Erasmus University Rotterdam
Erasmus School of Economics
Master Thesis
Data Science and Marketing Analytics
Hedonic Pricing Model for Rotterdam Housing Market
Empowered by Image Recognition and Text Analysis
Methods
Student name: Tomasz POTRAWA
Student ID number: 537444
Supervisor: Dr. A nastasija TETEREVA
Second assessor: Dr. Bas DONKERS
July 22, 2020
The views stated in this thesis are those of the author and not necessarily those of the supervisor, second
assessor, Erasmus School of Economics or Erasmus University Rotterdam.
Contents
1 Introduction 3
2 Theoretical background 6
2.1 Hedonic pricing theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Application of hedonic pricing for housing market . . . . . . . . . . . . . . . . . . . . 8
2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Methodology 15
3.1 Data extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Text analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Hedonic linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Random Forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Model-Agnostic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Variable Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Partial Dependence Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Local Interpretable Model-Agnostic Explanations (LIME) . . . . . . . . . . . . . . . 22
4 Data and feature extraction 24
4.1 Initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Image recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Text analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Analysis and results 35
5.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Explanatory analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Conclusions 49
7 Appendix 50
1
List of Figures
1 The accuracy of the model versus the number of epochs . . . . . . . . . . . . . . . . 28
2 Correlation plot for numerical variables. . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Residuals versus Fitted Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 The Partial Dependence Plot of Living Area . . . . . . . . . . . . . . . . . . . . . . . 44
5 The absolute value of a view on the city versus predicted rental cost . . . . . . . . . 47
6 The absolute value of insolation versus predicted rental cost . . . . . . . . . . . . . . 47
7 The absolute value of water body proximity versus predicted rental cost . . . . . . . 48
8 Auto-correlation plot for residuals of linear model . . . . . . . . . . . . . . . . . . . . 53
9 Normal QQ plot for linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
10 Residuals versus fitted values after Box-Cox transformation . . . . . . . . . . . . . . 54
11 Variable importance of the final random forest model . . . . . . . . . . . . . . . . . . 55
List of Tables
1 List of Commonly Used Housing Attributes in Hedonic Price Models (Chin & Chau,
2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Variables present in the initial data set . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Accuracy of the image recognition models . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Accuracy of the view image recognition model on the property level . . . . . . . . . . 30
5 Descriptive statistics 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Descriptive statistics 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 The coefficients of the linear model empowered by image and text features . . . . . . 38
8 The coefficients of the final linear model . . . . . . . . . . . . . . . . . . . . . . . . . 39
9 The summary of regression models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10 The comparison of variables’ impact on the prediction between the models . . . . . . 43
11 Frequency of the words appearing in the descriptions . . . . . . . . . . . . . . . . . 50
12 Generalized Variance Inflation Factor for linear regression . . . . . . . . . . . . . . . 52
2
Abstract
This study primarily aimed to address the question of whether using relevant images and
descriptions of rental apartments increases the performance of a hedonic pricing model for the
Rotterdam housing market. Secondarily, an attempt on deriving the hedonic prices of housing
attributes and their dependence on the used regression model was made. With the usage of
convolutional neural nets and text analysis methods, features related primarily to the external
attributes of properties were extracted and transformed into tabular data. Two models were
applied, the OLS regression and the random forest. In both cases using the extracted variables in
addition to the more traditional predictors led to increases in accuracy and explained variation. A
comparison of accuracy metric revealed significant superiority of the random forest over its linear
counterpart. On the contrary to the previous research conducted in the field, the comparison
between both methods was not limited solely to the accuracy metric and included the comparison
of models’ coefficients acquired with the model agnostic methods. It was found that the OLS
regression model, when compared to the random forest, is more likely to overestimate the value
of nonessential structural attributes such as garden or furnishings. This may be a reflection of
the model’s reduced capability of capturing locational aspects. Lastly, based on the obtained
data, and the usage of local interpretable model-agnostic explanations, it was concluded that
the hedonic price of a particular housing attribute is unlikely to be constant and its variability
is dependent on the total value of a property.
Keywords: hedonic price model, housing attributes, image recognition, model-agnostic methods
1 Introduction
The analysis of the real estate market has always been an object of remarkable interest among
researchers. The reason behind this trend is the fact that household prices affect directly the level
of life of a significant part of society. Furthermore, the fluctuations in the real estate market reflect
accurately the macroeconomic phenomena happening in the world, thus making them an interesting
benchmark in the analysis of other social and economical aspects. There have been many different
approaches taken in explaining the household prices so far. One of the most popular methods,
hedonic price modeling, claims that a good which in case of a housing market would be a real
estate, does not possess any utility by itself. Instead the good consists of characteristics that posses
constituted utility (Lancaster, 1966). As such, by comparing the prices of goods that do possess a
given attribute and the goods that do not, it becomes possible to derive the value of the attribute
in the eyes of consumers.
This characteristic of hedonic pricing made it a perfect tool in estimating the value of hardly
tangible assets. In numerous research, applying hedonic pricing on the housing market allowed
measuring the value of aspects such as air quality, green area proximity, a view on the ocean, or the
criminality level in the neighborhood. Hedonic models aim to quantify the importance of these less
tangible assets, therefore making the interpretability of the model a top priority. Consequently, the
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majority of papers related to hedonic pricing are based on simple statistical models such as linear
regression where the constituted value of the attribute of interest may be easily derived by analyzing
the regression coefficients. Models that could offer more predictive power like neural network or
random forest have remained relatively unpopular in this area of study, due to the complexity of
their interpretation, leaving a space for further research.
Additionally, most of the hedonic price modeling research have been based on simple, tabular
data, obtained often from institutions such as municipalities or real estate agencies. Only recently
the interest in more complex data extraction from sources such as texts and images started growing.
While the automation of the mentioned data extraction is definitely not an easy task, in the long run,
mastering it will allow decreasing the costs of gathering complex data and increase the versatility
and accuracy of future research.
Lastly, in comparison to the traditional housing market, there is relatively little research done
on the rental prices. It applies especially for the measurement of external and locational factors
impacting the rental price of a property such as the proximity of green areas, view from a property,
nearby services etc. While it may be argued that the mentioned type of variables has probably lower
impact on the rental prices than on the selling prices of real estates, it is rather unlikely that these
attributes do not contribute at all to the former.
As such, it has been concluded, that there are a few gaps in the hedonic pricing literature that
this paper aims to fill. First and foremost, the study addresses the functionality of images and texts
in the hedonic pricing of Rotterdam’s rental market. Therefore, the main research question may be
formulated in the following way:
Does including the features that are automatically extracted from the relevant image and text
sources significantly increase the accuracy of the hedonic pricing model of Rotterdam’s rental
housing market?
Additionally, the study focuses on measuring the impact of external factors such as the neighbor-
hood characteristics on the rental prices in Rotterdam. The analysis extends the previous research
of Law et al. (2019) as in order to address the question above, not only maps but also rental offer
photos serve as a source of information for the study.
Furthermore, the comparison between the accuracy of traditional hedonic linear regression and
complex machine learning methods is carried out. While this kind of comparison is definitely not
new in the research area of the housing market, to the best knowledge of the author none of the
previous papers studied the differences in the impact of external variables on the prediction across
the models. The lack of such an approach in the literature may be caused by the cumbersomeness
and the moderate reliability of the methods aiming at explaining the black box model behaviour.
As such it is understandable that the previous research opted to focus on the easily measurable and
comparable accuracy of the models.
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Nevertheless in the eyes of the author the blind pursuit of the best possible accuracy without
understanding how the prediction is being made is not completely relevant in the context of hedonic
pricing. One could argue that the ability to quantify the constituent value of goods’ characteristics
is the essence of the traditional hedonic model. Once this ability is lost it becomes infeasible to
use the model in deriving the value of assets such as previously mentioned air quality or green area
proximity. Therefore, with the usage of model agnostic methods the paper aims to prove that the
more complex, but often more accurate black box models may also be used to derive the value of
goods’ attributes. The importance of housing attributes in advanced machine learning algorithms
is compared with the traditional hedonic OLS regression coefficients. Furthermore, based on a non-
linear regression model, it is measured how hedonic prices of given attributes change, depending on
the price of a property.
It is worth mentioning here that the study has been designed in a way that for the most part
its methodology and findings should be relatively easily applicable to other cities. Rotterdam is a
major European city with diverse ethnic and religious groups, highly-developed services as well as
the transportation sector and reputable educational institutions. As such, the phenomena specific
for the Rotterdam’s housing market may at least partially be true for many other cities in the world.
Moreover, the data used in the study has been acquired from the rental websites and for the most
part, represents features and attributes valid for any other housing market. This further facilitates
conducting similar research for other cities.
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2 Theoretical background
In order to prepare for addressing the questions and problems defined for this research, proper
groundwork in the form of a broad literature review was laid. The following section provides an
in-depth overview of what has been already done in the literature with respect to hedonic methods
and real estate appraisal. The first part aims to provide the theoretical background and assumptions
of the hedonic price modeling. Subsequently, the summary of the most relevant real estate appraisal
papers and the insights gathered from them is presented. The last part of this chapter focuses on the
methodology used in particular research, and the comparison between statistical models commonly
used in the area of house market pricing.
2.1 Hedonic pricing theory
Hedonic pricing is a method rooted in the theory of consumer demand developed by Lancaster
(1966). In his work, the author breaks away from the back-then traditional assumption claiming
that a good by itself possesses utility. Lancaster’s innovative approach may be summarised by
three main statements. Firstly, a good, per se, does not provide any utility. Instead, each good
consists of characteristics that posses constituted utility. Secondly, a good is usually characterized
by many attributes, and a single attribute is most of the time shared by multiple goods. Lastly,
according to the author, the combination of goods may possess different characteristics than the
goods would have, if evaluated separately. The value of each attribute in Lancaster’s work is based
on the relationship between the observed prices of the goods and the number of attributes associated
with them (Chin & Chau, 2003).
Even though hedonic pricing is heavily influenced by Lancaster’s theory of demand from 1966, it
was officially formulated a few years later by Rosen (1974). Despite the fact that both Rosen’s and
Lancaster’s models are based on similar assumptions, they do have some significant differences. In the
Lancastrian model, it is claimed that goods, and consequently their characteristics, are aggregated
in groups, which subsequently are purchased by the customer. On the other hand, Rosen states that
goods are usually not acquired in combinations. Instead, for each good customer makes a separate
decision after analyzing the spectrum of brands offering the given good. While both models do not
necessarily contradict each other, they do seem to fit better different kinds of products. Chin & Chau
(2003) argues, that Rosen’s hedonic model is more suited to durable goods while the Lancastrian
model may explain better phenomena related to consumer goods.
Another dimension on which both authors do not come to similar conlusion is the type of rela-
tionship between the price of a good and its characteristics. In Lancaster’s work it is presumed that
the mentioned relation is linear, whereas Rosen postulated that it is more likely for this relationship
to be non-linear. The latter, implies that the implicit price of an attribute is non-constant and
according to Rosen depends on the interaction between the supply and the demand. More precisely,
6
in Rosen’s model the marginal implicit price of an attribute is presented as the partial derivative of
the price function over the given attribute, which always has to be equal to the marginal cost of an
extra unit of the attribute in the market and the individual consumer’s marginal willingness to pay
for the attribute. The latter aspect, which could also be simply called a demand price, is assumed
to be a function of the buyer’s income, the utility level and other variables describing particular
consumer such as tastes and preferences, age or education (Chin & Chau, 2003).
