# Artificial Neural Network, Quantile and Semi-Log Regression Modelling of Mass Appraisal in Housing

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{1}, X

_{2}, …, X

_{N}) and these inputs are weighted (W

_{1}, W

_{2}, …, W

_{3}). When the sum-product of the inputs and weights exceeds a threshold (θ

_{i}), the exceeded part is the input of the transfer function. This function is usually a sigmoid (Equation (1)) or tan-sigmoid function (Equation (2)).

- The weights and thresholds (t = 0) are randomly assigned.
- For any pattern (μ) of input data:
- Execute the network to obtain the output for the μ pattern.
- Obtain the errors in hidden and output layers.
- Calculate the increase of weight and threshold for each μ pattern.

- Calculate the total increase in all weights and the threshold for all patterns.
- Upgrade the weights and thresholds.

_{i}). Additionally, a median-based (quantile) estimator was also appealing, given that it is less sensitive to outliers than a mean-based estimator. Thus, the bias from unobserved characteristics (i.e., renovation, quality) should be smaller.

_{it}be the residual implied by the econometric model (Equation (4)). On the other hand, let q represent the target quantile from the distribution residuals. Thus, the quantile parameter estimates are the coefficients that minimise the following objective function:

- Mean squared error (MSE): the mean of the square distance between the target value and the estimated value. It measures the quality of the estimator by measuring the mean squared error of our estimations. The higher this value is, the worse the model is.

- Root mean squared error (RMSE): the square root of the MSE, i.e., it is calculated as the square root of the average of the quadratic differences between a variable and its estimation. RMSE is a measure of accuracy. It measures the amount of error between two datasets. To put it another way, it compares an estimated value and a known or observed value. This is one of the most commonly used statistics.

- Mean absolute error (MAE): the mean absolute distance between the target value and the estimated value, i.e., the average of the sum of the absolute differences between a variable and its estimation. The same scale as the data being measured is used in MAE. It is known as a scale-dependent measure of accuracy and, thus, it cannot be used to make comparisons between series using different scales.

- Mean absolute percentage error (MAPE): the mean absolute distance between the target value and the estimated value divided by the target value, i.e., the average sum of the relative difference between a variable and its estimation. It is a measure of the estimation’s accuracy. The mean absolute percentage error is an indicator of the performance of the demand estimation, which measures the size of the absolute error in percentage terms. It is useful even when the volume of demand for the product is not known since it is a relative measure.

- R-squared coefficient (R
^{2}): this provides information regarding to what extent the variance of a variable explains the variance of another variable. It is calculated as one minus the proportion between the square error from an estimation of a variable and the square error from the average of the same variable. It provides the measure of the accuracy of replication. The R^{2}is the indicator that allowed us to know how well these results can be estimated. Therefore, R^{2}is the variation percentage of the response variable that explains its relationship with one or more predictor variables. It can be said that, generally, the higher R^{2}is, the better the model fits the data.

## 3. Data

## 4. Results and Discussion

^{2}. We tested the models by means of a dataset of properties with transaction prices instead of appraisal prices, obtaining similar results (see Table A2).

^{2}coefficient, where only similar results were obtained using SLR methodology when the largest dataset is used. This is confirmed by Nghiep and Cripps [50], who determined that ANNs obtain results that are similar to those obtained using regression models when a large sample is used.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{i}, x

_{i}), i = 1, …, n, be a sample from some population, where x

_{i}is a K × 1 vector of regressors. It was assumed that:

## Appendix B

Property | Structure |
---|---|

Number of hidden neurons | The same number as the input layer |

Transfer function in the hidden layer | Tan-sigmoid |

Transfer function in the neuron of output layer | Linear |

Type of learning rule | Backpropagation with the Levenberg–Marquardt algorithm |

Control of overlearning | Early stop |

**Table A2.**Comparison of the performance of the ANNs, SLRs and QRs for the dataset with transaction prices.

Performance Measure | ANN 1 | ANN 2 | ANN 3 | ANN 4 | SLR 1 | SLR 2 | SLR 3 | SLR 4 | QR 1 | QR 2 | QR 3 | QR 4 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

