# RECORD 1daf1d20ba429ff68e39043313dece9a679c8c5d
# TITLE: A spatial semiparametric M-quantile regression for hedonic price modelling
# YEAR: 2023.0  VENUE: AStA Advances in Statistical Analysis  DOI: 10.1007/s10182-023-00476-w
# AUTHORS: Francesco Schirripa Spagnolo; R. Borgoni; A. Carcagnì; A. Michelangeli; N. Salvati
# PAGES: 25  CANDIDATE TABLE PAGES (1-based): [16]
# ABSTRACT: This paper proposes an M-quantile regression approach to address the heterogeneity of the housing market in a modern European city. We show how M-quantile modelling is a rich and flexible tool for empirical market price data analysis, allowing us to obtain a robust estimation of the hedonic price function whilst accounting for different sources of heterogeneity in market prices. The suggested methodology can generally be used to analyse nonlinear interactions between prices and predictors. In particular, we develop a spatial semiparametric M-quantile model to capture both the potential nonlinear effects of the cultural environment on pricing and spatial trends. In both cases, nonlinearity is introduced into the model using appropriate bases functions. We show how the implicit price associated with the variable that measures cultural amenities can be determined in this semiparametric framework. Our findings show that the effect of several housing attributes and urban amenities differs significantly across the response distribution, suggesting that buyers of lower-priced properties behave differently than buyers of higher-priced properties.


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Vol.:(0123456789)
AStA Advances in Statistical Analysis (2024) 108:159–183
https://doi.org/10.1007/s10182-023-00476-w
1 3
ORIGINAL PAPER
A spatial semiparametric M‑quantile regression for hedonic 
price modelling
Francesco Schirripa Spagnolo1   · Riccardo Borgoni2 · Antonella Carcagnì2 · 
Alessandra Michelangeli2 · Nicola Salvati1
Received: 11 July 2022 / Accepted: 11 March 2023 / Published online: 30 March 2023 
© The Author(s) 2023
Abstract
This paper proposes an M-quantile regression approach to address the heterogeneity 
of the housing market in a modern European city. We show how M-quantile mod-
elling is a rich and flexible tool for empirical market price data analysis, allowing 
us to obtain a robust estimation of the hedonic price function whilst accounting for 
different sources of heterogeneity in market prices. The suggested methodology can 
generally be used to analyse nonlinear interactions between prices and predictors. In 
particular, we develop a spatial semiparametric M-quantile model to capture both 
the potential nonlinear effects of the cultural environment on pricing and spatial 
trends. In both cases, nonlinearity is introduced into the model using appropriate 
bases functions. We show how the implicit price associated with the variable that 
measures cultural amenities can be determined in this semiparametric framework. 
Our findings show that the effect of several housing attributes and urban amenities 
differs significantly across the response distribution, suggesting that buyers of lower-
priced properties behave differently than buyers of higher-priced properties.
Keywords  Robust method · Hedonic regression · Penalised splines · Spatial data
 *	 Francesco Schirripa Spagnolo 
	
francesco.schirripa@unipi.it
	
Riccardo Borgoni 
	
riccardo.borgoni@unimib.it
	
Antonella Carcagnì 
	
antonella.carcagni@unimib.it
	
Alessandra Michelangeli 
	
alessandra.michelangeli@unimib.it
	
Nicola Salvati 
	
nicola.salvati@unipi.it
1	
Università di Pisa, Pisa, Italy
2	
Università Degli Studi di Milano-Bicocca, Milan, Italy


