asymptotic distributions of quasi maximum likelihood estimators for spatial autoregressive models this paper investigates asymptotic properties of the maximum likelihood estimator and the quasi maximum likelihood estimator for the spatial autoregressive model the rates of convergence of those estimators may depend on some general features of the spatial weights matrix of the model it is important to make the distinction with dif ferent spatial scenarios under the scenario that each unit will be influenced by only a few neighboring units the estimators may have n rate of convergence and be asymp totically normal when each unit can be influenced by many neighbors irregularity of the information matrix may occur and various components of the estimators may have different rates of convergence

adjusted qmle for the spatial autoregressive parameter abstract one simple and often very effective way to attenuate the impact of nuisance parameters on maximum likelihood estimation of a parameter of interest is to recenter the profile score for that parameter we apply this general principle to the quasi maximum likelihood estimator qmle of the autoregressive parameter in a spatial autoregression the resulting estimator for has better finite sample properties compared to the qmle for especially in the presence of a large number of covariates it can also solve the incidental parameter problem that arises for example in social interaction models with network fixed effects however spatial autoregressions present specific challenges for this type of adjustment because recentering the profile score may cause the adjusted estimate to be outside the usual parameter space for conditions for this to happen are given and implications are discussed for inference we propose confidence intervals based on a lugannani rice approximation to the distribution of the adjusted qmle of based on our simulations the coverage properties of these intervals are excellent even in models with a large number of covariates

central limit theorems for quadratic forms with time domain conditions we establish the central limit theorem for quadratic forms t s 1 n b t s p m n x t x s of the bivariate appell polynomials p m n x t x s under time domain conditions these conditions relate the weights b t and the covariances of the sequences p m n x t x s and x t the time domain approach together with the spectral domain approach developed earlier yields a general set of conditions for central limit theorems

finite sample properties of maximum likelihood estimator in spatial models we investigate the finite sample properties of the maximum likelihood estimator for the spatial autoregressive model a stochastic expansion of the score function is used to develop the second order bias and mean squared error of the maximum likelihood estimator we show that the results can be expressed in terms of the expectations of cross products of quadratic forms or ratios of quadratic forms in a normal vector which can be evaluated using the top order invariant polynomial our numerical calculations demonstrate that the second order behaviors of the maximum likelihood estimator depend on the degree of sparseness of the weights matrix

on the bootstrap for moran s i test for spatial dependence this paper is concerned with the use of the bootstrap for statistics in spatial econometric models with a focus on the test statistic for moran s i test for spatial dependence we show that for many statistics in spatial econometric models the bootstrap can be studied based on linear quadratic lq forms of disturbances by proving the uniform convergence of the cumulative distribution function for lq forms to that of a normal distribution we show that the bootstrap is generally consistent for test statistics that can be approximated by lq forms including moran s i possible asymptotic refinements of the bootstrap are most commonly studied using edgeworth expansions for spatial econometric models we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of lq forms when the disturbances are normal we prove the existence of the usual edgeworth expansions for lq forms when the disturbances are not normal we establish an asymptotic expansion of lq forms based on martingales these results are applied to show the second order correctness of the bootstrap for moran s i test

diagnostic tests for homoskedasticity in spatial cross sectional or panel models abstract we propose an adjusted quasi score aqs method for constructing tests for homoskedasticity in spatial econometric models we first obtain an aqs function by adjusting the score type function from the given model to achieve unbiasedness and then develop an outer product of martingale difference opmd estimate of its variance in standard problems where a genuine quasi score vector is available the aqs opmd method leads to finite sample improved tests over the usual methods more importantly in non standard problems where a genuine quasi score is not available and the usual methods fail the proposed aqs opmd method provides feasible solutions the aqs tests are formally derived and asymptotic properties examined for three representative models spatial cross sectional static and dynamic panel models monte carlo results show that the proposed aqs tests have good finite sample properties

