Using machine learning to identify spatial market segments. A reproducible study of major Spanish markets

David Rey Blanco (Idealista) Pelayo Arbués (Idealista) Fernando A. López (Universidad Politécnica de Cartagena) Antonio Páez (McMaster University)

Keywords:

• Hedonic prices
• market segments
• decision trees
• machine learning
• spatial econometrics
• reproducible research
• Spain

Abstract

Identifying market segments can improve the fit and performance of hedonic price models. In this paper, we present a novel approach to market segmentation based on the use of machine learning techniques. Concretely, we propose a two-stage process. In the first stage, classification trees with interactive basis functions are used to identify non-orthogonal and non-linear sub-market boundaries. The market segments that result are then introduced in a spatial econometric model to obtain hedonic estimates of the implicit prices of interest. The proposed approach is illustrated with a reproducible example of three major Spanish real estate markets. We conclude that identifying market subsegments using the approach proposed is a relatively simple and demonstrate the potential of the proposed modelling strategy to produce better models and more accurate predictions

Introduction

Hedonic price analysis is one of the most widely-used approaches for the study and valuation of properties in real estate markets. This approach is attractive due to its strong theoretical grounding and appealing interpretation (Rosen 1974). Indeed, when hedonic price models are estimated using multiple linear regression the coefficients of the model are thought to capture the implicit prices of attributes in a bundled good. In this way, while a room may lack an explicit price in the valuation of a property, the coefficient of a hedonic price model quantifies its implicit value. Such decomposition of the price of a bundled good into the implicit prices of its constituent parts is important for multiple reasons: this analysis is the industry standard for property assessment for tax purposes (Morillo, Cepeda, and Martinez-Cuevas 2017); these models are used to quantify the willingness to pay for non-market environmental amenities (Montero, Fernandez-Aviles, and Minguez 2018), including air quality and open spaces and similarly they can be used to assess the cost of disamenities (e.g., Graevenitz 2018).

The need to assess property values in a transparent, accurate, and precise way has led to numerous developments. A strand of research has aimed at enhancing the performance of models by incorporating spatial information. Geographic Information Systems (GIS) in particular have been used to make explicit some attributes of properties and their environments that might otherwise be overlooked (Paterson and Boyle 2002). The use of spatial data, in turn, has brought increased attention to the question of statistical sufficiency and therefore the need for approaches that appropriately consider the issues of spatial association and spatial heterogeneity in hedonic price analysis (Pace and Gilley 1997; A. Paez, Uchida, and Miyamoto 2001). As a result, there has been a proliferation of studies that apply spatial statistical or econometric methods to the issue of property valuation (A. Paez 2009). Recent applications include the use of hierarchical spatial autoregressive models (Cellmer, Kobylinska, and Belej 2019), moving windows approaches (Páez, Long, and Farber 2008), spatial filtering (Helbich and Griffith 2016), and kriging techniques (Montero-Lorenzo, Larraz-Iribas, and Paez 2009), among others.

In addition to interest in spatial data, the application of machine learning techniques for hedonic price analysis has also become an active topic of research. There are at least two distinct ways in which machine learning can be used for hedonic price analysis. In some studies, the role of machine learning algorithms is to process information that would otherwise be difficult or impossible to obtain using non-automated means. The information obtained is then used as an input in econometric hedonic models. For example, Humphreys, Nowak, and Zhou (2019) and Nowak and Sayago-Gomez (2018) used machine learning classifiers to ethnically profile buyers and sellers based on last names to understand whether potential cultural biases and/or discrimination issues exist in property transactions. In other research, machine learning algorithms replace the conventional hedonic price model (Hu et al. 2019; Yoo, Im, and Wagner 2012; Füss and Koller 2016). The evidence available shows that machine learning methods can perform remarkably well, but can also be seen as black boxes¹ with low interpretability (see page 25 in James et al. 2013).

Our objective in this paper is to introduce a novel approach that retains the interpretability of econometric approaches, but is enhanced by the identification of spatial market segments obtained from the use of machine learning techniques. We propose a two-stage approach. In the first stage, classification trees are implemented to identify homogeneous spatial market segments. The number of market segments is endogenous, and, compared to Füss and Koller (2016), the use of interactive basis functions (see Antonio Paez et al. 2019) can accommodate non-orthogonal and non-linear decision boundaries. The market segments are then introduced as covariates in an econometric model. This approach can potentially enhance the model without compromising its interpretability.

A reproducible case study of property values in three major markets in Spain helps to illustrate the proposed approach. Following recommendations for openness and reproducibility in geospatial research (Antonio Paez 2021), this paper is accompanied by a fully documented and open data product (see Arribas-Bel et al. 2021), and the code is embedded in a self-contained Rmarkdown document. The results show that modeling prices using the approach proposed to identify spatial market segments improves the fit of the models and can in addition enhance the quality of predictions.

Spatial Market Segmentation

The importance of housing submarkets has long been recognized in the literature (e.g. Rapkin, Winnick, and Blank 1953). Market differentiation can be the result of a variety of processes operating separately or in conjunction, including substitution, differentiation, and variations in consumer preferences (Galster 1996). In principle, this implies a degree of homogeneity within the market segment that differentiates it from other segments. According to Thibodeau (2003, 4–5) a spatial housing submarket “defines a geographic area where the price of housing per unit of housing service is constant”. Given the non-tradeable nature of location, research has shown the relevance of spatial market segments (Bourassa, Cantoni, and Hoesh 2007; Royuela and Duque 2013; Usman, Lizam, and Adekunle 2020).