Despite the fact, that Rosen’s model is still seen as a landmark paper, allowing a better under-
standing of multiple fields of economics such as urban, environmental, or labor economics (Green-
stone, 2017), it does suffer from multiple empirical issues. The first is the choice of the functional
form. Throughout the years, multiple types of forms have been applied to hedonic models e.g. lin-
ear, semi-log, or Box-Cox transformation-based. Nevertheless, there is little research on how the
functional function should be chosen (Butler, 1982). Moreover, even with applying more complex
techniques such as Box-Cox transformation aiming at normalizing the distribution of the data, it is
not guaranteed that the transformed data will suit the model and its assumptions. On the other
hand, applying more advanced machine learning methods with less strict requirements may lead
to a situation where measuring the marginal implicit price of an attribute may be challenging or
sometimes even impossible.
Another issue related to hedonic pricing models is the problem of incorrect choice of the at-
tributes, also knows as the misspecification of the variables. In cases when the non-relevant predictor
is included in the model we refer to it as an over-specification, whereas the opposite situation, when
a key variable is not included in the analysis is known as an under-specification. While both cases
are obviously not welcome, it may be argued which one causes more concern and consequently bias
more the results of the hedonic model. Chin & Chau (2003) argues that as a result of an over-
specification predictors are still unbiased and consistent, however, some of them are inefficient. An
under-specification on the other hand results in an estimated biased and inconsistent coefficients.
In the opinion of Butler (1982) as all the estimates of a hedonic price model are in some degree
misspecified, in general, the models with a small number of key attributes should perform well
enough. Overall, Butler suggests to use in the regression only the predictors that are expensive in
producing and are likely to yield additional value in the form of utility. To similar conclusion came
Mok et al. (1995) who concurred that the bias introduced to a model due to missing attribute of a
good is usually small, and does not cause a significant drop in model’s performance in the terms of
prediction and explanatory power. In addition, the author discourages using proxy predictors in a
hedonic price model as they result in biased and inconsistent outcome of an analysis.
The mentioned drawbacks of hedonic pricing are not the only issues related to this method.
The description of each one of them is not in the scope of this paper, however, the last one worth
mentioning is the unrealistic assumption of perfect competition. It implies among the others, that
the information flow between consumers and suppliers is instant and not disrupted in any way. Con-
7
sequently, the model does not take into account the delayed reaction of a market to any changes
or an imperfect estimation of a value (or in this case also utility) of given attributes. As far as
the assumption of the perfect competition is not unique to hedonic models, and therefore, men-
tioned limitations are relatively popular in economical models, hedonic pricing models distinguish
themselves with one major limitation. In real life, it is completely possible that consumers do not
have a full knowledge of every attribute of a good they are considering to purchase. Nevertheless,
they still have to estimate the value/utility of a good on the basis of the knowledge they possess.
Therefore, even the best hedonic model may fail in estimating the constituent price of an attribute
if the consumers fail to perceive it.
All the mentioned limitations did not stop, hedonic pricing from becoming a popular tool in
multiple aspects of economics. It is still widely used in numerous research especially among those
focusing on deriving the value of hardly tangible environmental and sociological aspects such crimi-
nality rate or air pollution. Even though the above summary may seem like an oversimplification of
a complex phenomenon which is hedonic pricing, hopefully, it is sufficient to prove that a hedonic
model is not just a regression analysis aiming at measuring the magnitude of attributes’ coefficients,
but also a theory-based marketing analysis.
2.2 Application of hedonic pricing for housi ng market
Hedonic pricing has been extensively applied in the real estate appraisal research due to the char-
acteristics of a housing market. The fact that real estates may be treated as goods consisting of
multiple separate attributes such as a living area, number of rooms or localization, perfectly matches
the requirements of a hedonic approach. Furthermore, the changes in the housing market prices ac-
curately reflect the macro-economical phenomena happening in the society, which puts a housing
market in the position of a popular benchmark used in other types of analysis. Moreover, the com-
bined versatility of a housing market and hedonic approach in addition to the easily accessible data,
makes it a perfect tool e.g. in estimating the values of environmental features such as air quality
over the world.
The attributes of a property in the hedonic approach tend to be divided into multiple categories
in the literature. In their review of hedonic pricing papers Chin & Chau (2003) propose the division
of characteristics into three main groups. The first one, the locational group consists of real estate
characteristics such as the distance to the central business district (CBD) and the type of view that
a given location has to offer e.g. view on the lake or the golf course. The accessibility to the center of
a city has been found to have an impact on the price of a house in multiple research (McMillan, 1992;
Palmquist, 1992). Similarly, numerous types of views from a property have been proven to influence
the price of an estate (Gillard, 1981; Mok et al., 1995). Moreover, according to Benson et al. (1998)
not only the type of view affects the value of a house, but the quality of it also plays a crucial role.
As an example, in their work, Benson et al. (1998) concluded that the ocean frontage adds 147% to
8
the property’s sales price, while the obstructed, partial view on the ocean only contributes with an
increase of 10% of the price.
The second category, namely the structural attributes, describes the traditional characteristics
of a house such as a living area, the number of bedrooms, or the age of a building. Throughout the
years a plethora of research aimed to quantify the importance of particular construction aspects of
a building on a price. Numerous papers agree on the fact that one of the most important structural
attributes is a floor area followed by the number of rooms, bedrooms, and bathrooms (Ball, 1973;
Garrod, 1992). While the age of building according to multiple research such as D. E. Clark &
Herrin (2000) is negatively related to the estate price, in some analysis (H. J. Li M M Brown, 1980)
it has been proved to have an opposite effect, probably due to a historical value older buildings may
possess. Lastly, additional areas of an estate such as a garage, a basement, a patio, or a storage
in multiple research were also classified as significant predictors in the regression analysis of house
prices (Forrest, 1996; Garrod, 1992).
The last class of attributes relates to a broad range of features characterizing the neighborhood
of a real estate. According to Linneman (1980), these attributes stand for 15% up to 50% of the
standardized variation of a site evaluation model. Previous research divide further the neighbor-
hood characteristics into three subgroups: socio-economic variables, local government/municipality
services, and externalities such as crime rate or traffic noise (Chin & Chau, 2003).
In the past studies, the level of income in the neighborhood (Kain, 1970) and the dominating
ethnicity in the area (Ketkar, 1992) have been found as significant variables impacting the predicted
housing price. In terms of municipality services, the proximity to good schools (D. E. Clark &
Herrin, 2000) and places of worship such as mosques (Carroll, 1996) proved to positively impact
the value of a house. On the contrary, the high criminality rate in an area undoubtedly lowers the
average value of an estate (D. E. Clark & Herrin, 2000). The slightly modified summary of the most
popular variables used in hedonic price models, and their impact on the real estates’ prices prepared
by Chin & Chau (2003) is presented in Table 1 below.
The external factors impacting the housing price have been much more widely covered by
Karanikolas et al. (2011) where the authors provide a summary of insights gathered from numerous
research in that area. In general, the features analyzed in this paper may be divided into four
categories: green areas, water areas, topography, and environmental risks. The first category was
a matter of interest for Bishop (2005) where it was concluded that there is an evident correlation
between the existence of green spaces and the market value of houses. It was estimated that the
houses located nearby green areas may be even up 20% more expensive than their counterparts
without that access.
Similarly, in Wolf (2007) the author found out that potential buyers are willing to pay much
more for a house if it is located nearby an existing park. Moreover, the type of a green area had
a significant impact on the estate price. The traditional parks located in a distance of up to 400
9
Table 1: List of Commonly Used Housing Attributes in Hedonic Price Models (Chin & Chau, 2003).
Attribute Expected effect on housing price
Distance from CBD negative
View of the sea, lakes or rivers positive
View of hills/valley/golf course positive
Obstructed view negative
Locational
Length of land lease positive
Structural
Number of rooms, bedrooms, bathrooms positive
Floor area positive
Basement, garage, storage and patio positive
Building services (e.g. lift, AC) positive
Floor level positive
Structural quality positive
Facilities (e.g. swimming pool,
gym, tennis court)
positive
Age of the building rather negative
Income of residents positive
Proximity to good schools positive
Proximity to hospitals unknown
Proximity to places of worship
(e.g. churches or mosques)
positive
Crime rate negative
Traffic/airport noise negative
Proximity to shopping centers unknown
Proximity to forest unknown
Neighbourhood
Environmental quality
(e.g. landscape, garden, playground)
positive
10
meters increased on average the price by 10% while the existence of nearby parks that were not free
for the public lead to an increase of 20%.
In terms of water areas, their impact on household prices in Arizona have been investigated by
Colby (2003). In general, the prices of estates were higher in cases where water area such as a lake
or a river were located up to 2 kilometers from a house. An estimated increase in the price of 5.9%
was the highest for properties located around 160 meters from the water. On the other hand, it is
worth mentioning that the properties located at a distance of less than 150 meters were cheaper,
probably due to the risk of floods. Nevertheless, the area of Arizona for which the research was
made, is characterized by dry and desert climate giving the paper’s result a specific context.
Another type of external features impacting the price of households refers to the topography
of the real estate’s neighborhood. While the natural landform is not an object of primary focus
in this research, the impact of anthropogenic features of the neighborhood cannot be omitted.
According to Klein (2003) the proximity of a highway negatively impacts the price of real estates
by approximately 8 to 10%. It could mean, that the easiness of communication is not enough to
compensate for noise pollution caused by high-speed roads. Interestingly, the impact of railway
proximity has been estimated to be smaller in Brinckerhoff (2001) and was equal to 6.7% decrease
in the market value of a property.
Due to a much wider range of literature, the gathered insights mostly relate to the value of
real estate, not its rental price. As far as these two are obviously strongly correlated, they may
differ significantly in some aspects, especially in the context of hedonic pricing. In the end, the
hedonic method implies that the value of a good comes from the consumers’ willingness to pay for
its particular attributes. Therefore, the total utility of the same property may be different for a
potential tenant and a potential buyer. As an example, a poorly equipped, small flat close to the
university may be worth much more in the eyes of a student interested in renting a property than
of a working person looking for a property to buy.
The ratio between house price and rental cost has been an object of analysis for S. Clark &
Lomax (2019). The authors reach in the paper quite a few remarkable conclusions on the rent/price
ratio. Firstly, the ratio is the highest for flats, followed by terraced houses, leaving (semi)detached
houses behind. Secondly, most probably due to a handful of rental offers with large living areas,
the ratio has been found to raise with the increase in the surface and the number of bedrooms. In
terms of neighborhood attributes, the ratio turns out to be lower for properties being in proximity
to "healthy retail environment (away from fast-food restaurants, tobacconists and gambling) and
access to health services" (S. Clark & Lomax, 2019). On the contrary, the neighborhood with
high-quality environmental features such as low air pollution and access to green areas, raises the
mentioned price/rent ratio. Finally, S. Clark & Lomax (2019) concluded, that some attributes that
increase the price of a real estate, do not impact its rental price e.g. the distance to a good school
or proximity of amenities.
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2.3 Methodology
Most of the studies analyzed in the previous section have been based on a traditional, linear hedonic
pricing regression model. While the simplification in the form of linearity assumption, allows an
easy determination of the importance of each attribute characterizing a given good (by analyzing
the coefficients of an OLS regression), it disallows the model to capture more complex, nonlinear
patterns in the data. While it is difficult to distinguish the complexity of patterns in a given data set
prior to the analysis, in his work Rosen (1974) argues that in general, the nonlinearity between the
price of goods and their inherent attributes is likely to happen. The problem can be addressed by
applying advanced machine learning methods. This part of the literature review provides a summary
of research where this type of approach has been used in the context of the housing market.
