Matrix Exponential Spatial Specification Research Articles

Empirical likelihood for spatial cross-sectional data models with matrix exponential spatial specification

Empirical likelihood for spatial cross-sectional data models with matrix exponential spatial specification

Model selection and model averaging for matrix exponential spatial models

In this paper, we focus on a model specification problem in spatial econometric models when an empiricist needs to choose from a pool of candidates for the spatial weights matrix. We propose a model selection (MS) procedure for the matrix exponential spatial specification (MESS), when the true spatial weights matrix may not be in the set of candidate spatial weights matrices. We show that the selection estimator is asymptotically optimal in the sense that asymptotically it is as efficient as the infeasible estimator that uses the best candidate spatial weights matrix. The proposed selection procedure is also consistent in the sense that when the data generating process involves spatial effects, it chooses the true spatial weights matrix with probability approaching one in large samples. We also propose a model averaging (MA) estimator that compromises across a set of candidate models. We show that it is asymptotically optimal. We further flesh out how to extend the proposed selection and averaging schemes to higher order specifications and to the MESS with heteroscedasticity. Our Monte Carlo simulation results indicate that the MS and MA estimators perform well in finite samples. We also illustrate the usefulness of the proposed MS and MA schemes in a spatially augmented economic growth model.

Unified M-estimation of matrix exponential spatial dynamic panel specification

In this paper, a unified M-estimation method in Yang (2018) is extended to the matrix exponential spatial dynamic panel specification (MESDPS) with fixed effects in short panels. Similar to the STLE model which includes the spatial lag effect, the space-time effect and the spatial error effect in Yang (2018), the quasi-maximum likelihood (QML) estimation for MESDPS also has the initial condition specification problem. The initial-condition free M-estimator in this paper solves this problem and is proved to be consistent and asymptotically normal. An outer product of martingale difference (OPMD) estimator for the variance-covariance (VC) matrix of the M-estimator is also derived and proved to be consistent. The finite sample property of the M-estimator is studied through an extensive Monte Carlo study. The method is applied to US outward FDI data to show its validity.

Fast estimation of matrix exponential spatial models

The matrix exponential spatial specification (MESS) is an alternative to the spatial autoregressive-type (SAR-type) specifications with several attractive properties. The spatial dependence in the MESS-type models is formulated through a matrix exponential term, and the estimation of these models may require the computation of the matrix exponential terms many times in an estimation procedure. In the literature, it is well documented that the computation of the matrix exponential terms can pose challenges in terms of reliability, stability, accuracy, and efficiency. We propose a matrix-vector products approach based on the truncation of Taylor series expansion of the matrix exponential terms for the fast estimation of MESS-type models. We show how to efficiently implement this approach for a first-order MESS model, and provide extensive simulation evidence for its computational advantage over the default method utilized by a popular statistical software.

Quasi Maximum Likelihood for MESS Varying Coefficient Panel Data Models with Fixed Effects

The study of spatial econometrics has developed rapidly and has found wide applications in many different scientific fields, such as demography, epidemiology, regional economics, and psychology. With the deepening of research, some scholars find that there are some model specifications in spatial econometrics, such as spatial autoregressive (SAR) model and matrix exponential spatial specification (MESS), which cannot be nested within each other. Compared with the common SAR models, the MESS models have computational advantages because it eliminates the need for logarithmic determinant calculation in maximum likelihood estimation and Bayesian estimation. Meanwhile, MESS models have theoretical advantages. However, the theoretical research and application of MESS models have not been promoted vigorously. Therefore, the study of MESS model theory has practical significance. This paper studies the quasi maximum likelihood estimation for matrix exponential spatial specification (MESS) varying coefficient panel data models with fixed effects. It is shown that the estimators of model parameters and function coefficients satisfy the consistency and asymptotic normality to make a further supplement for the theoretical study of MESS model.

