Estimation Of Spatial Autoregressive Models Research Articles

Estimation of spatial autoregressive models for origin–destination flows: A partial likelihood approach

We extend LeSage and Pace (2008)’s spatial autoregressive model for origin–destination flows by accommodating two-way fixed effects. A partial likelihood approach is used for estimation by applying an orthogonal transformation to remove fixed effects in the model. The quasi-maximum likelihood (QML) estimator of the partial log-likelihood function is consistent and asymptotically centered normal. Monte Carlo experiments verify this advantage in finite samples. From the U.S. migration flows, significant spatial influences are captured with smaller magnitudes than those from the model without fixed effects.

Estimation of spatial autoregressive models with covariate measurement errors

In this paper, linear spatial autoregressive (SAR) models with covariate measurement errors are studied. A three-stage least squares (3SLS) estimation method both with Berkson’s and classical types of instrumental variables is proposed and asymptotic normality of the proposed estimator using each type of instrumental variables is derived under mild conditions. Simulation studies are conducted to investigate the finite sample performance of the proposed estimator. A real data example is used to illustrate the developed method.

QML and Efficient GMM Estimation of Spatial Autoregressive Models with Dominant (Popular) Units

This article investigates QML and GMM estimation of spatial autoregressive (SAR) models in which the column sums of the spatial weights matrix might not be uniformly bounded. We develop a central limit theorem in which the number of columns with unbounded sums can be finite or infinite and the magnitude of their column sums can be if . Asymptotic distributions of QML and GMM estimators are derived under this setting, including the GMM estimators with the best linear and quadratic moments when the disturbances are not normally distributed. The Monte Carlo experiments show that these QML and GMM estimators have satisfactory finite sample performances, while cases with a column sums magnitude of O(n) might not have satisfactory performance. An empirical application with growth convergence in which the trade flow network has the feature of dominant units is provided. Supplementary materials for this article are available online.

Reduced-bias estimation of spatial autoregressive models with incompletely geocoded data

The application of spatial Cliff–Ord models requires information about spatial coordinates of statistical units to be reliable, which is usually the case in the context of areal data. With micro-geographic point-level data, however, such information is inevitably affected by locational errors, that can be generated intentionally by the data producer for privacy protection or can be due to inaccuracy of the geocoding procedures. This unfortunate circumstance can potentially limit the use of the spatial autoregressive modelling framework for the analysis of micro data, as the presence of locational errors may have a non-negligible impact on the estimates of model parameters. This contribution aims at developing a strategy to reduce the bias and produce more reliable inference for spatial models with location errors. The proposed estimation strategy models both the spatial stochastic process and the coarsening mechanism by means of a marked point process. The model is fitted through the maximisation of a doubly-marginalised likelihood function of the marked point process, which cleans out the effects of coarsening. The validity of the proposed approach is assessed by means of a Monte Carlo simulation study under different real-case scenarios, whereas it is applied to real data on house prices.

Identification and Estimation of Spatial Autoregressive Models with Common Factors

This paper considers panel data models with cross-sectional dependence arising from both spatial autocorrelation and unobserved common factors. It proposes estimation methods that employ cross-sectional averages as factor proxies, including the 2SLS, Best 2SLS, and GMM estimations. The proposed estimators are robust to unknown heteroskedasticity and serial correlation in the disturbances, unrequired to estimate the number of unknown factors, and computationally tractable. The paper establishes the asymptotic distributions of these estimators and compares their consistency and efficiency properties. Extensive Monte Carlo experiments lend support to the theoretical findings and demonstrate the satisfactory finite sample performance of the proposed estimators. The empirical section of the paper finds strong evidence of spatial dependence of real house price changes across 377 Metropolitan Statistical Areas in the US from 1975Q1 to 2014Q4.

SEMIPARAMETRIC ESTIMATION OF CENSORED SPATIAL AUTOREGRESSIVE MODELS

This study considers the estimation of spatial autoregressive models with censored dependent variables, where the spatial autocorrelation exists within the uncensored latent dependent variables. The estimator proposed in this paper is semiparametric, in the sense that the error distribution is not parametrically specified and can be heteroskedastic. Under a median restriction, we show that the proposed estimator is consistent and asymptotically normally distributed. As an empirical illustration, we investigate the determinants of the risk of assault and other violent crimes including injury in the Tokyo metropolitan area.