One of the attempts to compare the performance between the traditional hedonic pricing regres-
sion model and more complex algorithms has been made by Limsombunc et al. (2004). According to
the results, the artificial neural network outperforms the hedonic price model in terms of sheer pre-
dictive power. Moreover, the hedonic model has been criticized by authors, due to its assumptions
and common problems such as data multicollinearity and heteroscedasticity or inability to capture
non-linear patterns.
On the other hand, the authors emphasized the complexity of neural network interpretation and
did not perform any type of variable importance analysis. Furthermore, the paper includes references
to previous research such as Lenk et al. (1997) and Do & Grudnitski (1992) which argue that the
results of a neural network model may be inconsistent and not always outperform regression models.
Another comparison between the performance of hedonic regression and machine learning algo-
rithms has been done by Neloy et al. (2019). On contrary to most of the research in the area of
housing pricing, the authors opted to base their study on rental cost data. The comparison between
linear regression, penalized linear regression, support vector machine, neural network, and multiple
versions of decision tree-based ensemble learning methods, shows the empirical superiority of the
latter. While the differences in the average RMSE between tree-based methods and other machine
learning models were of high magnitude, the differences between variations of random forest in the
form of Bagged Trees, Gradient Boosted and XGBoosted were rather negligible.
Recently Hong et al. (2020) applied a random forest model in price evaluation of Seoul households.
The results of the research were surprisingly good as the average percentage deviation between the
predicted and actual market price was equal to 5.5 (out-of-sample). In comparison, applying the
traditional OLS-based hedonic regression on the same data led to an average percentage deviation
of almost 20. As such, it can be concluded, that decision tree models can be more successful in
predicting house prices, than their traditional linear regression counterparts. Nevertheless, it may
be worth mentioning that the data used in the research was of extremely good quality as it consisted
of 40% of all the transactions made in the area within the last 10 years. Moreover, the data was
12
limited only to one, not diversified district of Seoul. Therefore, achieving such good results with the
usage of a decision tree model only, may not be feasible in most of the cases.
The choice of a regression technique for this research is not an easy task as both, probably most
popular advanced machine learning techniques neural networks and random forest have proved to be
effective in the previous research. Nevertheless, decision tree-based methods seem more reasonable.
As non-linear models, they can capture more complicated patterns in the data than the traditional
hedonic regression. On the other hand, the regression will be based on a relatively simple tabular data
where extremely complicated relations between variables are not expected. In his study Rossbach
(2018) provides empirical results of a comparison between the performance of 179 neural net and
random forest models. In most of the cases, random forest did not come in short in comparison
with its deep learning counterparts. Moreover, the author emphasizes the advantage of tree-based
methods over neural nets in terms of robustness, benefits in cost and time, and especially the easiness
of interpretability.
The superiority of artificial neural networks is usually visible in areas such as text or image anal-
ysis where connections between different predictors may be extremely abstruse. Therefore, random
forest seems like a well-suited model for this research, due to fairly advanced pattern recognition
and the ease of use. The latter will be of most benefit in the later stage of research, where multiple
models will be compared to see if including features gathered from images and other sources signifi-
cantly improves the prediction accuracy. On the other hand, neural networks are probably the best
method to be used in the mentioned image recognition part of the study.
The number of applications where image recognition is being used has been constantly growing
over the last decade. The ability of software to identify patterns, people, and objects from the image
is not only directly used in areas such as driver assistance systems but has also proved to be helpful
in increasing the accuracy of statistical models. In their research Bajari et al. (2019) approached
the topic of measuring inflation based on quality-adjusted prices of different products available at
Amazon. The features of the mentioned goods have not been limited just to conventional ones as
both text and image analysis have been performed. The authors have found that including features
extracted from images as well as texts leads to a significant improvement in model’s performance.
The usage of image recognition in the analysis of the real estate market has been a popular area
of study in recent years. Law et al. (2019) successfully used the Street View and satellite images
to improve the performance of house price evaluation models. Nonetheless, the authors opted to
focus on the accuracy of pricing by using black box model, while the relative importance of different
predictors has not been their top priority. Moreover, Law et al. (2019) work could also be extended
by using text analysis, environmental attributes usually present in hedonic pricing models and images
of property interior.
The last aspect has been studied in papers such as Poursaeed et al. (2018) and You et al. (2017).
In the former, the convolutional neural network has been trained to rank the photos on the scale
13
of one to eight based on the luxury of a property. In the latter, the authors have taken quite an
innovative approach of using recurrent neural network LSTM model in order to predict the price of
the house solely on the basis of photos and location. Typical data e.g. size or number of rooms has
not been used in the research which did not stay in the way of reaching promising results.
Unfortunately, there is one major shortcoming of all the papers cited in the methodological part
of the literature review. All the mentioned research, do apply advanced machine learning algorithms
to the housing market data, nevertheless, their focus is solely put on the prediction accuracy of the
created models. Almost no attention is given to the estimation of the housing attributes importance
and their utility. Therefore, it could be argued that the comparison between the hedonic linear
regression and machine learning methods presented in the cited papers is not fully appropriate, as
the essence of the hedonic price modelling, the value of each attribute of a given good, is being lost
in the pursuit of achieving the best possible accuracy of a prediction.
14
3 Methodology
The methodology used in the research may be divided into three main categories. The first one,
consists of methods used to extract features from complex data sources such as images and text,
and transform the gathered insights into low-dimensional tabular data. The second class includes
the methodology describing the regression models used in the research. The purpose of the methods
present in the last group is to explain the behaviour of the black box model described in the second
category of the section, and determine the importance as well as the impact of particular predictors
on the prediction.
3.1 Data extraction
Image analysis. Using a convolutional neural network (CNN) is a standard approach while build-
ing an image recognition model (F. Li et al., 2019). Convolutional neural nets are constructed
similarly as regular neural networks, they also consist of the input layer, hidden layers, and the
output. The main difference lies in the architecture of the layers. In image recognition models, the
input usually takes the form of A x B x 3 matrices, where A stands for the picture width, B for the
picture height while 3 represents color channel values (RGB). Let’s assume that the pictures fed to
the model have a small size of 100x100 pixels. In regular neural net each neuron in the first fully
connected hidden layer would already have 30000 weights. It is easy to imagine how the number
would scale for larger images. As a result of the regular net architecture, not only the computational
time of the model would be enormous but the number of adjustable weights could easily lead to
overfitting of the algorithm.
Convolutional neural nets deal with this problem by introducing three-dimensional hidden layers,
which size is being reduced in a subsequent processing. The first layer in the model is called
convolutional layer. Its input size is equal to the input image A (width) x B (height) x 3 (depth),
which in consequence usually requires all the images to be scaled to the same size. At this part, the
model analyzes one part of the picture, multiplies its values by a pre-defined smaller matrix known
as a filter or kernel, and then moves to the new part until the whole picture is scanned. The size of
the part of a picture being scanned at one moment is equal to the kernel size, which is one of the
parameters that may be tuned. However, the sizes of 3 x 3 x 3 and 5 x 5 x 3 are the most popular,
mostly due to the fact that they are used in some of the best performing CNNs: GoogleNet and
VGG (Szegedy et al., 2015; Simonyan & Zisserman, 2014). The process of multiplying the input
values with a filter and moving to a new part is in the literature referred to as convolving the filter
with the image (F. Li et al., 2019). The number of times the filter has to convolve with the picture
depends on the kernel size and a parameter known as a stride, which indicates by how many pixels
the filter should "slide".
As the output of the numerous multiplication, a two-dimensional activation map is being created
15
for each filter used. The output of a convolutional layer, created by stacking activation maps along
the depth dimension takes a form of a D x E x C sized matrix, where C substitutes the previous
input image depth of 3, with the value equal to the number of activation maps. D and E again
stand for the width and height of a matrix, however on the contrary to A and B they do not always
match the image size. Instead, their value depends on the number of times the kernel may fit in the
input matrix. Therefore, D and E tend to have smaller values than A and B unless the technique
known as padding is being used. The classic way of padding called "same padding", augments the
input matrix by adding column(s) and row(s) on the matrix borders with imputed values of 0. The
aim of this approach as the name suggests is to equalize the sizes of input and output matrices, so
no data would be lost in the process of transferring data to consecutive layers.
The number of filters used in the convolutional neural net is yet another parameter that may be
tuned. As one could think of filters as feature detectors, it could be argued that the more complex
the patterns that the model aims to capture, the more filters it will need to do it properly. On the
other hand, each filter drastically increases the output of a convolutional layer, thus, increasing the
computational power needed to train a model. As the training process continues, the network learns
filters that are being activated when some specific features of the image are being captured. The
first convolutional layer is responsible for extracting low-level features of an image such as edges
or colors. With each convolutional layer added, the higher-level features start to be captured by
the model. The output of a convolutional layer is subsequently passed to an activation function
layer which in case of a CNN is usually a ReLU function defined as f(x) = max(0, x) which applies
elementwise non-linearity (F. Li et al., 2019). However, in cases of too complex architecture of a
model, the problem with overfitting may appear. Srivastava et al. (2014) propose a relatively simple
way of addressing that problem with a technique called dropout. The key idea of this approach is to
drop out a random set of activations, by setting them to zero. As counter-intuitive as the method
may seem, it forces the model to learn how to correctly classify an image, even when some of its key
attributes are lost.
The next type of layer used in convolutional neural nets is called a pooling layer. Its main
function is to reduce the size of convolved features, and therefore, the number of parameters and
computational time. This goal is again accomplished by analyzing the input matrix by parts. In
case of the most popular approach known as max pooling, for each submatrix which size depends
on the pre-defined kernel, only the maximum value is returned. Then, the process is repeated until
the whole image is traversed and the original matrix is transformed into less-dimensional one.
In each model depending on the needs of the analyst, multiple convolutional, ReLU, and pooling
layers may be used. No matter how many of them are included in the architecture of the model and
what is their order, the final output is being transferred to the fully connected layer (also known as a
dense layer). At this point, the data is being transformed into a column vector, which subsequently
is fed to a regular feed-forward neural net, which in the end returns probability values for each class.
16
Similarly, as in traditional, artificial neural network, the loss and activation functions have to be set
separately for the dense layer.
Convolutional neural networks are also analogous to artificial neural nets in terms of data re-
quirements. Both methods usually require a large amount of training data to perform well. However,
in case of CNN the process of gathering additional data in the form of images is often more cum-
bersome than collecting extra tabular data for ANN. Image augmentation is one of the techniques
allowing artificially increasing the training sample size. The additional observations are created on
the basis of various transformations applied to original photos such as random rotation, shifts in
width and height, zooming or flips. The choice of transformations used in a particular model should
always be adjusted individually, as their functionality depends on the context of images and the
patterns that the model is aiming to recognize.
Text analysis. After the initial analysis of apartment search sites and housing market research,
it has been concluded that there is a considerable amount of information gathered in rental offers’
description that would be difficult to extract from other sources such as images or maps. Therefore,
the text analytics methods are used in the research, however to a limited degree. It appears that
most of the text analytics method e.g. sentiment analysis would probably not bring any additional
value to the study, due to a distinctive nature of the real estates’ descriptions and the type of
language they are written in. As such, the focus of this part of the research is put on a simple
information extraction from the rental offer description.
