170
October 2020

Dynamic Hedonic Analysis Using Time-Varying
Coefficients:
Application to Dubaiʼs Housing Market

Hiroki Baba（Center for Spatial Information Science, The University of Tokyo）

Hayato Nishi（Graduate School of Engineering, The University of Tokyo）

Ashoka Mahabala Seetharamapura（Emirates Real Estate Solutions, Dubai Land Department）

Chihiro Shimizu（Center for Spatial Information Science, The University of Tokyo/
Nihon University）

0

Dynamic Hedonic Analysis Using Time-Varying
Coefficients:
Application to Dubai’s Housing Market
Hiroki Baba∗, Hayato Nishi†, Ashoka Mahabala Seetharamapura‡,
and Chihiro Shimizu§
October 8, 2020

Abstract
In the housing market analysis, capturing the temporal change in market structure isof importance. We aim at proposing housing price index estimation models employing
time-varying coefficients and compare the models considering the lengths of the time
periods. We firstly employ ordinary least squares for each time period: a separate hedonic model, and then, a time window is applied for a specific length of time period:a rolling hedonic model. We, thirdly, consider coefficient-wise stochastic innovation
terms: a dynamic hedonic model, since random walks in the Kalman filter enable us toestimate time-varing coefficients that are robust against exogenous shocks. We study
the Dubai’s housing market, where housing prices have been very volatile for political reasons. One of our findings indicates that the dynamic hedonic model shows
the highest predictive power. Moreover, since the trends of the coefficients differ foreach variable, observing them helps identify the factors that are more important for
explaining housing price indexes. A stability analysis shows that the separate hedonic
model is far more unstable than the others, whereas the dynamic hedonic model is themost stable method for all the variables. We consider that the dynamic hedonic model
is applicable even when only a small number of samples have been obtained, because
the method refers to all the samples for smoothing the coefficients.
Keywords: housing price index; hedonic model; Kalman filtering; Dubai; housing market.
JEL Classification: C43; E31; R31.

∗

Center for Spatial Information Science, The University of Tokyo, hbaba@csis.u-tokyo.ac.jp
Graduate School of Engineering, The University of Tokyo, h.nishi@ua.t.u-tokyo.ac.jp‡
Emirates Real Estate Solutions, Dubai Land Department§
Center for Spatial Information Science, The University of Tokyo and College of Sports Sciences, Nihon
University, cshimizu@csis.u-tokyo.ac.jp
†

1

1

Introduction

In a housing market analysis, it is important to capture temporal change in the market
structure. A fail to predict the market conditions can cause a financial crisis (Kaplan, Mitman, and Violante, 2017). A great number of papers, thus, have constructed a housing
price index (HPI) to improve the reliability and timeliness (Case and Shiller, 1989; Hill and
Scholz, 2018; Shimizu et al., 2010; Shimizu, Nishimura, and Watanabe, 2010; Diewert andShimizu, 2015, 2016).
One of the fundamental ways to construct an HPI is to employ a hedonic method,whose theoretical framework is provided by Rosen (1974). The hedonic method includes
a market clearing function within the interaction between households’ bid functions andsuppliers’ offer functions (see Chapter 2, Diewert et al. (2020) for details). The frameworkof a hedonic approach is applied extensively not only to HPIs (Goodman, 1978), but also
to measure the demand for housing (Shefer, 1986) and the impact of neighborhood externalities on housing prices (Michaels and Smith, 1990).
Hedonic methods have been developed to deal with changes in time series (Hill, 2013).
One of the basic methods dealing with time is adding time fixed dummies, but the idea ofrolling time dummies has been getting more attention in recent years(Shimizu et al., 2010).
Hill et al. (2020) explore the optimal length of the rolling window and the optimal linking
method by minimizing the X metric proposed by Diewert (2002, 2009). Nevertheless, thosestudies use samples within time periods of a fixed length to estimate the coefficients andregard the coefficients as non-stochastic variables, so that those models are not always
robust against exogenous shocks.
We aim at proposing housing price index estimation models employing time-varying
coefficients and compare the models considering the lengths of time periods. We firstly
employ ordinary least squares for each time period, and secondly, a time window is applied for a specific length of time period. We, thirdly, consider coefficient-wise stochastic
innovation terms, since random walks in the Kalman filter enable us to estimate timevaring coefficients that are robust against exogenous shocks. The idea of stochastic coefficients was proposed by Guirguis, Giannikos, and Anderson (2005) with the use oftime-varying Kalman filtering. However, since they explore various methods from theperspective of macro finance, the micro-structure of the housing market has not been well2

investigated using their framework.
Another novelty in the present paper is the study area: Dubai, one of the most prosperous real estate markets in the Arab countries. Despite the success of its real estatemarket, Dubai has sometimes experienced large volatility in its housing prices, due togeopolitical issues. It is true that Hepşen and Vatansever (2011, 2012) have previously
tried to forecast HPIs in Dubai, but they focus on macroeconomic trends, not considering
the micro-structure of the buildings and property owners. We thus think that validating
our proposed method fits well in Dubai and well predicts the rise and fall in the HPIs.