MSE | 0.1666 | 0.1334 | 0.1056 | 0.1675 | 0.2221 | 0.1416 | 0.0671 | 0.0663 | 0.5646 | 0.4756 | 0.4170 | 0.4072 |

RMSE | 0.4082 | 0.3652 | 0.3249 | 0.4092 | 0.4345 | 0.3469 | 0.2389 | 0.2374 | 0.6927 | 0.6358 | 0.5954 | 0.5883 |

MAE | 0.3187 | 0.2819 | 0.2436 | 0.3163 | 0.3391 | 0.2758 | 0.1903 | 0.1721 | 0.5769 | 0.5436 | 0.4984 | 0.4875 |

MAPE | 0.0268 | 0.0237 | 0.0204 | 0.0274 | 0.0297 | 0.0199 | 0.0163 | 0.0148 | 0.0503 | 0.0480 | 0.0430 | 0.0423 |

R^{2} | 0.3072 | 0.4449 | 0.5609 | 0.4934 | 0.1783 | 0.4161 | 0.6436 | 0.6562 | 0.1429 | 0.1946 | 0.2269 | 0.2278 |

## References

- Akin, O.; Montalvo, J.G.; Villar, J.G.; Peydró, J.-L.; Raya, J.M. The real estate and credit bubble: Evidence from Spain. SERIEs
**2014**, 5, 223–243. [Google Scholar] [CrossRef] [Green Version] - San-José, L.; Retolaza, J.L.; Torres-Pruñonosa, J. Efficiency in Spanish banking: A multi-stakeholder approach analysis. Journal of International Financial Markets. Instit. Money
**2014**, 32, 240–255. [Google Scholar] [CrossRef] [Green Version] - San-José, L.; Retolaza, J.L.; Torres-Pruñonosa, J. Eficiencia social en las cajas de ahorro españolas transformadas en bancos [Social Efficiency in Savings Banks Transformed into Commercial Banks in Spain]. Trimest Econ.
**2020**, 87, 759–787. [Google Scholar] [CrossRef] - Griliches, Z. Price Indexes and Quality Change; Harvard University Press: Cambridge, MA, USA, 1971. [Google Scholar]
- Court, L.M. Entrepreneurial and consumer demand theories for commodity spectra. Econometrica
**1941**, 9, 135–162. [Google Scholar] [CrossRef] - Tinbergen, J. Some remarks on the distribution of labour incomes. Int. Econ. Pap.
**1951**, 1, 195–207. [Google Scholar] - Rosen, S. Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition. J. Politi Econ.
**1974**, 82, 34–55. [Google Scholar] [CrossRef] - Lancaster, K.T. A New Approach to Consumer Theory. J. Political Econ.
**1966**, 74, 132–157. [Google Scholar] [CrossRef] - Bartik, T.J. The Estimation of Demand Parameters in Hedonic Price Models. J. Politi Econ.
**1987**, 95, 81–88. [Google Scholar] [CrossRef] - Bin, O. A semiparametric hedonic model for valuing wetlands. Appl. Econ. Lett.
**2005**, 12, 597–601. [Google Scholar] [CrossRef] - Bover, O.; Velilla, P. Hedonic house prices without characteristics: The case of new multiunit housing. In ECB Working Paper 117; European Central Bank: Frankfurt, Germany, 2002. [Google Scholar]
- Garcia, J.; Raya, J.M. Price and Income Elasticities of Demand for Housing Characteristics in the City of Barcelona. Reg. Stud.
**2011**, 45, 597–608. [Google Scholar] [CrossRef] [Green Version] - Mendelsohn, R. Estimating the Structural Equations of Implicit Markets and Household Production Functions. Rev. Econ. Stat.
**1984**, 66, 673–677. [Google Scholar] [CrossRef] - Mills, E.S.; Simenauer, R. New Hedonic Estimates of Regional Constant Quality House Prices. J. Urban Econ.
**1996**, 39, 209–215. [Google Scholar] [CrossRef] - Palmquist, R.B. Estimating the Demand for the Characteristics of Housing. Rev. Econ. Stat.
**1984**, 66, 394–404. [Google Scholar] [CrossRef] - Kuminoff, N.V.; Parmeter, C.F.; Pope, J.C. Which hedonic models can we trust to recover the marginal willingness to pay for environmental amenities? J. Environ. Econ. Manag.
