1# RECORD 91a343b71c11d4139d2606d6c4d1c5fc64cbc6c32# TITLE: Estimation of Housing Price Variations Using Spatio-Temporal Data3# YEAR: 2019.0 VENUE: Sustainability DOI: 10.3390/su110615514# AUTHORS: Jorge Chica-Olmo; Rafael Cano-Guervos; M. Chica-Rivas5# PAGES: 21 CANDIDATE TABLE PAGES (1-based): [12, 14]6# 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.789===== PAGE 1 (score 6, numeric-paren 0, table-words 1, var-words 4) =====1011sustainability12Article13Estimation of Housing Price Variations Using14Spatio-Temporal Data15Jorge Chica-Olmo 1,*16, Rafael Cano-Guervos 1 and Mario Chica-Rivas 217118Department of Quantitative Methods for Economics and Business, University of Granada, 18011 Granada,19Spain; rcano@ugr.es20221Department of Mathematical Analysis, University of Granada, 18071 Granada, Spain; mcrivas@ugr.es22*23Correspondence: jchica@ugr.es24Received: 7 February 2019; Accepted: 11 March 2019; Published: 14 March 2019252627Abstract: This paper proposes a hedonic regression model to estimate housing prices and the spatial28variability of prices over multiple years. Using the model, maps are obtained that represent areas of the29city where there have been positive or negative changes in housing prices. The regression-cokriging30(RCK) method is used to predict housing prices. The results are compared to the cokriging with external31drift (CKED) model, also known as universal cokriging (UCK). To apply the model, heterotopic data32of homes for sale at different moments in time are used. The procedure is applied to predict the spatial33variability of housing prices in multi-years and to obtain isovalue maps of these variations for the city34of Granada, Spain. The research is useful for the fields of urban studies, economics, real estate, real35estate valuations, urban planning, and for scholars.36Keywords: hedonic regression model; cokriging; spatio-temporal; housing price variation371. Introduction38In the real estate field, multivariate spatial data models have been widely explored [1–5],39while multivariate spatio-temporal data models have received relatively less attention. The hedonic40regression model is used to make real estate valuations and to determine the characteristics that affect41property prices [6]. Classical estimation methods, such as ordinary-least-squares (OLS), ignore the42presence of spatial autocorrelation, since they assume the independence of the observations [7].43Two approaches are commonly used in spatial hedonic modeling: spatial econometrics [8] and44the geostatistical approach [9]. Spatial econometrics uses the spatial weight matrix and requires an45a priori specification of the matrix, which may affect the final results [10]. However, geostatistics is46a method in which the variance–covariance matrix depends on the Euclidean distance [11]. In the47case of spatial autocorrelation of the error terms, the spatial error model (SEM) is not better than the48geostatistical approach [12]. The literature suggests that the kriging geostatistical method improves49the prediction performance of spatial hedonic models [13]. Moreover, kriging models are superior to50other methods (OLS, SEM, spatial lag model, etc.) in terms of their prediction accuracy [14].51Several methods that consider both spatial and temporal dependences that simultaneously affect52housing prices have been proposed for the study of real estate. A classical method for considering53time-fixed effects is to include dummy variables indicating the year a house is being sold [15,16].54The spatio-temporal autoregressive (STAR) model and its variations [17–20] is another method that55consists of a spatio-temporal model based on a generalization of the traditional spatial autoregressive56model (SAR) [21]. This method has also been used to analyze housing prices in Spain [22,23].57Several authors have used geographically and temporally weighted regression to capture spatial58and temporal heterogeneity in real estate market data [24–27].59Sustainability 2019, 11, 1551; doi:10.3390/su1106155160www.mdpi.com/journal/sustainability616263===== PAGE 12 (score 42, numeric-paren 28, table-words 6, var-words 2) =====6465Sustainability 2019, 11, 15516612 of 2167The global drift multi-equation model ( ˆm(s0)) was estimated by GLS. The results of the estimation68are presented in Table 4. In the estimated equations, all the variables are significant at a confidence69level above or equal to 90%, with most being significant at 99%. All the signs of the coefficients are as70expected (negative relation in the AGE, DIST, REHAB, and FLOOR variables, and a positive