1230 Spatial Hedonic Models
whereWi(c)are the weights that depend on the location(c). The solution to
the minimization problem in equation (26.38) yields the standard weighted least
squares expression, in matrix notation:
βˆ=(X′WX)−^1 (X′Wy). (26.39)In this expression,Wis not a spatial weights matrix, but a matrix that extracts the
observations used in the estimation of the parameter for each locationi. Several
approaches have been suggested, such as a straightforward k-nearest neighbors
weighting scheme (Pavlov, 2000), or a kernel smoother. The latter is the common
approach taken in GWR, where a Nadaraya–Watson-type kernel smoother ensures
that those observations near the point where the parameters are being estimated
have more influence than those observations further away. Using such a kernel
function in GWR then yields the function to be minimized as:
minp,q∑i⎧
⎪⎨⎪⎩
Kh(d 0 i)⎡
⎣P−β 0 (c)−
∑kXkβk(c)⎤
⎦2
⎫
⎪⎬⎪⎭
, (26.40)whereKh(·)=K(·/h),Kis a given kernel function andha bandwidth parameter.
Common choices for the kernel are the bi-square function:
K(t)={
( 1 −t^2 )^2 ,if|t|≤1,
0, otherwise,(26.41)and the Gaussian kernel:
K(t)=exp(
−1
2
t^2). (26.42)
Since the term “GWR” was first introduced in Brunsdonet al. (1996), an extensive
set of papers has been published treating various theoretical issues related to model
estimation, specification testing and cross-validation (see, among others, Fother-
inghamet al., 1998, 2002; Paezet al., 2002a, 2002b). Specific empirical applications
to spatial hedonic specifications are reviewed in section 26.6.2.
26.4 Methodological challenges
Spatial hedonic analysis not only considers the specification of spatial relation-
ships in the model, but also the estimation of relevant parameters on the basis of
spatial data. In this section, we briefly point out some important methodological
issues that need to be accounted for, especially the problem of spatial scale and the
treatment of endogeneity.
26.4.1 Spatial scale
Spatial scale is important in the empirical implementation of hedonic models in a
number of ways. The standard assumption is that the spatial units of observation
match the process under consideration. However, with spatial data, this is not
necessarily the case, and errors due to aggregation or interpolation need to be