Abstract: This paper reviews the literature on spatial panel data models in econometrics.In recent decades,panel data models with spatial interactions have become increasingly important in empirical research,as they account for dynamic and spatial dependencies and control for unobservable heterogeneity.With panel data,we can not only have a larger sample size to improve the efficiency of the estimators,but also investigate some problems that cross-sectional data cannot handle,such as heterogeneity and state dependence across time.
This paper first introduces various spatial panel data model specifications,which are divided into two categories:static spatial panels and dynamic spatial panels.For static spatial panel data models,the regressors do not include a time-lagged term,but the disturbances can have serial correlation along with the spatial correlation.Depending on whether the individual effects are correlated with regressors,we have fixed effects models and random effects models.For spatial dynamic panel data models,we need to consider the influence of the initial period,and the length of time periods is important for asymptotic analysis.Depending on the eigenvalue structure of the dynamic process,we have stable,spatial cointegration,unit root,and explosive processes.Besides these two benchmark models,various model specifications are proposed in the literature,such as the semiparametric approach,common factors,endogenous spatial weights matrix,simultaneous equations model,and structural change.
We then introduce corresponding estimation methods in detail for the two benchmarks,including quasi-maximum likelihood estimation and generalized moment method.For the static spatial panel data models,we present likelihood approaches for the fixed and random effects models.For the fixed effects model,we can either estimate those fixed effects directly along with the regression coefficients,or transform the data to eliminate those fixed effects and then perform the estimation.The latter transformation approach can avoid the incidental parameter problem and yield consistent estimation for all parameters.For the random effects model,we do not have the incidental parameter problem and the estimators are more efficient than those from the fixed effects model.In the literature for spatial panel data,the Hausman test is proposed for the model specification,which is also applicable to a general static spatial panel model with serially and spatially correlated disturbances.For the dynamic spatial panel data models,we first present likelihood estimation for various spatial dynamic panel data depending on their eigenvalues.Even though the maximum likelihood estimator is consistent over a long time period,asymptotic bias will still invalidate the statistical inference.Thus,a bias correction procedure is recommended to eliminate the asymptotic bias,where the bias formula might take different forms depending on the stability feature of the data-generating process.We then review the GMM estimation utilizing both linear and quadratic moments.Compared with QML estimation,the GMM is computationally convenient,is valid regardless of short and long time periods,and is applicable when the spatial weight matrix is not row-normalized under the model with time effects.The GMM estimation can be as efficient as QML estimation,and more efficient if the disturbances are not normally distributed.For both QML and GMM estimation,estimation and inference might be invalid under cross-sectional heteroskedasticity.We review recent work on this issue,including adjusted QML estimation and recentered method of the moment.The adjusted QML makes the bias correction for the score vector of the likelihood function,while the recentered method of moments investigates the correlation of endogenous regressors and disturbances.
Finally,the semi-parametric estimation of spatial panel data models in recent years is reviewed,where spatial weights matrix,exogenous regressor,spatial lag,or regression coefficients could be nonparametrically specified.In conclusion,we expect that dyadic data and nonparametric model specification tests for the spatial panel data can be two promising fields in future research.