances based on different living circle ranges. By combining the panel threshold model (PTM) and the spatial panel Durbin model (SPDM), this study constructs a spatial autoregressive threshold model to explore the spatial nonlinear effects of street vitality, considering various spatial weight matrices and thresholds of construction intensity and functional diversity. Our results show the following: (1) Street vitality exhibits significant spatial spillover effects, which gradually weaken as the living circle range expands (Moran indices are 0.178***, 0.160***, and 0.145*** for the 500 m, 1000 m, and 1500 m spatial weight matrices, respectively). (2) Construction intensity has a threshold, which is 0.1466 under spatial matrices of different distances. Functional diversity has two thresholds: 0.6832 and 2.2065 for the 500 m spatial weight matrix, and 0.6832 and 1.4325 for the 1000 m matrices, and 0.6832 and 1.2724 for 1500 m matrices. (3) As an international metropolis, street accessibility in Shenzhen has a significant and strong positive impact on its street vitality. This conclusion provides stakeholders with spatial patterns that influence street vitality, offering a theoretical foundation to further break down barriers to street vitality.

Enhanced Spatio-Temporal Modeling for Rainfall Forecasting: A High-Resolution Grid Analysis

Rainfall serves as a lifeline for crop cultivation in many agriculture-dependent countries including India. Being spatio-temporal data, the forecasting of rainfall becomes a more complex and tedious process. Application of conventional time series models and machine learning techniques will not be a suitable choice as they may not adequately account for the complex spatial and temporal dependencies integrated within the data. This demands some data-driven techniques that can handle the intrinsic patterns such as non-linearity, non-stationarity, and non-normality. Space–Time Autoregressive Moving Average (STARMA) models were highly known for its ability to capture both spatial and temporal dependencies, offering a comprehensive framework for analyzing complex datasets. Spatial Weight Matrix (SWM) developed by the STARMA model helps in integrating the spatial effects of the neighboring sites. The study employed a novel dataset consisting of annual rainfall measurements spanning over 50 (1970–2019) years from 119 different locations (grid of 0.25 × 0.25 degree resolution) of West Bengal, a state of India. These extensive datasets were split into testing and training groups that enable the better understanding of the rainfall patterns at a granular level. The study findings demonstrated a notable improvement in forecasting accuracy by the STARMA model that can exhibit promising implications for agricultural management and planning, particularly in regions vulnerable to climate variability.

Public debt, investment and economic growth dynamics: Do geographical proximity and spatial spillover effects matter?

This study employed an innovative spatial econometrics approach to analyze the spillover effects of foreign public debt and foreign direct investment (FDI) on regional economic growth dynamics of Eastern Africa. Specifically, we utilize Spatial Durbin Fixed Effect (SDM-FE) Model, drawing on balanced panel data from the World Development Indicators (WDI) for the period 1992–2019. This study contributes by utilizing a spatial weight matrix and investigating its interactions with economic growth and various other macro-regional factors, thereby unveiling the presence of spatial spillover effects and the impact of geographical proximity of countries within the region. Our results indicate that foreign public debt, FDI, gross fixed capital formation, human capital development, inflation, and government expenditure, significantly have spatial spillover effects on regional economic growth. Additionally, the results underscore the presence of regional growth divergence and provide further evidence of substantial regional spatial dependence in economic growth. Based on these outcomes, it is evident that foreign public debt has a negative impact on economic growth, which is consistent with the debt overhang theory and the crowding-out hypothesis. Moreover, the foreign public debt and regional growth policies should be devised considering the geographical interconnectedness between countries in the region, as the research output has revealed the impacts of growth and debt spillovers among the countries studied. Policies that prioritize these objectives would promote long-term regional economic growth and prevent the effects of excessive foreign public debt by creating a favorable macroeconomic climate.

The Impact of Green Finance and Financial Technology on Regional Green Energy Technological Innovation Based on the Dual Machine Learning and Spatial Econometric Models

Regional green energy technological innovation is an important means to alleviate economic–environmental contradictions. The purpose of this study was to explore the mechanisms of green finance, financial technology, and regional green energy technological innovation. In this study, we constructed dual machine learning models, spatial econometric models, and panel threshold effect models to investigate the effects of green finance and financial technology on regional green energy technological innovation, using panel data from 266 cities nationwide from 2009 to 2021. The research findings are as follows: (1) Both green finance and financial technology significantly promote regional green energy technological innovation. (2) Based on a spatial weight matrix embedded in economic geography, both green finance and financial technology generate positive spatial spillover effects on regional green energy technological innovation. (3) The interaction between green finance and financial technology significantly contributes to regional green energy technological innovation. Financial technology can strengthen the positive local and neighboring effects of green finance on regional green energy technological innovation. (4) Based on the threshold effect of financial technology, green finance cannot significantly promote regional green energy technological innovation when financial technology is in an underdeveloped stage. With the advancement of financial technology, green finance continues to have a positive impact on regional green energy technological innovation. Based on this analysis and our conclusions, we propose practical policy recommendations that can provide a more sustainable approach to green energy technology innovation.