Submarket analysis is often implemented in a pragmatic way, encompassing regional boundaries, for instance those of metropolitan regions, cities, or municipalities. It has long been recognized, though, that sub-markets may exist at smaller scales (e.g., Rapkin, Winnick, and Blank 1953). In particular, the pioneering work of Alonso (Alonso 1964) on urban structure led to the realization of the importance of geography in terms of differentiation of real estate property. Since then, vast amounts of empirical evidence have contributed to demonstrate just how commonplace differences in hedonic prices are at the intraurban scale. Concurrently, market segmentation has been shown to be not only a conceptually sound practice (see Watkins 2001), but also conducive to higher quality models and improved predictive performance, in particular when geography is explicitly taken into consideration (Páez, Long, and Farber 2008).

Numerous approaches have been proposed to identify market segments. Some are based on expert opinion, such as from appraisers (Wheeler et al. 2014). Many others are data-driven, using statistical or machine learning techniques (e.g., Helbich et al. 2013; Wu et al. 2018). Heuristic approaches also exist that exploit the latent homogeneity in values (Royuela and Duque 2013). Implementation of market segments in hedonic price models can be accomplished by means of fixed effects (i.e., dummy variables) for sub-regions (e.g., Bourassa, Cantoni, and Hoesh 2007), spatial drift by means of a trend surface (e.g. Pace and Gilley 1997), spatially autoregressive models (e.g., Pace et al. 1998), switching regressions (e.g., Islam and Asami 2011; A. Paez, Uchida, and Miyamoto 2001), multilevel and/or Bayesian models (e.g., Wheeler et al. 2014), or by means of spatially moving windows or non-parametric techniques to obtain soft market segments (Páez, Long, and Farber 2008; Hwang and Thill 2009). As is commonly the case, there is no one technique that performs consistently better than the alternatives in every case, since performance depends to some extent on the characteristics of the process being modeled (Usman, Lizam, and Adekunle 2020). It is therefore valuable to explore alternative approaches to identify and model market segments, to further enrich the repertoire of techniques available to analysts.

A recent proposal along these lines is due to Füss and Koller (2016), who suggest using decision trees to identify and model market segments. James et al. (2013) list some attractive features of decision trees. They are relatively simple to estimate and intuitive to interpret. They divide attribute space into a set of mutually exclusive and collectively exhaustive regions, and thus are ideally suited for market segmentation. By design, the regions generated are spatially compact and internally homogeneous. And they can outperform other regression techniques. Market segments derived from a decision tree can be used in combination with other modeling techniques, such as a second-stage tree regression (with fixed effects for the market segments from the preliminary tree regression), linear models, or models with spatial or spatio-temporal effects, such as space-time autoregression. Füss and Koller (2016) compare several different modeling techniques. Their findings confirm that introducing a form of market segmentation greatly improves prediction accuracy, and the use of tree-based market segments does so more than the use of an a priori zoning system defined by ZIP codes. Furthermore, accounting for residual spatial pattern in the form of a spatial autoregressive model further improves the accuracy of estimation.

The results reported by Füss and Koller (2016) are appealing. However, the modeling strategy that they implement inherits a limitation of tree regression, namely the relatively inflexible way in which attribute space is partitioned using recursive binary splits. What this means is that market segments obtained in this way are limited to linear and orthogonal boundaries (see page 1359 in Füss and Koller 2016). While prediction accuracy reportedly improves with tree-based segmentation of the market, it might be desirable to define market segments more flexibly, so that they are not constrained to rectangular shapes. Secondly, estimates of a regression tree are the mean of the values contained in the volume of a leaf, which means they are constants for each leaf. In a geographical application the leaves are mutually exclusive and collectively exhaustive partitions of geographical space. Using the residuals in the second step of the modelling strategy induces spatial autocorrelation, since all properties in the same segment will be given estimated residuals that are constants in each market segment. The issue here is that by introducing spatial autocorrelation in the second step some of the spatial information about location is obscured since there is zero spatial variation in the estimated residuals for a given market segment.

We address these two issues by using interactive basis functions (Antonio Paez et al. 2019) to induce non-orthogonal and non-linear decision boundaries in our models of market segments. Further, by moving the analysis of market segments to the first stage of the analysis, we obtain market segments with good homogeneity properties, and any spatial autocorrelation is dealt with by means of the spatial econometric model in the second step. The modelling strategy is described in more detail next.

Modeling Strategy and Methods

Modeling Strategy

We propose a two-stage modelling strategy, as follows:

1. Estimate a first stage classification tree using the prices and the coordinates of the observations only (similar to trend surface analysis, see Unwin 1978).

• Map the regions $R_m$ that result: these are the $m=1,\cdots,M$ submarkets.
• Overlay the observations on the tree-based regions and create a set of $m$ indicator variables for submarket membership: $I_m=I(y_i\in R_m)$; when the argument of the indicator function is true (i.e., when observation $y_i$ is in $R_m$) then $I_m=1$, otherwise $I_m=0$.

2. Estimate a second-stage hedonic price model that incorporates the indicator variables for submarkets obtained in first stage including spatial interaction effects and other relevant covariates.

