Luc Anselin and Nancy Lozano-Gracia 1225In the next iteration, the linear part of the model is estimated again using OLS, but
now usingYˆ^1 as the dependent variable, where:
Yˆ^1 ≡Xβˆ^0 +ˆε^0. (26.22)Iterations between the parametric and nonparametric portions of the model
continue until theβˆs change by no more than 5%.
A different estimation approach for this model is described in Gibbons (2003)
and Dayet al. (2007), who usespatially filteredvariables to consistently estimate the
βcoefficients, following the steps first outlined in Robinson (1988). To implement
this, the model in equation (26.18) is rewritten as:
P−E[P|Xc,Yc]=(X−E[X|Xc,Yc])β+ε. (26.23)Estimates for the marginal prices are obtained by using the spatially weighted
means of the explanatory variables. First, the conditional meansE[P|Xc,Yc]and
E[X|Xc,Yc]are estimated using nonparametric regression. Estimates of these func-
tions are then substituted into equation (26.23) and, following Robinson (1988),
consistent estimates for theβcoefficients are obtained from OLS on equation
(26.23).
Dayet al. (2007) consider an additional complication and allow for the presence
of spatial error autocorrelation in the form of a spatial autoregressive process (as in
equation 26.11). Rather than applying OLS, estimates for theβin equation (26.23)
are obtained using the Kelejian and Prucha (1999) GM approach.
Alternatively, Clappet al. (2002) utilize Bayesian methods to remove any remain-
ing spatial autocorrelation from the model. In standard Bayesian fashion, the error
terms are specified as consisting of two components; one being spatial, the other
a white-noise process. Formally:
ε(ˆci)=δ(ci)+ψ(ci), (26.24)in whichδ(ci)is assumed to come from a stationary Gaussian spatial process with
mean 0 and spatial covariance functioncov(δ(c),δ(c′))=σ^2 exp(−ψ‖c−c′‖)andψ
is assumedi.i.d∼N(0,τ). Bayesian fitting is then applied to the first stage residuals
using Gibbs sampling combined with a Metropolis–Hasting procedure to account
for remaining spatial autocorrelation in the residuals.
26.3.1.4 Models for space-time dependence
The temporal dimension has not received much attention in spatial hedonic mod-
els. There are both theoretical as well as practical reasons for this omission. First,
using data for several time periods would require the assumption that the marginal
prices stay constant through time. While this assumption may seem appropriate
for a short period of time, it is unlikely to hold when several years are considered.
As a result, hedonic analyses have tended to favor pure cross-sectional approaches.
Furthermore, explicitly including both the temporal and spatial dimension
requires complex estimation methods. Most applications have used Bayesian meth-
ods to tackle this complexity. Some examples of spatio-temporal hedonic analyses