# RECORD 91a343b71c11d4139d2606d6c4d1c5fc64cbc6c3
# TITLE: Estimation of Housing Price Variations Using Spatio-Temporal Data
# YEAR: 2019.0  VENUE: Sustainability  DOI: 10.3390/su11061551
# AUTHORS: Jorge Chica-Olmo; Rafael Cano-Guervos; M. Chica-Rivas
# PAGES: 21  CANDIDATE TABLE PAGES (1-based): [12, 14]
# ABSTRACT: This paper proposes a hedonic regression model to estimate housing prices and the spatial variability of prices over multiple years. Using the model, maps are obtained that represent areas of the city where there have been positive or negative changes in housing prices. The regression-cokriging (RCK) method is used to predict housing prices. The results are compared to the cokriging with external drift (CKED) model, also known as universal cokriging (UCK). To apply the model, heterotopic data of homes for sale at different moments in time are used. The procedure is applied to predict the spatial variability of housing prices in multi-years and to obtain isovalue maps of these variations for the city of Granada, Spain. The research is useful for the fields of urban studies, economics, real estate, real estate valuations, urban planning, and for scholars.


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sustainability
Article
Estimation of Housing Price Variations Using
Spatio-Temporal Data
Jorge Chica-Olmo 1,*
, Rafael Cano-Guervos 1 and Mario Chica-Rivas 2
1
Department of Quantitative Methods for Economics and Business, University of Granada, 18011 Granada,
Spain; rcano@ugr.es
2
Department of Mathematical Analysis, University of Granada, 18071 Granada, Spain; mcrivas@ugr.es
*
Correspondence: jchica@ugr.es
Received: 7 February 2019; Accepted: 11 March 2019; Published: 14 March 2019





Abstract: This paper proposes a hedonic regression model to estimate housing prices and the spatial
variability of prices over multiple years. Using the model, maps are obtained that represent areas of the
city where there have been positive or negative changes in housing prices. The regression-cokriging
(RCK) method is used to predict housing prices. The results are compared to the cokriging with external
drift (CKED) model, also known as universal cokriging (UCK). To apply the model, heterotopic data
of homes for sale at different moments in time are used. The procedure is applied to predict the spatial
variability of housing prices in multi-years and to obtain isovalue maps of these variations for the city
of Granada, Spain. The research is useful for the fields of urban studies, economics, real estate, real
estate valuations, urban planning, and for scholars.
Keywords: hedonic regression model; cokriging; spatio-temporal; housing price variation
1. Introduction
In the real estate field, multivariate spatial data models have been widely explored [1–5],
while multivariate spatio-temporal data models have received relatively less attention. The hedonic
regression model is used to make real estate valuations and to determine the characteristics that affect
property prices [6]. Classical estimation methods, such as ordinary-least-squares (OLS), ignore the
presence of spatial autocorrelation, since they assume the independence of the observations [7].
Two approaches are commonly used in spatial hedonic modeling: spatial econometrics [8] and
the geostatistical approach [9]. Spatial econometrics uses the spatial weight matrix and requires an
a priori specification of the matrix, which may affect the final results [10]. However, geostatistics is
a method in which the variance–covariance matrix depends on the Euclidean distance [11]. In the
case of spatial autocorrelation of the error terms, the spatial error model (SEM) is not better than the
geostatistical approach [12]. The literature suggests that the kriging geostatistical method improves
the prediction performance of spatial hedonic models [13]. Moreover, kriging models are superior to
other methods (OLS, SEM, spatial lag model, etc.) in terms of their prediction accuracy [14].
Several methods that consider both spatial and temporal dependences that simultaneously affect
housing prices have been proposed for the study of real estate. A classical method for considering
time-fixed effects is to include dummy variables indicating the year a house is being sold [15,16].
The spatio-temporal autoregressive (STAR) model and its variations [17–20] is another method that
consists of a spatio-temporal model based on a generalization of the traditional spatial autoregressive
model (SAR) [21]. This method has also been used to analyze housing prices in Spain [22,23].
Several authors have used geographically and temporally weighted regression to capture spatial
and temporal heterogeneity in real estate market data [24–27].
Sustainability 2019, 11, 1551; doi:10.3390/su11061551
www.mdpi.com/journal/sustainability