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 model 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

panel data models with spatially correlated error components abstract in this paper we consider a panel data model with error components that are both spatially and time wise correlated the model blends specifications typically considered in the spatial literature with those considered in the error components literature we introduce generalizations of the generalized moments estimators suggested in kelejian and prucha 1999 a generalized moments estimator for the autoregressive parameter in a spatial model international economic review 40 509 533 for estimating the spatial autoregressive parameter and the variance components of the disturbance process we then use those estimators to define a feasible generalized least squares procedure for the regression parameters we give formal large sample results for the proposed estimators we emphasize that our estimators remain computationally feasible even in large samples

gmm estimation of spatial autoregressive models with unknown heteroskedasticity abstract in the presence of heteroskedastic disturbances the mle for the sar models without taking into account the heteroskedasticity is generally inconsistent the 2sls estimates can have large variances and biases for cases where regressors do not have strong effects in contrast gmm estimators obtained from certain moment conditions can be robust asymptotically valid inferences can be drawn with consistently estimated covariance matrices efficiency can be improved by constructing the optimal weighted estimation the approaches are applied to the study of county teenage pregnancy rates the empirical results show a strong spatial convergence among county teenage pregnancy rates

exact and higher order properties of the mle in spatial autoregressive models with applications to inference abstract the quasi maximum likelihood estimator for the autoregressive parameter in a spatial autoregression usually cannot be written explicitly in terms of the data a rigorous analysis of the first order asymptotic properties of the estimator under some assumptions on the evolution of the spatial design matrix is available in lee 2004 but very little is known about its exact or higher order properties in this paper we first show that the exact cumulative distribution function of the estimator can under mild assumptions be written in terms of that of a particular quadratic form simple examples are used to illustrate important exact properties of the estimator that follow from this representation in general models a complete exact analysis is not possible but a higher order saddlepoint approximation is made available by the main result we use this approximation to construct confidence intervals for the autoregressive parameter coverage properties of the proposed confidence intervals are studied by monte carlo simulation and are found to be excellent in a variety of circumstances

estimation of fixed effects panel regression models with separable and nonseparable space time filters abstract this paper considers a quasi maximum likelihood estimation for a linear panel data model with time and individual fixed effects where the disturbances have dynamic and spatial correlations which might be spatially stable or unstable we first consider both separable and nonseparable space time filters for the stable model the separable space time filter is subject to a parametric restriction which results in relative computational simplicity in contrast to the spatial econometrics literature we expose economic restrictions imposed by the separable space time filter model and explore computational tractability of the nonseparable filter model throughout this paper the effect of initial observations is taken into account which results in an exact likelihood function for estimation this is important when the span of time periods is short we then investigate spatial unstable cases where we propose to apply a spatial differencing to all variables in the regression equation as a data transformation which may eliminate unstable or explosive spatial components in order to achieve a robust estimator for estimates of the parameters in both the regression part and the disturbance process they are n t consistent and asymptotically centered normal regardless of whether t is large or not and whether the process is stable or not

a general method for third order bias and variance corrections on a nonlinear estimator motivated by a recent study of bao and ullah 2007a on finite sample properties of mle in the pure sar spatial autoregressive model a general method for third order bias and variance corrections on a nonlinear estimator is proposed based on stochastic expansion and bootstrap working with concentrated estimating equation simplifies greatly the high order expansions for bias and variance a simple bootstrap procedure overcomes a major difficulty in analytically evaluating expectations of various quantities in the expansions the method is then studied in detail using a more general sar model with its effectiveness in correcting bias and improving inference fully demonstrated by extensive monte carlo experiments compared with the analytical approach the proposed approach is much simpler and has a much wider applicability the validity of the bootstrap procedure is formally established the proposed method is then extended to the case of more than one nonlinear estimator

on the asymptotic distribution of the moran i test statistic with applications by far the most popular test for spatial correlation is the one based on moran s 1950 i test statistic despite this the available results in the literature concerning the large sample distribution of this statistic are limited and have been derived under assumptions that do not cover many applications of interest in this paper we 3rst give a general result concerning the large sample distribution of moran i type test statistics we then apply this result to derive the large sample distribution of the moran i test statistic for a variety of important models in order to establishth ese results we also give a new central limit theorem for linear quadratic forms 2001 elsevier science s a all rights reserved jel classi cation c12 c21

a likelihood ratio test for spatial model selection abstract 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

