1228 Spatial Hedonic Models
hedonic models, discrete spatial heterogeneity is taken into account in the form of
separate models forsub-markets. For example, inelasticities in supply and demand
may lead to market segmentation that results in spatial heterogeneity in the form
of varying marginal prices (Goodman and Thibodeau, 1998). Sub-markets can be
defined spatially or non-spatially but, in practice, preference goes to delineations
that follow clearly observed spatial boundaries (for examples, see section 26.6.1).
More formally, discrete spatial heterogeneity can be expressed as:
yi=fi(Xi,βi, (^) i), (26.33)
withias an index corresponding to a given discrete sub-set of the data. This gen-
eral formulation includes as special cases functional instability,fi =fj(e.g., linear
model for one region, log-linear in another), parameter variation,βi =βj(e.g., dif-
ferent parameter values for house characteristics in different sub-markets), as well
as heteroskedasticity,Var[ (^) i] =Var[ (^) j].
These examples represent fairly standard methodological issues that can readily
be addressed and do not require an explicit spatial econometric treatment. How-
ever, in practice, in many instances heterogeneity and spatial dependence occur
together, or, are difficult to identify separately (Anselin and Griffith, 1988). For
example, spatial spillover may not be constrained to each specific spatial sub-set
of the data, but may reach across the boundaries. In those cases, the treatment of
the heterogeneity becomes complicated by the presence of spatial dependence, and
extensions of the standard spatial lag and error models and associated specification
tests are in order. One example of such tests is the so-called spatial Chow test of
coefficient stability, which is an extension of the standard case that incorporates
spatial dependence (Anselin, 1990).
In spatial hedonic models, attention has focused primarily on the delineation of
sub-markets, and the acknowledgment of spatial dependence between sub-markets
has only received limited attention. We provide specific examples in section 26.6.1.
26.3.2.2 Continuous spatial heterogeneity
As an alternative to considering discrete spatial sub-sets of the data, heterogene-
ity can be viewed as a smooth continuous process of varying parameters. One of
the earliest applications of this perspective to spatial analysis was in the so-called
spatial expansion methodproposed by Casetti in the early 1970s (see, e.g., Casetti,
1972, 1997).
Spatial expansion is a special case of a varying coefficients model and also shows
great similarity to the approach taken in multi-level modeling (e.g., Goldstein,
1995). Using Casetti’s terminology, the first step is a so-called initial equation,
which is a simple linear regression specification for each observationi:
yi=
∑
k
xkiβki+ (^) i, (26.34)
fork explanatory variables, including a constant term. Next, an expansion
equation expresses the variability of the regression coefficient overias a function