Luc Anselin and Nancy Lozano-Gracia 1221where, under standard regularity conditions, the inverse(I−ρW)−^1 can be
expressed as a power expansion:
(I−ρW)−^1 =I+ρW+ρ^2 W^2 +.... (26.7)The reduced form thus expresses the house price as a function of its own charac-
teristics (X), but also of the characteristics of neighboring properties, (WX,W^2 X),
albeit subject to a distance decay operator (the combined effect of powering the
spatial autoregressive parameter and the spatial weights matrix). In addition, omit-
ted variables, both property-specific as well as related to neighboring properties, are
encompassed in the error term. In essence, this reflects a scale mismatch between
the property location and the spatial scale of the attributes that enter into the
determination of the equilibrium price.
From a purely pragmatic perspective, one can also argue that the spatial lag spec-
ification allows for afilteringof a strong spatial trend (similar to detrending in the
time domain), i.e., ensuring the proper inference for theβcoefficients when there
is insufficient variability across space. Formally, the spatial filter interpretation
stresses the estimation ofβin:
y−ρWy=Xβ+u. (26.8)
In most spatial hedonic applications, the use of a spatial lag specification follows
as the result of a specification search based on specialized Lagrange multiplier tests
that indicate the preference of this alternative over an error specification (Anselin
and Bera, 1998; Anselin, 2001a; Floraxet al., 2003). In such instances, the selection
of this model is mostly pragmatic, without necessarily implying a theoretical model
of social interaction.
The most commonly applied estimation method for the parameters of the spatial
lag model is maximum likelihood, following the principles outlined by Ord (1975)
(for additional technical details, see, e.g., Anselin, 1988; Anselin and Bera, 1998;
Anselin, 2006). More recently, an instrumental variables or spatial two stage least
squares approach has gained greater popularity, because it lends itself more readily
to application in the large datasets characteristic of hedonic studies. Early results
were given in Anselin (1988) and Kelejian and Robinson (1993), but more recently
interest has focused on formal proofs of asymptotic properties and the choice
of optimal instruments, e.g., in Lee (2003, 2007), Daset al. (2003) and Kelejian
et al. (2004). Bayesian estimation of spatial autoregressive models is covered in
LeSage (1997).
We leave a more detailed discussion of specific applications of spatial hedonic
models to section 26.5.
26.3.1.2 Spatial error model
From a theoretical viewpoint, a spatial error specification is the more natural way to
include spatial effects in a hedonic model. Unobserved neighborhood effects will be
shared by housing units in the same area and naturally lead to spatially correlated
error terms. This results in a non-diagonal error variance-covariance matrix:
Var[uu′]=E[uu′]=, (26.9)