profile quasi maximum likelihood estimation of partially linear spatial autoregressive models abstract we propose profile quasi maximum likelihood estimation of spatial autoregressive models that are partially linear the rate of convergence of the spatial parameter estimator depends on some general features of the spatial weight matrix of the model the estimators of other finite dimensional parameters in the model have the regular n rate of convergence and the estimator of the nonparametric component is consistent but with different restrictions on the choice of bandwidth parameter associated with different natures of the spatial weights monte carlo simulations verify our theory and indicate that our estimators perform reasonably well in finite samples

the continued fraction algorithm and regulator for quadratic function fields of characteristic 2 the continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied recently these ideas have even been used to produce cryptosystems much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic we will explore these ideas and show that the situation is very similar to the odd characteristic case this exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2

estimation of simultaneous systems of spatially interrelated cross sectional equations in this paper we consider a simultaneous system of spatially interrelated cross sectional equations our speci cation incorporates spatial lags in the endogenous and exogenous variables in modelling the disturbance process we allow for both spatial correlation as well as correlation across equations the data set is taken to be a single cross section of observations the model may be viewed as an extension of the widely used single equation cli3 ord model we suggest computationally simple limited and full information instrumental variable estimators for the parameters of the system and give formal large sample results

estimation of spatial autoregressive panel data models with fixed effects this paper establishes asymptotic properties of quasi maximum likelihood estimators for sar panel data models with fixed effects and sar disturbances a direct approach is to estimate all the parameters including the fixed effects because of the incidental parameter problem some parameter estimators may be inconsistent or their distributions are not properly centered we propose an alternative estimation method based on transformation which yields consistent estimators with properly centered distributions for the model with individual effects only the direct approach does not yield a consistent estimator of the variance parameter unless t is large but the estimators for other common parameters are the same as those of the transformation approach we also consider the estimation of the model with both individual and time effects

on the stability of a classical second order method for solving a class of two point boundary value problems abstract we consider the stability of a classical second order method for the class of two point boundary value problems y 4 f x y 0 y 0 a1 y 1 b1 y 0 a2 y 1 b2 a1 a2 b1 and b2 being constants it is shown that for sufficiently small h this well known method is stable for all f y satisfying 4

inference with dependent data using cluster covariance estimators this paper presents a novel way to conduct inference using dependent data in time series spatial and panel data applications our method involves constructing t and wald statistics utilizing a cluster covariance matrix estimator cce we then use an approximation that takes the number of clusters groups as xed and the number of observations per group to be large and calcu late limiting distributions of the t and wald statistics this approximation is analogous to xed b asymptotics of kiefer and vogelsang 2002 2005 kv for heteroskedasticity and autocorrelation consistent inference but in our case yields standard t and f distributions where the number of groups essentially plays the role of sample size we provide simulation evidence that demonstrates our procedure outperforms conventional inference procedures and performs comparably to kv

nonparametric spatial regression under near epoch dependence abstract this paper establishes asymptotic normality and uniform consistency with convergence rates of the local linear estimator for spatial near epoch dependent ned processes the class of the ned spatial processes covers important spatial processes including nonlinear autoregressive and infinite moving average random fields which generally do not satisfy mixing conditions apart from accommodating a larger class of dependent processes the proposed asymptotic theory allows for triangular arrays of heterogeneous random fields located on unevenly spaced lattices and sampled over regions of arbitrary configuration all these features make the results applicable in a wide range of empirical settings

partial maximum likelihood estimation of spatial probit models this paper analyzes spatial probit models for cross sectional dependent data in a binary choice context observations are divided by pairwise groups and bivariate normal distributions are specified within each group partial maximum likelihood estimators are introduced and they are shown to be consistent and asymptotically normal under some regularity conditions consistent covariance matrix estimators are also provided estimates of average partial effects can also be obtained once we characterize the conditional distribution of the latent error finally a simulation study shows the advantages of our new estimation procedure in this setting our proposed partial maximum likelihood estimators are shown to be more efficient than the generalized method of moments counterparts