Note that the modeling strategy proposed here differs from the one proposed by Füss and Koller (2016) in that the market areas are identified by these authors based on the residuals of a preliminary regression, whereas we identify them based on the prices directly. It is worth noting that these two strategies reflect different heuristics. Identification of market areas based on the prices implies that market areas are formed based on unitary properties before properties are assessed as bundles of attributes. Identification of market areas based on the residuals, on the other hand, implies that properties are first seen as bundles of attributes and that submarkets form based on other non-identified attributes.

Methods

Two methodologies are combined in the modeling strategy. For first-stage we apply the well-known algorithm of classification trees with the objective of identify spatial submarkets. The algorithm is applied using the variation suggested by Antonio Paez et al. (2019) to obtain non-orthogonal and non-linear boundaries via interactive basis functions. A short description of this method is found in the supplementary material. For the second-stage we apply spatial econometric methods to solve the presence of spatial autocorrelation in the residual of the classical hedonic models. The patial econometric models considered are also briefly described in the supplementary material. These methods are implemented in a number of open-source R packages. The tree R package (Ripley 2021) was used in first-stage and spsur (Lopez, Minguez, and Mur 2020) and spatialreg (Bivand et al. 2013) in second-stage to estimate spatial regression models. Finally, with the objective of evaluate the forecasting accuracies of the different models and avoid overfitting the data set is split in training and test subsamples. The training subsample is used to obtain the model and the test subsample to evaluate the forecasting. The R package spatialreg is used to get the out-of-sample predictions is a spatial econometric framework.

Data

The empirical examples to follow correspond to large cities in Spain. The real estate market is one of the most important sectors of the Spanish economy, and the largest urban areas in Spain are important points of reference for the real estate market in the country. The three largest markets are Madrid (the national capital with 3.2 million inhabitants), Barcelona (1.6 million), and Valencia (0.8 million inhabitants). The focus of our application is on property prices in these cities. Micro-data from official sources are not available in Spain; instead, we draw our data from an online real estate database, Idealista.com (the leading real estate portal in Spain).

The data are documented and prepared for sharing publicly in the form of an open data product (Arribas-Bel et al. 2021) under the structure of a R package free available from a repository and a data paper describe the full data set. The database is for postings during 2018, and the analysis uses the last quarter of the year. We use the asking price as a proxy for the selling price; this is common practice in many real estate studies (e.g., López, Chasco, and Gallo 2015; Chasco, Le Gallo, and López 2018). For the three data sets we consider the most frequent type of property in Spain, namely the flat (hereon termed “houses”); this excludes other types of properties, such as duplex, chalets, and attics, which conform separate real estate markets.

The data sets used in the analysis correspond to the last quarter of 2018, and include a total of $n=$ 44,270 for Madrid, $n=$ 23,334 for Barcelona, and $n=$ 14,018 for Valencia. The distribution of prices displays a long tail in all three cities, and following conventional practice it is log-transformed. The coordinates are converted from latitude and longitude to northing and easting in meters, and then rescaled and centered using the corresponding city’s Central Business District as a false origin. These transformations have no impact on the analysis, and rescaling and centering of the coordinates is necessary for the correct implementation of the interactive basis functions in decision trees (see Antonio Paez et al. 2019, 188–89).

For this research we select thirteen explanatory variables. Of these, ten attributes are data provided by Idealista.com and represent key structural attributes of the properties. These are whether the property is a studio (a small type of bachelor apartment), whether it is on the top floor of the building, and its built area, number of rooms, number of baths, and presence of a terrace. In addition, there are variables for elevator in the building, air conditioner, swimming pool, and parking spaces. We augment these attributes with locational variables derived from the coordinates of the property, including distance to nearest major transit station (metro), distance to the city center (central business district; CBD), and distance to major avenues. These locational attributes are frequently advertised by real estate agents and often capitalized in housing prices. Table gives the definitions of these variables and the descriptive statistics of the data.

Empirical Examples

Experimental Design

Each city’s data set is split into a training sample and a testing sample using a 7:3 proportion. The training samples are used to estimate the models and the testing samples are used to assess the out-of-sample performance of the models.

We consider four models. First is a Base Model:

The second is a base model with market segments (Base Model + MS):

The third is a spatial lag model (Spatial Model):

And finally, the most general is a spatial lag model with market segments (Spatial Model + MS):

Please note that all models nest in the Spatial Model + MS depending on what restrictions are placed on the parameters. In the Base Model:

In the Base Model + MS:

And in the Spatial Model:

The weights in the spatial weight matrices are calculated using the inverse of the distance between neighboring observations, so that closer observations receive a higher weight. To avoid increasing the density of the matrices [which has computational and also estimation effects], we combine this criterion with a cutoff of $k=6$ nearest neighbors. Given the distribution of distances in the sample, beyond these neighbors the inverse distance results in extremely small contributions to the autocorrelation effect. With respect to the interactive basis functions for the decision trees, we consider the following functions (see supplementary material) with $u$ and $v$ as the planar coordinates of the observations, easting and northing respectively:

We first look at the estimated models before discussing the in- and out-of-sample predictive performance of the models.

Modelling Results

The first model in each of Tables , , and is the Base Model. The fit of these models is reasonably high: the adjusted coefficients of determination are $R^2=$ 0.796, $R^2=$ 0.777, and $R^2=$ 0.795 for Barcelona, Madrid, and Valencia, respectively. These models are relatively naive in that they disregard both the possibility of spatial autocorrelation and spatial heterogeneity (in the form of spatial market sub-segments). They do provide a useful benchmark to compare the proposed modelling strategy.

