1226 Spatial Hedonic Models
include Paceet al. (1998), Paceet al. (2000), Gelfandet al. (2004), Sunet al. (2005)
and Huanget al. (2006).
As an example, consider Paceet al. (1998) and Paceet al. (2000). They propose
a spatio-temporal specification in which a filtering approach is carried out across
both dimensions. The model is:
(I−W)Y=(I−W)Xβ+u, (26.25)whereWis a row-standardizedN×Nlower triangular spatio-temporal weights
matrix. The weight matrixWis used to filter out the spatio-temporal correla-
tion and the model is then estimated for the uncorrelated variables(I−W)X
and(I−W)Y.
Data are ordered by time period, so that the first row in the data refers to the
earliest observation. Therefore, only previous sales are allowed to influence cur-
rent house prices. This explicit relation between time periods contrasts with the
approach taken in cross-sectional studies, where all sales are assumed to have taken
place during the same period. As a result, in a cross-sectional set-up, there is a simul-
taneous feedback between all house prices in the sample, even though technically
the sales may have taken place at different times during the period (so, conceiv-
ably, a later sale could affect an earlier sale). The explicit inclusion of space-time
correlation allows for a more realistic model of the actual timing of real estate
transactions.
In this approach,Wis not assumeda prioribut rather defined through a flexible
form, where(I−W)=(I−φSS−φTT−φTSTS−φSTST), withTandSrespec-
tively defined as temporal and spatial weight matrices. The expanded definition
of(I−W)is then plugged back into equation (26.25) and the model is estimated
using Bayesian methods for the unfiltered variableY.
Gelfandet al. (2004) introduce time and space in the model by allowing the
coefficients to change over time. They define three possible forms for the error term
in a separable form, which avoids explicit specification of space-time interactions:
U(s,t)=α(t)+W(s)+ε(s,t), (26.26)
U(s,t)=αs(t)+ε(s,t), (26.27)
U(s,t)=Wt(s)+ε(s,t), (26.28)withε(s,t)asi.i.d.N(0,σε^2 )error terms,αs(t)as the temporal effect andW(s)as the
spatial effect.
Equation (26.26) provides a structure that is additive in spatial and temporal
effects. In contrast, equation (26.27) suggests that the temporal effects are local,
changing from one site to the other. Finally, the form for the error term suggested
in equation (26.28) pertains to the case where the spatial effects are specific to the
time period. Gelfandet al. (2004) then outline a fully Bayesian approach and specify
the associated likelihoods together with the prior distributions for all parameters
in the model.