Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

One important goal of this study is to develop 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 generalized moments (GM) estimator suggested in Kelejian and Prucha (1998, 1999) for the spatial autoregressive parameter in the disturbance process. We prove the consistency of our estimator; unlike in our earlier paper we also determine its asymptotic distribution, and discuss issues of efficiency. We then define instrumental variable (IV) estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GM estimator under reasonable conditions. Much of the theory is kept general to cover a wide range ofsettings. We note the estimation theory developed by Kelejian and Prucha (1998, 1999) for GM and IV estimators and by Lee (2004) for the quasi-maximum likelihood estimator under the assumption of homoskedastic innovations does not carry over to the case of heteroskedastic innovations. The paper also provides a critical discussion of the usual specification of the parameter space.spatial dependence, heteroskedasticity, Cliff-Ord model, two-stage least squares,generalized moments estimation, asymptotics

Estimation of Spatial Regression Models with Autoregressive Errors by Two Stage Least Squares Procedures: A Serious Problem

Various two stage least squares procedures have been suggested for the estimation of the autoregressive parameter in the spatial autoregressive model of order one. These procedures are computationally convenient and so their use is "tempting". In this paper we show that these procedures are, in general, not consistent and therefore should not be used.Spatial Models, Autocorrelation, Two Stage Least Squares

Estimation of Spatial Models with Endogenous Weighting Matrices and an Application to a Demand Model for Cigarettes

Weighting matrices are typically assumed to be exogenous. However, in many cases this exogeneity assumption may not be reasonable. In these cases, typical model specifications and corresponding estimation procedures will no longer be valid. In this paper we specify a spatial panel data model which contains a spatially lagged dependent variable in terms of an endogenous weighting matrix. We suggest an estimator for the regression parameters, and demonstrate its consistency and asymptotic normality. We also suggest an estimator for the large sample variance-covariance matrix of that distribution. We then apply our results to an interstate panel data cigarette demand model which contains an endogenous weighting matrix. Among other things, our results suggest that, if properly accounted for, the bootlegging effect of buyers, or “agents” for them, crossing state borders to purchase cigarette turns out to be positive and significant

A J-test for Panel Models with Fixed Effects, Spatial and Time

In this paper we suggest a J-test in a spatial panel framework of a null model against one or more alternatives. The null model we consider has fixed effects, along with spatial and time dependence. The alternatives can have either fixed or random effects. We implement our procedure to test the specifications of a demand for cigarette model. We find that the most appropriate specification is one that contains the average price of cigarettes in neighboring states, as well as the spatial lag of the dependent variable. Along with formal large sample results, we also give small sample Monte Carlo results. Our large samples results are based on the assumption N → ∞ and T is fixed. Our Monte Carlo results suggest that our proposed J-test has good power, and proper size even for small to moderately sized samples

Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances

One important goal of this study is to develop 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 generalized moments (GM) estimator suggested in Kelejian and Prucha (1998, 1999) for the spatial autoregressive parameter in the disturbance process. We prove the consistency of our estimator; unlike in our earlier paper we also determine its asymptotic distribution, and discuss issues of efficiency. We then define instrumental variable (IV) estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GM estimator under reasonable conditions. Much of the theory is kept general to cover a wide range of settings. We note the estimation theory developed by Kelejian and Prucha (1998, 1999) for GM and IV estimators and by Lee (2004) for the quasi-maximum likelihood estimator under the assumption of homoskedastic innovations does not carry over to the case of heteroskedastic innovations. The paper also provides a critical discussion of the usual specification of the parameter space

A Generalized Spatial Two Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances

Cross sectional spatial models frequently contain a spatial lag of the dependent variable as a regressor, or a disturbance term which is spatially autoregressive. In this paper we describe a computationally simple procedure for estimating cross sectional models which contain both of these characteristics. We also give formal large sample results.Spatial Models, Autocorrelation, Two Stage Least Squares, Generalized Moments Estimator

A Spatial Cliff-Ord-type Model with Heteroskedastic Innovations: Small and Large Sample Results

In this paper we specify a linear Cliff and Ord-type spatial model. The model allows for spatial lags in the dependent variable, the exogenous variables, and disturbances. The innovations in the disturbance process are assumed to be heteroskedastic with an unknown form. We formulate a multi-step GMM/IV type estimation procedure for the parameters of the model. We then establish the limiting distribution of our suggested estimators, and give consistent estimators for their asymptotic variance covariance matrices, utilizing results given in Kelejian and Prucha (2007b). Monte Carlo results are given which suggest that the derived large sample distribution provides a good approximation to the actual small sample distribution of our estimators.

A spatial Cliff-ord-type model with heteroskedastic innovations: small and large sample results

In this paper we specify a linear Cliff and Ord-type spatial model. The model allows for spatial lags in the dependent variable, the exogenous variables, and disturbances. The innovations in the disturbance process are assumed to be heteroskedastic with an unknown form. We formulate a multi-step GMM/IV type estimation procedure for the parameters of the model. We then establish the limiting distribution of our suggested estimators, and give consistent estimators for their asymptotic variance covariance matrices, utilizing results given in Kelejian and Prucha (2007b). Monte Carlo results are given which suggest that the derived large sample distribution provides a good approximation to the actual small sample distribution of our estimators.

Heteroskedasticity of Unknown Form in Spatial Autoregressive Models with Moving Average Disturbance Term

In this study, I investigate the necessary condition for the consistency of the maximum likelihood estimator (MLE) of spatial models with a spatial moving average process in the disturbance term. I show that the MLE of spatial autoregressive and spatial moving average parameters is generally inconsistent when heteroskedasticity is not considered in the estimation. I also show that the MLE of parameters of exogenous variables is inconsistent and determine its asymptotic bias. I provide simulation results to evaluate the performance of the MLE. The simulation results indicate that the MLE imposes a substantial amount of bias on both autoregressive and moving average parameters