The first stage of the modelling strategy is to train a decision tree on the property values using only the coordinates of the observations. The spatial market sub-segments derived from the decision trees are shown in Figure . It can be seen there that the algorithm detects seven market sub-segments in Barcelona, nine market sub-segments in Madrid, and eight in Valencia. These submarkets are compact, mutually exclusive, and collectively exhaustive. The smallest market segment is found in Valencia and has 331 recorded transactions; the largest market segment, in contrast, has 7,816 recorded transactions and is found in Madrid. The maps in the figure show how the use of interactive basis functions leads to non-orthogonal/non-linear boundaries for the sub-markets. In the case of Barcelona, there are some distinctive diagonal shapes reminiscent of the street pattern in the city. In Madrid there is a clear distinction given by the M-30 orbital that surrounds the central almond of the city; in addition, there is Paseo de la Castellana, a major north-south avenue that crosses the city. This avenue divides two zones in the north that tend to include more expensive real estate, whereas the south tends to be lower income and less expensive. In Valencia, the sub-markets identify several zones in the historical center of the city, and then larger regional patterns depending on proximity to the waterfront to the west of the city.

The spatial market sub-segments are coded as dummy variables in the data sets before re-estimating the Base Model with market segments (Base Model + MS). The second model reported in Tables , , and shows that the market segments tend to be highly significant, and also improve the fit of the model. In the case of Barcelona, the adjusted coefficient of determination changes to $R^2=$ 0.821, for a modest increase of 3.19%. The introduction of the market segments into the Base Model for Madrid results in an adjusted coefficient of determination of $R^2=$ 0.878, which represents a change of 13.09% relative to the adjusted coefficient of determination of the Base Model. In Valencia, the Base Model with market segments has an adjusted coefficient of determination of $R^2=$ 0.83, for an increase with respect to the Base Model of 4.49%.

It is well-known that spatial heterogeneity and association can co-exist (e.g., Bourassa, Cantoni, and Hoesh 2007; A. Paez, Uchida, and Miyamoto 2001). Sub-market identification can assist with spatial heterogeneity, but a process of spatial association could result from the common heuristic of comparative sales used by real estate agents. This process is appropriately represented by a spatial lag model. The third model reported in Tables , , and is the Spatial Model, that is the Base Model with a spatial lag (i.e., Equation ). Spatial lag models, being non-linear, lack the coefficient of determination of linear regression. Instead, their goodness of fit is evaluated using likelihood measures. It can be seen that there is a substantial improvement in this regard in all three cities.

The spatial lag parameter $\rho$ represents the proportion of the mean of the neighboring prices that is reflected in the price of the property at $i$. In Barcelona, this parameter suggests that approximately 32.16% of the mean of the price of the $k=6$ nearest neighbors is reflected in the price at $i$. This “comparative sales” effect is markedly stronger in Madrid, where it amounts to 47.89% of the mean price of the neighbors. In Valencia, this effect is 40.31%. The spatial lag parameter is significant in all three cases, and the results suggest that comparisons with other properties play a larger role in the determination of prices in Madrid.

The last model that we consider for these case studies is a spatial lag model with market segments. This is the most general of the four models, and we see that the combination of market segments and a spatial lag variable gives the best fit in terms of the log-likelihood, and also reduces the size of the spatial lag coefficient, shifting some of the spatial effect from spatial autocorrelation to spatial heterogeneity.

At this point, it is important to note that the coefficients of models with spatial lags cannot be interpreted as marginal effects due to the ripple effects of lagging variables (i.e., the multiplier effect of the lag). Instead, the direct, indirect, and total impacts need to be considered. The impacts of our best models (spatial models with market segments) are presented in Tables , , and .

Models Barcelona (Dependent Variable is log of Price)

	Base Model		Base Model + MS		Spatial Model		Spatial Model + MS
Variable	Estimate	p-val	Estimate	p-val	Estimate	p-val	Estimate	p-val
Property attributes
(Intercept)	11.9207	0.001	11.5877	0.001	7.9308	0.001	8.4258	0.001
CONSTRUCTEDAREA	0.0058	0.001	0.0053	0.001	0.0048	0.001	0.0048	0.001
ROOMNUMBER	0.0233	0.001	0.0255	0.001	0.0239	0.001	0.0243	0.001
BATHNUMBER	0.1351	0.001	0.1164	0.001	0.1026	0.001	0.1004	0.001
HASTERRACE	0.0785	0.001	0.0799	0.001	0.074	0.001	0.0752	0.001
HASLIFT	0.2302	0.001	0.1962	0.001	0.165	0.001	0.1583	0.001
HASAIRCONDITIONING	0.1095	0.001	0.1093	0.001	0.1017	0.001	0.1024	0.001
HASSWIMMINGPOOL	0.1572	0.001	0.1508	0.001	0.129	0.001	0.127	0.001
ISSTUDIO	-0.2628	0.001	-0.2568	0.001	-0.237	0.001	-0.2386	0.001
ISINTOPFLOOR	0.0408	0.0045	0.0476	0.001	0.0441	0.001	0.0456	0.001
HASPARKINGSPACE	0.1385	0.001	0.0806	0.001	0.0726	0.001	0.0585	0.001
DISTANCE_TO_CITY_CENTER	-0.1023	0.001	-0.0611	0.001	-0.0664	0.001	-0.0469	0.001
Market segments
market_segmentZ2			0.1572	0.001			0.0885	0.001
market_segmentZ3			0.2761	0.001			0.1697	0.001
market_segmentZ4			0.3654	0.001			0.2191	0.001
market_segmentZ5			0.4102	0.001			0.2381	0.001
market_segmentZ6			0.5074	0.001			0.2739	0.001
market_segmentZ7			0.5605	0.001			0.2541	0.001
Spatial lag parameter
rho					0.3216	0.001	0.2647	0.001
Model diagnostics
R-squared		0.8		0.82
adj-R-squared:		0.8		0.82
log-likelihood:		-781.44		303.91		989.45		1247.72