HETEROSKEDASTICITY ROBUST SPECIFICATION TESTING IN SPATIAL AUTOREGRESSION

Spatial autoregressive (SAR) and related models offer flexible yet parsimonious ways to model spatial and network interactions. SAR specifications typically rely on a particular parametric functional form and an exogenous choice of the so-called spatial weight matrix with only limited guidance from theory in making these specifications. Also, the choice of a SAR model over other alternatives, such as spatial Durbin (SD) or spatial lagged X (SLX) models, is often arbitrary, raising issues of potential specification error. To address such issues, this paper develops a new specification test within the SAR framework that can detect general forms of misspecification including that of the spatial weight matrix, the functional form and the model itself. The test is robust to the presence of heteroskedasticity of unknown form in the disturbances and the approach relates to the conditional moment test framework of Bierens ([1982, Journal of Econometrics 20, 105–134], [1990, Econometrica 58, 1443–1458]). The Bierens test is shown to be inconsistent in general against spatial alternatives and the new test introduces modifications to achieve test consistency in the spatial setting. A central element is the infinite-dimensional endogeneity induced by spatial linkages. This complexity is addressed by introducing a new component to the omnibus test that captures the effects of potential spatial matrix misspecification. With this modification, the approach leads to a simple pivotal test procedure with standard critical values that is the first test in the literature to have power against misspecifications in the spatial linkages. We derive the asymptotic distribution of the test under the null hypothesis of correct SAR specification and prove consistency. A Monte Carlo study is conducted to study its finite sample performance. An empirical illustration on the performance of the test in modeling tax competition in Finland is included.

Unveiling land use-carbon Nexus: Spatial matrix-enhanced neural network for predicting commercial and residential carbon emissions

Carbon emissions play a pivotal role in driving global warming, with commercial and residential sectors ranking as the third-largest source after industrial and transportation sectors. Consequently, establishing a framework for spatial characterizing and predicting carbon emissions from these sectors is vital for effective carbon reduction planning. This study proposes an optimized Backpropagation Neural Network, incorporating a Spatial Weight Matrix, to model the relationship between land use and carbon emissions, taking spatial effects into account. Additionally, an optimized Random Forest model with variable search step lengths is developed for large-scale and multi-class land use prediction. Beijing, China serves as the study site, yielding the following findings: (1) The optimized Backpropagation Neural Network outperforms traditional models, achieving goodness of fit of 0.999 and 0.997 in training and testing datasets, respectively. (2) The optimized Random Forest model, while reducing prediction precision for ecological areas (Root Mean Square Error increases by 4.12 %), enhances overall model performance by 73.95 % in a single iteration. (3) By 2025, commercial and residential carbon emissions in Beijing are projected to reach 8.91 million tons. (4) The study characterizes the spatial patterns of commercial and residential carbon emissions at a 1 km resolution for the next 15 years, revealing statistical features. (5) Expansion of mixed commercial and residential land relies on service-oriented development, positively impacting regional carbon emission intensity control.

Integrated deviance information criterion for spatial autoregressive models with heteroskedasticity

In this study, we introduce the integrated deviance information criterion (DIC) for nested and non-nested model selection problems in heteroskedastic spatial autoregressive models. In a Bayesian estimation setting, we assume that the idiosyncratic error terms of our spatial autoregressive model have a scale mixture of normal distributions, where the scale mixture variables are latent variables that induce heteroskedasticity. We first derive the integrated likelihood function by analytically integrating out the scale mixture variables from the complete-data likelihood function. We then use the integrated likelihood function to formulate the integrated DIC measure. We investigate the finite sample performance of the integrated DIC in selecting the true model in a simulation study. The simulation results show that the integrated DIC performs satisfactorily and can be useful for selecting the correct model in specification search exercises. Finally, in a spatially augmented economic growth model, we use the integrated DIC to choose the spatial weights matrix that leads to better predictive accuracy.

Spatial Weight Matrix Comparison of SAR-X Model using Casetti Approach

The Spatial Autoregressive Exogenous (SAR-X) model with the Casetti approach is used to describe the influence of location and exogenous variables in the description and prediction of spatial observations, namely, people's habits and behavior towards culture in Java Island. The SAR-X model with the Casetti approach is characterized by a spatial weight matrix that describes the coordinates of the region at each location. The spatial weight matrix is determined outside the model. This study examines the spatial weight matrix determined based on rook contiguity, bishop contiguity, queen contiguity, inverse distance and inverse distance squared, and compares the application of the spatial weight matrix to the SAR-X model with the Casetti approach for the description and prediction of people's habits and behavior towards culture in Java Island. The description and prediction results obtained are measured using the Root Mean Square Error (RMSE) value. The results of data processing show that the best spatial weight matrix in the SAR-X model with the Casetti approach to community habits and behavior in Java Island is the inverse distance squared spatial weight matrix, supported by the calculation of the minimum RMSE value and the coefficient of determination above 60%.