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The global drift multi-equation model ( ˆm(s0)) was estimated by GLS. The results of the estimation
are presented in Table 4. In the estimated equations, all the variables are significant at a confidence
level above or equal to 90%, with most being significant at 99%. All the signs of the coefficients are as
expected (negative relation in the AGE, DIST, REHAB, and FLOOR variables, and a positive relation in
the rest of the variables).
Table 4. Estimation results of multi-equation model by RCK (p-values in brackets).
1988
1991
1995
2005
Intercept
1.042 × 101
1.084 × 101
1.041 × 101
1.148 × 101
(0.000)
(0.000)
(0.000)
(0.000)
AGE
−2.119 × 10−2
−7.828 × 10−3
−8.497 × 10−3
−6.176 × 10−3
(0.000)
(0.000)
(0.000)
(0.000)
BATH
8.929 × 10−2
1.176 × 10−1
6.929 × 10−2
9.256 × 10−2
(0.020)
(0.000)
(0.000)
(0.000)
AREA
7.949 × 10−3
5.782 × 10−3
8.361 × 10−3
5.951 × 10−3
(0.000)
(0.000)
(0.000)
(0.000)
DIST
−3.853 × 10−4
−2.532 × 10−4
−2.111 × 10−4
−2.257 × 10−4
(0.000)
(0.000)
(0.000)
(0.000)
REHAB
–
−1.517 × 10−1
–
−6.733 × 10−2
(0.000)
(0.008)
ELEV
–
–
1.552 × 10−1
1.187 × 10−1
(0.000)
(0.000)
HEAT
–
–
7.528 × 10−2
4.679 × 10−2
(0.000)
(0.060)
FLOOR
–
–
–
−1.056 × 10−1
(0.059)
SPORT
–
–
–
1.278 × 10−1
(0.011)
R2cv
0.8507
0.8727
0.9332
0.8001
MAE
0.1793
0.1371
0.1362
0.1545
MSE
0.0503
0.0318
0.0335
0.0416
n
260
247
293
207
In this paper, the CKED and RCK methods are applied. In order to compare both methods, a
cross-validation method has been used, specifically leave-one-out cross-validation (LOOCV). LOOCV
allows for comparison of predicted and observed values using only the information available in
the sample dataset. The LOOCV procedure consists of temporarily discarding a sample value at
a particular location from the sample data set, and then estimating the value at the same location
using the remaining samples [55]. This procedure is repeated for all the experimental points in
order to compare the observed values to the predicted values using statistical and visual tools.
The cross-validation statistics used are mean absolute error (MAE), mean squared error (MSE), and
R-squared of cross-validation, which is obtained from the square of the correlation between the model’s
predicted values and the observed values (R2
cv). As can be seen in Table 5, the two methods show
similar cross-validation statistics, although those of the RCK method are slightly better, as the R2
cv value
is closer to one and the other statistics are closer to zero. Figure 5 shows the regression between the
predicted and observed data with RCK and CKED. As can be observed, the two scatterplots are quite
similar and the regression line is very close to the 1:1 line. Moreover, Figure 6 shows the RCK versus
the CKED predictions. As can be seen, the predictions obtained with both methods are very similar.


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Figure 5. Regression between predicted and observed data. Note: The solid straight line represents the
1:1 line and the dashed line represents the regression line.
Figure 6. Comparison of RCK and CKED predictions.
Table 5.
Cross-validation statistics of CKED and RCK (R-squared of cross-validation, R2cv;
mean absolute error, MAE; mean squared error, MSE and sample size, n).
CKED
RCK
R2cv
0.9277
0.9331
MAE
0.1400
0.1355
MSE
0.0343
0.0317
n
1007
1007
Additionally, in order to compare the predictions of traditional OLS and cokriging, Table 6 shows
the OLS estimates for each of the equations. This permits quantification of the added value of the
cokriging method versus OLS. As can be observed in the table, all the variables are significant, as was
the case with RCK.