**2010**, 60, 145–160. [Google Scholar] [CrossRef] [Green Version] - Li, H.; Wei, Y.D.; Yu, Z.; Tian, G. Amenity, accessibility and housing values in metropolitan USA: A study of Salt Lake County, Utah. Cities
**2016**, 59, 113–125. [Google Scholar] [CrossRef] - Li, H.; Wei, Y.D.; Wu, Y.; Tian, G. Analyzing housing prices in Shanghai with open data: Amenity, accessibility and urban structure. Cities
**2019**, 91, 165–179. [Google Scholar] [CrossRef] - Bruegge, C.; Carrión-Flores, C.; Pope, J.C. Does the housing market value energy efficient homes? Evidence from the energy star program. Reg. Sci. Urban Econ.
**2016**, 57, 63–76. [Google Scholar] [CrossRef] - Wu, C.; Ye, X.; Du, Q.; Luo, P. Spatial effects of accessibility to parks on housing prices in Shenzhen, China. Habitat Int.
**2017**, 63, 45–54. [Google Scholar] [CrossRef] - Raya, J.M.; García-Estévez, P.; Prado-Román, C.; Torres-Pruñonosa, J. Living in a smart city affects the value of a dwelling? In Sustainable Smart Cities: Creating Spaces for Technological, Social and Business Development Innovation, Technology, and Knowledge Management; Peris-Ortiz, M., Bennett, D., Yábar, D.P.-B., Eds.; Springer: Berlin/Heidelberg, Germany, 2017; pp. 193–198. [Google Scholar]
- Pérez-Sánchez, V.R.; Mora-García, R.T.; Pérez-Sánchez, J.C.; Céspedes-López, M.F. La influencia de las caracte-rísticas de las viviendas de segunda mano en sus precios de venta: Evidencias en el mercado alicantino. Infor. Constr.
**2020**, 72, e345. [Google Scholar] [CrossRef] - Coulson, N.E.; McMillen, D.P. The Dynamics of Intraurban Quantile House Price Indexes. Urban Stud.
**2007**, 44, 1517–1537. [Google Scholar] [CrossRef] - García, J.; Raya, J.M. Use of a Gini index to examine housing price heterogeneity: A quantile approach. J. Hous. Econ.
**2015**, 29, 59–71. [Google Scholar] - McMillen, D.P. Changes in the distribution of house prices over time: Structural characteristics, neighborhood, or coefficients? J. Urban Econ.
**2008**, 64, 573–589. [Google Scholar] [CrossRef] - McMillen, D.P.; Thorsnes, P. Housing Renovations and the Quantile Repeat-Sales Price Index. Real. Estate Econ.
**2006**, 34, 567–584. [Google Scholar] [CrossRef] - Nicodemo, C.; Raya, J.M. Change in the distribution of house prices across Spanish cities. Reg. Sci. Urban Econ.
**2012**, 42, 739–748. [Google Scholar] [CrossRef] [Green Version] - Deng, Y.; McMillen, D.P.; Sing, T.F. Private residential price indices in Singapore: A matching approach. Reg. Sci. Urban Econ.
**2012**, 42, 485–494. [Google Scholar] [CrossRef] - Liao, W.; Wang, X. Hedonic house prices and spatial quantile regression. J. Hous. Econ.
**2012**, 21, 16–27. [Google Scholar] [CrossRef] - Kholodilin, K.A.; Ulbricht, D. Urban House Prices: A Tale of 48 Cities. Econ. Open-Access E-J.
**2015**, 9, 1–43. [Google Scholar] [CrossRef] [Green Version] - Waltl, S.R. Variation Across Price Segments and Locations: A Comprehensive Quantile Regression Analysis of the Sydney Housing Market. Real Estate Econ.
**2016**, 47, 723–756. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.; Yi, Y. What contributes to the rising house prices in Beijing? A decomposition approach. J. Hous. Econ.
**2018**, 41, 72–84. [Google Scholar] [CrossRef] - Peng, C.-W.; Tsai, I.-C. The long- and short-run influences of housing prices on migration. Cities
**2019**, 93, 253–262. [Google Scholar] [CrossRef] - Mora-Garcia, R.-T.; Cespedes-Lopez, M.-F.; Perez-Sanchez, V.R.; Marti, P.; Perez-Sanchez, J.-C. Determinants of the Price of Housing in the Province of Alicante (Spain): Analysis Using Quantile Regression. Sustainability
**2019**, 11, 437. [Google Scholar] [CrossRef] [Green Version] - Chien, M.