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174
	
F. Schirripa Spagnolo et al.
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for penalised regression splines once a certain minimum value is reached (Pratesi 
et al. 2009; Ruppert et al. 2003).
All covariates act in an a priori predictable manner. Most of them are statistically 
significant and are valued differently at different points of the conditional distribu-
tion of house prices. This is also clearly displayed in Fig. 4, which shows the effect 
Table 3   Results of SSPMQ ( q = 0.10, 0.25, 0.50, 0.75, 0.90 ) estimated using 20 knots for f1q and 40 
knots for f2q - 훿= 3†
† Point estimates with standard errors in parentheses and the associated p-value: ∗∗∗p < 0.01 ; ∗∗p < 0.05 ; 
∗p < 0.1
Variable
훽0.10
훽0.25
훽0.50
훽0.75
훽0.90
Housing-specific characteristics
Intercept
3755.36***
4712.36***
5951.43***
6987.73***
7475.13***
(582.01)
(435.49)
(418.10)
(553.04)
(773.76)
Floor low
− 282.06*
− 278.25***
− 245.67**
− 181.39
− 59.53
(148.31)
(103.94)
(100.43)
(139.40)
(273.26)
Floor high
− 218.79
− 258.78***
− 227.98***
− 101.96
124.35
(145.36)
(102.90)
(100.03)
(139.94)
(276.12)
Lift
562.95***
579.63***
532.38***
575.58***
862.57***
(170.38)
(119.65)
(114.56)
(156.44)
(302.79)
Heating System
125.97
184.53
− 1.57
− 25.19
− 567.16
(195.78)
(135.17)
(30.98)
(181.81)
(345.35)
Parking area
1227.97*
1940.22***
2325.26***
1913.61***
1508.37
(721.31)
(462.08)
(441.55)
(602.36)
(1103.27)
Bathroom
841.78***
924.98***
896.58***
742.37***
824.39**
(178.24)
(126.16)
(123.13)
(174.75)
(355.76)
Housing Area
13793.34***
2229.88
− 18234.58***
− 38649.82***
− 59740.72***
(3749.06)
(2575.02 )
(2402.27)
(3718.76)
(7237.17)
Housing Area2
26441.33**
88902.99***
201195.88***
312160.71***
450974.91***
(12661.30)
(8845.19)
(8073.78)
(13545.17)
(25960.66)
Housing Area3
59418.29***
21609.05***
− 70502.26***
− 146289.17***
− 281400.47***
(11060.53)
(7778.60)
(6841.90)
(13132.12)
(24340.79)
Age
385.45**
467.61***
536.03***
662.14***
1622.88***
(162.86)
(115.82)
(112.99)
(159.36)
(316.73)
Urban amenities
Abandoned area
− 191.62
− 356.68***
− 506.70***
− 667.02***
− 1056.59***
(184.95)
(132.17)
(127.00)
(174.07)
(325.83)
University
808.94***
896.77***
927.38***
763.42***
513.88*
(146.14)
(107.38)
(106.50)
(154.85 )
(297.03 )
Metro
140.45
− 39.91
− 242.94**
− 402.89***
− 304.77
(143.41)
(103.14)
(101.00 )
(141.32)
(275.28)
Year
− 440.49***
− 428.92***
− 377.85***
− 356.10***
− 283.69
(130.27 )
(92.56 )
(89.97 )
(125.08 )
(246.91 )


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175
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A spatial semiparametric M‑quantile regression for hedonic…
of each variable at different M-quantiles. Grey bands represent the 95% confidence 
interval of the parameter of interest. We have also added a blue line to each graph to 
represent the regression coefficients obtained by fitting the standard semiparamet-
ric additive model on the mean specified in Equation (1). This allows us to assess 
whether and for which variables there is a heterogeneous effect on the distribution of 
response.
Some housing-specific attributes have a different impact at different levels of the 
price distribution, suggesting that their value is different at different points of the 
housing price distribution. The floor at which the house is located has been found 
to be statistically significant only for very low and medium-value houses. A non-
constant effect of floor level along the price distribution has also been found by 
Fig. 4   Plot of regression coefficients with their confidence intervals of all the covariates included in the 
SSPMQ model: Intercept (a), Floor low (b), Floor high (c), Lift (d), Heating system (e), Parking Area 
(f), Bathroom (g), Building’s age (h), Abandoned area (i), University (j), Metro (k) and Year (l). The 
plots are drawn using the estimates obtained for the five M-quantiles considered in table 3 and interpolate 
them linearly. The blue lines represent the regression coefficients obtained fitted a standard semiparamet-
ric linear model on the mean