1220 Spatial Hedonic Models
26.3.1 Spatial dependence
26.3.1.1 Spatial lag model
The spatial lag specification is characterized by the inclusion of a new variable
on the right-hand side of the equation. This variable, referred to as a spatially
lagged dependent variable (Anselin, 1988) captures the spatial interaction effect as
a weighted average of neighboring observations. This is most commonly applied
in a linear form, as:
y=ρWy+Xβ+u, (26.5)whereyis ann×1 vector of observations on the dependent variable,Xis ann×k
matrix of observations on explanatory variables,Wis ann×nspatial weights
matrix,uis ann×1 vector of independent and identically distributed (i.i.d.) error
terms,ρis the spatial autoregressive coefficient, andβis ak×1 vector of regression
coefficients.
Then×nspatial weights matrix defines the neighbor set for each individual
location. Its positive elementswijare non-zero when observationsiandjareneigh-
bors, and zero otherwise. By convention, self-neighbors are excluded, such that
the diagonal elements ofWare zero. In addition, in practice the weights matrix is
typically row-standardized, such that
∑
jwij=1. Many different definitions of the
neighbor relation are possible, and there is little formal guidance on the choice of
the “correct” spatial weights.^1 The termWyin equation (26.5) is referred to as a
spatially lagged dependent variable, or spatial lag. For a row-standardized weights
matrix, it consists of a weighted average of the values ofyin neighboring locations,
with weightswij.
As stated in Anselin and Bera (1998), there are two main interpretations for a
significant spatial autoregressive coefficientρ. First, this may suggest a contagion
process or the presence of spatial spillovers. However, this interpretation is valid
only if the process takes place at the spatial unit used in the analysis, and is sup-
ported by a theoretical model. In the context of spatial hedonic models, this is often
difficult to maintain, since it is unlikely that economic agents simultaneously take
into account the prices of neighboring units. An alternative explanation for a sig-
nificant spatial autocorrelation coefficient is the existence of a mismatch between
the observed spatial unit and the true spatial scale of the process being studied.
The theoretical motivation for a spatial lag specification is based on the literature
on interacting agents and social interaction. For example, a spatial lag follows as the
equilibrium solution of aspatial reaction function(Brueckner, 2003) that includes the
decision variable of other agents in the determination of the decision variable of an
agent (see also Manski, 1993, 2000). In hedonic models, however, where a purely
cross-sectional setting is more common, it is often difficult to maintain such a
theoretical motivation, since it would imply that buyers and sellers simultaneously
take into account prices obtained in other transactions.
An alternative interpretation is provided by focusing on the reduced form of the
spatial lag model:
y=(I−ρW)−^1 Xβ+(I−ρW)−^1 u, (26.6)