-S.; Setyowati, N. The effects of uncertainty shocks on global housing markets. Int. J. Hous. Mark. Anal.
**2020**, 14, 218–242. [Google Scholar] [CrossRef] - McMillen, D.; Shimizu, C. Decompositions of house price distributions over time: The rise and fall of Tokyo house prices. Real. Estate Econ.
**2020**. [Google Scholar] [CrossRef] - Ekeland, I.; Heckman, J.J.; Nesheim, L. Identifying Hedonic Models. Am. Econ. Rev.
**2002**, 92, 304–309. [Google Scholar] [CrossRef] - Selim, H. Determinants of house prices in Turkey: Hedonic regression versus artificial neural network. Expert Syst. Appl.
**2009**, 36, 2843–2852. [Google Scholar] [CrossRef] - White, H. Economic prediction using neural networks: The case of IBM daily stock returns. In Proceedings of the IEEE International Conference on Neural Networks, San Diego, CA, USA, 24–27 June 1988; pp. 451–459. [Google Scholar]
- Din, A.; Hoesli, M.; Bender, A. Environmental Variables and Real Estate Prices. Urban Stud.
**2001**, 38, 1989–2000. [Google Scholar] [CrossRef] - Do, A.Q.; Grudnitski, G. A neural network approach to residential property appraisal. Real Estate Apprais.
**1992**, 58, 38–45. [Google Scholar] - Kauko, T. On current neural network applications involving spatial modelling of property prices. Neth. J. Hous. Environ. Res.
**2003**, 18, 159–181. [Google Scholar] [CrossRef] - Landajo, M.; Bilbao, C.; Bilbao, A. Nonparametric neural network modeling of hedonic prices in the housing market. Empir. Econ.
**2011**, 42, 987–1009. [Google Scholar] [CrossRef] - Limsombunchai, V.; Gan, C.; Lee, M. House Price Prediction: Hedonic Price Model vs. Artificial Neural Network. Am. J. Appl. Sci.
**2004**, 1, 193–201. [Google Scholar] [CrossRef] - Peterson, S.; Flanagan, A. Neural Network Hedonic Pricing Models in Mass Real Estate Appraisal. J. Real Estate Res.
**2009**, 31, 147–164. [Google Scholar] [CrossRef] - Tay, D.P.; Ho, D.K. Artificial Intelligence and the Mass Appraisal of Residential Apartments. J. Prop. Valuat. Invest.
**1992**, 10, 525–540. [Google Scholar] [CrossRef] - Curry, B.; Morgan, P.; Silver, M. Neural networks and non-linear statistical methods: An application to the modelling of price–quality relationships. Comput. Oper. Res.
**2002**, 29, 951–969. [Google Scholar] [CrossRef] - McGreal, S.; Adair, A.; McBurney, D.; Patterson, D. Neural networks: The prediction of residential values. J. Prop. Valuat. Invest.
**1998**, 16, 57–70. [Google Scholar] [CrossRef] - Worzala, E.; Lenk, M.; Silva, A. An Exploration of Neural Networks and Its Application to Real Estate Valuation. J. Real Estate Res.
**1995**, 10, 185–201. [Google Scholar] [CrossRef] - Nghiep, N.; Cripps, A. Predicting Housing Value: A Comparison of Multiple Regression Analysis and Artificial Neural Networks. J. Real Estate Res.
**2001**, 3, 313–336. [Google Scholar] [CrossRef] - Liu, J.-G.; Zhang, X.-L.; Wu, W.-P. Application of Fuzzy Neural Network for Real Estate Prediction. In International Symposium on Neural Networks; Springer: Berlin/Heidelberg, Germany, 2016; pp. 1187–1191. [Google Scholar]
- Abidoye, R.B.; Chan, A.P.C. Improving property valuation accuracy: A comparison of hedonic pricing model and artificial neural network. Pac. Rim Prop. Res. J.