Network Centrality and Road Freight Efficiency: A Spatial Econometric Analysis of South Korea’s Transport Regions

Purpose - The purpose of this research is to investigate the road freight efficiency in 15 regions of South Korea using a super-efficiency SBM model and to understand the role of network centrality in influencing this efficiency. It aims to analyze how in-degree and out-degree centrality within the road freight network impacts the efficiency of road freight and how these effects manifest spatially across different regions. Design/Methodology – The study applied a super-efficiency Slacks-Based Measure (SBM) model to evaluate the efficiency of road freight across various regions. It incorporated two indices, indegree and out-degree centrality, to measure the network centrality of regions within the road freight network. Furthermore, it utilized the Spatial Durbin Model (SDM) to decompose and analyze the direct and indirect effects of network centrality on road freight efficiency, considering different spatial weight matrix. Findings – The study found that road freight efficiency in a region positively correlates with its centrality within the road transport network, indicating regions that are central to the network tend to utilize their inputs more effectively and improve transport efficiency. It also discovered positive spillover effects of network centrality, suggesting that a region’s central role in the network benefits not only itself but also adjacent regions. Originality/value - The research is valuable for its application of the super-efficiency SBM and SDM models to the context of road transport networks in South Korea, a relatively underexplored area in transportation studies. It contributes original insights into the spatial dynamics of road freight efficiency, emphasizing the role of network centrality and its variable effects across different regions.

Nonlinear Spatial Dynamic Panel Data Models with Endogenous Dominant Units: An Application to Share Data

This article develops a nonlinear spatial dynamic panel data model with one particularly interesting application to a structural interaction model for share data. To account for effects from dominant (popular) units, the spatial weights matrix in our model can allow for unbounded column sums. To account for heterogeneity, our model includes two-way fixed effects and heteroscedastic errors. We further consider the potential endogeneity of the spatial weight matrix constructed from socioeconomic distance. We investigate the quasi-maximum likelihood estimator (QMLE), generalized methods of moments estimator (GMME), and root estimator (RTE), and establish their consistency and asymptotic normality based on the near epoch dependence (NED) framework. The RTE can derive a relatively computationally simple and closed-form solution without evaluating the QMLE’s Jacobian matrix as well as the iterations by GMME. We consider both n , T → ∞ , and the strength of the dominant units is equal to 1 when T → ∞ . For the purpose of empirical analysis, we derive the marginal effects and their limiting distributions based on the proposed estimators. In an empirical application, we apply our model to China’s prefecture city-level data, revealing significant spillover effects of the tertiary industry share. These findings suggest that the development of the tertiary sector in large cities can foster its growth in small cities.

Spatial difference analysis of water and sanitation in China's counties based on a spatial econometric model

Access to safe drinking water and sanitation are preconditions for achieving the Sustainable Development Goals, which is beneficial to promote new urbanization construction accurately. However, the urbanization rate is still less than 25%. Comprehensive governance capacity and carrying capacity are still relatively weak. This study seeks to increase knowledge of spatial differences in water and sanitation in China's counties, which can reveal the living environment and health conditions of residents in different areas. Furthermore, studying the spatial distribution of water and sanitation facilities in different counties can help understand the development differences between urban and rural areas and narrow the gap between urban and rural. Therefore, four indicators of water supply, sewage disposal, garbage disposal, and public toilets as proxy indicators of water and sanitation were selected, and Moran's I was used to test the spatial correlation between county-level sanitation and economic conditions, government fiscal behavior, and social and geographical factors. This study exploited the inverse distance spatial weight matrix to implement the spatial spillover effect of water and sanitation according to the division of the east, middle, and west and four aspects of water supply, sewage disposal, garbage disposal, and public toilets in terms of per capita occupancy. We observed that there was no significant correlation between population density and county-level water, sanitation, and hygiene enhancement. Economic growth can promote the improvement of water and sanitation to a certain extent. The comparison results indicated that the level of urbanization and education had a positive spatial spillover effect on water and sanitation. The water and sanitation of the demonstration county was characterized by siphon, which occupied the water and sanitation development space of neighboring counties. The results of this study are of great significance to improve county-level water and sanitation facilities and narrow the gap between urban and rural.

The distance decay effect and spatial reach of spillovers

This paper quantifies and graphically illustrates the distance decay effect and spatial reach of spillover effects derived from a spatial Durbin (SD) model with parameterized spatial weight matrices. Building on attributes of the concept of spatial autocorrelation developed by Arthur Getis, we adopt a distance-based negative exponential spatial weight matrix and parameterize it by a decay parameter that is different for each spatial lag in this model, both of the regressand and of all regressors. The quantification and illustration are applied to the spatially augmented neoclassical growth framework, which we estimate using data for 266 NUTS-2 regions in the EU over the period 2000–2018. We find distance decay parameters ranging from 0.233 to 2.224 and spatial reaches ranging from 700 to more than 1500 km for the different growth determinants in this model. These wide ranges highlight the restrictiveness of the conventional SD model based on one common spatial weight matrix for all spatial lags.