non nested testing of spatial correlation we develop non nested tests in a general spatial spatio temporal or panel data context the spatial aspect can be interpreted quite generally in either a geographical sense or employing notions of economic distance or when parametric modelling arises in part from a common factor or other structure in the former case observations may be regularly spaced across one or more dimensions as is typical with much spatio temporal data or irregularly spaced across all dimensions both isotropic models and non isotropic models can be considered and a wide variety of correlation structures in the second case models involving spatial weight matrices are covered such as spatial autoregressive models the setting is sufficiently general to potentially cover other parametric structures such as certain factor models and vector valued observations and here our preliminary asymptotic theory for parameter estimates is of some independent value the test statistic is based on a gaussian pseudo likelihood ratio and is shown to have an asymptotic standard normal distribution under the null hypothesis that one of the two models is correct this limit theory rests strongly on a central limit theorem for the gaussian pseudo maximum likelihood parameter estimates a small monte carlo study of finite sample performance is included

identification and qml estimation of multivariate and simultaneous equations spatial autoregressive models abstract this paper investigates a simultaneous equations spatial autoregressive model which incorporates simultaneity effects own variable spatial lags and cross variable spatial lags as explanatory variables and allows for correlation between disturbances across equations in exposition we also discuss a multivariate spatial autoregressive model that can be treated as a reduced form of the simultaneous equations model we study parameter spaces parameter identification asymptotic properties of the quasi maximum likelihood estimation and computational issues monte carlo experiments illustrate the advantages of the qml broader applicability and efficiency compared to instrumental variables based estimation methods in the existing literature

gmm estimation of social interaction models with centrality this paper considers the specification and estimation of social interaction models with network structures and the presence of endogenous contextual correlated and group fixed effects when the network structure in a group is captured by a graph in which the degrees of nodes are not all equal the different positions of group members as measured by the bonacich 1987 centrality provide additional information for identification and estimation in this case the bonacich centrality measure for each group can be used as an instrument for the endogenous social effect but the number of such instruments grows with the number of groups we consider the 2sls and gmm estimation for the model the proposed estimators are asymptotically efficient respectively within the class of iv estimators and the class of gmm estimators based on linear and quadratic moments when the sample size grows fast enough relative to the number of instruments

unified m estimation of fixed effects spatial dynamic models with short panels abstract it is well known that quasi maximum likelihood qml estimation of dynamic panel data dpd models with short panels depends on the assumptions on the initial values and a wrong treatment of them will result in inconsistency and serious bias the same issues apply to spatial dpd sdpd models with short panels in this paper a unified m estimation method is proposed for estimating the fixed effects sdpd models containing three major types of spatial effects namely spatial lag spatial error and space time lag the method is free from the specification of the distribution of the initial observations and robust against nonnormality of the errors consistency and asymptotic normality of the proposed m estimator are established a martingale difference representation of the underlying estimating functions is developed which leads to an initial condition free estimate of the variance of the m estimators monte carlo results show that the proposed methods have excellent finite sample performance

specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances this study develops a methodology of inference for a widely used cliff ord type spatial model containing spatial lags in the dependent variable exogenous variables and the disturbance terms while allowing for unknown heteroskedasticity in the innovations we first generalize the gmm estimator suggested in kelejian and prucha 1998 1999 for the spatial autoregressive parameter in the disturbance process we also define iv estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the gmm estimator much of the theory is kept general to cover a wide range of settings

an efficient gmm estimator of spatial autoregressive models in this paper we consider gmm estimation of the regression and mrsar models with sar disturbances we derive the best gmm estimator within the class of gmm estimators based on linear and quadratic moment conditions the best gmm estimator has the merit of computational simplicity and asymptotic efficiency it is asymptotically as efficient as the ml estimator under normality and asymptotically more efficient than the gaussian qml estimator otherwise monte carlo studies show that with moderate sized samples the best gmm estimator has its biggest advantage when the disturbances are asymmetrically distributed when the diagonal elements of the spatial weights matrix have enough variation incorporating kurtosis of the disturbances in the moment functions will also be helpful

gmm and 2sls estimation of mixed regressive spatial autoregressive models the gmm method and the classical 2sls method are considered for the estimation of mixed regressive spatial autoregressive models these methods have computational advantage over the conventional maximum likelihood method the proposed gmm estimators are shown to be consistent and asymptotically normal within certain classes of gmm estimators best ones are derived the proposed gmm estimators improve upon the 2sls estimators and are applicable even if all regressors are irrelevant a best gmm estimator may have the same limiting distribution as the ml estimator with normal disturbances

a central limit theorem for endogenous locations and complex spatial interactions we provide a new central limit theorem clt for spatial processes under weak conditions that are plausible for many economic applications in which location is endogenous in particular our clt is designed for problems that have some but not necessarily all of the following features i agents choose the locations of observations to maximize profits welfare or some other objective ii the objects that are chosen e g stores or brands interact with one another for example they can be substitutes or complements iii interaction can be complex in particular interaction between i and j need not depend only on the distance between the locations of i and j but can also depend on distance to or location of other observations k or possibly on the number of other such observations

