1222 Spatial Hedonic Models
where = I, withIas the identity matrix. Typically,contains “nuisance”
parameters that need to be estimated consistently. This, in turn, yields consistent
estimates for the regression coefficients by means of a feasible generalized least
squares (FGLS) estimation. The interpretation of the nuisance parameters is very
different from the spatial autoregressive coefficient in the spatial lag model, in that
there is no particular relation to a substantive model of spatial interaction. These
parameters are only included in order to obtain better estimates for the regression
slope coefficients.
The particular structure offollows from a spatial ordering of the observations
(e.g., as argued in Kelejian and Robinson, 1992). In practice, the most commonly
used specification assumes a spatial autoregressive process for the error terms:
y=Xβ+ε, (26.10)with:
ε=λWε+u, (26.11)withu∼i.i.d., andλas the spatial autoregressive coefficient.
The resulting error variance-covariance matrix is as follows:
E[εε′]=σ^2 [(I−λW)(I−λW′)]−^1. (26.12)A commonly used alternative in hedonic analyses is to base the structure of the
error variance-covariance matrix on principles from geostatistics. Early work by
Dubin (1988) (see also Dubin, 1992; Basu and Thibodeau, 1998; Dubinet al., 1999;
Miltinoet al., 2004) suggested a so-calleddirect representationfor the elements of
the variance-covariance matrix.
In this approach, the covariance between each pair of error terms is specified as
an inverse function of the distance between them. Formally:
E[εiεj]=σ^2 f(dij,φ), (26.13)withεi,εjas the regression error terms,σ^2 the error variance, anddijthe distance
separatingiandj. The functionfshould be a distancedecayfunction that ensures a
positive definite covariance matrix. This requires∂f/∂d<0 and|f(dij,φ)|≤1, with
φ∈#as ap×1 vector of parameters on an open sub-set#ofRp. This approach is
closely related to the variogram models used in geostatistics, and requires assump-
tions of stationarity and isotropy (see Cressie, 1993, for an extensive review). The
complete variance-covariance is then:
E[εε′]=σ^2 (dij,φ). (26.14)A commonly used specification is based on a negative exponential distance
decay:
E[εε′]=σ^2 [I+γ!], (26.15)