In general, a rental offer description may contain a lot of useful information about the neighbor-
hood, close services, and potential rental restrictions. On the other hand, the narrative is expected
to be biased as owners do not mention the negative characteristics of their real estates. Moreover,
the descriptions differ significantly between each other, and the information mentioned in some is
missing in the others. Therefore, where possible, the extraction is done in a way that gathered
attributes are easy to categorize or have a dummy form in which, in case of missing data, a de-
fault value may be set. The examples of features and data extraction methodology should help in
understanding that logic:
• Rental restrictions: some landlords do not allow pets, children, students, or smoking persons
in their properties. In order to find if it is a case for a particular property, the description
is searched for the mentioned keywords (smoking, pets, etc.). After finding the position of a
keyword in a description, the surrounding words are searched for contradictions. The range of
the search, usually referred to as a window, is set after the initial analysis of the description
examples, and depends on the most popular grammar structures used. The size of a window
and its (a)symmetry may also vary between the keywords searched for.
• Nearby services: as the type of services mentioned in the descriptions differs, and it is less
17
probable to assume that e.g. if the owner does not mention nearby restaurants there are
indeed no restaurants in the neighborhood, creating dummy variables may not be the best
idea. To overcome this problem, a set of keywords based on the literature review, and common
sense such as hairdresser, market, restaurant, etc. are created. Then, the number of existing
keywords in each description are summed up.
• Additional information: sometimes additional requirements have to be met to rent a prop-
erty. Similarly, with the usage of keywords, sentences with the words “guarantor”, “minimal
income” etc. are found and the window of keywords is searched for the amount of required
money/income.
Additionally, in order not to miss any popularly mentioned features in the descriptions that
could be specific for the Netherlands or the area of Rotterdam, the frequency analysis of words is
performed. Firstly, the popular stop words are removed from the descriptions. Subsequently, the
most popular words are manually screened to see if any potentially important characteristic of a
property or neighborhood has not been omitted.
3.2 Regression analysis
Hedonic linear model. Most of the research using hedonic pricing are based on an Ordinary
Least Squared regression, most commonly called linear regression. The OLS regression model takes
the form of::
y
i
= β
0
+ β
1
x
i1
+ ... + β
p
x
ip
+
i
, (1)
where:
y
i
is the value of a dependent variable for i
th
observation,
β
0
is the intercept term,
β
p
is the regression coefficient of the p
th
independent variable and
i
is the model’s error term.
Let’s recall that in the hedonic model presented by Rosen (1974) the marginal implicit price
of an attribute is presented as the partial derivative of the price function over the given attribute,
which in case of an OLS regression is equal to the value of an attribute’s coefficient. Therefore,
following the linearity of the model the total value of an attribute p for an i
th
observation (good) is
a product of the observation’s value of p and β
p
coefficient.
However, in order to rely on a linear model’s results five main assumptions have to be met:
• The linearity of the data: the relationship between the dependent and independent variables
has to be linear. This assumption can be checked by using scatterplot comparing residuals of
the model with fitted values.
18
• Multivariate normality: the residuals of the model should be normally distributed which can
be checked with a Q-Q plot or with a goodness of fit test such as the Kolmogorov-Smirnov
test.
• Little to no multicollinearity in the data: the independent variables should not be highly
correlated with each other. This assumption can be tested by calculating Pearson’s correlation
coefficients for each pair of predictors and with a Variance Inflation Factor (VIF).
• Little to no auto-correlation: the residuals of the model should not be correlated with each
other which can be checked with the Durbin-Watson test. However, the mentioned test only
checks the correlation between directly neighboring residuals, therefore auto-correlation func-
tion should also be applied to assure that there are no higher-order effects.
• Homoscedasticity: the variance of error terms should be similar at all levels of the independent
variable. Any significant indications of heteroscedasticity may be checked by analyzing the plot
of standardized residuals versus predicted values.
While the interpretation and the ease of use are undoubtedly substantial advantages of an OLS
regression, it is rather a rare case that data in their original form meet all the above assumptions.
Instead, the data transformation using methods such as Box-Cox transformation is often needed.
Nevertheless, the more transformed the data, the less interpretable the results of the model become.
Moreover, it is argued that Box-Cox transformation reduces the accuracy of any single coefficient
(Cassel, 1985). Lastly, no data transformation technique guarantees that the relationship between
variables will become linear, thus, bending or breaking the assumptions of an OLS regression is
sometimes inevitable. If this is the case, applying an OLS regression to such data leads to the
situation where the forecasts, confidence intervals and the insights provided by the model become
inefficient and misleading. To conclude, capturing non-linear patterns in the data with an OLS
regression is a difficult or sometimes even an impossible task. This is problematic in the context of
hedonic price modeling as even in his early work Rosen (1974) claims that some variables such as
budget constraints are likely to be nonlinear.
Random Forest. As argued in the methodological section of the literature review, the decision
tree-based method, random forest seems to be a suitable method for this research. The decision
tree is a non-linear model consisting of multiple conditional statements that separate the data into
smaller nodes. The variable on which the split is performed in a given part of the regression model
is based on the decrease in RSS it would cause. Overall, at each moment the algorithm chooses the
split that leads to the greatest decrease in RSS, then repeats the process for newly created nodes
until a stopping rule is met, or the number of observations in each node is equal to one.
19
The common problem with decision trees is their tendency to grow deep and therefore overfit the
data. Random forest is one of the methods that help in dealing with this problem with the usage of
bootstrapping. Bootstrap method simulates N new data sets by randomly drawing observations with
replacement from the original data set. Subsequently, for each N
th
bootstrapped sample decision
tree model is built. The prediction of the random forest model is an average prediction value among
all N regression trees. The fact that the final prediction is the average score of multiple predictions
helps in dealing with overfitting by reducing the overall variation of the model in comparison to a
single tree (James et al., 2013).
One of the imperfections of this approach is the fact that by default bootstrapped-based trees
are highly correlated. Random forest addresses this problem by allowing only a random subset of
original variables to be used in a single bootstrapped tree. This approach not only decorrelates the
trees, but it also disallows the most important variables to completely dominate the final model. As
each variable appears only in a subset of trees, predictors which would seem to have less impact on
the prediction has more space to act in the final model. The optimal number of predictors used in
each tree is relatively easy to define by applying the grid search and choosing the model with the
best performance in a chosen metric such as RMSE.
The quite important flaw of traditional random forest is its tendency to rank continuous variables
or categorical variables with multiple levels as more important than the others. That is because these
types of variables have more possible options to split the data and therefore are more likely to be
chosen at higher level splits than e.g. binary variable. In case of this research, a significant part
of the data set, especially the features extracted from images have a form of dummy variables.
Therefore, in order to limit the mentioned bias, conditional decision tree-based random forest are
also used. In contrast to regular trees, conditional trees base the splits in the model on the results
of permutation-based significance tests performed for each variable.
3.3 Model-Agnostic methods
In numerous research random forest has proved its superiority over an OLS regression in terms
of prediction accuracy. Nevertheless, this advantage comes at the cost of much more complicated
architecture of a model which in consequence disallows a simple determination of the variables
importance and their impact on the prediction. As previously mentioned in this paper, the issue
is especially troublesome for hedonic price modeling, in which the determination of the marginal
implicit price of an attribute may be treated as the essence of the method. Fortunately, there is a
wide range of methods aiming to uncover the functioning of black box models.
Variable Importance. One of the most standard approaches in determining the importance of
variables used in a black box model is the analysis of the mean decrease in accuracy. The idea of the
method is to permute one predictor in order to decouple its relation with the dependent variable.
20
Subsequently, a new model with the permuted predictor is fitted and its accuracy is measured. The
more the accuracy of the new model decreases in comparison to its original counterpart, the more
important the predictor is deemed. As such by repeating the procedure for each predictor in the
data set, their relative importance can be gathered. While the method may be criticised for its
moderate robustness and unreliability in case of strongly correlated data, it is still an easy and a
fast way to acquire the basic overview of the black box model.
Partial Dependence Plots. The partial dependence plot (PDP) shows the marginal effect a
feature has on the predicted outcome of a machine learning model (Friedman, 2001). In addition, it
allows us to measure in approximation the type of relation the independent and dependent variables
have e.g. linear, monotonic, or complex. PDP is based on a partial dependence function, which in
case of regression is defined as:
ˆ
f
x
S
(x
S
) = E
x
C
h
ˆ
f(x
S
, x
C
)
i
=
R
ˆ
f(x
S
, x
C
)dP (x
C
), (2)
where:
x
S
is the feature for which partial dependence function should be calculated, and
x
C
are all the other predictors featuring in the machine learning model.
While the set S may consist of multiple variables, usually it is limited only to one, as the
PDP visualization in more than two dimensions is difficult to analyze. Partial dependence works
by marginalizing the machine learning model output over the distribution of the features in set C
(Molnar, 2019). That way, the function estimates the relationship between the x
S
feature and the
predicted outcome of the machine learning model. According to Friedman’s approach, the partial
dependence plots are obtained by calculating the following average and plotting it over a range of
x
S
values:
ˆ
f
x
S
(x
S
) =
1
n
P
n
i=1
ˆ
f(x
S
, x
(i)
C
), (3)
where:
n is the number of observations in the train set, and
x
(i)
C
is the value of the i
th
sample for the x
C
features.
An important assumption of a partial dependence function is no serious correlation between
feature(s) x
S
and features x
C
. If this assumption is not satisfied, the averages calculated in the
formula 3 will include data points that are unlikely or even impossible to reach (Molnar, 2019). An-
other drawback of partial dependence plots is the fact that they do not contain feature distribution.
Therefore, the analysis of PDP may be misleading in some parts where not enough data is available.
The partial dependence method may be described as a global function. Based on all the obser-
vations that were fed to a machine learning model, it provides a global relationship between their
21
predictions and a particular feature for which function was estimated.
Local Interpretable Model-Agnostic Explanations (LIME). While the globality of a partial
dependence function may be seen more as a method’s characteristic and not necessarily as a disad-
vantage or limitation, it does not allow us to understand the reasoning of a machine learning method
behind a single prediction. One of the methods which allows this type of analysis is known as Local
Interpretable Model-agnostic Explanations or simply LIME. Originally introduced by Ribeiro et al.
(2016), LIME aims to explain the functionality of a black box method for a specific observation by
fitting a simpler, easier to interpret model also known as a glass-box model at a local scale.
LIME is rooted in the assumption that even the most complex model is linear on a local scale.
In consequence, the assumption implies the key idea of LIME which may be formulated in the
following way: if two observations possess very similar characteristics, they should behave similarly
in a machine learning model. Therefore, if multiple similar observations behave similarly in a black
box model it is possible to fit a prediction model on their basis, that would mimic and consequently
explain how the original model behaves at that locality (Pedersen & Benesty, 2019).
The surrogate model may take numerous forms such as linear regression, LASSO or decision tree.
The only limitation for the chosen type of a model is to be easily interpretable. The accuracy with
which the surrogate model explains the black box algorithm’s behaviour is known in the literature
as a local fidelity (Molnar, 2019). Local fidelity may be used as a metric on the basis of which the
optimal type of local model is chosen.
Mathematically, the local approximation of a black box model may be defined as:
ˆg = arg min
g∈G
L(f, g, π
x
) + Ω(g), (4)
where:
G represents the class of interpretable models,
g is an interpretable model belonging to the class G,
L is the fidelity measure measured with a chosen metric e.g. mean squared error,
f is the black box model for which the local approximation is sought,
π
x
is the proximity measure defining a neighborhood of an instance x in which the approximation
is sought, and
Ω(g) is the g model’s penalty of complexity (Biecek & Burzykowski, 2020).
While the above formula already shed some light on the theory behind LIME, the application
side has been described in an approachable way by Pedersen & Benesty (2019). Firstly for each
prediction to explain the observation is permuted n times. Then, the prediction for each permuted
observation is run with the usage of the original black box model. Subsequently, the distances from
all permutations to the original observation are calculated and converted into similarity score. After
22
selecting the number of features best describing the black box model with Ω(g), a surrogate model
is fitted to the permuted data. While training the local explanatory model the outcome is weighted
for a permuted observation by its similarity to the original observation. Finally, by extracting the
feature weights from the simple model it becomes possible to use these as explanations for the
complex model’s local behavior.