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Table 6. OLS estimation for each equation (p-values in brackets).
1988
1991
1995
2005
Intercept
1.027 × 101 e+01
(0.000)
1.074 × 101
(0.000)
1.446 × 101
(0.000)
1.650 × 101
(0.000)
AGE
−2.033 × 10−2
(0.000)
−6.484 × 10−3
(0.000)
−1.035 × 10−2
(0.000)
−6.583 × 10−3
(0.001)
BATH
1.269 × 10−1
(0.004)
1.788 × 10−1
(0.000)
9.536 × 10−2
(0.002)
1.107 × 10−1
(0.002)
AREA
8.506 × 10−3
(0.000)
5.766 × 10−3
(0.000)
1.188 × 10−2
(0.000)
6.125 × 10−3
(0.000)
DIST
−3.521 × 10−4
(0.000)
−2.510 × 10−4
(0.000)
−2.685 × 10−4
(0.000)
−2.139 × 10−4
(0.000)
REHAB
–
−1.981 × 10−1
(0.000)
–
−6.925 × 10−2
(0.048)
ELEV
–
–
2.540 × 10−1
(0.000)
1.875 × 10−1
(0.000)
HEAT
–
–
1.550 × 10−1
(0.000)
7.201 × 10−2
(0.053)
FLOOR
–
–
–
−1.482 × 10−1
(0.079)
SPORT
–
–
–
1.376 × 10−1
(0.041)
R-squared
0.8176
0.8278
0.9234
0.7745
R2cv
0.8060
0.8173
0.9180
0.7442
MAE
0.2056
0.1664
0.1567
0.1767
MSE
0.0655
0.0457
0.0411
0.0556
n
260
247
293
207
This study has considered accessibility to the CBD, which is one of the main locational variables.
However, other relevant variables have been omitted (i.e., provision of public and private services,
socio-economic and environmental variables, etc.). Therefore, the differences observed between the
OLS coefficients (see Table 6) and RCK (see Table 4) may be due to this omission. It is difficult to specify
these locational factors and quantify their radius of influence, since they usually refer to areas whose
sizes and shapes tend to be subjective [30,74]. In fact, numerous studies on hedonic models have
mostly included structural variables and omitted relevant characteristics of the location [8,9,31,43,75].
Nevertheless, models that correct this omission by considering the presence of spatial autocorrelation
in disturbances, such as the SEM and RCK models, provide better results than OLS [4]. Therefore, an
advantage of the RCK model is that it improves the estimates by indirectly considering the omission of
relevant variables through modeling the correlation between disturbances.
Table 6 shows the results of the cross-validation. It is important to keep in mind that in traditional
OLS, all observations are used to fit the model, while this does not occur in cross-validation. Since the
R-squared of the OLS model is not directly comparable to the R2
cv of the RCK model, the R2
cv and other
statistics for the OLS models have been obtained using cross-validation (MAE and MSE). Table 7 shows
the improvement in % by RCK over OLS. Specifically, the R2
cv in the RCK model shows an almost 8%
improvement over OLS, while the MAE shows an improvement of approximately 18% and the MSE
of more than 30%. As can be seen, there is a clear improvement in the predictive ability of the RCK
model, thus supporting the added value of the spatial effect and the cross between space and time.
Table 7. Improvement in % by RCK over OLS.
1988
1991
1995
2005
R2cv
5.5459
6.7784
1.6557
7.5114
MAE
12.7918
17.6082
13.0823
12.5637
MSE
23.2061
30.4157
18.4915
25.1798


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The co-dispersion coefficients have also been obtained. These coefficients provide an interpretive
tool to analyze the correlation between the variation between two dates [76]:
ccij(h) =
γij(h)
q
γii(h) γjj(h)
If coefficients ccij(h) are constant, the correlation of the variable in two dates does not depend
on the spatial scale, which is referred to as intrinsic correlation [52]. However, if the correlation is
affected by spatial scale (ccij(h)) are not constant), it is necessary to cokrige the variable, as suggested
by the right-hand path [70]. The experimental co-dispersion function (ccij(h)) is represented in Figure 7
and shows the correlation coefficients between two dates by h-increments. Since a constant function
behavior is not observed, it is appropriate to use cokriging.
Figure 7. Co-dispersion coefficients between different years.
3.3. Estimation of Spatial Price Variation in Multi-Years
From the standpoint of the housing market, it is clearly of interest to determine spatio-temporal
variation in housing prices [24,77] at any location, and thus obtain flat isovalue maps of these
changes [18,78]. To do so, it is necessary to first estimate the price of housing for each of the years
observed at any point on the map. Since it is not possible to know the explanatory characteristics at
each point of the map, with the exception of the variable DIST (which was calculated for each point
to be estimated), a standard dwelling on which to make the prediction must be defined. A standard
dwelling is obtained by assigning the numerical value of the sample average of these characteristics to
each of the structural characteristics of the dwelling. Because the model estimates show the estimated
value of a standard dwelling in different locations in space, the spatial distribution of the prices
estimated with the proposed model are caused by the value of the location. In this work, a standard
dwelling is defined according to the following values:
AGE = 16, AREA = 113, BATH = 1, FLOOR = 0, ELEV = 1, HEAT = 1, SPORT = 0, and REHAB = 0.