# An Anisotropic Model for Spatial Processes

## Article excerpt

One of the key assumptions in spatial econometric modeling is that the spatial process is isotropic, which means that direction is irrelevant in the specification of the spatial structure. On the one hand, this assumption largely reduces the complexity of the spatial models and facilitates estimation and interpretation; on the other hand, it appears rather restrictive and hard to justify in many empirical applications. In this article a very general anisotropic spatial model, which allows for a high level of flexibility in the spatial structure, is proposed. This new model can be estimated using maximum likelihood and its asymptotic properties are derived at length. When the model is applied to the well-known 1970 Boston housing prices data, it significantly outperforms the isotropic spatial lag model. It also provides interesting additional insights into the price determination process in the properties market. Finally, a Monte Carlo simulation study is used to confirm the optimal properties of the model.

Introduction

The most important aspect of spatial econometric modeling is the incorporation of a dependence structure between cross-sectional observations. With N spatial units, potentially there are up to ([N.sup.2] - N) unique spatial relationships. The problem of incidental parameters arises, and, unless the phenomenon of interest can be observed for the same set of cross-sectional units over a large number of time periods, it is impossible to separately identify these ([N.sup.2] - N) spatial relationships.

To ensure identifiability of the spatial models, typically rather restrictive assumptions are made with regard to the extent and form of the spatial dependence structure. One of the key assumptions is that of isotropy, which requires that spatial effects are of equal strength in all directions between all spatially contiguous observations. For example, in the well-known spatial lag model (Anselin 1988):

y = [rho]Wy + X[beta] + [epsilon] (1)

where y is an (N x 1) vector of observed values of the dependent variable; X is an (N x K) matrix of observed values of K explanatory variables; and [epsilon] is an (N x 1) vector of independently and identically distributed normal disturbances, isotropy is implied. The spatial weights matrix W is an (N x N) matrix that identifies spatial relations between cross-sectional units in the sample. The most commonly employed spatial weights matrices, as proposed by Cliff and Ord (1973), are either of a binary contiguity design or of a distance decay design and do not take direction into account. Moreover, the spatial parameter p is a scalar that is multiplied to each and every spatial neighbor and makes it impossible to allow for varying degrees of spatial influence to originate from different spatial neighbors.

While the assumption of isotropy is a standard practice in spatial econometric modeling, it seems overly restrictive in many empirical applications. Just as in our personal relationships not every acquaintance is a best friend, it is unlikely that every spatial neighbor is of equal importance. For example, in the study of crime patterns, it is reasonable to expect neighborhoods with severe poverty to export crimes to other neighborhoods, but not the other way round. In the study of water-borne diseases, it is reasonable to expect the diffusion of the disease to closely follow the flow of the water system. In hedonic house-price studies, neighbors with strong characteristics, such as high crime rates, are expected to attract greater attention. Therefore, it appears more reasonable to allow the strength of spatial effects to differ depending on the relative characteristics of the two spatial neighbors.

Over the years, some attempts have been made to incorporate more information and greater flexibility into the spatial structure, and most of them involve some form of parameterization of the spatial weights matrix. …

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