In theory, following the above procedure should allow us to explain the mechanics of any black
box model for every single prediction. Nevertheless, LIME is still a method in a development phase
and as promising as the above statement sounds, it does not always end up being that functional.
As a relatively new method, LIME suffers from a number of issues that have not been yet resolved.
Firstly, there is not much literature available on the tuning of LIME parameters, especially the width
of a smoothing kernel π
x
, which for now has to be tuned with a trial and error approach.
Another problem, the robustness of explanatory models created with LIME causes even more
concern. In Alvarez-Melis & Jaakkola (2018) the authors show that the explanation provided with
LIME for two similar observations may vary significantly. Molnar (2019) also criticizes the stability
of LIME, claiming that the results may change after repeating the sampling process.
23
4 Data and feature extraction
The research has been based not on the prices of properties but the rental costs of real estates
instead. There are numerous reasons for choosing this approach. Firstly, the number of research on
the prices of house rents is relatively limited in comparison to the house prices, while the results of
both may differ significantly (S. Clark & Lomax, 2019). Secondly, obtaining the data in English on
the Dutch market, which is a primary interest of the study, is much easier in case of rentals. In terms
of data, it has been collected only from rental websites and Google Maps, on the contrary to most
of the research mentioned in the literature review, which are based on data received from real estate
agencies. One could argue that the web-scraped data may be of lower quality in some aspects. For
example, there is no certainty that the rental price shown in the offer has not been negotiated, and
is the final rent that the customer agreed to pay. Nevertheless by taking into account the relatively
fast-moving nature of the rental market in large cities limiting the negotiating space, making this
kind of assumptions seems reasonable. On the other hand, gathering data from the rental websites
allows for a more flexible approach in its usage, as the research would not be limited to predefined
data set. Moreover, basing the study on accessible data leads to a situation where the model can be
easily applied to any other city or area.
Another reason for using only data from rental websites is linked to the theory of hedonic
models. One of the most often mentioned characteristics (or even limitations) of hedonic models is
the customers’ willingness to pay more or less for a product depending on the utility of its attributes,
but only the ones that they can perceive. It may be argued that in case of renting an estate, potential
customers do less in-depth research than they would if they wanted to buy it. Therefore, it makes a
rental offer a primary source of information for customers not only in terms of house characteristics
but external factors as well. Nevertheless, it would be naive to assume that people willing to rent a
house do not make any research on their own. As one of the most important features of a real estate
is its location, Google Maps seems like an obvious tool that in a short amount of time, allows us
to gather relatively a lot of knowledge about the neighborhood of a property. Therefore, extracting
features from satellite images plays a significant role in this research.
The core of the data gathered for the research comes from one of the leading rental websites
in the Netherlands. The main reason for limiting the data source only to one web page is the
fact that all rental offers there are presented in English, which is rather uncommon for the Dutch
market. Moreover, the website is well structured and allows for a relatively easy process of web
scraping. Furthermore, as the web page is administrated by real estate brokers the quality of the
data presented in the rental offers is on a very high level. Therefore, the estimated values of the
properties advertised on the website are assumed to be accurate and in consequence, should limit
the number of outliers in the data. Last but not least, it appears that most rental offers are unique,
which is quite a phenomenon when compared to other websites where duplicates appear quite often.
24
4.1 Initial data
The most crucial part of the data describing the main house characteristics has been web-scraped
from the mentioned rental website. Typical variable examples of such data are rental price, type
of a house or living area of a property. As far as most of the data in this section according to the
nomenclature proposed by Chin & Chau (2003) would be defined as structural, some information
such as the name of a street or postal code would match more the Chin’s category locality.
Nevertheless, after investigating multiple rental offers it has been found out, that many offers
differ in the amount of information fields presented to a visitor of the website e.g. energy rating field
is present only in a relatively small number of offers. Therefore, in order to create a reliable data
set some variables have been excluded from the process of web scraping.
The process of gathering data has been fully automated with the usage of R and respective
packages allowing scrapping the data based on HTML and XML code of the website. Firstly, the
links to all the rental offers have been scrapped from the search page with the only filter of the
city being Rotterdam. Over 2000 real estate rental offers have been found and crawled since the
beginning of April till the end of May 2020.
After performing data cleaning the data set consisting of 1844 unique observations has been
created. Moreover, as one of the most important factors in predicting the rental cost of a house is
its location, with the usage of Google Maps API, a set of variables have been collected: the distance
from a house street to the Rotterdam Central Station and the time it takes to travel the distance by
walking, by biking and by public transport. Similarly, the geographical coordinates of each street
appearing in the initial data set in the form of longitude and latitude have been gathered.
Table 2 contains a detailed description of the variables present in the initial data set. Out of
1844 observations, the only missing values appear in variables Construction_year, Bedrooms and
Bathrooms. For the last two variables, they do miss only in case of house type being a single room.
Therefore, it has been decided to substitute the missing values in these cases with 1. In terms of
construction year 721 observations out of 1844 miss this information.
4.2 Image recognition
Simultaneously with scraping typical house characteristics data, images of each rental offer have
been collected. With an average of 22 photos per one offer, over 40000 images have been gathered
in total. Moreover, as one of the goals of the research is to use image recognition models in order
to extract features from the mentioned photos, additional images have been gathered to use them
in training classification models. Around 2000 images from other rental offer websites have been
gathered, this time however, the data came not only from Rotterdam but other Dutch cities as well.
Such an approach allows limiting the potential bias in classification models’ accuracy, which would
be otherwise caused by manual labeling the part of collected photos.
25
Table 2: Variables present in the initial data set
Variable Type Description
House.ID Numeric Unique ID of a house
Street Character Street
URL Character URL
Postal_code Character Postal code
District Factor (64 levels) District
Price Numeric Price in euros
Living_area Numeric Living area in squared meters
Rooms Numeric Number of rooms
Construction_year Numeric Construction year
House_type Factor (3 levels) Is a property a house, a room or a flat?
Bedrooms Numeric Number of bedrooms
Bathrooms Numeric Number of bathrooms
Balcony Factor (2 levels) Does a property have a balcony?
Garden Factor (2 levels) Does a property have a garden?
Storage Factor (2 levels) Does a property have a storage?
Garage Factor (2 levels) Does a property have a garage?
Shower Factor (2 levels) Does a property have a shower?
Bath Factor (2 levels) Does a property have a bath?
Lift Factor (2 levels) Does a property have a lift?
Toilet Factor (2 levels) Does a property have a separate toilet?
Furnished Factor (2 levels) Is a property fully furnished?
Service_cost Factor (2 levels) Are service costs included in the price?
Description Character Description of a property
Time_walking Numeric Travel time by walking in seconds
Time_biking Numeric Travel time by biking in seconds
Time_public Numeric Travel time by public transport in seconds
Longitude Numeric Street’s longitude coordinate
Latitude Numeric Street’s latitude coordinate
26
Subsequently, the process of extracting features that according to the previous research may turn
out significant in predicting housing price has been started. Firstly, the rental offers’ photos have
been initially analyzed. Although unsurprisingly a majority of photos present the inside of a given
estate, it has been found out that most of the offers do feature photos showing the outside of the
building and view from the property as well. Therefore, as this study focuses mostly on the external
factors impacting the rental price in Rotterdam, the decision to analyze the view from properties has
been made. It may be argued to which degree the view from a property is an external, an internal,
or a locational factor. Nevertheless, it is rather safe to assume that view is affected by external
factors such as the height and density of the buildings in the direct neighborhood, thus making it
at least partially externally dependant.
The image recognition process for different types of views has been divided into two sequential
convolutional neural net models. The first model aims to filter the property images and classify them
whether as outside or inside. The model has been trained in a way that outside label is given to
photos of balconies, views from the windows, street images, etc. All the other photos including not
only interior but e.g. graphics illustrating the layout of a flat are classified as inside. Even though
the added value this classification model brings to the data set and the future prediction model is
low, it allows us to simplify the subsequent model.
The goal of a second model is to analyze the outside photos and classify them into four categories:
1. View on the city: category featuring photos with relatively unbroken view on the charismatic
panorama of Rotterdam
2. Green view: category featuring images with a view on a park, a canal or another green area
3. Enjoyable view: arbitrary category featuring images with a pleasant, above the average view
e.g. photos presenting an unbroken view on a neighborhood, river or open areas
4. Other: category featuring all the other images.
Both models have been built with the usage of Keras neural network library with the TensorFlow
backend and are based on the same architecture. Firstly, images were scaled to the same size and di-
vided into three sets: train, validation, and test. The first model has been trained with 1500 images
while the validation and training sets had 300 photos. The second model was trained on 900 obser-
vations while the test and validation sets consisted of 200 images. Subsequently, the train images
were augmented by applying width and height shifts, horizontal flips, and shear transformations.
The overall architecture of the models is fairly simple. The first convolutional layer consists of
16 filters and the kernel size applied to the layer has a standard size of 3 x 3 pixels. Additionally to
preserve the data same padding has been used. The number of filters has been set to a rather small
number as it was expected of the model to capture mostly low-level features of an image e.g. sky in
the first model or vegetation in the second one.
27
Figure 1: The accuracy of the model versus the number of epochs
Subsequently, the output of the first layer is transferred to the ReLU layer, which after applying
the activation function transfers the data to the next convolutional layer. The second convolutional
layer differs from the first one only in terms of the number of filters, which according to the best
practice has been increased to 32. The reasoning behind this practice is the fact that it is more
profitable to increase the number of filters in the deeper parts of the model, where calculations are
performed on the initially filtered, less noisy data.
Afterward, the outcome is passed to the next ReLU layer, which subsequently transfers the data
to the pooling layer. Except for reducing the size of an input matrix, the pooling layer additionally
applies the dropout method with the parameter equal to 0.25 which stands for 25% chance of
deactivating each active node. Lastly, the output of the pooling layer is transferred to the dense
layer for which commonly used functions have been set in the form of activation function being
softmax and loss function being categorical cross-entropy. To ensure the stability of the prediction,
the number of epochs which is the number of times the entire dataset is passed forward and backward
through the neural net has been set to 2000. With each epoch added, the model has an additional
opportunity to adjust its weights aiming at improving the accuracy. Therefore, the higher number of
epochs usually leads to better predictions, nevertheless at the cost of increased computational time.
As such a compromise between these two aspects had to be found leading to the chosen number of
epochs where the quality of predictions became acceptable. The progress of training the first model
is presented in Figure 1.
After performing the analysis of the rental offers’ photos, the Google Maps have been put to
use. Out of 1844 observations in the data set, 578 unique street names have been filtered. For each
street, two types of Google Maps images have been collected. The first one, the map at the zoom
of 17 (scale used in Google Maps) presents the street and its neighborhood in a 300m radius. The
second map at the zoom of 15 shows the surrounding of a street with a radius of 1km. Similarly as
28
Table 3: Accuracy of the image recognition models
Model Validation Test
Outside/Inside 97% 95%
Water body 95% 94%
Park 87% 86%
View 78% 76%
with rental offers’ photos, to decrease the accuracy bias additional maps have been scrapped from
other cities in the Netherlands.
Following the conclusions drawn from the literature review, it seemed appropriate to analyze
the impact of neighborhood features on rental property prices. By taking into account the specific
nature of Rotterdam’s urban layout, it has been concluded to focus on two categories:
1. Parks: category indicating whether a given property lies in a direct neighborhood (300m) of
a park or other green area.