Spatial inequalities of COVID-19 mortality rate in relation to socioeconomic and environmental factors across England

In this study, we aimed to examine spatial inequalities of COVID-19 mortality rate in relation to spatial inequalities of socioeconomic and environmental factors across England. Specifically, we first explored spatial patterns of COVID-19 mortality rate in comparison to non-COVID-19 mortality rate. Subsequently, we established models to investigate contributions of socioeconomic and environmental factors to spatial variations of COVID-19 mortality rate across England (N = 317). Two newly developed specifications of spatial regression models were established successfully to estimate COVID-19 mortality rate (R2 = 0.49 and R2 = 0.793). The level of spatial inequalities of COVID-19 mortality is higher than that of non-COVID-19 mortality in England. Although global spatial association of COVID-19 mortality and non-COVID-19 mortality is positive, local spatial association of COVID-19 mortality and non-COVID-19 mortality is negative in some areas. Expectedly, hospital accessibility is negatively related to COVID-19 mortality rate. Percent of Asians, percent of Blacks, and unemployment rate are positively related to COVID-19 mortality rate. More importantly, relative humidity is negatively related to COVID-19 mortality rate. Moreover, among the spatial models estimated, the ‘random effects specification of eigenvector spatial filtering model’ outperforms the ‘matrix exponential spatial specification of spatial autoregressive model’.

Spatial Patterns of Childhood Obesity Prevalence in Relation to Socioeconomic Factors across England

To examine to what extent spatial inequalities in childhood obesity are attributable to spatial inequalities in socioeconomic characteristics across a country, we aimed to investigate the spatial associations of socioeconomic characteristics and childhood obesity. We first explored spatial patterns of childhood obesity prevalence, and subsequently investigated the spatial associations of socioeconomic factors and childhood obesity prevalence across England by selecting and estimating appropriate spatial regression models. As the data used are geospatial data, we used two newly developed specifications of spatial regression models to investigate the spatial association of socioeconomic factors and childhood obesity prevalence. As a result, among the two newly developed specifications of spatial regression models, the fast random effects specification of eigenvector spatial filtering (FRES-ESF) model appears to outperform the matrix exponential spatial specification of spatial autoregressive (MESS-SAR) model. Empirical results indicate that positive spatial dependence is found to exist in childhood obesity prevalence across England; and that socioeconomic factors are significantly associated with childhood obesity prevalence across England. In England, children living in areas with lower socioeconomic status are at higher risk of obesity. This study suggests effectively reducing spatial inequalities in socioeconomic status will plays a vital role in mitigating spatial inequalities in childhood obesity prevalence.

Büyük Veri Setlerinde Üstel Mekânsal Matris Tanımı Yaklaşımı: İstanbul Ofis Piyasası Analizi

Çalışmada, İstanbul Ofis Piyasası’nda ofis kira değerlerinin belirlenmesinde etkili olan faktörleri tespit etmek amacıyla mekânsal bağımlılık etkisini dikkate alan hedonik ofis kira modelinin geliştirilmesi amaçlanmıştır. Bu amaçla, mekânsal gecikme modeli ile birlikte mekânsal hata ve mekânsal Durbin modeli tahmin edilmiş ve çeşitli kriterlere göre ilişkiyi diğer mekânsal modellere göre daha iyi açıklayan modelin mekânsal gecikme modeli olduğuna karar verilmiştir. Mekânsal modeller 2018’in ilk çeyreği boyunca İstanbul’un 28 ilçesinden elde edilen 2348 ofise ait veri seti kullanılarak tahmin edilmiştir. Uygun model olarak belirlenen mekânsal gecikme modelinin tahmin sonuçlarına göre, ofis kiraları üzerindeki en etkili ilk üç değişken sırasıyla ofis binalarındaki ortalama boşluk oranı, bina tipi ve boğaz manzarasıdır. Veri setinin büyük boyutlu olması mekânsal gecikme modelinin tahmininde yanıltıcı tahminlere neden olacağından, üstel mekânsal matris modeli tahmin edilmiştir. Her iki modelin tahmin sonuçları karşılaştırıldığında katsayı tahminlerinin neredeyse aynı olduğu gözlemlenmiştir.

