Luc Anselin and Nancy Lozano-Gracia 1241between 1995 and 1997 and derive aggregates of school zones as sub-markets. The
procedure starts by estimating a hierarchical model for two adjacent school zones.
Then, if the coefficient associated with the sub-market is significant, those school
zones are considered to pertain to different sub-markets. If, on the other hand,
the coefficients are not significant, the two zones are merged. One by one, each
school zone is added until all zones have been included. To avoid sub-market def-
initions that are path dependent, sensitivity checks are included of how the final
sub-market definition depends on the starting point.
Comparative evidence is provided in the evaluation of the hierarchical approach
to a sub-market definition using zip codes and census tracts in Goodman and
Thibodeau (2003). In terms of prediction accuracy, the Goodman and Thibodeau
(1998) hierarchical approach outperforms the other two methods.
In more recent work, Ugarteet al. (2004) propose the use of a mixture of linear
models. This first provides a classification of the observations into groups (sub-
markets), and then estimates the parameters for the hedonic price equilibrium in
each group. The data are allowed to determine the group structure and coefficients
are estimated jointly. A linear mixed model with random effects is estimated by
means of nonparametric ML. However, due to the small number of properties con-
sidered (293 in Pamplona, Spain), the degree of generality of this approach remains
a topic for further research.
In the empirical literature, the discussion of sub-market definition and evaluation
of the performance of different methods has tended to focus primarily on the
out-of-sample predictive precision. The effect of different sub-market definitions
on the substantive interpretation of the value and MWTP of specific house (and
environmental) characteristics largely remains to be investigated.
26.6.2 Continuous spatial heterogeneity: spatially varying coefficients
Early efforts to allow for continuous variation of the parameters in a hedonic spec-
ification were based on applications of Casetti’s expansion method. This was first
implemented by Can (1990) as a way to address spatial heterogeneity in house
prices at the neighborhood level. In her specification, the marginal price of housing
attributes are assumed to vary as a function of neighborhood characteristics. Specif-
ically, Can defined a composite neighborhood quality (NQ) index and introduced
linear and quadratic functions of NQ as the expansion equation. The hedonic price
equilibrium is then specified as:
P=α+∑kβkSk+ε, (26.43)where theSkare the structural characteristics of the house andβkis a function of
a neighborhood variableNQ. For example, for the linear case:
βk=βk 0 +βk 1 NQ+u. (26.44)Other examples of applications of the expansion method to hedonic specifica-
tions can be found in Can (1992), Theriaultet al. (2003), Fiket al. (2003) and