Least squares estimation of spatial autoregressive models for large-scale social networks

Due to the rapid development of various social networks, the spatial autoregressive (SAR) model is becoming an important tool in social network analysis. However, major bottlenecks remain in analyzing large-scale networks (e.g., Facebook has over 700 million active users), including computational scalability, estimation consistency, and proper network sampling. To address these challenges, we propose a novel least squares estimator (LSE) for analyzing large sparse networks based on the SAR model. Computationally, the LSE is linear in the network size, making it scalable to analysis of huge networks. In theory, the LSE is $\sqrt{n}$-consistent and asymptotically normal under certain regularity conditions. A new LSE-based network sampling technique is further developed, which can automatically adjust autocorrelation between sampled and unsampled units and hence guarantee valid statistical inferences. Moreover, we generalize the LSE approach for the classical SAR model to more complex networks associated with multiple sources of social interaction effect. Numerical results for simulated and real data are presented to illustrate performance of the LSE.

GMM estimation of the spatial autoregressive model in a system of interrelated networks

This paper considers efficient estimation of spatial autoregressive models in a system of interrelated networks. An example describes a market situation with several chain stores competing against each other. The strategy of a store in the chain does not only involve coordination with the other stores in the same chain, but also competition against opponent stores in other chains. To estimate the system, we extend the generalized method of moments framework based on linear and quadratic moment conditions proposed by Lee (2007) and Lin and Lee (2010). We show that under some regularity assumptions the proposed GMM estimator is consistent and asymptotically normal. We derive the best GMM estimator under normality and propose a robust GMM estimator against unknown heteroskedasticity. Monte Carlo experiments are conducted to study the finite sample performance of the GMM estimation. We also provide an empirical application of the model on the spatial competition between chain stores in the market of prescription drugs.

Estimation of spatial autoregressive models with measurement error for large data sets

Maximum likelihood (ML) estimation of spatial autoregressive models for large spatial data sets is well established by making use of the commonly sparse nature of the contiguity matrix on which spatial dependence is built. Adding a measurement error that naturally separates the spatial process from the measurement error process are not well established in the literature, however, and ML estimation of such models to large data sets is challenging. Recently a reduced rank approach was suggested which re-expresses and approximates such a model as a spatial random effects model (SRE) in order to achieve fast fitting of large data sets by fitting the corresponding SRE. In this paper we propose a fast and exact method to accomplish ML estimation and restricted ML estimation of complexity of $$O(n^{3/2})$$ operations when the contiguity matrix is based on a local neighbourhood. The methods are illustrated using the well known data set on house prices in Lucas County in Ohio.

Pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension

Pseudo maximum likelihood estimates are developed for higher-order spatial autoregressive models with increasingly many parameters, including models with spatial lags in the dependent variables both with and without a linear or nonlinear regression component, and regression models with spatial autoregressive disturbances. Consistency and asymptotic normality of the estimates are established. Monte Carlo experiments examine finite-sample behaviour.

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

ABSTRACTThis paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.

Semiparametric Estimation of Censored Spatial Autoregressive Models

This study considers the estimation of spatial autoregressive models with censored dependent variables, where the spatial autocorrelation exists within the uncensored latent dependent variables. The estimator proposed in this paper is semiparametric in the sense that the error distribution is not parametrically specified and can be heteroscedastic. Under a median restriction, we show that the proposed estimator is consistent and asymptotically normally distributed. A set of Monte Carlo simulations is conducted to evaluate the finite sample performance of the estimator. As an empirical illustration, we investigate the determinants of household burglary risk in the Tokyo metropolitan area.

Modified QML estimation of spatial autoregressive models with unknown heteroskedasticity and nonnormality

In the presence of heteroskedasticity, Lin and Lee (2010) show that the quasi-maximum likelihood (QML) estimator of the spatial autoregressive (SAR) model can be inconsistent as a ‘necessary’ condition for consistency can be violated, and thus propose robust GMM estimators for the model. In this paper, we first show that this condition may hold in certain situations and when it does the regular QML estimator can still be consistent. In cases where this condition is violated, we propose a simple modified QML estimation method robust against unknown heteroskedasticity. In both cases, asymptotic distributions of the estimators are derived, and methods for estimating robust variances are given, leading to robust inferences for the model. Extensive Monte Carlo results show that the modified QML estimator outperforms the GMM and QML estimators even when the latter is consistent. The proposed methods are then extended to the more general SARAR models.

Spatial autoregressive models with unknown heteroskedasticity: A comparison of Bayesian and robust GMM approach

Most of the estimators suggested for the estimation of spatial autoregressive models are generally inconsistent in the presence of an unknown form of heteroskedasticity in the disturbance term. The estimators formulated from the generalized method of moments (GMM) and the Bayesian Markov Chain Monte Carlo (MCMC) frameworks can be robust to unknown forms of heteroskedasticity. In this study, the finite sample properties of the robust GMM estimator are compared with the estimators based on the Bayesian MCMC approach for the spatial autoregressive models with heteroskedasticity of an unknown form. A Monte Carlo simulation study provides evaluation of the performance of the heteroskedasticity robust estimators. Our results indicate that the MLE and the Bayesian estimators impose relatively greater bias on the spatial autoregressive parameter when there is negative spatial dependence in the model. In terms of finite sample efficiency, the Bayesian estimators perform better than the robust GMM estimator. In addition, two empirical applications are provided to evaluate relative performance of heteroskedasticity robust estimators.