Models Madrid (Dependent Variable is log of Price)

	Base Model		Base Model + MS		Spatial Model		Spatial Model + MS
Variable	Estimate	p-val	Estimate	p-val	Estimate	p-val	Estimate	p-val
Property attributes
(Intercept)	11.8006	0.001	11.318	0.001	5.9368	0.001	8.1051	0.001
CONSTRUCTEDAREA	0.0055	0.001	0.0045	0.001	0.0038	0.001	0.0039	0.001
ROOMNUMBER	-0.0068	0.0087	0.0345	0.001	0.0223	0.001	0.0368	0.001
BATHNUMBER	0.1653	0.001	0.1163	0.001	0.0999	0.001	0.0958	0.001
HASTERRACE	-0.0098	0.0265	0.0459	0.001	0.0202	0.001	0.0436	0.001
HASLIFT	0.3809	0.001	0.2527	0.001	0.2175	0.001	0.2022	0.001
HASAIRCONDITIONING	0.1036	0.001	0.0878	0.001	0.0871	0.001	0.0841	0.001
HASSWIMMINGPOOL	0.2119	0.001	0.1961	0.001	0.0895	0.001	0.129	0.001
ISSTUDIO	-0.1746	0.001	-0.1827	0.001	-0.1497	0.001	-0.1671	0.001
ISINTOPFLOOR	0.0256	0.0565	0.0225	0.0233	0.0361	0.001	0.0296	0.0012
HASPARKINGSPACE	0.0885	0.001	0.1197	0.001	0.0598	0.001	0.0915	0.001
DISTANCE_TO_METRO	0.033	0.001	-0.0414	0.001	-0.0065	0.001	-0.0394	0.001
DISTANCE_TO_CITY_CENTER	-0.0474	0.001	-0.0484	0.001	-0.0301	0.001	-0.0394	0.001
DISTANCE_TO_CASTELLANA	-0.0631	0.001	0.0195	0.001	-0.0215	0.001	0.0175	0.001
Market segments
market_segmentZ2			0.0663	0.001			0.047	0.001
market_segmentZ3			0.2286	0.001			0.1643	0.001
market_segmentZ4			0.4721	0.001			0.3385	0.001
market_segmentZ5			0.4969	0.001			0.3501	0.001
market_segmentZ6			0.5343	0.001			0.366	0.001
market_segmentZ7			0.7173	0.001			0.4814	0.001
market_segmentZ8			0.7399	0.001			0.4986	0.001
market_segmentZ9			0.9617	0.001			0.6203	0.001
Spatial lag parameter
rho					0.4789	0.001	0.275	0.001
Model diagnostics
R-squared		0.78		0.88
adj-R-squared:		0.78		0.88
log-likelihood:		-12024.16		-2623.16		-4050.73		-338.92
Note:
0.001 in the p-values represents any value less than 0.001

Models Valencia (Dependent Variable is log of Price)

	Base Model		Base Model + MS		Spatial Model		Spatial Model + MS
Variable	Estimate	p-val	Estimate	p-val	Estimate	p-val	Estimate	p-val
Property attributes
(Intercept)	11.4979	0.001	10.8973	0.001	6.5688	0.001	7.1013	0.001
CONSTRUCTEDAREA	0.0066	0.001	0.006	0.001	0.0052	0.001	0.0051	0.001
ROOMNUMBER	-0.0528	0.001	-0.0392	0.001	-0.0393	0.001	-0.0339	0.001
BATHNUMBER	0.1619	0.001	0.1461	0.001	0.128	0.001	0.1269	0.001
HASTERRACE	0.0832	0.001	0.0872	0.001	0.0723	0.001	0.0756	0.001
HASLIFT	0.3022	0.001	0.3118	0.001	0.2231	0.001	0.2457	0.001
HASAIRCONDITIONING	0.1139	0.001	0.1047	0.001	0.096	0.001	0.093	0.001
HASSWIMMINGPOOL	0.3738	0.001	0.3418	0.001	0.1742	0.001	0.1937	0.001
ISSTUDIO	-0.0519	0.1331	-0.0727	0.0206	-0.042	0.1493	-0.0589	0.0342
ISINTOPFLOOR	0.074	0.0023	0.092	0.001	0.1104	0.001	0.1157	0.001
HASPARKINGSPACE	0.1301	0.001	0.1482	0.001	0.0961	0.001	0.1135	0.001
DISTANCE_TO_CITY_CENTER	-0.1556	0.001	-0.057	0.001	-0.0695	0.001	-0.0271	0.001
DISTANCE_TO_METRO	-0.1506	0.001	-0.1061	0.001	-0.0651	0.001	-0.0585	0.001
DISTANCE_TO_BLASCO	-0.1263	0.001	-0.0806	0.001	-0.0679	0.001	-0.0514	0.001
Market segments
market_segmentZ2			0.2071	0.001			0.1018	0.001
market_segmentZ3			0.3544	0.001			0.2164	0.001
market_segmentZ4			0.4348	0.001			0.2514	0.001
market_segmentZ5			0.3271	0.001			0.164	0.001
market_segmentZ6			0.6119	0.001			0.3907	0.001
market_segmentZ7			0.6346	0.001			0.3635	0.001
market_segmentZ8			0.7518	0.001			0.3814	0.001
Spatial lag parameter
rho					0.4031	0.001	0.3314	0.001
Model diagnostics
R-squared		0.79		0.83
adj-R-squared:		0.79		0.83
log-likelihood:		-2282.65		-1343.2		-830.93		-449.46
Note:
0.001 in the p-values represents any value less than 0.001

