1# RECORD 41c05e4c190bfd812f576bb4bc53a0c147331a7d2# TITLE: Accounting for Spatial Autocorrelation in Algorithm-Driven Hedonic Models: A Spatial Cross-Validation Approach3# YEAR: 2022.0 VENUE: Journal of real estate finance and economics DOI: 10.1007/s11146-022-09915-y4# AUTHORS: Juergen Deppner; Marcelo Cajias5# PAGES: 39 CANDIDATE TABLE PAGES (1-based): [32, 33]6# ABSTRACT: Data-driven machine learning algorithms have initiated a paradigm shift in hedonic house price and rent modeling through their ability to capture highly complex and non-monotonic relationships. Their superior accuracy compared to parametric model alternatives has been demonstrated repeatedly in the literature. However, the statistical independence of the data implicitly assumed by resampling-based error estimates is unlikely to hold in a real estate context as price-formation processes in property markets are inherently spatial, which leads to spatial dependence structures in the data. When performing conventional cross-validation techniques for model selection and model assessment, spatial dependence between training and test data may lead to undetected overfitting and overoptimistic perception of predictive power. This study sheds light on the bias in cross-validation errors of tree-based algorithms induced by spatial autocorrelation and proposes a bias-reduced spatial cross-validation strategy. The findings confirm that error estimates from non-spatial resampling methods are overly optimistic, whereas spatially conscious techniques are more dependable and can increase generalizability. As accurate and unbiased error estimates are crucial to automated valuation methods, our results prove helpful for applications including, but not limited to, mass appraisal, credit risk management, portfolio allocation and investment decision making.789===== PAGE 1 (score 3, numeric-paren 0, table-words 0, var-words 3) =====1011Vol.:(0123456789)12The Journal of Real Estate Finance and Economics (2024) 68:235–27313https://doi.org/10.1007/s11146-022-09915-y141 315Accounting for Spatial Autocorrelation in Algorithm‑Driven 16Hedonic Models: A Spatial Cross‑Validation Approach17Juergen Deppner1 · Marcelo Cajias1,2 18Accepted: 17 June 2022 / Published online: 13 July 2022 19© The Author(s) 202220Abstract21Data-driven machine learning algorithms have initiated a paradigm shift in hedonic 22house price and rent modeling through their ability to capture highly complex and 23non-monotonic relationships. Their superior accuracy compared to parametric 24model alternatives has been demonstrated repeatedly in the literature. However, the 25statistical independence of the data implicitly assumed by resampling-based error 26estimates is unlikely to hold in a real estate context as price-formation processes 27in property markets are inherently spatial, which leads to spatial dependence struc-28tures in the data. When performing conventional cross-validation techniques for 29model selection and model assessment, spatial dependence between training and 30test data may lead to undetected overfitting and overoptimistic perception of predic-31tive power. This study sheds light on the bias in cross-validation errors of tree-based 32algorithms induced by spatial autocorrelation and proposes a bias-reduced spatial 33cross-validation strategy. The findings confirm that error estimates from non-spatial 34resampling methods are overly optimistic, whereas spatially conscious techniques 35are more dependable and can increase generalizability. As accurate and unbiased 36error estimates are crucial to automated valuation methods, our results prove helpful 37for applications including, but not limited to, mass appraisal, credit risk manage-38ment, portfolio allocation and investment decision making.39Keywords Hedonic modeling · Machine learning · Spatial autocorrelation · Spatial 40cross-validation · Mass appraisal · Automated valuation models41 * Juergen Deppner 42 43juergen.deppner@irebs.de44 45Marcelo Cajias 46 47marcelo.cajias@irebs.de481 49University of Regensburg, IRE|BS International Real Estate Business School, Universitätsstraße 5031, 93053 Regensburg, Germany512 52PATRIZIA AG, Fuggerstraße 26, 86150 Augsburg, Germany535455===== PAGE 32 (score 44, numeric-paren 27, table-words 2, var-words 13) =====565726658 59J. Deppner, M. Cajias 601 361Appendix 162Table 6 OLS Regression Output63Panel A: OLS Model including Spatial Controls64Variable65Estimate66Std. Error67t-value68p value69Significance70(Intercept)71−74.91725.9273−12.65740.0075***76Continuous77 Living Area [log]780.80790.0180 81.98810.0082***83 Age [years]840.00850.0086−20.11870.0088***89 