GMM estimation of spatial autoregressive models with moving average disturbances

In this paper, we introduce the one-step generalized method of moments (GMM) estimation methods considered in Lee (2007a) and Liu, Lee, and Bollinger (2010) to spatial models that impose a spatial moving average process for the disturbance term. First, we determine the set of best linear and quadratic moment functions for GMM estimation. Second, we show that the optimal GMM estimator (GMME) formulated from this set is the most efficient estimator within the class of GMMEs formulated from the set of linear and quadratic moment functions. Our analytical results show that the one-step GMME can be more efficient than the quasi maximum likelihood (QMLE), when the disturbance term is simply i.i.d. With an extensive Monte Carlo study, we compare its finite sample properties against the MLE, the QMLE and the estimators suggested in Fingleton (2008a).

Generalized Spatial Two Stage Least Squares Estimation of Spatial Autoregressive Models with Autoregressive Disturbances in the Presence of Endogenous Regressors and Many Instruments

This paper studies the generalized spatial two stage least squares (GS2SLS) estimation of spatial autoregressive models with autoregressive disturbances when there are endogenous regressors with many valid instruments. Using many instruments may improve the efficiency of estimators asymptotically, but the bias might be large in finite samples, making the inference inaccurate. We consider the case that the number of instruments K increases with, but at a rate slower than, the sample size, and derive the approximate mean square errors (MSE) that account for the trade-offs between the bias and variance, for both the GS2SLS estimator and a bias-corrected GS2SLS estimator. A criterion function for the optimal K selection can be based on the approximate MSEs. Monte Carlo experiments are provided to show the performance of our procedure of choosing K.

Two-Stage Least Squares Estimation of Spatial Autoregressive Models with Endogenous Regressors and Many Instruments

This paper considers the IV estimation of spatial autoregressive models with endogenous regressors in the presence of many instruments. To improve asymptotic efficiency, it may be desirable to use many valid instruments. However, finite sample properties of IV estimators can be sensitive to the number of instruments. For a spatial model with endogenous regressors, this paper derives the asymptotic distribution of the two-stage least squares (2SLS) estimator when the number of instruments grows with the sample size, and suggests a bias-correction procedure based on the leading-order many-instrument bias. The paper also gives the Nagar-type approximate mean square errors (MSEs) of the 2SLS estimator and the bias-corrected 2SLS estimator, which can be minimized to choose instruments as in Donald and Newey (2001). A limited Monte Carlo experiment is carried out to study the finite sample performance of the instrument selection procedure.

GMM Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

We consider a spatial econometric model containing a spatial lag in the dependent variable and the disturbance term with an unknown form of heteroskedasticity in innovations. We first prove that the maximum likelihood (ML) estimator for spatial autoregressive models is generally inconsistent when heteroskedasticity is not taken into account in the estimation. We show that the necessary condition for the consistency of the ML estimator of spatial autoregressive parameters depends on the structure of the spatial weight matrices. Then, we extend the robust generalized method of moment (GMM) estimation approach in Lin and Lee (2010) for the spatial model allowing for a spatial lag not only in the dependent variable but also in the disturbance term. We show the consistency of the robust GMM estimator and determine its asymptotic distribution. Finally, through a comprehensive Monte Carlo simulation, we compare finite sample properties of the robust GMM estimator with other estimators proposed in the literature.

Estimation of spatial autoregressive models with randomly missing data in the dependent variable

Summary We suggest and compare different methods for estimating spatial autoregressive models with randomly missing data in the dependent variable. Aside from the traditional expectation-maximization (EM) algorithm, a nonlinear least squares method is suggested and a generalized method of moments estimation is developed for the model. A two-stage least squares estimation with imputation is proposed as well. We analytically compare these estimation methods and find that generalized nonlinear least squares, best generalized two-stage least squares with imputation and best method of moments estimators have identical asymptotic variances. These methods are less efficient than maximum likelihood estimation implemented with the EM algorithm. When unknown heteroscedasticity exists, however, EM estimation produces inconsistent estimates. Under this situation, these methods outperform EM. We provide finite sample evidence through Monte Carlo experiments.

Estimation of spatial autoregressive models with boundary specification problem

This article examines the impact of the boundary specification problem upon the estimation of spatial autoregressive models within an instrumental variable (IV) framework. We show the usual IV estimator remains consistent and asymptotically normal, but incurs an asymptotic bias of order Op(n2/ν), where n2 is the number of boundary units and ν is the total sample size. Our results show some promise for IV based inference even in the presence of the boundary problem.