Impacts Spatial Model + MS Barcelona (Dependent Variable is log of Price)

Variable	Direct	p-val	Indirect	p-val	Total	p-val
Property attributes
CONSTRUCTEDAREA	0.005	0.001	0.002	0.001	0.007	0.001
ROOMNUMBER	0.025	0.001	0.009	0.001	0.033	0.001
BATHNUMBER	0.101	0.001	0.035	0.001	0.137	0.001
HASTERRACE	0.076	0.001	0.026	0.001	0.102	0.001
HASLIFT	0.160	0.001	0.055	0.001	0.215	0.001
HASAIRCONDITIONING	0.103	0.001	0.036	0.001	0.139	0.001
HASSWIMMINGPOOL	0.128	0.001	0.044	0.001	0.173	0.001
ISSTUDIO	-0.241	0.001	-0.084	0.001	-0.324	0.001
ISINTOPFLOOR	0.046	0.001	0.016	0.001	0.062	0.001
HASPARKINGSPACE	0.059	0.001	0.021	0.001	0.080	0.001
DISTANCE_TO_CITY_CENTER	-0.047	0.001	-0.016	0.001	-0.064	0.001
Market segments
market_segmentZ2	0.089	0.001	0.031	0.001	0.120	0.001
market_segmentZ3	0.171	0.001	0.059	0.001	0.231	0.001
market_segmentZ4	0.221	0.001	0.077	0.001	0.298	0.001
market_segmentZ5	0.240	0.001	0.083	0.001	0.324	0.001
market_segmentZ6	0.277	0.001	0.096	0.001	0.373	0.001
market_segmentZ7	0.257	0.001	0.089	0.001	0.346	0.001
Note:
0.001 in the p-values represents any value less than 0.001

Impacts Spatial Model + MS Madrid (Dependent Variable is log of Price)

Variable	Direct	p-val	Indirect	p-val	Total	p-val
Property attributes
CONSTRUCTEDAREA	0.004	0.001	0.001	0.001	0.005	0.001
ROOMNUMBER	0.037	0.001	0.014	0.001	0.051	0.001
BATHNUMBER	0.096	0.001	0.036	0.001	0.132	0.001
HASTERRACE	0.044	0.001	0.016	0.001	0.060	0.001
HASLIFT	0.203	0.001	0.076	0.001	0.279	0.001
HASAIRCONDITIONING	0.085	0.001	0.031	0.001	0.116	0.001
HASSWIMMINGPOOL	0.130	0.001	0.048	0.001	0.178	0.001
ISSTUDIO	-0.168	0.001	-0.062	0.001	-0.230	0.001
ISINTOPFLOOR	0.030	0.002	0.011	0.002	0.041	0.002
HASPARKINGSPACE	0.092	0.001	0.034	0.001	0.126	0.001
DISTANCE_TO_METRO	-0.040	0.001	-0.015	0.001	-0.054	0.001
DISTANCE_TO_CITY_CENTER	-0.040	0.001	-0.015	0.001	-0.054	0.001
DISTANCE_TO_CASTELLANA	0.018	0.001	0.007	0.001	0.024	0.001
Market segments
market_segmentZ2	0.047	0.001	0.018	0.001	0.065	0.001
market_segmentZ3	0.165	0.001	0.061	0.001	0.227	0.001
market_segmentZ4	0.340	0.001	0.127	0.001	0.467	0.001
market_segmentZ5	0.352	0.001	0.131	0.001	0.483	0.001
market_segmentZ6	0.368	0.001	0.137	0.001	0.505	0.001
market_segmentZ7	0.484	0.001	0.180	0.001	0.664	0.001
market_segmentZ8	0.501	0.001	0.186	0.001	0.688	0.001
market_segmentZ9	0.624	0.001	0.232	0.001	0.856	0.001
Note:
0.001 in the p-values represents any value less than 0.001

Impacts Spatial Model + MS Valencia (Dependent Variable is log of Price)