2. Water bodies: category indicating whether a given property lies in a direct neighborhood
(1km) of the Meuse river (the main river in Rotterdam, flowing through its city center) or a
large lake.
For each of the aspects with the usage of Google Maps images, a convolutional neural net model
has been created. Due to a limited amount of time and resources, the architecture of the model has
been copied from the previous models. The models analyzing park and water body proximity have
been trained on 500 maps while the test and the validation sets had 100 observations. Moreover,
the data augmentation was more limited in case of these models as all the map images from Google
Maps keep the same formatting and orientation. The accuracy of all the models described above is
presented in Table 3.
While the accuracy of the "Water body", “Outside” and “Park” models is relatively high, the
score for the “View” model may cause some concern. However, it has to be remembered that the
accuracy is calculated on the photo level, while the accuracy that is the most important for the
research lies on the property level. The classification thresholds of the model have been set in a way,
that the model is more likely to underestimate the view e.g. by categorizing the “Enjoyable view”
image as “Other” than to overestimate it. As each house has multiple photos of the outside, it is
sufficient if the model confidently classify just one photo instead of risking the error in case of images
being on the verge between classes. The accuracy on the property level is represented in Table 4.
29
Table 4: Accuracy of the view image recognition model on the property level
Variable Test
City 90%
Green 86%
Enjoyable 76%
4.3 Text analysis
The text analysis part of the research as mentioned in the methodological section, has been limited
to a fairly simple information extraction. The first step of the analysis was to identify the most
commonly used words in rental offer descriptions. The preparation part included the transformation
of all the descriptions into lowercase and removing stop words. Subsequently, the words with the
frequency of above 150 appearances have been analyzed. The frequency count is presented in Table
11 in the appendix.
Unsurprisingly, most of the words are related to the previously created variables based on the
featured fields on the rental website such as the number of rooms, location, or the surface of an
estate. Nevertheless, three additional categories of words which do not depend completely on the
structural attributes of a property have been distinguished:
• Services: words such as restaurant, shop, cinema,
• Owner’s restrictions: words indicating the restrictions under which the rent is allowed e.g.
pets, income, deposit,
• Insolation: words indicating that a property is exposed to sunlight e.g. bright, sunny, south.
Undoubtedly, insolation is an attribute strongly related to the structural attributes of a house.
Nonetheless, similar to the view from a property, it is dependent on the height and density of the
buildings in the direct neighborhood thus making the variable at least partially external, which
is why it has been included in the research. For each of the featured categories slightly different
approach of variable creation has been applied.
• Services On the basis of Table 11 and common sense, a list of keywords relating to various
types of services has been created. The list featured the following strings: shop, restaurant,
hairdres, barber, supermarket, bar, school, university, erasmus, markthal, market, cafe, food,
gym, and sport. Subsequently, the description of each property has been scanned for the
mentioned keywords. The final value for a given observation is the sum of unique services
found in its description.
30
• Owner’s restrictions based on this category two variables have been created. The first one,
Income refers to the financial requirements for renting a property. In most of the observations
containing keywords income and/or guarantor, the financial stipulation is set to three times
the amount of rental cost. Therefore, the final variable income takes the form of a dummy,
indicating whether a financial condition is mentioned or not.
The second variable in this category refers to pet restriction. Each description has been scanned
for the presence of keyword pet. After finding the position of the keyword, the window of one
word before and two words after has been searched for strings "no", "n’t", "aren’t". As such
the most popular grammar forms "No pets", "pets are not allowed" and "pets aren’t allowed"
have been included. If in the window no contradictions are found, the variable takes the form
of 0, otherwise is given the value of 1. For observations in which the keyword "pet" is not
present, the value is set to 0.
• Insolation Each description has been scanned for keywords light, bright, sun, and south. In
case of the presence of at least one of the first three keywords, the variable insolation takes
the form of 1. However, in case of the word south, additional requirements have been set to
distinguish descriptions relating to the south in the context of location and direction of the
flat. As in the context of the former, the word south is usually in the direct neighborhood
of the word Rotterdam ("south Rotterdam", "Rotterdam south", "south of Rotterdam"), the
window of one word before and two words after has been applied. If the word Rotterdam is
present in the window, the variable takes the form of 0, otherwise it is set to 1.
4.4 Summary
The final dataset consists of 27 variables which descriptive statistics are presented in Tables 5 and
6. According to the data, an average property rented in Rotterdam has the living area of 73 squared
meters, divided into three separate rooms. While the average rental price for such a property equals
1295AC, the variable price is characterized by its wide range with a minimum of 295AC, the maximum
of 4995AC with the standard deviation of 542AC. Another continuous variable that draws attention
is the variable Services extracted from the descriptions of rental offers. By taking into account the
mean of less than two services mentioned in the average offer and the possible maximum of fifteen
services, it seems that the owners do not share information about the neighborhood as often as it
was initially deemed. Consequently, the variable can be seen as of rather low quality and may not
be particularly useful in the modeling process at least in its original interpretation. As numerous
descriptions do not feature information about the nearby services, the potential significance of the
variable could be only interpreted directly; as the number of services mentioned in the description,
not as the approximated number of services in the property’s neighborhood.
In terms of dummy variables presented in Table 6 the majority of features represented by them are
31
rather uncommon for an average property in Rotterdam. The only characteristics which appear in
the majority of real estates are furnishing and proximity to some water body. The overall frequency
of the variables is little wonder in most of the cases and generally seems reasonable. Nevertheless,
a problem has been noticed with variables Shower and Bath, which even if summed up, do not
exceed the total number of observations. This implies that there are real estates without access to
any of these objects. As such a situation seems extremely unlikely, it has been concluded that the
owners simply did not treat shower as a characteristic worth mentioning in numerous rental offers.
Therefore, the variable Shower has been excluded from further analysis while the variable Bath has
been kept in its original form.
Table 5: Descriptive statistics 1
Variable Min. Mean Median Max St. dev.
Price 295 1347.00 1295 4995 542.38
Living_Area 6 75.61 73 935 41.29
Rooms 1 2.72 3 11 1.13
Bedrooms 1 1.20 1 5 0.81
Time_walking 123 2275.00 1754 9037 1512.49
Time_biking 32 704.70 540 2530 458.37
Time_public 109 1085.70 996 3085 530.86
Longitude 4.41 4.48 4.48 4.58 0.027
Latitude 51.87 51.92 51.92 51.98 0.017
Services 0.00 1.82 1.00 9.00 1.863
32
Table 6: Descriptive statistics 2
Variable Not present Present
Balcony 1152 692
Garden 1638 206
Storage 1382 462
Garage 1759 85
Shower 1291 553
Bath 1371 473
Lift 1456 388
Toilet 1053 791
Furnished 562 1282
Park 1359 485
View_on_the_city 1617 227
Enjoyable_view 1204 640
Green_view 1589 255
Water_body 892 952
Pets_not_allowed 1696 148
Income 1693 151
Insolation 1171 673
Before moving to the analysis part, this seems like a proper moment to provide the reader with
the reasoning behind the choice of variables used in the research. Let’s recall that in the literature
review a handful of external aspects impacting the value of a real estate has been described e.g. air
and noise pollution, while they did not find their place in the final data set. The initial analysis of
air pollution shown, that as a windy city located nearby a sea, Rotterdam in general does not suffer
greatly from air quality problems. Moreover, air quality in the city seems to be rather homogenous,
without any districts standing out notably from the average. Furthermore, as stated in the theoretical
part of the study, according to the hedonic modeling theory, the customers’ willingness to pay for a
particular good is based only on the attributes that they may perceive. Therefore aspects such as
air quality or environmental risks seemed unlikely to affect the rental price. In the end, it is rather
uncommon for the potential tenants to conduct research on these areas. Instead, it is much more
probable that the evaluation of a property is usually based on more easily noticeable features such
as the proximity to a park or the view from the window.
Nonetheless, as in any other big city, the problem of noise pollution should not be omitted. In
this study, an attempt of addressing this problem has been made with the usage of Google Maps. An
33
image recognition model has been trained to derive if a given property lies in a direct neighborhood
of any of Rotterdam’s busy roads. Unfortunately, the accuracy of the initial model has not been
satisfying, and therefore, with the limited time and resources taken into account, the idea has been
abandoned.
Due to limited data sources, it is a difficult task to include in the study all the structural and
locational attributes studied in the previous research. However, numerous aspects are indirectly in-
cluded in the analysis through the usage of geographical coordinates which serve as a proxy variable.
On the one hand, it is not possible to quantify with this approach the value of the neighborhood
characteristics such as ethnicity in the area or criminality rate. On the other hand, it is rather safe
to assume that if these characteristics do impact the rental price in some of the Rotterdam’s districts
it will be reflected in the regression model through locational variables.
34
5 Analysis and results
The analysis has been divided into two parts. Firstly, multiple regression models have been created.
After comparing the accuracy of the models, the one with the best performance served as a base for
the explanatory analysis of the housing market in Rotterdam, which is presented in the second part
of the section.
5.1 Regression
Before proceeding with the regression analysis, the correlation of continuous variables in the data set
have been checked. The result in the form of a correlation plot is presented in Figure 2. At the first
glance, it may be seen that there are two aggregations of linearly correlated variables. Living area,
the number of rooms and bedrooms are unsurprisingly highly correlated with each other. In order
to diminish the problems caused by multicollinearity of the explanatory variables in the regression
model, the variable Bedrooms has been dropped. Nevertheless, even with the high correlation taken
into account, it is safe to assume that the number of rooms and living area are crucial characteristics
of a property and should not be excluded from the analysis.
The second aggregation of correlation consists of features describing the localization of a prop-
erty. While there is no point in using all three measures of time needed to travel the distance to
Rotterdam Central Station in the regression, their correlation provides some valuable insights. As
all the variables are similarly correlated it may be assumed that the biking infrastructure and public
transport are evenly distributed over the area of Rotterdam. Therefore, after additionally taking in
to account the extraordinary popularity of biking in Rotterdam, the variables Time_walking and
Time_public have been excluded from further analysis.
35
Figure 2: Correlation plot for numerical variables.
Lastly, even though the longitude and latitude are not linearly correlated with the variable
Time_biking it is safe to assume that in the non-linear model, the multicollinearity problem would
appear. Therefore, the decision on choosing only one of the locational approaches had to be made.
Overall, in case of classic hedonic models, the variable Time_biking seems more appropriate than
the geographical coordinates, which are not expected to perform well in linear conditions. The linear
model allows geographical variable only to indicate through the sign of the coefficient in which part
of the city (e.g west or east in case of longitude) the rental costs are on average higher. However, it
is completely possible that while the majority of expensive estates are located on the western side of
a city, eastern suburbs feature some luxurious districts. Such a phenomenon could not be captured
through the usage of geographical coordinates in the traditional OLS regression.
On the contrary, in non-linear models such as decision trees, the combination of longitude and
latitude is expected to capture such patterns for all the possible geographical directions. On the
other hand, by including the geographical coordinates in the non-linear model, the variables extracted
from Google Maps: Water_body and Park would coincide, which consequently could decrease their
importance and impact on prediction accuracy.
As such multiple models have been created and compared to determine the most effective one
in terms of prediction accuracy and reliability. For all the models the same train and test sets have
36
been used, where the former consisted of 70% and the latter of 30% of observations. The comparison
between the models has been based on the RMSE metric calculated on the test set.