A likelihood ratio test for spatial model selection

This paper develops a nondegenerate likelihood-ratio test for model selection between two competitive spatial econometrics models. It generalizes the test of Vuong (1989) to models with spatial near-epoch dependent (NED) data. We do not make any structural assumption on the true model specification and allow for the cases where both or one of the two competing models are mis-specified. The test is valid whether two models are nested or non-nested. As a prerequisite of the test, we first show that quasi-maximum likelihood estimators (QMLE) of spatial econometrics models are consistent estimators of their pseudo-true values and are asymptotically normal under regularity conditions. In particular, we study spatial autoregressive models with spatial autoregressive errors (SARAR) and matrix exponential spatial specification (MESS) models. We derive the limiting null distribution of the test statistic. A spatial heteroskedastic and autoregressive consistent estimator of asymptotic variance of the test statistic under the null, which is necessary to implement the test, is constructed. Monte Carlo experiments are designed to investigate finite sample performance of QMLEs for SARAR and MESS models, as well as the size and power of the proposed test.

QML estimation of the matrix exponential spatial specification panel data model with fixed effects and heteroskedasticity

This paper studies a spatial panel data model with fixed effects and heteroskedasticity, where the spatial effects in the dependent variable and disturbances are in the form of matrix exponential spatial specification (MESS). The asymptotic properties of quasi maximum likelihood (QML) estimators with large n and finite or large T are established. We show that the QML estimator (QMLE) can be consistent and asymptotically normal under unknown heteroskedasticity when the spatial weights matrices in the two MESS processes are commutative. We provide a consistent estimator for the standard deviation of the QMLE under regularity conditions, which can be used for inference.

Evaluating the applicability of MESS (matrix exponential spatial specification) model to assess water quality using GIS technique in agricultural mountain catchment (Western Carpathian)

The formation of many sources of pollution in a short period of time is due to mountain soil erosion by water. One of the major mechanisms decisive in the intensification of such erosion is the loosening of soil material on the slope. Water quality studies show the impact of diversified spatial management and allow making the right decisions in environmental management in mountain areas with high variability of use and land cover. The research undertaken as part of the paper was carried out in order to determine the dependency between total suspended solids (TSS) and the physicochemical parameters of surface waters and the amount of soil losses in the use structure within the mountain catchment. The paper focused on the frequency of phenomena in time and the possibility of stopping the surface runoff on the slope and on the soil’s susceptibility to water erosion. The dependencies between multipoint sampling and the concentration of material washed off the slope due to precipitation were verified with a multivariate analysis. Sampling took place in hydrometric sections, and during small floods, in the waterbed cross section. Research shows that such sampling is the basis for the calculation of the transported load, reflecting the average variation in concentration. The variation in the volume of the load from the individual parts of the catchment was assessed by the spatial autoregressive model. It was found that the use of river basin areas affects water chemistry. Water reservoirs are an important ecological barrier for the migration of nitrate nitrogen (N-NO3) and phosphate phosphorus (P-PO4), which is marked by changes in the growing season. Water along the sections of the river near the quarry with a high degree of sodding showed good quality condition. Despite significant differences between measurement sampling sites, high total dissolved solid (TDS) values were found in communities adjacent to forests and meadows. However, the highest electrical conductivity (EC) and TSS concentrations were found in the interface with cultivated areas. Biogenic indices showed variation depending on the way the adjacent areas were used. GIS linked spatial variables with the formation of water pollution. The analysis of spatial autoregression pointed to the impact of arable land. Moreover, the analysis of spatial autoregression with the MESS function designated a connection between agricultural land use and nitrite nitrogen (N-NO2), EC, TSS, and dissolved oxygen (DO).Graphical abstractᅟ

Irregular N2SLS and LASSO estimation of the matrix exponential spatial specification model

In this paper, we consider estimation of the matrix exponential spatial specification model with the Durbin and endogenous regressors. We find that the nonlinear two-stage least squares (N2SLS) estimator is in general consistent and asymptotically normal. However, when the Durbin and endogenous regressors are irrelevant, the gradient vector of the N2SLS criterion function has a singular covariance matrix with probability approaching one (w.p.a.1.). Some components of the N2SLS estimator have slower rates of convergence and their asymptotic distributions are nonstandard. The distance difference and gradient test statistics, which have irregular asymptotic distributions, are derived to test for the irrelevance of the Durbin and endogenous regressors. As an alternative estimation and model selection approach, we propose the adaptive group LASSO, which penalizes the coefficients of the Durbin and endogenous explanatory variables. We show that the estimator has the oracle properties, so the true model can be selected w.p.a.1. and the estimator always has the n-rate of convergence and asymptotic normal distribution. We propose to select the tuning parameter for the adaptive group LASSO by minimizing an information criterion.