**2017**, 24, 71–83. [Google Scholar] [CrossRef] - Štubňová, M.; Urbaníková, M.; Hudáková, J.; Papcunová, V. Estimation of Residential Property Market Price: Comparison of Artificial Neural Networks and Hedonic Pricing Model. Emerg. Sci. J.
**2020**, 4, 530–538. [Google Scholar] [CrossRef] - Mayer, M.; Bourassa, S.C.; Hoesli, M.; Scognamiglio, D. Estimation and updating methods for hedonic valuation. J. Eur. Real Estate Res.
**2019**, 12, 134–150. [Google Scholar] [CrossRef] [Green Version] - Jiang, C.; Jiang, M.; Xu, Q.; Huang, X. Expectile regression neural network model with applications. Neurocomputing
**2017**, 247, 73–86. [Google Scholar] [CrossRef] - Xu, C.; Chen, H. A hybrid data mining approach for anomaly detection and evaluation in residential buildings energy data. Energy Build.
**2020**, 215, 109864. [Google Scholar] [CrossRef] - Tyrväinen, L.; Miettinen, A. Property Prices and Urban Forest Amenities. J. Environ. Econ. Manag.
**2000**, 39, 205–223. [Google Scholar] [CrossRef] [Green Version] - Geoghegan, J. The value of open spaces in residential land use. Land Use Policy
**2002**, 19, 91–98. [Google Scholar] [CrossRef] - Cropper, M.L.; Deck, L.B.; McConnell, K.E. On the Choice of Funtional Form for Hedonic Price Functions. Rev. Econ. Stat.
**1988**, 70, 668–675. [Google Scholar] [CrossRef] [Green Version] - Tabales, J.N.; Ocerín, J.M.C.; Carmona, F.R. Artificial Neural Networks for Predicting Real Estate Prices. Rev. Métodos Cuantitativos Econ. Empresa
**2013**, 15, 29–44. [Google Scholar] - Tabales, J.N.; Carmona, F.R.; Ocerín, J.M.C. Precios implícitos en valoración inmobiliaria urbana. Rev. Constr.
**2013**, 12, 116–126. [Google Scholar] - Tabales, J.M.N.; Carmona, F.J.R.; Ocerin, J.M.C.Y. Redes neuronales (RN) aplicadas a la valoración de locales comerciales. Infor. Constr.
**2017**, 69, 179. [Google Scholar] [CrossRef] [Green Version] - Baldominos, A.; Blanco, I.; Moreno, A.J.; Iturrarte, R.; Bernardez, O.; Afonso, C. Identifying Real Estate Opportunities Using Machine Learning. Appl. Sci.
**2018**, 8, 2321. [Google Scholar] [CrossRef] [Green Version] - Edelstein, R.H.; Quan, D.C. How Does Appraisal Smoothing Bias Real Estate Returns Measurement? J. Real Estate Financ. Econ.
**2006**, 32, 41–60. [Google Scholar] [CrossRef] [Green Version] - García-Montalvo, J.; Raya, J.M. Constraints on LTV as a Macroprudential Tool: A Precautionary Tale. Oxf. Econ. Pap.
**2018**, 70, 821–845. [Google Scholar] [CrossRef] - Wang, D.; Li, V.J. Mass Appraisal Models of Real Estate in the 21st Century: A Systematic Literature Review. Sustainability
**2019**, 11, 7006. [Google Scholar] [CrossRef] [Green Version] - Torres-Pruñonosa, J.; Retolaza, J.L.; San-José, L. Gobernanza multifiduciaria de stakeholders: Análisis comparado de la eficiencia de bancos y cajas de ahorros. Revesco. Rev. Estud. Coop.
**2012**, 108, 152–172. [Google Scholar] [CrossRef] [Green Version] - San-José, L.; Retolaza, J.L.; Torres-Pruñonosa, J. Empirical evidence of Spanish banking efficiency: The stakeholder theory perspective. In Soft Computing in Management and Business Economics Studies in Fuzziness and Soft Computing; Gil-Lafuente, A.M., Gil-Lafuente, J., Merigó-Lindahl, J.M., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 153–165. [Google Scholar]
- de La Paz, P.T.; Gabrielli, L. Housing Supply and Price Reactions: A Comparison Approach to Spanish and Italian Markets. Hous. Stud.
