Palgrave Handbook of Econometrics: Applied Econometrics

Luc Anselin and Nancy Lozano-Gracia 1229

of additional explanatory variableszhi(again, including a constant term):

βki=

∑

h

zhiγh, (26.35)

where theγhare an additional set of parameters. Note that, unlike the standard
multilevel model, the observational unit for thexandzvariables is the same (i).
The combination of the initial model with the expansion equation yields the so-
called final equation, which contains the original explanatory variables, as well as
interaction variables, the product of eachxkiwith all thezhi:

yi=

∑

k

xki

⎛

⎝

∑

h

zhiγh

⎞

⎠+ (^) i. (26.36)
A slight generalization is obtained when the expansion equation includes an error
term, which yields a heteroskedastic model (Anselin, 1992). A common problem
in the implementation of this approach is a high degree of multicollinearity.
In spatial hedonic specifications, the expansion method is used to model het-
erogeneity in the form of so-called neighborhood drift (Can, 1992). This may also
account for some omitted variables at the neighborhood level and therefore reduce
the intensity of the spatial autocorrelation problem. However, it is important to
note that, even if a parametric drift is introduced, spatial autocorrelation and
heterogeneity may remain as a problem.
An alternative to the parametric specification of the expansion equation is a
nonparametric approach, in which the variability of the model parameters is deter-
mined by the data. The best known among these approaches is arguably the geo-
graphically weighted regression (GWR), popularized in the work of Fotheringham
and collaborators (for an overview, see Fotheringhamet al., 2002).
GWR is essentially a special case of a local regression model (LRM) (e.g., as
proposed in Cleveland and Devlin, 1988), in which the weighting scheme that
determines the variability in the parameters is based on the spatial closeness
of observations (for examples of spatial hedonic applications, see Pavlov, 2000;
Gelfandet al., 2003; Choet al., 2006; Kestenset al., 2006, among others). In this
approach, a model parameter is defined as a function of the location of individual
observations. In addition, a weighting scheme is designed such that greater weight
is given to locations that are closer in space. To illustrate this approach, consider a
hedonic model specified as:
P=β 0 (c)+
∑
k
Xkβk(c)+ε, (26.37)
wherecis a vector ofXc,Yccoordinates that define the location of the data points.
The parameters are estimated by minimizing a weighted residual sum of squares:
minp,q
∑
i
⎧
⎪⎨
⎪⎩
Wi(c)
⎡
⎣P−β 0 (c)−
∑
k
Xkβk(c)
⎤
⎦
2
⎫
⎪⎬
⎪⎭
, (26.38)