Fast and scalable variational Bayes estimation of spatial econometric models for Gaussian data

Spatial econometric models have been widely used for analyzing cross-sectional data in which spatial dependence is of primary interest. Although proven successful, Bayesian estimation via Markov chain Monte Carlo (MCMC) for spatial econometric models can be computationally demanding as the size of data and complexity of models grow. This paper proposes two variational Bayes methods that are more scalable and computationally faster in estimating general spatial autoregressive and matrix exponential spatial specification models: the hybrid mean-field variational Bayes (MFVB) method and the integrated nonfactorized variational Bayes (INFVB) method. The hybrid MFVB method assumes posterior independence and, when applicable, can yield accurate results but tends to underestimate posterior variances. In comparison, the INFVB method provides more robust results by accounting for posterior dependence and is computationally appealing due to parallelization. We demonstrate that variational Bayesian inference can be a faster and more scalable alternative to the MCMC approach for Bayesian spatial econometric modeling. The effectiveness of our proposed methods for spatial econometric models is demonstrated through simulated examples and a real-world data application.

A Matrix Exponential Spatial Panel Model with Heterogeneous Coefficients

We extend the heterogeneous coefficients spatial autoregressive panel model (HSAR) from Aquaro, Bailey, and Pesaran (2015) to the case of a heterogeneous coefficients matrix exponential spatial specification (HMESS). The HSAR is capable of producing parameter estimates for each region in the sample, that follow a spatial autoregressive process. Spatial autoregressive processes apply geometric decay of influence to higher‐order neighboring regions. The HMESS takes a similar approach as the HSAR to produce estimates for each region in the sample, but relies on a matrix exponential function to apply exponential decay to higher‐order neighbors. The MESS introduced by LeSage and Pace (2007) for the case of cross‐sectional spatial data samples has some potential computational advantages over the spatial autoregressive specification. In addition, the spatial dependence parameter in the MESS ranges from minus to plus infinity, which allows for use of normal priors assigned to this parameter in a Bayesian setting. We extend the cross‐sectional MESS to the case of a heterogeneous coefficients model, and describe Bayesian Markov Chain Monte Carlo estimation. We illustrate the HMESS model with a panel wage curve relationship using quarterly unemployment and wage rates from 261 counties centered on the Bakken shale oil region in North Dakota and Montana.

Large sample properties of the matrix exponential spatial specification with an application to FDI

This paper studies large sample properties of the matrix exponential spatial specification (MESS). We find that the quasi-maximum likelihood estimator (QMLE) for the MESS is consistent under heteroskedasticity, a property not shared by the QMLE of the SAR model. For the general model that has MESS in both the dependent variable and disturbances, labeled MESS(1,1), the QMLE can be consistent under unknown heteroskedasticity when the spatial weights matrices in the two MESS processes are commutative. We also consider the generalized method of moments estimator (GMME). In the homoskedastic case, we derive a best GMME that is as efficient as the maximum likelihood estimator under normality and can be asymptotically more efficient than the QMLE under non-normality. In the heteroskedastic case, an optimal GMME can be more efficient than the QMLE asymptotically. The QML approach for the MESS has the computational advantage over that of a SAR model. The computational simplicity carries over to MESS models with any finite order of spatial matrices. No parameter range needs to be imposed in order for the model to be stable. Results of Monte Carlo experiments for finite sample properties of the estimators are reported. Finally, the MESS(1,1) is applied to Belgium’s outward FDI data and we observe that the dominant motivation of Belgium’s outward FDI lies in finding cheaper factor inputs.