**2015**, 30, 1036–1063. [Google Scholar] [CrossRef] - Dol, K.; Mazo, E.C.; Llop, N.L.; Hoekstra, J.; Fuentes, G.C.; Etxarri, A.E. Regionalization of housing policies? An exploratory study of Andalusia, Catalonia and the Basque Country. Neth. J. Hous. Environ. Res.
**2016**, 32, 581–598. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Instituto Nacional de Estadística. Available online: https://www.ine.es/ (accessed on 27 February 2021).
- Ministerio de Agricultura, Alimentación y Medio Ambiente del Gobierno de España. Tercer Inventario Forestal Nacional. Available online: https://www.miteco.gob.es/es/biodiversidad/servicios/banco-datos-naturaleza/informacion-disponible/ifn3.aspx (accessed on 27 February 2021).
- Instituto Geográfico Nacional. Available online: https://www.ign.es/web/ign/portal/inicio (accessed on 27 February 2021).
- Ministerio de Transportes, Movilidad y Agenda Urbana. Estimación de Precios de Suelo Urbano. Available online: https://www.fomento.gob.es/BE2/?nivel=2&orden=36000000 (accessed on 27 February 2021).
- Funahasi, K.I. On the approximate realization of continuous mapping by neural networks. Neural Netw.
**1989**, 3, 183–192. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer feedforward networks are universal approximators. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - del Brío, B.M.; Sanz, A. Redes Neuronales y Sistemas Borrosos; Ra–ma Editorial: Madrid, Spain, 1997. [Google Scholar]
- Minsky, M.; Papert, S. Perceptrons; The MIT Press: Cambridge, UK, 1969; pp. 1–20. [Google Scholar]
- Widrow, B.; Hoff, M.E. Adaptive Switching Circuits; Stanford Univ Ca Stanford Electronics Labs: Stanford, CA, USA, 1960. [Google Scholar]
- Vesanto, J.; Alhoniemi, E. Clustering of the self-organizing map. IEEE Trans. Neural Netw.
**2000**, 11, 586–600. [Google Scholar] [CrossRef] - Rosenblatt, F. The perceptron: A probabilistic model for information storage and organization in the brain. Psychol. Rev.
**1958**, 65, 386–408. [Google Scholar] [CrossRef] [Green Version] - Hecht-Nielsen, R. Theory of the backpropagation neural network. In Neural Networks for Perception; Academic Press: Cambridge, MA, USA, 1989; pp. 593–605. [Google Scholar]
- Sánchez-Serrano, J.R.; Alaminos, D.; García-Lagos, F.; Callejón-Gil, A.M. Predicting Audit Opinion in Consolidated Financial Statements with Artificial Neural Networks. Mathematics
**2020**, 8, 1288. [Google Scholar] [CrossRef] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Nat. Cell Biol.
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Li, Y.; Chen, W. A Comparative Performance Assessment of Ensemble Learning for Credit Scoring. Mathematics
**2020**, 8, 1756. [Google Scholar] [CrossRef] - Kohonen, T. Self-organized formation of topologically correct feature maps. Biol. Cybern.
**1982**, 43, 59–69. [Google Scholar] [CrossRef] - Demuth, H.; Beale, M.; Hagan, M. Neural Network ToolBox TM 6. User’s Guide; The MathWorks, Inc.: Natick, MA, USA, 2009. [Google Scholar]
- Wooldridge, J.M. Introductory Econometrics. A Modern Approach, 7th ed.; Cengage Learning: Boston, MA, USA, 2020. [Google Scholar]
- Gujarati, D.N.; Porter, D.C. Econometría, 5th ed.; McGraw Hill: New York, NY, USA, 2010. [Google Scholar]
- Buchinsky, M. Recent Advances in Quantile Regression Models: A Practical Guideline for Empirical Research. J. Hum. Resour.
**1998**, 33, 88–126. [Google Scholar] [CrossRef] - Vilchez, J.R. Destination and Seasonality Valuations: A Quantile Approach. Tour. Econ.
**2013**, 19, 835–853. [Google Scholar] [CrossRef] - Koenker, R.; Bassett, G., Jr. Regression quantiles. Econometrica
**1978**, 1, 33–50. [Google Scholar] [CrossRef] - Ministerio de Transportes, Movilidad y Agenda Urbana. Transacciones Inmobiliarias (Compraventa). Available online: https://www.fomento.gob.es/be2/?nivel=2&orden=34000000 (accessed on 27 February 2021).