Age squared900.00910.0092 19.59930.0094***95 Entry date [years]96 −0.02970.0198 −2.12990.03100*101 Latitude11021.451030.12104 12.171050.00106***107 Longitude11080.671090.06110 11.681110.00112***113Discrete114 Rooms1150.041160.00117 10.051180.00119***120 Floor1210.001220.00123 5.291240.00125***126Binary [1 = yes, 0 = no]127 Bathtub128 −0.041290.00130 −9.781310.00132***133 Refurbished1340.021350.01136 3.031370.00138**139 Built-in kitchen1400.111410.01142 21.651430.00144***145 Balcony1460.021470.00148 3.841490.00150***151 Parking1520.031530.01154 4.881550.00156***157 Elevator1580.041590.01160 7.641610.00162***163 Terrace1640.041650.01166 5.991670.00168***169Distances170 NUTS centroid [km]171−0.021720.00173−13.021740.00175***176 Bakery [km]177−0.011780.01179 −1.221800.22181 Bar [km]182−0.021830.00184 −3.731850.00186***187 Biergarten [km]188−0.051890.00190−14.711910.00192***193 Café [km]194−0.021950.01196 −2.111970.03198*199 School [km]200−0.032010.01202 −2.352030.02204*205 Supermarket [km]2060.022070.01208 1.202090.23210 Bus station [km]211 −0.042120.00213−17.162140.00215***216Panel B: OLS model excluding Spatial Controls217(Intercept)2183.122190.04220 83.322210.00222***223Continuous224 Living Area [log]2250.842260.01227 76.622280.00229***230 Age [years]2310.002320.00233−18.162340.00235***236 Age squared2370.002380.00239 21.452400.00241***242 Entry date [years]243 −0.022440.01245 −2.252460.02247*248249250===== PAGE 33 (score 34, numeric-paren 8, table-words 6, var-words 14) =====2512522672531 3254Accounting for Spatial Autocorrelation in Algorithm‑Driven…2551. As modeling choices may differ depending on whether analysis or prediction is the main objective 256of a study, we concentrate primarily on prediction and do not wish to draw any causal inference or 257conclusions of the market under investigation.2582. Although the use of asking rents can been criticized since they may deviate from actual contract 259rents, multiple listing systems (MLS) provide a valuable data source for statistical learning 260applications due to their high frequency of occurrence and timely availability and have been 261repeatedly used in the algorithmic hedonic literature (e.g., Chiarazzo et al., 2014; Park and Bae, 2622015; Baldominos et al., 2018; Gröbel and Thomschke, 2018; Hu et al., 2019; Pérez-Rave et al., 2632019; Pace and Hayunga, 2020; Rico-Juan and Taltavull de La Paz, 2021). Considering vacancy 264rates for dwellings in Frankfurt well below 1 %, we can follow the rationale of Gröbel (2019) and 265assume that renters are price takers, such that there should be no notable differences between asking 266and contract rents. Besides that, deviations between asking and contract rents “[…] are not expected 267to lead to an error bias”, especially when hedonic characteristics are controlled for, as stated by 268Cajias (2018). We hence do not see any reason to question the validity of asking rents, especially in 269view of the objective of our study.2703. The systematic variation of rent formation processes across space should not introduce bias into 271cross-validation errors since machine learning algorithms do not assume fixed hedonic pricing 272coefficients but have the flexibility to differentiate between spatially heterogeneous environments. In 273this study, spatial heterogeneity is, therefore, put aside and the focus lies on spatial autocorrelation 274only.2754. Evidently, housing data not only exhibit high levels of spatial but also temporal dependence 276when pooled across time (Pace et al., 2000). In this study, we leave temporal aspects aside for future 277research and focus solely on space.278This table reports the ordinary least squares (OLS) regression outputs for the model including spatial 279controls in panel A and the model excluding spatial controls in panel B. The dependent variable is the 280log(rent), independent variables are listed in the left column accordingly. Significance codes: p < 0.001 281‘***’, p < 0.01 ‘**’, p < 0.05 ‘*’, p < 0.1 ‘.’, p > 0.1 “. Std. Error: standard error. Panel A: F-statistic 2822245.00 (p value: 0.0000), AIC -4450.02, BIC -4276.96. Panel B: F-statistic 2968.00 (p value: 0.0000), 283AIC -2592.44, BIC -2488.612841 The coefficients of the coordinates indicate that the rent in the study area increases on average ceteris 285paribus by 67% for one degree of longitude to the east and analogous by 145% for on the degree of lati-286tude to the north. We acknowledge that, in a linear context, interpretation of coordinates is very abstract 287and has limitations such as anisotropy across space. A more practical interpretation