Variable	Direct	p-val	Indirect	p-val	Total	p-val
Property attributes
CONSTRUCTEDAREA	0.005	0.001	0.002	0.001	0.008	0.001
ROOMNUMBER	-0.035	0.001	-0.016	0.001	-0.051	0.001
BATHNUMBER	0.130	0.001	0.060	0.001	0.190	0.001
HASTERRACE	0.078	0.001	0.035	0.001	0.113	0.001
HASLIFT	0.252	0.001	0.115	0.001	0.367	0.001
HASAIRCONDITIONING	0.095	0.001	0.044	0.001	0.139	0.001
HASSWIMMINGPOOL	0.199	0.001	0.091	0.001	0.290	0.001
ISSTUDIO	-0.060	0.041	-0.028	0.041	-0.088	0.041
ISINTOPFLOOR	0.119	0.001	0.054	0.001	0.173	0.001
HASPARKINGSPACE	0.116	0.001	0.053	0.001	0.170	0.001
DISTANCE_TO_CITY_CENTER	-0.028	0.001	-0.013	0.001	-0.040	0.001
DISTANCE_TO_METRO	-0.060	0.001	-0.027	0.001	-0.087	0.001
DISTANCE_TO_BLASCO	-0.053	0.001	-0.024	0.001	-0.077	0.001
Market segments
market_segmentZ2	0.104	0.001	0.048	0.001	0.152	0.001
market_segmentZ3	0.222	0.001	0.102	0.001	0.324	0.001
market_segmentZ4	0.258	0.001	0.118	0.001	0.376	0.001
market_segmentZ5	0.168	0.001	0.077	0.001	0.245	0.001
market_segmentZ6	0.401	0.001	0.183	0.001	0.584	0.001
market_segmentZ7	0.373	0.001	0.171	0.001	0.544	0.001
market_segmentZ8	0.391	0.001	0.179	0.001	0.570	0.001
Note:
0.001 in the p-values represents any value less than 0.001

Predictive Performance: Comparison of Models

Prediction is a relevant concern in hedonic price analysis. Inspection of the results in Tables , , and suggest that the introduction of spatial market segments leads to markedly improved model fits. The measures of performance reported in these tables are based on the training sample exclusively. To conclude this investigation, in this section the predictive performance of the models is compared based on their performance using training (in-sample) as well as testing (out-of-sample) data sets. It is important to recall at this point that test data were not used in the calibration of the models discussed in the preceding sections.

The models without a spatially lagged dependent variable assume that the process is not spatially autocorrelated and therefore prediction requires only observations of the exogenous explanatory variables for the property to be assessed, since the price setting mechanism does not include information about the neighbors. In contrast, prediction with the models with a spatial lagged dependent variable require information regarding neighboring dependent and explanatory variables. This increases the data requirements and increase the computational complexity of prediction. Several approaches to spatial prediction with models that include spatially autocorrelated components are discussed in the literature (e.g., Goulard, Laurent, and Thomas-Agnan 2017); these are discussed briefly next.

In case of model () two types of prediction based on the data can be considered: in- and out-of-sample predictions. In this paper we follow Goulard, Laurent, and Thomas-Agnan (2017) proposal, as follows: we can reorder the observations in equation () to obtain the block matrix form below, where the subscript $S$ denote in-sample (training) data, and the subscript $O$ out-of-sample (testing) data:

The best predictor (BP) approach is:

where ${\hat Q_{SS}}={1 \over \hat \sigma^2}(I-\hat \rho {W'{SS}})(I - \hat \rho W´{SS})$, $\hat \rho$ is the in-sample spatial dependence estimate parameter and $\hat \sigma^2$ is the estimate variance.

There are four alternatives for out-of-sample prediction:

Of the four out-of-sample prediction methods we use the Best Predictor (BP) approach. Further detail on these alternatives can be found in Goulard, Laurent, and Thomas-Agnan (2017). These prediction methods are implemented in the R package spatialreg (Bivand et al. 2013).

We use several metrics of performance for comparison. Tables , , report the mean absolute error (mae), median absolute prediction error (mdae), root mean squared error (rmse), mean absolute prediction error (mape), median absolute prediction error (medape), bias, percent bias (pc_bias) and hit rates. The latter are the proportion of predictions smaller than a given absolute deviation in percentage. For instance, the 5% hit rate (hit_rate_5) of the linear model for Barcelona is a 98%, therefore 98% of all observations have an absolute percent error smaller than a 5%.

The results indicate that adding market segments and/or a spatially lagged variable improve the linear base model. The spatial model with market segments is comparable to or better than the spatial model without market segments. For example, the in- and out-of-sample predictions in Valencia perform very similarly in these two models. In Madrid and Valencia the results of the spatial model with market segment are superior for both the in-sample and the out-of-sample predictions.

Model performance comparison: Barcelona

estimator	split	n	mae	mdae	rmse	mape	medape	bias	pc_bias	hit_ratio_5	hit_ratio_10
Base Model
Base LM	Train	16325	0.191	0.152	0.254	0.015	0.012	0.000	0	0.980	0.999
Base LM	Test	6997	0.196	0.152	0.266	0.015	0.012	0.002	0	0.977	0.998
Base Model + Market Segments
LM MS	Train	16325	0.179	0.142	0.238	0.014	0.011	0.000	0	0.983	0.999
LM MS	Test	6997	0.184	0.144	0.249	0.014	0.011	0.002	0	0.981	0.998
Spatial Model
Spatial Model (BP)	Train	16325	0.162	0.126	0.218	0.013	0.010	0.000	0	0.986	0.999
Spatial Model (BP)	Test	6997	0.168	0.130	0.229	0.013	0.010	0.002	0	0.984	0.999
Spatial Model + Market Segments
Spatial Model + MS (BP)	Train	16325	0.161	0.124	0.216	0.013	0.010	0.000	0	0.987	0.999
Spatial Model + MS (BP)	Test	6997	0.167	0.130	0.227	0.013	0.010	0.002	0	0.984	0.999