**Figure 1.**Price per square meter in the free market of dwellings (1995–2020). Source: Ministry of Transport, Mobility and Urban Agenda (Ministerio de Transportes, Movilidad y Agenda Urbana) [74].

**Figure 2.**Housing Price Index (2007–2020). Source: Instituto Nacional de Estadística (INE) [71].

**Figure 3.**Representation of a neural network with three input data neurons, four hidden ones and one output data neuron

Variable | Type | Definition | |
---|---|---|---|

ln(Price_{A}) | Quantitative | Natural logarithm of the appraisal price | |

Hedonic variables | Height | Qualitative | The height of the house ranging from −2 to 19 |

Elevator | Dichotomous | Whether the access to the house is by means of an elevator (1 = yes, 0 = no) | |

Heating | Dichotomous | Whether the house uses a heating system (1 = yes, 0 = no) | |

Pool | Dichotomous | Whether the house or the residents’ association property includes a swimming pool (1 = yes, 0 = no) | |

Gardens | Dichotomous | Whether the house or the residents’ association property includes a garden (1 = yes, 0 = no) | |

Size | Quantitative | Constructed area of the house in square meters | |

Condition | Dichotomous | Physical state of the house (meaning 1 = good, 0 = bad) | |

Baths | Quantitative | Number of baths per house | |

Rooms | Quantitative | Number of rooms per house | |

PC | Qualitative | Postal code | |

Year | Quantitative | Year when the house was priced |

Variable | Dataset 1 | Dataset 2 | ||
---|---|---|---|---|

Mean | Std. Dev. | Mean | Std. Dev. | |

ln(Price_{A}) | 12.025 | 0.683 | 11.839 | 0.553 |

Size | 131.025 | 80.575 | 85.672 | 63.464 |

Baths | 1.595 | 0.703 | 1.295 | 0.660 |

Rooms | 3.127 | 0.855 | 2.533 | 1.233 |

N | 163,871 | 24,781 |

Variable | Dataset 1 | Dataset 2 | ||
---|---|---|---|---|

Median | Mode | Median | Mode | |

Height | 2.000 | 0.000 | 1.000 | 0.000 |

Elevator | 0.000 | 0.000 | 0.000 | 0.000 |

Heating | 1.000 | 1.000 | 0.000 | 0.000 |

Pool | 0.000 | 0.000 | 0.000 | 0.000 |

Gardens | 0.000 | 0.000 | 0.000 | 0.000 |

Condition | 1.000 | 1.000 | 0.000 | 0.000 |

N | 163,871 | 24,781 |

**Table 4.**Comparison of the performances of artificial neural networks (ANNs), semi-log regressions (SLRs) and quantile regressions (QRs) for dataset 1.