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 with cross sectional dependence this paper presents a spatial model of dependence among agents using a metric of economic distance measurements of this economic distance provide cross sectional data with a structure similar to that provided by the time index in time series data generalized method of moments estimators using such dependent data are shown to be consistent and asymptotically normal this paper presents a class of non parametric positive semi definite covariance matrix estimators that allow for general forms of dependence characterized by economic distance these covariance matrix estimators are shown to remain consistent when economic distances are not precisely observed

autoregressive spatial spectral estimates abstract nonparametric spectral density estimates find many uses in econometrics for stationary random fields on a regular spatial lattice we propose an autoregressive nonparametric spectral density estimate that is guaranteed positive even when suitable edge effect correction is employed and is simple to compute using least squares our estimate is based on truncating a true half plane infinite autoregressive representation while also allowing the truncation length to diverge in all dimensions to avoid the potential bias due to truncation at a fixed lag length uniform consistency of the proposed estimate is established and new criteria for order selection are also suggested and studied in practical settings the asymptotic distribution of the estimate is shown to be zero mean normal and independent at fixed distinct frequencies mirroring the behaviour for time series a small monte carlo experiment examines finite sample performance technically the key to the results is the covariance structure of stationary random fields defined on regularly spaced lattices we show the covariance matrix to satisfy a generalization of the toeplitz property familiar from time series analysis

inference on higher order spatial autoregressive models with increasingly many parameters this paper develops consistency and asymptotic normality of parameter estimates for a higher order spatial autoregressive model whose order and number of regressors are allowed to approach infinity slowly with sample size both least squares and instrumental variables estimates are examined and the permissible rate of growth of the dimension of the parameter space relative to sample size is studied besides allowing the number of parameters to increase with the data this has the advantage of accommodating some asymptotic regimes that are suggested by certain spatial settings several of which are discussed a small empirical example is also included and a monte carlo study analyses various implications of the theory in finite samples

refinements in maximum likelihood inference on spatial autocorrelation in panel data in a panel data model with fixed effects possible cross sectional dependence is investigated in a spatial autoregressive setting an edgeworth expansion is developed for the maximum likelihood estimate of the spatial correlation coefficient the expansion is used to develop more accurate interval estimates for the coefficient and tests for cross sectional independence that have better size properties than corresponding rules of statistical inference based on first order asymptotic theory comparisons of finite sample performance are carried out using monte carlo simulations

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

large panels with common factors and spatial correlation this paper considers the statistical analysis of large panel data sets where even after condi tioning on common observed eects the cross section units might remain dependently distrib uted this could arise when the cross section units are subject to unobserved common eects and or if there are spill over eects due to spatial or other forms of local dependencies the paper provides an overview of the literature on cross section dependence introduces the con cepts of time speci c weak and strong cross section dependence and shows that the commonly used spatial models are examples of weak cross section dependence it is then established that the common correlated eects cce estimator of panel data model with a multifactor error structure recently advanced by pesaran 2006 continues to provide consistent estimates of the slope coe cient even in the presence of spatial error processes small sample properties of the cce estimator under various patterns of cross section dependence including spatial forms are investigated by monte carlo experiments results show that the cce approach works well in the presence of weak and or strong cross sectionally correlated errors we also explore the role of certain characteristics of spatial processes in determining the performance of cce estimators such as the form and intensity of spatial dependence and the sparseness of the spatial weight

the bootstrap applied to power spectral density function estimation summary a shortcoming in the use of autoregressive spectral estimation has been the lack of methods for placing confidence intervals or bands on estimated spectral density the bootstrap method is applied to obtain nonparametric confidence bands and intervals it is the aim of this paper to indicate how this can be done and to report on the outcome of some monte carlo studies of the properties of the resulting procedures some asymptotic theory concerning the properties of the bootstrap confidence band is also discussed