is that, in general, 288rents tend to increase towards the east/north of the city289Table 6 (continued)290Panel A: OLS Model including Spatial Controls291Variable292Estimate293Std. Error294t-value295p value296Significance297Discrete298 Rooms299 0.033000.00301 6.323020.00303***304 Floor305 0.013060.00307 6.433080.00309***310Binary [1 = yes, 0 = no]311 Bathtub312−0.053130.00314−10.783150.00316***317 Refurbished3180.033190.01320 4.533210.00322***323 Built-in kitchen3240.153250.01326 26.573270.00328***329 Balcony3300.023310.01332 2.943330.00334**335 Parking3360.013370.01338 1.523390.13340 Elevator3410.103420.01343 16.783440.00345***346 Terrace3470.033480.01349 4.063500.00351***352353354===== PAGE 34 (score 10, numeric-paren 0, table-words 2, var-words 6) =====355356268357 358J. Deppner, M. Cajias 3591 33605. One limitation of the selected k-means clustering cross-validation strategy (Brenning, 2012) is that 361full independence between training and test data can only be achieved if the distance between each pair 362of training and test observation exceeds the spatial autocorrelation range (Brenning, 2005; Le Rest et al., 3632014). This is unlikely to be the case for all observations in our resampling instantiation, especially for 364data points located at the borders of the spatial clusters. Nonetheless, we believe that the number of 365observations in each test fold is large enough to counteract structural overfitting, such that the impact on 366aggregated results should be minor. Following the suggestion of Roberts et al. (2017), we refrain from 367further reducing the number of k folds in the cross-validation procedure since this would withhold too 368much information during training and may introduce unnecessary extrapolation into the models.3696. All analyses and model estimations were executed using the open-source statistical programming 370language R under the version 4.0.4 (R Core Team, 2021). All machine learning algorithms and 371resampling techniques were employed using the mlr3 framework implemented by Lang et al. (2019), 372which is an ecosystem that facilitates a standardized interface to many existing packages in the R 373environment. To obtain reproducible results and to ensure that the instantiation of the resampling 374calls do not vary between the different models, which could distort the results, all outputs were 375produced with the same random number generator using the set.seed function in R.3767. Estimations were executed on a standard 1.80GHz processor with four cores, eight logical 377processors and eight gigabytes of RAM using a 64-bit Windows operating system. After 378parallelization, the in-sample estimation of the random forest required between 11 and 20 hours for 379each of the four estimated resampling alternatives, whereby spatial tuning in the inner resampling 380loop reduced estimation time by up to 43%. The much more efficient extreme gradient boosting 381algorithm needed only about 3 to 3.5 hours respectively with the spatial tuner being slightly less 382time-consuming. During the one quarter ahead prediction, where cross-validation only needs to 383be performed for hyperparameter tuning in the inner loop, estimation time dropped to 3.5 hours 384for the bagging algorithm and to approximately 30 minutes for the boosting algorithm. The spatial 385linear models required less than 30 minutes each and the least squares estimator less than a second. 386Estimation time was significantly lower for the alternative model specification B for all models.387Acknowledgments The authors especially thank PATRIZIA AG for contributing to this study. All state-388ments of opinions are those of the authors and do not necessarily reflect the opinion of PATRIZIA AG or 389its associated companies.390Authors’ Contributions All authors contributed to the study conception and design, analysis, and material 391preparation. All authors read and approved the final manuscript.392Funding Open Access funding enabled and organized by Projekt DEAL.393Data Availability The data were provided by PATRIZIA AG and are confidential.394Code Availability Custom code using R open-source software and packages (R Core Team, 2021).395Declarations 396Conflicts of Interest/Competing Interests The authors have no conflicts of interest to declare that are relevant to 397the content of this article. All authors certify that they have no affiliations with or involvement in any organiza-398tion or entity with any financial interest or non-financial interest in the subject matter or materials discussed in 399this manuscript. 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