relation in71the rest of the variables).72Table 4. Estimation results of multi-equation model by RCK (p-values in brackets).73198874199175199576200577Intercept781.042 × 101791.084 × 101801.041 × 101811.148 × 10182(0.000)83(0.000)84(0.000)85(0.000)86AGE87−2.119 × 10−288−7.828 × 10−389−8.497 × 10−390−6.176 × 10−391(0.000)92(0.000)93(0.000)94(0.000)95BATH968.929 × 10−2971.176 × 10−1986.929 × 10−2999.256 × 10−2100(0.020)101(0.000)102(0.000)103(0.000)104AREA1057.949 × 10−31065.782 × 10−31078.361 × 10−31085.951 × 10−3109(0.000)110(0.000)111(0.000)112(0.000)113DIST114−3.853 × 10−4115−2.532 × 10−4116−2.111 × 10−4117−2.257 × 10−4118(0.000)119(0.000)120(0.000)121(0.000)122REHAB123–124−1.517 × 10−1125–126−6.733 × 10−2127(0.000)128(0.008)129ELEV130–131–1321.552 × 10−11331.187 × 10−1134(0.000)135(0.000)136HEAT137–138–1397.528 × 10−21404.679 × 10−2141(0.000)142(0.060)143FLOOR144–145–146–147−1.056 × 10−1148(0.059)149SPORT150–151–152–1531.278 × 10−1154(0.011)155R2cv1560.85071570.87271580.93321590.8001160MAE1610.17931620.13711630.13621640.1545165MSE1660.05031670.03181680.03351690.0416170n171260172247173293174207175In this paper, the CKED and RCK methods are applied. In order to compare both methods, a176cross-validation method has been used, specifically leave-one-out cross-validation (LOOCV). LOOCV177allows for comparison of predicted and observed values using only the information available in178the sample dataset. The LOOCV procedure consists of temporarily discarding a sample value at179a particular location from the sample data set, and then estimating the value at the same location180using the remaining samples [55]. This procedure is repeated for all the experimental points in181order to compare the observed values to the predicted values using statistical and visual tools.182The cross-validation statistics used are mean absolute error (MAE), mean squared error (MSE), and183R-squared of cross-validation, which is obtained from the square of the correlation between the model’s184predicted values and the observed values (R2185cv). As can be seen in Table 5, the two methods show186similar cross-validation statistics, although those of the RCK method are slightly better, as the R2187cv value188is closer to one and the other statistics are closer to zero. Figure 5 shows the regression between the189predicted and observed data with RCK and CKED. As can be observed, the two scatterplots are quite190similar and the regression line is very close to the 1:1 line. Moreover, Figure 6 shows the RCK versus191the CKED predictions. As can be seen, the predictions obtained with both methods are very similar.192193194===== PAGE 13 (score 8, numeric-paren 0, table-words 4, var-words 0) =====195196Sustainability 2019, 11, 155119713 of 21198Figure 5. Regression between predicted and observed data. Note: The solid straight line represents the1991:1 line and the dashed line represents the regression line.200Figure 6. Comparison of RCK and CKED predictions.201Table 5.202Cross-validation statistics of CKED and RCK (R-squared of cross-validation, R2cv;203mean absolute error, MAE; mean squared error, MSE and sample size, n).204CKED205RCK206R2cv2070.92772080.9331209MAE2100.14002110.1355212MSE2130.03432140.0317215n21610072171007218Additionally, in order to compare the predictions of traditional OLS and cokriging, Table 6 shows219the OLS estimates for each of the equations. This permits quantification of the added value of the220cokriging method versus OLS. As can be observed in the table, all the variables are significant, as was221the case with RCK.222223224===== PAGE 14 (score 55, numeric-paren 28, table-words 12, var-words 3) =====225226Sustainability 2019, 11, 155122714 of 21228Table 6. OLS estimation for each equation (p-values in brackets).2291988230199123119952322005233Intercept2341.027 × 101 e+01235(0.000)2361.074 × 101237(0.000)2381.446 × 101239(0.000)2401.650 × 101241(0.000)242AGE243−2.033 × 10−2244(0.000)245−6.484 × 10−3246(0.000)247−1.035 × 10−2248(0.000)249−6.583 × 10−3250(0.001)251BATH2521.269 × 10−1253(0.004)2541.788 × 10−1255(0.000)2569.536 × 10−2257(0.002)2581.107 × 10−1259(0.002)260AREA2618.506 × 10−3262(0.000)2635.766 × 10−3264(0.000)2651.188 × 10−2266(0.000)2676.125 × 10−3268(0.000)269DIST270−3.521 × 10−4271(0.000)272−2.510 × 10−4273(0.000)274−2.685 × 10−4275(0.000)276−2.139 × 10−4277(0.000)278REHAB279–280−1.981 × 10−1281(0.000)282–283−6.925 × 10−2284(0.048)285ELEV286–287–2882.540 × 10−1289(0.000)2901.875 × 10−1291(0.000)292HEAT293–294–2951.550 × 10−1296(0.000)2977.201 × 10−2298(0.053)299FLOOR300–301–302–303−1.482 × 10−1304(0.079)305SPORT306–307–308–3091.376 × 10−1310(0.041)311R-squared3120.81763130.82783140.92343150.7745316R2cv3170.80603180.81733190.91803200.7442321MAE3220.20563230.16643240.15673250.1767326MSE3270.06553280.04573290.04113300.0556331n332260333247334293335207336This study has considered accessibility to the CBD, which is one of the main locational variables.337However, other relevant variables have been omitted (i.e., provision of public and private services,338socio-economic and environmental variables, etc.). Therefore, the differences observed between the339OLS coefficients (see Table 6) and RCK (see Table 4) may be due to this omission. It is difficult to specify340these locational factors and quantify their radius of influence, since they usually refer to areas whose341sizes and shapes tend to be subjective [30,74]. In fact, numerous studies on hedonic models have342mostly included structural variables and omitted relevant characteristics of the location [8,9,31,43,75].343Nevertheless, models that correct this omission by considering the presence of spatial autocorrelation344in disturbances, such as the SEM and RCK models, provide better results than OLS [4]. Therefore, an345advantage of the RCK model is that it improves the estimates by indirectly considering the omission of346relevant variables through modeling the correlation between disturbances.347Table 6 shows the results of the cross-validation. It is important to keep in mind that in traditional348OLS, all observations are used to fit the model, while this does not occur in cross-validation. Since the349R-squared of the OLS model is not directly comparable to the R2350cv of the RCK model, the R2351cv and other352statistics for the OLS models have been obtained using cross-validation (MAE and MSE). Table 7 shows353the improvement in % by RCK over OLS. Specifically, the R2354cv in the RCK model shows an almost 8%355improvement over OLS, while the MAE shows an improvement of approximately 18% and the MSE356of more than 30%. As can be seen, there is a clear improvement in the predictive ability of the RCK357model, thus supporting the added value of the spatial effect and the cross between space and time.358Table 7. Improvement in % by RCK over OLS.3591988360199136119953622005363R2cv3645.54593656.77843661.65573677.5114368MAE36912.791837017.608237113.082337212.5637373MSE37423.206137530.415737618.491537725.1798378379380===== PAGE 15 (score 3, numeric-paren 0, table-words 0, var-words 3) =====381382Sustainability 2019, 11, 155138315 of 21384The co-dispersion coefficients have also been obtained. These coefficients provide an interpretive385tool to analyze the correlation between the variation between two dates [76]:386ccij(h) =387γij(h)388q389γii(h) γjj(h)390If coefficients ccij(h) are constant, the correlation of the variable in two dates does not depend391on the spatial scale, which is referred to as intrinsic correlation [52]. However, if the correlation is392affected by spatial scale (ccij(h)) are not constant), it is necessary to cokrige the variable, as suggested393by the right-hand path [70]. The experimental co-dispersion function (ccij(h)) is represented in Figure 7394and shows the correlation coefficients between two dates by h-increments. Since a constant function395behavior is not observed, it is appropriate to use cokriging.396Figure 7. Co-dispersion coefficients between different years.3973.3. Estimation of Spatial Price Variation in Multi-Years398From the standpoint of the housing market, it is clearly of interest to determine spatio-temporal399variation in housing prices [24,77] at any location, and thus obtain flat isovalue maps of these400changes [18,78]. To do so, it is necessary to first estimate the price of housing for each of the years401observed at any point on the map. Since it is not possible to know the explanatory characteristics at402each point of the map, with the exception of the variable DIST (which was calculated for each point403to be estimated), a standard dwelling on which to make the prediction must be defined. A standard404dwelling is obtained by assigning the numerical value of the sample average of these characteristics to405each of the structural characteristics of the dwelling. Because the model estimates show the estimated406value of a standard dwelling in different locations in space, the spatial distribution of the prices407estimated with the proposed model are caused by the value of the location. In this work, a standard408dwelling is defined according to the following values:409AGE = 16, AREA = 113, BATH = 1, FLOOR = 0, ELEV = 1, HEAT = 1, SPORT = 0, and REHAB = 0.410