Matrix Powers in Spatial Econometric Models

This paper demonstrates several implications of the Cayley-Hamilton Theorem for specification, estimation, and identification of spatial autoregressive models. First, the minimal polynomial of the spatial weights matrix implies an upper bound on the admissible lag order in spatial autoregressive models when powers of the weights matrix are used to form higher-order spatial lags. Second, knowledge of the minimal polynomial facilitates the derivation of closed-form solutions of the marginal effects. Compared to the common practice of numerically inverting the spatial lag polynomial, the use of closed-form solutions enhances the interpretation of the structural model parameters and can be computationally attractive in large samples. Moreover, I demonstrate that the numerical inversion approach may result in biased finite-sample estimates of the marginal effects due to the sample size dependence of the spatial weights matrix even if all coefficient estimates are unbiased. The closed-form solutions can be used to obtain an exact expression of the bias and to avoid it in the first place. Third, I show that the matrix exponential spatial specification has a restricted spatial autoregressive model representation. Finally, I highlight that the order of the minimal polynomial of the spatial weights (or social interaction) matrix is related to the identification of endogenous and exogenous interaction effects in social networks.

A matrix exponential spatial specification approach to panel data models

This paper extends the matrix exponential spatial specification to panel data models. The matrix exponential spatial panel specification produces estimates and inferences comparable to those from conventional spatial panel models, but has computational advantages. We present maximum likelihood approach to the estimation of this spatial model specification and compare the results with the fixed effects spatial autoregressive panel model.

The impact of weight matrices on parameter estimation and inference: A case study of binary response using land-use data

This paper develops two new models and evaluates the impact of using different weight matrices on parameter estimates and inference in three distinct spatial specifications for discrete response. These specifications rely on a conventional, sparse, inverse-distance weight matrix for a spatial auto-regressive probit (SARP), a spatial autoregressive approach where the weight matrix includes an endogenous distance-decay parameter (SARPα), and a matrix exponential spatial specification for probit (MESSP). These are applied in a binary choice setting using both simulated data and parcel-level land-use data. Parameters of all models are estimated using Bayesian methods.
 
 In simulated tests, adding a distance-decay parameter term to the spatial weight matrix improved the quality of estimation and inference, as reflected by a lower deviance information criteriaon (DIC) value, but the added sampling loop required to estimate the distance-decay parameter substantially increased computing times. In contrast, the MESSP model’s obvious advantage is its fast computing time, thanks to elimination of a log-determinant calculation for the weight matrix. In the model tests using actual land-use data, the MESSP approach emerged as the clear winner, in terms of fit and computing times. Results from all three models offer consistent interpretation of parameter estimates, with locations farther away from the regional central business district (CBD) and closer to roadways being more prone to (mostly residential) development (as expected). Again, the MESSP model offered the greatest computing-time savings benefits, but all three specifications yielded similar marginal effects estimates, showing how a focus on the spatial interactions and net (direct plus indirect) effects across observational units is more important than a focus on slope-parameter estimates when properly analyzing spatial data.

Bayesian estimation and model selection for spatial Durbin error model with finite distributed lags

In this paper we investigate a spatial Durbin error model with finite distributed lags and consider the Bayesian MCMC estimation of the model with a smoothness prior. We study also the corresponding Bayesian model selection procedure for the spatial Durbin error model, the spatial autoregressive model and the matrix exponential spatial specification model. We derive expressions of the marginal likelihood of the three models, which greatly simplify the model selection procedure. Simulation results suggest that the Bayesian estimates of high order spatial distributed lag coefficients are more precise than the maximum likelihood estimates. When the data is generated with a general declining pattern or a unimodal pattern for lag coefficients, the spatial Durbin error model can better capture the pattern than the SAR and the MESS models in most cases. We apply the procedure to study the effect of right to work (RTW) laws on manufacturing employment.

Model selection using J-test for the spatial autoregressive model vs. the matrix exponential spatial model

We consider using the J-test procedure for the non-nested model selection problem between the spatial autoregressive (SAR) model and the matrix exponential spatial specification (MESS) model. The 2SLS and GMM methods are used to implement the J-test procedure and derive several test statistics under the GMM framework. We investigate the behavior of those J-test statistics in terms of pseudo true values. We extend the J-test procedure into the setting when error terms in the model are with unknown heteroskedasticity. Monte Carlo results suggest with strong spatial dependence the J-test statistics can have good power to distinguish the SAR and MESS models.