Spatial Autoregressive Models for Geographically Hierarchical Data Structures

Abstract

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.

On Spatial and Platial Dependence: Examining Shrinkage in Spatially-Dependent Multilevel Models

“…However, spatiallyvarying intercept models only fall into this class in some cases. If the varying-intercept is modeled using a mixed regressive-autoregressive "spatial lag" model, it does not decompose into simple variance components (Dong and Harris, 2015). Using a J × 1 vector of intercepts, α J , the varying intercept model can be stated:…”

“…While there still are highly-complex interactions between regional effects in the spatial multilevel variance components model that we discuss later, the marginal effect estimates from spatially-correlated variance components models can be interpreted directly. However, models like that of Dong and Harris (2015) or the general spatial model of Anselin (1988) are not easily D R A F T restated into variance components and require a different analysis that accounts for the direct/indirect spillover structure.…”

“…Thus, it is likely that spatial multilevel models with lag specifications have even more complex shrinkage behavior. The systematic formal analysis shown here must be extend to pathbreaking studies of Lacombe et al (2014), Dong and Harris (2015), and Lacombe and McIntyre (2016). Ideally, diagnostics should target an even more general spatial model capable of expressing the complex interactions between multilevel and spatial dependence structures.…”

“…Spatial econometricians and statisticians have indeed been working on models to extend the notions of spatial dependence that are available in multilevel models (Corrado and Fingleton, 2012;Lacombe et al, 2014;Dong and Harris, 2015;Dong and Wu, 2016;Lacombe and McIntyre, 2016). This work typically focuses on multilevel extensions or adaptations of simultaneous autoregressive model structures (Whittle, 1954;Cliff and Ord, 1973;Haining, 1978;Anselin, 1988;Hepple, 2003;LeSage and Pace, 2009).…”

Comparing definitions of spatial relations for the analysis of geographic disparities in mortality within a Bayesian mixed-effects framework

“…In the context of the geographical classification of people, hierarchical MM captures vertical dependence (hierarchy) but omits horizontal dependence (proximity) between geographic units 4 . In other words, these models do not take into consideration the spatial configuration of geographic units.…”