The initial models have been based on a classic hedonic pricing approach in the form of an
OLS regression. The first two models aimed to determine which locational approach performs
better in linear conditions: Time_biking or longitude and latitude. The regression models except
for locational variables consisted of the following predictors: Living_Area, Rooms, House_Type,
Bathrooms, Balcony, Garden, Storage, Garage, Bath, Lift, Toilet and Furnished. It may be recalled
that all the mentioned variables are part of the initial data set, therefore they have been directly
scrapped from featured fields of rental offers website. As such, this set of variables has been used
as a benchmark in addressing the research question asking if image and text features impact the
accuracy of the hedonic model. For the simplicity of visualization, the set of variables used will be
referred to as basic housing attributes (BHC) in the following parts of the analysis.
The comparison of the models proved the slight superiority of Time_biking over geographical
coordinates in linear conditions. The model with variable Time_biking surpassed its counterpart
in terms of both RMSE (321 vs 329) and R
2
(0.59 vs 0.56). Therefore, it has been decided that
the Time_biking model will serve as a benchmark for more complex linear models. Subsequently,
an OLS model featuring all the BHC and features extracted from images and descriptions have
been built. The coefficients of the model are presented in Table 7. Out of basic housing attributes
living area, the number of rooms and bathrooms, presence of garden, garage and bath as well as the
furnishing turned out to be significant, positive predictors of a rental cost prediction. Moreover, it
has been found out that the single rooms are 194 euros cheaper than the flats, leaving all the other
variable values constant. Lastly, the farther from the city center the property is located, the less
expensive its rent is.
37
Table 7: The coefficients of the linear model empowered by image and text features
Variable Estimate Pr(>|t|)
(Intercept) 420.62 < 0.001
Living_Area 6.06 < 0.001
Rooms 88.88 < 0.001
House_Type House 39.58 0.422
House_Type Room -194.04 < 0.001
Bathrooms 238.40 < 0.001
Balcony Present -5.36 0.817
Garden Present 108.09 0.002
Storage Present -39.66 0.158
Garage Present 252.32 < 0.001
Bath Present 56.31 0.027
Lift Present 36.65 0.211
Toilet Present -11.57 0.608
Furnished Yes 81.46 < 0.001
View_on_the_city 1 122.41 < 0.001
Park 1 5.26 0.822
Pets_not_allowed 1 -60.26 0.160
Enjoyable_view 1 8.29 0.711
Green_view 1 29.48 0.314
Water_body 1 47.67 0.022
Income 1 -159.12 < 0.001
Services 13.22 0.125
Insolation 1 106.76 < 0.001
Time_biking -0.26 < 0.001
In terms of features extracted from images and descriptions, the view on Rotterdam, the prox-
imity of a water body (either the Meuse river or a large lake), and insolation positively impact the
rental price of a house. On the other hand, the financial requirements for renting a property are
linked with an average drop in rental price of 159 euros. With regards to the overall performance of
the model, RMSE decreased in comparison to the previous OLS model to 308, while R
2
increased by
0.02 to the level of 0.61. Therefore, the initial analysis provides strong foundations to claim that the
additionally extracted features do positively impact the accuracy of a hedonic model. Nevertheless
let’s recall that according to Butler (1982) and Mok et al. (1995) the under-specification of variables
38
is more welcome than the over-specification in case of hedonic regression models. Therefore, the
last linear model has been built, this time however the insignificant features extracted from images
and descriptions have been dropped. The performance of the last model, almost did not change in
comparison with its predecessor as its R
2
and RMSE scores were equal to 0.61 and 309 respectively.
Moreover, only slight changes in the size of coefficients has been found which may be seen in Table
8.
Table 8: The coefficients of the final linear model
Coefficients Estimate Pr(>|t|)
(Intercept) 433.37 < 0.001
Living_Area 6.09 < 0.001
Rooms 89.80 < 0.001
House_Type House 26.52 0.589
House_Type Room -192.16 < 0.001
Bathrooms 238.77 < 0.001
Balcony Present -5.44 0.813
Garden Present 111.29 0.002
Storage Present -33.27 0.234
Garage Present 256.53 < 0.001
Bath Present 64.62 0.011
Lift Present 38.26 0.188
Toilet Present -9.34 0.678
Furnished Yes 87.34 < 0.001
View_on_the_city 1 115.21 < 0.001
Water_body 1 46.30 0.025
Income 1 -165.19 < 0.001
Insolation 1 117.83 < 0.001
Time_biking -0.26 < 0.001
Nevertheless, as mentioned previously in the methodological part of the study, an OLS model is
not fully reliable if it does not meet its five main assumptions. Therefore, the verification of the linear
requirements has been performed. Firstly, the assumption of linearity of the data has been checked
with the usage of plot comparing residuals of the model with fitted values presented in Figure 3.
For smaller predictions, the residuals are fairly equally located around the 0 line. Nevertheless, by
following a red line (showing the average values of residuals at given fitted values) the decreasing
trend may be noticed. This proves that the relationships in the data are not perfectly linear, as
39
for larger predictions the model underestimates the values of a property. Moreover, on the basis
of Figure 3 it may be concluded that the model suffers from a heteroscedasticity, as the larger the
fitted values are, the larger the spread of residuals becomes.
Figure 3: Residuals versus Fitted Values
Afterward, the multicollinearity in the model has been checked with the usage of Generalized
Variance Inflation Factor (GVIF), a variation of VIF, which is applicable also for categorical vari-
ables. While the rule of thumb in interpreting the VIF is not to have a single value exceeding 5,
in case of generalized metric it is GV IF
(1/(2∗Df ))
that should not reach that threshold (Buteikis,
2019). Therefore, the results presented in Table 12 in the appendix show no serious problem with
multicollinearity in the model. The next assumption, lack of auto-correlation in residuals of the
model is met, as seen in Figure 8 in the appendix. The last assumption, claiming that residuals
should be roughly normally distributed is violated in this particular model. In the QQ plot presented
as a Figure 9 in the appendix the heavy tails may be noticed, indicating that the distribution of
residuals is not of gaussian type.
Two main approaches may be taken to overcome the problems with violated linearity assump-
tions. The first one, the data manipulation using methods such as Box-Cox transformation has been
frequently applied in other hedonic research papers. While this approach is not a primary focus of
the study, some standard data transformations have been applied in order to verify if they would
help with meeting the above assumptions. Nonetheless, neither logarithmic nor Box-Cox transfor-
mation led to a notable improvement in the area. In both cases there were still assumptions violated
which example may be seen in Figure 10 in the appendix. Moreover, the accuracy of the transformed
40
models have dropped in comparison to the original counterpart.
As such, more attention has been given to the second approach: addressing the above problems
through the usage of a non-parametric model, that does not have such strong assumptions as the
OLS regression. Moreover, as discussed in the previous sections of the paper, applying random
forest or any other advanced machine learning method, allows capturing more complex patterns in
the data which consequently, improves the model’s performance in terms of accuracy. However, as
the linear models did perform relatively well in this study, and their assumptions were not extremely
violated, the random forest models have been built in a similar manner.
In total five random forest models have been built. Before proceeding to their description, it
is crucial to recall, that random forest is a method based on bootstrapping. This method, brings
an essential problem, in the form of its randomness. By default in R, setting the random number
generator does not impact the results of bootstrapping which is always random. Consequently,
even with random number generator set, the results of random forest may slightly change in each
run. Therefore, all the results shown below are presented in the form of average model’s accuracy,
calculated on the basis of 20 runs.
The first two models, similarly as in case of an OLS regression, aimed to distinguish which
locational approach is more suited: Time_biking or geographical coordinates. On the contrary to
linear models, the latter approach indisputably proved to be superior over its counterpart. The third
model, consisted of basic housing attributes (BHC), geographical coordinates and features extracted
from images and text, that turned out significant in the linear conditions. The fourth model, shared
the same predictors with the third one, however instead of traditional random forest, conditional
random forest was used. The fifth model was created with the sole goal of being used as a benchmark
and consisted only of three predictors: Living_area, Longitude and Latitude. The performance of
all the created models is presented in Table 9.
Overall, the random forest model including the significant variables extracted from images and
texts turned out to be the most accurate model. Nevertheless in order to formally address the main
research question, the difference in accuracy between this model and the random forest that did
not use the mentioned predictors has been statistically tested. As both samples, consisted only of
twenty observations with each one representing model’s accuracy in a given run, the non-parametric
Mann-Whitney U test was performed. It resulted in a p-value of 1.451e-11 implying, that there is a
significant difference between the accuracy of both models that is not caused by the randomness of
the random forest method.
41
Table 9: The summary of regression models
Model R Squared RMSE
OLS: BHC + Long|Lat 0.56 329
OLS: BHC + Time_biking 0.59 321
OLS: BHC + Time_biking + Image and Text Variables 0.61 308
OLS: BHC + Time_biking + Significant Image and Text Variables 0.61 309
RF: BHC + Time_biking 0.70 258
RF: BHC + Long|Lat 0.71 246
RF: BHC + Long|Lat + Significant Image and Text Variables 0.74 240
CRF: BHC + Long|Lat + Significant Image and Text Variables 0.65 290
RF: Living_area + Long|Lat 0.69 249
5.2 Explanatory analysis
The results presented in the previous section, provide us with a fairly reasonable conclusion that
the (semi)external factors do impact the rental price of a property in Rotterdam to a small extent.
Before proceeding with the more in-depth explanatory analysis of the random forest model, its
variable importance based on the mean decrease in accuracy has been calculated. The results
presented in Figure 11 in the appendix indicate that the model’s accuracy is mostly based on the
variables living area, the number of rooms and the geographical coordinates of the real estate. While
such results are definitely not surprising in the context of the housing market and especially its rental
subset, the variable importance does show that the other predictors, both external and structural,
also contribute to the final prediction of the model.
Therefore, the last step in the research aims to quantify this impact and determine how the
estimation changes depending on the model used. In case of linear model, estimating the attributes’
effect on the price of a good may be easily done by analyzing the coefficients of an OLS regression.
This cannot be said about the random forest model, where in order to determine the effect of a single
predictor on the prediction, slightly more advanced methods have to be used. As such, in order to
derive the magnitude of the mentioned effects, partial dependence function has been calculated for
the majority of variables used in the model. Table 10 presents the comparison between the values
obtained with partial dependence functions and the coefficients of the final linear model.
42
Table 10: The comparison of variables’ impact on the prediction between the models
Variable OLS coefficient RF Partial Dependence
House_Type Room -192.16 -281.73
Garden 111.29 9.53
Garage 256.53 104.96
Bath 64.62 37.59
Furnished 87.34 57.12
View_on_the_city 115.21 62.56
Water_body 46.30 10.88
Income -165.19 -15.44
Insolation 117.82 51.61
The results of the comparison are somewhat surprising. While both the random forest and the
OLS regression models agree on the sign of variables’ coefficients, there are notable differences in
their magnitude. For the majority of predictors, especially for Garden, Garage, and Income, their
impact on the prediction is much lower in case of a random forest model. The only variable which the
random forest model evaluated as more influential than the OLS regression did is the House_Type
Room.
Apart from the categorical variables presented in Table 10 the partial dependence has been also
applied to the numerical variables. Probably the most interesting conclusion may be drawn from the
analysis of the living area presented in Figure 4. On the contrary to the OLS regression where each
additional squared meter of the property has been connected to the increase in rental price of 6.09AC,
the dependence between both variables in the random forest has been found as not constant. For
the properties with the living area between 6 and 136 meters, the average value of each additional
squared meter has been estimated to 7.26AC. Surprisingly, this value drops drastically to 1.65AC after
reaching the threshold of 136 squared meters, in order to eventually rise again to 5.54AC after passing
the threshold of 191 squared meters. However, the average price of each squared meter calculated
based on the whole set is equal to 5.70AC which is not that different from the OLS coefficient.