Model performance comparison: Madrid

estimator	split	n	mae	mdae	rmse	mape	medape	bias	pc_bias	hit_ratio_5	hit_ratio_10
Base Model
Base LM	Train	30976	0.278	0.229	0.357	0.022	0.018	0.000	-0.001	0.924	0.998
Base LM	Test	13275	0.276	0.227	0.354	0.022	0.018	-0.003	-0.001	0.928	0.997
Base Model + Market Segments
LM MS	Train	30976	0.199	0.157	0.263	0.016	0.013	0.000	0.000	0.976	0.999
LM MS	Test	13275	0.199	0.156	0.266	0.016	0.012	-0.002	-0.001	0.975	0.998
Spatial Model
Spatial Model (BP)	Train	30976	0.181	0.136	0.246	0.014	0.011	0.000	0.000	0.978	0.999
Spatial Model (BP)	Test	13275	0.190	0.143	0.259	0.015	0.011	-0.002	-0.001	0.974	0.999
Spatial Model + Market Segments
Spatial Model + MS (BP)	Train	30976	0.172	0.130	0.233	0.014	0.010	0.000	0.000	0.983	0.999
Spatial Model + MS (BP)	Test	13275	0.176	0.133	0.242	0.014	0.011	-0.002	-0.001	0.980	0.998

Model performance comparison: Valencia

estimator	split	n	mae	mdae	rmse	mape	medape	bias	pc_bias	hit_ratio_5	hit_ratio_10
Base Model
Base LM	Train	9802	0.235	0.188	0.305	0.020	0.016	0.000	-0.001	0.950	0.998
Base LM	Test	4201	0.237	0.189	0.311	0.020	0.016	-0.001	-0.001	0.943	0.998
Base Model + Market Segments
LM MS	Train	9802	0.212	0.170	0.278	0.018	0.014	0.000	-0.001	0.965	0.999
LM MS	Test	4201	0.216	0.176	0.285	0.018	0.015	0.000	-0.001	0.963	0.998
Spatial Model
Spatial Model (BP)	Train	9802	0.185	0.143	0.247	0.015	0.012	0.000	0.000	0.972	0.999
Spatial Model (BP)	Test	4201	0.194	0.150	0.258	0.016	0.013	0.002	0.000	0.974	0.999
Spatial Model + Market Segments
Spatial Model + MS (BP)	Train	9802	0.183	0.143	0.243	0.015	0.012	0.000	0.000	0.976	0.999
Spatial Model + MS (BP)	Test	4201	0.191	0.148	0.253	0.016	0.012	0.002	0.000	0.976	0.999

Conclusions

Market segmentation is a topic of interest in the literature on real estate appraisal and valuation. In addition to being conceptually sound, numerous studies throughout the years have demonstrated that the practice of identifying market segments for hedonic price analysis can lead to higher quality models and enhanced performance.

The contribution of this paper has been to demonstrate a modelling strategy to obtain flexible tree-based market segments for use in spatial hedonic price modeling. Implementation of regression trees for market segmentation was proposed in a recent paper by Füss and Koller (2016). Our modelling strategy differs to the one proposed by these authors in two respects: 1) the use of decision trees with flexible (i.e. non-orthogonal and possibly non-linear market boundaries); and 2) the timing of the estimations of the market segments, which in the case of Füss and Koller (2016) is based on the residuals of an initial regression model, whereas in our case it is done in the first step of the modelling strategy.

The results using three large data sets from cities in Spain indicate that modelling the market segments can improve the fit of the models, as well as their predictive performance. The best model consistently included a spatially lagged dependent variable and market segments. The market segments in addition to improving the fit and the predictive performance also reduced the magnitude of the spatial lag parameter, thus allocating some of the spatial effect to regional heterogeneity that would otherwise be assumed to be micro-scale information spillovers. Overall, the results serve to demonstrate the potential of the proposed modelling strategy to produce better models and more accurate predictions.

One direction for future research is to investigate the temporal stability of spatial market segments. It is well known that there are seasonal effects in housing markets, but an open research question is whether spatial market segments experience seasonal variations, both in terms of their geographical extent as well as the magnitude of their effects. Another possibility is that there are longer term trends (e.g., gentrification) that could affect the spatial configuration of the market segments. Both seasonality and/or longer term trends would require multi-year data sets, compared to the single-year data set that we used for this research. For the time being, it is important to note that the results presented in this paper support the argument that the two-step method described in this paper performs well for now-casting or relatively short term forecasts. Given the dearth of information about seasonality and temporal stability of spatial market segments, any attempt to use them for longer term forecasts should be done with caution.

Finally, the study was designed as an example of reproducible research: all code and data used in this research is publicly available which should allow other researchers reproduce our results or expand them in other directions.

Supplemental material

Supplemental material for this article is available.

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Footnotes

1. An emergent body of research aims at increasing the interpretability of machine learning methods, including Du, Liu, and Hu (2019) and Murdoch et al. (2019), among others. This is an area of research that is quickly evolving, although it is not without critics (e.g., Rudin 2019). Currently, existing approaches depend on fairly strong assumptions. For example, the causal forest framework (Wager and Athey 2018; Knaus, Lechner, and Strittmatter 2021) assumes that the leaves of trees are sufficiently small to mimic a randomized experiment. Assuming independence is often inappropriate in the analysis of spatial data, and econometric techniques that correctly treat spatial dependencies are mature. It is possible that in the future interpretable machine learning techniques will address spatial dependencies as well, so we are advised to pay attention to this stream of research. ↩