Higher-order spatial econometric models that include more than one weights matrix have seen increasing use in the spatial econometrics literature. There are two distinct issues related to the specification of these extended models. The first issue is what form the higher-order spatial econometric model takes, i.e. higher-order polynomials in the spatial weights matrices vs. higher-order spatial autoregressive processes. The second issue relates to the parameter space in such models and how this can affect the choice of model specification, estimation, and inference. We outline a procedure that is simple both mathematically and computationally for finding the stationary region for spatial econometric models with up to K weights matrices for higher-order spatial autoregressive processes. We also compare and contrast this approach with the parameter space for models that incorporate higher-order polynomials in the spatial weights matrices. Regardless of the model utilized in empirical practice, ignoring the relevant parameter region can lead to incorrect inferences regarding both the nature of the spatial autocorrelation process and the effects of changes in covariates on the dependent variable.
Higher-order spatial econometric models that include more than one weights matrix have seen increasing use in the spatial econometrics literature, e.g. Brandsma and Ketellapper, 1979, Sherrell, 1990, Hepple, 1995, Bell and Bocksteal, 2000, Bordignon et al., 2003, Lacombe, 2004, Allers and Elhorst, 2005, McMillen et al., 2007, Ward and Gleditsch, 2008, Dall'Erba et al., 2008, Elhorst and Fréret, 2009, Lee and Liu, 2010 and Badinger and Egger, 2011. These so-called higher order spatial econometric models allow for a richer dependence structure that is not capable of being captured in a standard single weights matrix spatial econometric model framework. Generally, there are two ways in which the dependence structure in these higher-order spatial econometric models can be incorporated. The first, which has seen the most application in applied studies, is a simple extension of the single-W spatial autoregressive model to the case of multiple-W weights matrix models, i.e. the higher-order spatial autoregressive model case. The second form, which has seen limited application in applied research, is the incorporation of higher-order polynomials in the spatial weights matrices. However, issues related to the appropriate parameter space in these multiple spatial weights matrix models, regardless of the form they take, have been either ignored or erroneously assumed to be simple extensions of the single weights matrix case. This may be due to mathematical and computational difficulties in determining the parameter space or to the presumption that certain forms of spatial autocorrelation (e.g. negative spatial autocorrelation) are unlikely to occur.
In this paper, we outline a simple procedure for finding the stationary region for models with multiple spatial weights matrices (i.e. K ≥ 2) for the higher-order spatial autoregressive model case as well as the higher-order polynomials in the spatial weights matrices case. Finding the appropriate admissible parameter space is important for at least two reasons. First, recent advances in the literature have shown an increased interest in both the theoretical and empirical aspects of negative spatial autocorrelation, e.g. Griffith and Arbia (2010). In the present paper, we will show that negative spatial autocorrelation may also occur as a result of choosing two spatial weights matrices that partly overlap. Second, obtaining proper parameter estimates, which are based to some extent on identification of the proper parameter space, is required for making correct inferences regarding the effect of changes in independent variables on the dependent variable. These so-called “effects estimates”, as developed by LeSage and Pace (2009), provide summary measures of the direct, indirect, and total effects and are conditional upon the values of the spatial autocorrelation parameters. We illustrate the different methods of calculating these effects estimates with a numerical example and show that regardless of the type of higher-order spatial econometric model utilized in applied practice, completely separating out the effects estimates for each individual weights matrix is not feasible.
Higher order spatial econometric models are seeing increasing use in various applications because of their ability to address empirical questions that are not readily addressable via traditional single weights matrix spatial econometric models. However, issues regarding the identification of the proper parameter space have received little attention thus far, possibly because of computational or theoretical issues surrounding their calculation.
We present two different specifications the researcher may choose from: higher-order polynomials in the spatial weights matrices and higher-order spatial autoregressive processes. Whereas the stationarity region of the parameters of the first model is easy to determine, the stationarity region of the second model is not. In this paper, we propose a mathematically and computationally simple procedure for finding the stationary region for higher order spatial autoregressive processes. The identification of the proper parameter space in these models is important for at least two reasons. First, negative spatial autocorrelation may be of theoretical or empirical interest or be the result of choosing spatial weights matrices that partly overlap and the a priori exclusion of certain parameter values may be unwarranted. Second, obtaining accurate estimates of the spatial autocorrelation parameters is crucial in terms of the calculation of the direct, indirect, and total effects estimates. Arbitrarily restricting the allowable parameter space can lead to improper inferences about the nature and magnitude of any spatial autocorrelation present in the data.