43
Figure 4: The Partial Dependence Plot of Living Area
Similarly to the living area, the number of rooms also has a comparable value between two
models. On the basis of partial dependence function it has been concluded that each additional
room increases on average the rental price of a property by 87.16AC. However it only applies until
the threshold of seven rooms after which no further increase in the price has been noticed. When
compared to the OLS coefficient of 89.80AC the difference again is quite negligible and is definitely
much smaller than in case of categorical variables. This phenomena suggests that the differences
between both models may be partially caused by the nature of random forest which in general favors
the continuous variables in terms of their impact and importance. Nonetheless, if both living area
and the number of rooms do share similar coefficients across the models, the drop in the importance
of categorical variables in the random forest has to be compensated by other predictors.
The difference in the locational approach of both models at least partially explains the presented
behavior. In the random forest model longitude and latitude have been used, on the contrary to the
OLS model where Time_biking variable is present. As shown in the previous section, in non-linear
conditions geographical coordinates perform much better than the distance to the city center. As
such, it may be suspected that as longitude and latitude are much more functional in the prediction
(as proxy variables they indirectly reflect numerous aspects that Time_biking does not) they affect
the other predictors in the model to a further degree than the Time_biking does in the OLS case.
There is no certainty in establishing which of the models describes better the real-life values of
properties’ characteristics. However, the much better performance of a random forest model in terms
of accuracy and the variance explained together with the violated assumptions of the linear model do
44
prompt to conclude, that the obtained coefficients may be closer to reality than the ones of the OLS
regression, even with the drawbacks of the partial dependence function taken into account. This
claim however implies that the OLS model may have a tendency to overestimate the value of the
less important structural attributes which is caused by its incapability of capturing the locational
aspects well.
For the most part, the results of the random forest regression model come to similar conclusions
as previous research presented in the literature review part. The most important attributes of a
property in terms of its rental price are the living area, the number of rooms, and the general
location in the city. Nevertheless, on the contrary to numerous studies, the proximity of parks or
other green areas does not seem to affect the rental prices in Rotterdam. Moreover, only a small
increase in the price of 10.88AC has been associated with the proximity of any major water body. As
such it may be concluded that the importance of location is caused by some other aspects contained
in the geographical coordinates which is an interesting aspect to be studied in the future.
Additionally, it has been found out that the decisions made by landlords are not important
in the prediction. Neither pet prohibition, nor the number of services mentioned in the property
description impact the rental price, while the income requirement has only a small negative impact
of -15.44AC.
In regards to additional structural attributes of a property, a garage seems like a desirable perk
among Rotterdam’s residents. On average they are willing to pay for it 104.96AC. Similarly, the
existence of a bath in a property is associated with an increase in rent by 37.59AC. However many
attributes which in accordance with the summary provided in Table 1 should theoretically increase
the rental price, have been found as insignificant in case of Rotterdam. These attributes are storage
room, lift, separate toilet, and balcony. Moreover, gardens have been evaluated with only 9.53AC
which stands in strong contrast with the results of the OLS approach. Furthermore, it seems that
for the tenants in Rotterdam the furnishing has very little value as on average the rent of fully
furnished properties is only 57.12AC more expensive.
Previous research indicate that the view from a property impacts the housing price positively as
long as the view is unbroken and features some enjoyable objects such as sea, river, or hill. On the
other hand, obstructed views do have a tendency to lower the value of a property (Chin & Chau,
2003). The insignificance of variables Green_view and Enjoyable_view shows that even the view
that subjectively has been rated as above the average, it does not suffice for an average Rotterdam’s
tenant to pay a premium for it. In case of this study it appears that in order for a view to have an
impact on the predicted rental price, it has to feature a spectacular panorama of Rotterdam. Even
though the value of the view on Rotterdam has been derived with the usage of partial dependence
function, the more detailed analysis of the variable, together with Insolation and Water_body is
presented down below in the paper.
Even in his early work Rosen (1974) argued that the marginal willingness to pay for the attribute
45
of a good changes for consumers, depending on their (nonlinear) budget constraints and preferences.
Therefore, not only it further supports the decision of using a non-linear machine learning model
which is the random forest, but it also encourages checking how the hedonic price of household
attributes varies in Rotterdam. Nevertheless, due to the globality of a partial dependence function,
the analysis is not possible with this particular method. Instead, Local Interpretable Model-Agnostic
Explanations (LIME) have been applied.
As mentioned in the methodological part of the paper, there is not much literature on parameter
tuning for LIME. As LIME is a relatively computationally expensive method, the tuning has been
performed with a trial and error approach, resulting in the following parameters:
• Local interpretable model type: decision tree
• The width of smoothing kernel π
x
: 2
• Distance function: Manhattan
• Number of explained features: 18.
For each observation in the train set, the local surrogate model has been built, allowing an ap-
proximation of the impact of each random forest’s predictor on each prediction separately. However,
as for some observations, the explanatory model featured extremely low R
2
value, only the obser-
vations with the R
2
of above 0.3 were taken into account in the further analysis. Figures 5, 6 and
7 below represent the changes in the value of housing attributes depending on the rental cost of
properties.
The results of LIME analysis indicate that indeed the hedonic price of an attribute may not
be linear in many cases. For example, the value a view on the city presented in Figure 5 for the
flats/rooms with rent below 1000AC stays at the level of around 60AC, in order to rise above 100AC for
the properties costing more than 1500AC per month.
A similar growing tendency has been noted for the insolation of a property. In this case however,
the hedonic price of an attribute lowers after reaching the rental cost threshold of 1500AC. Such a
behavior may be explained by the fact that in the data set detached houses and larger apartments
start appearing after that threshold. As it may be expected from such properties to be exposed to
more than one geographical direction, the more sunny such places usually are by default.
The implicit price of proximity to a water body presented in Figure 7 also has a rising pattern.
While similarly to the partial dependence results for the most common, relatively cheap properties
the value is expected to be below 20AC, the value of the trait reaches 50AC for more expensive estates.
46
Figure 5: The absolute value of a view on the city versus predicted rental cost
Figure 6: The absolute value of insolation versus predicted rental cost
47
Figure 7: The absolute value of water body proximity versus predicted rental cost
Nevertheless, if we treated the rental cost of a property as the value of a budget constraint
function, the interpretation of the figures could be linked to the hedonic pricing theory of Rosen
(1974). As according to Rosen, hedonic (implicit) price of an attribute depends on the function of
utility level, budget constraint, consumers preferences and other variables such as age or education,
what the graphs above represent, is in fact a simplified example of such function with just one
predictor used. Unfortunately, it is nearly impossible to gather data relating to matters such as age
or education level of tenants without cooperating with real estate brokers or municipality. Therefore,
deriving a more advanced hedonic price function for the given attributes was not feasible in this
research, however it is an interesting aspect to be analysed in the future studies.
48
6 Conclusions
For almost 50 years hedonic price models have been extensively used in numerous research. How-
ever, despite their undeniable popularity, in most cases, the hedonic models have been based on
relatively simple regression methods and tabular data. This study proves that such an approach can
be successfully developed with the usage of complex data sources and advanced machine learning
algorithms. The variables extracted from images and descriptions of rental offers allowed to improve
the performances of both traditional hedonic regression and random forest. Moreover, an accu-
racy comparison between the models revealed an undeniable superiority of the decision tree-based
method.
The problem with the interpretability of the black box model has been addressed with model-
agnostic methods which allowed the comparison of variables’ importance among the models. As
long as the models agreed on some of the most important predictors namely the living area and the
number of rooms, the locational aspect has been more emphasized by the random forest model. The
results have also revealed that the OLS regression model, when compared to the random forest, is
more likely to overestimate the value of nonessential structural attributes. Lastly, with the innovative
usage of LIME, it has been found out that the hedonic prices of housing attributes are not likely to
be constant and depend on the total value of a property. Such results seem to be in accordance with
the original hedonic theory published by Rosen (1974), who argued that in general, the nonlinearity
between the price of goods and their inherent attributes is likely to happen.
49
7 Appendix
Table 11: Frequency of the words appearing in the descriptions
Word Frequency Word Frequency Word Frequency
apartment 3052 space 424 shared 221
floor 2194 month 420 contact 220
rotterdam 2043 property 416 windows 219
bedroom 1922 facilities 400 short 217
kitchen 1919 ground 395 erasmus 215
located 1752 income 389 service 213
living 1715 building 386 neighborhood 211
spacious 1551 shopping 371 furniture 210
rent 1440 m2 370 lovely 210
bathroom 1401 hob 368 3rd 207
equipped 1265 metro 365 students 207
furnished 1206 tram 363 master 205
access 1150 layout 357 accessible 204
toilet 1072 microwave 355 upholstered 203
balcony 1006 light 354 south 202
shower 993 price 352 terrace 201
center 917 dining 344 requirement 199
entrance 908 rear 342 basin 198
house 893 period 340 combi 197
city 806 deposit 337 derived 194
bedrooms 799 street 333 maximum 194
distance 780 corner 327 details 192
beautiful 764 garden 321 rights 192
public 725 de 312 extra 191
double 717 bright 307 free 186
walking 715 luxury 307 tiled 185
hall 688 excluding 306 bike 183
washing 673 electricity 302 popular 183
transport 661 private 302 person 180
bed 655 completely 295 rented 180
shops 647 dryer 295 offers 179
Continued on next page
50
Table 11 – Continued from previous page
Word Frequency Word Frequency Word Frequency
storage 641 extractor 293 staircase 179
station 619 apartments 290 sunny 179
district 601 laminate 286 connection 178
rental 591 studio 286 stay 178
central 581 minimum 276 cozy 177
front 580 cupboard 275 hallway 177
machine 579 washbasin 273 information 177
separate 564 hood 272 accommodation 172
minutes 563 situated 270 kralingen 170
modern 555 including 264 glazing 169
view 536 wardrobe 264 basement 168
walk 524 monthly 263 bus 168
gas 509 door 260 renting 168
months 509 centre 256 cleaning 162
water 506 built 255 entire 162
fridge 504 ca 254 mailboxes 162
renovated 493 offer 252 design 161
sink 493 close 249 pets 161
parking 488 costs 247 text 161
tv 484 bath 246 reached 160
heating 457 stairs 243 refrigerator 160
restaurants 457 mirror 240 west 160
freezer 449 reach 228 quiet 159
dishwasher 444 appliances 225 home 158
oven 440 luxurious 225 allowed 157
nice 436 suitable 223 radiator 156
approx 433 closet 222 elevator 155
location 428 complex 221 closed 154
internet 425 roads 221 doors 154
51
Table 12: Generalized Variance Inflation Factor for linear regression
Variable GVIF Df GVIF
ˆ
(1/(2*Df))
Living_Area 2.233044 1 1.494337
Rooms 2.496504 1 1.580033
House_Type 2.359207 2 1.239343
Bathrooms 1.136956 1 1.066282
Balcony 1.303721 1 1.141806
Garden 1.22419 1 1.106431
Storage 1.524984 1 1.234902
Garage 1.150035 1 1.072397
Bath 1.277762 1 1.130381
Lift 1.433762 1 1.197398
Toilet 1.270875 1 1.127331
Furnished 1.165545 1 1.079604
View_on_the_city 1.066187 1 1.032564
Water_body 1.086581 1 1.042392
Income 1.074908 1 1.036778
Light 1.064132 1 1.031568
Time_biking 1.237806 1 1.112567
52
Figure 8: Auto-correlation plot for residuals of linear model
Figure 9: Normal QQ plot for linear model
53
Figure 10: Residuals versus fitted values after Box-Cox transformation
54
Figure 11: Variable importance of the final random forest model
55
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