Performance Measure | ANN 1 | ANN 2 | ANN 3 | ANN 4 | SLR 1 | SLR 2 | SLR 3 | SLR 4 | QR 1 | QR 2 | QR 3 | QR 4 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

MSE | 0.2933 | 0.2750 | 0.1107 | 0.1096 | 0.3141 | 0.1622 | 0.1162 | 0.1162 | 0.8533 | 0.8332 | 0.3120 | 0.2994 |

RMSE | 0.5416 | 0.5244 | 0.3326 | 0.3311 | 0.5604 | 0.4028 | 0.3409 | 0.3409 | 0.9237 | 0.9128 | 0.5586 | 0.5472 |

MAE | 0.4273 | 0.4128 | 0.2269 | 0.2278 | 0.4458 | 0.3982 | 0.1983 | 0.1969 | 0.7911 | 0.7808 | 0.4763 | 0.4642 |

MAPE | 0.0361 | 0.0349 | 0.0192 | 0.0193 | 0.0371 | 0.0328 | 0.0188 | 0.0186 | 0.0772 | 0.0667 | 0.0432 | 0.0398 |

R^{2} | 0.3712 | 0.4105 | 0.7628 | 0.7651 | 0.3267 | 0.4680 | 0.8180 | 0.8200 | 0.2358 | 0.2452 | 0.4084 | 0.4124 |

Performance Measure | ANN 1 | ANN 2 | ANN 3 | ANN 4 | SLR 1 | SLR 2 | SLR 3 | SLR 4 | QR 1 | QR 2 | QR 3 | QR 4 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

MSE | 0.2088 | 0.1706 | 0.1238 | 0.1273 | 0.2294 | 0.1190 | 0.0866 | 0.0853 | 0.6363 | 0.5491 | 0.4900 | 0.4742 |

RMSE | 0.4569 | 0.4131 | 0.3519 | 0.3568 | 0.4790 | 0.3450 | 0.2943 | 0.2920 | 0.7977 | 0.7410 | 0.7000 | 0.6886 |

MAE | 0.3571 | 0.3196 | 0.2595 | 0.2633 | 0.3730 | 0.2711 | 0.2172 | 0.2159 | 0.6851 | 0.6311 | 0.5022 | 0.4979 |

MAPE | 0.0305 | 0.0272 | 0.0221 | 0.0224 | 0.0321 | 0.0218 | 0.0181 | 0.0175 | 0.0580 | 0.0550 | 0.0455 | 0.0445 |

R^{2} | 0.3171 | 0.4419 | 0.5951 | 0.5835 | 0.2492 | 0.5900 | 0.7081 | 0.7150 | 0.2129 | 0.2470 | 0.5351 | 0.5360 |

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**MDPI and ACS Style**

Torres-Pruñonosa, J.; García-Estévez, P.; Prado-Román, C.
Artificial Neural Network, Quantile and Semi-Log Regression Modelling of Mass Appraisal in Housing. *Mathematics* **2021**, *9*, 783.
https://doi.org/10.3390/math9070783

**AMA Style**

Torres-Pruñonosa J, García-Estévez P, Prado-Román C.
Artificial Neural Network, Quantile and Semi-Log Regression Modelling of Mass Appraisal in Housing. *Mathematics*. 2021; 9(7):783.
https://doi.org/10.3390/math9070783

**Chicago/Turabian Style**

Torres-Pruñonosa, Jose, Pablo García-Estévez, and Camilo Prado-Román.
2021. "Artificial Neural Network, Quantile and Semi-Log Regression Modelling of Mass Appraisal in Housing" *Mathematics* 9, no. 7: 783.
https://doi.org/10.3390/math9070783