2

Dubai’s housing market

Dubai is one of the most attractive cities in the world. Surrounded by desert, a large partof the land is artificially developed and significant tall buildings are agglomerated. The
characteristic of the city is the high density of buildings in its central business district. In
the field of economics, high density encourages knowledge spillover and drives growthof a city Glaeser et al. (1992). While such a high density in Dubai is appreciated amongexperts and policy makers, the citizens in Dubai consider that the other areas also represent
their cultural context (Alawadi and Benkraouda, 2019). For the middle-class population,
considering not only the high density area but also the other areas enhances the value ofthe city (Alawadi, Khanal, and Almulla, 2018).
From a bird’s eye view, a developed land that seems to be a palm tree is called Palm
Jumeirah and Palm Jebel Ali, where all the lands in the island are reclaimed and developed
for the rich (Figure 1 1 ). The reason why they have developed the island is that the acquisition of an ocean view is even worth developing the reclaimed land. The tallest building in
Dubai is called Burj Khalı̄fah, with a height of 828.0 m and 206 stories. This height provides
both buyers and sellers with a rent premium for a good view.
Knowing the characteristics of the property rights in Dubai helps us understand thenature of the housing market. Due to the influence of the British housing market system,
Dubai’s property rights are categorized into freehold and non-freehold 2 . Although in the1

The base information of the map is retrieved from OpenStreetMap (https://www.openstreetmap.
org).
2
British property law originally categorized property rights into freehold and leasehold. While in the freehold market, any buyers and sellers are able to transact as they like, in the leasehold market, either the individuals or the corporate bodies own the property right and they lease the right for a specific term.

3

Figure 1: Map of Dubai
freehold market, any foreign buyers and sellers are able to transact as they like, this area
is clearly delineated. In the non-freehold market, only residents of the Gulf Cooperation
Council 3 are allowed to make property transactions. Since foreign capital is invested in
the freehold market, innumerable skyscrapers have been built in the area (Bagaeen, 2007).
For further prominent successes in Dubai’s housing market, dealing with risks fromgeopolitical issues is of importance. For instance, diplomatic relations with adjacent countries would affect the stability of the financial status, leading housing price volatility.
Focusing on the freehold market, such exogenous shocks are amplified due to the specific market mechanism. Most properties in the freehold market start appearing on themarket together with the beginning of development, and developers collect the expenseof construction. Since the development lasts for two to three years, most buyers expect topurchase the properties for the purpose of investment. Actually, real estate prices in Dubai
is severely influenced by the 2007–2008 financial crisis, causing discontinuation of many
development projects since then.
Considering the uncertainty, investors carefully observe whether and when economic
3

The Gulf Cooperation Council consists of the United Arab Emirates, Bahrain, Kuwait, Oman, Qatar, andSaudi Arabia.

4growth stagnates. One of the risks in the market is housing oversupply. Since foreigners
expected to invest properties, substantial housing properties were supplied. Moreover,housing market, not only in Dubai but in countries worldwide, is always facing exogenous shocks, such as COVID-19 crisis. Thus, predicting the trend of the housing market isimportant, for the sake of reducing the uncertainty in the market.

3

Method

3.1

Hedonic models and HPI

We start by explaining the hedonic approach theorized by Rosen (1974), one of the mostpopular models to estimate HPIs. The hedonic method deals with a price as a bundle ofhousing characteristics and estimates the value using regression analysis. When we obtainpooled data that includes comprehensive housing characteristics, it enables us to calculatequality-adjusted price indexes.
The estimation of HPIs with hedonic models has two major problems: omitted variables bias and structural change. The former problem occurs due to the difficulty in collecting all the variables required for the estimation of the functions as well as the presenceof unobservable variables (Case and Quigley, 1991; Clapp, 2003). The latter involves thenecessity of accommodating changes in the house price structure, since HPIs change overlong periods of time (Case, Pollakowski, and Wachter, 1991; Clapp, Giaccotto, and Tirtiroglu, 1991; Clapp and Giaccotto, 1998).
Possible ways to avoid the problem of omitted variables bias include collecting othervariables or using the repeat-sales method. The repeat-sales method estimates HPIs bycalculating changes in property prices based on the same property within a specific time
period (Case and Quigley, 1991; Case and Shiller, 1989). While this method is convenient
because the fixed effects cancel out, the sample size for properties that are traded morethan twice is likely to be small, especially in a thin market like Japan, which may cause asample selection bias. In the case of Dubai’s data, the sample period is limited to 2007–2018, so that we decided to collect as many variables as possible, instead of applying therepeat-sales method.
Under such circumstances, the importance of estimating HPIs with high accuracy is

5associated with changes in the market structure. To consider changes in the market structure, we introduce two models: the restricted hedonic model and the overlapping-period
hedonic model, and then propose hedonic models with time-varying coefficients. The restricted hedonic model fixes the market structure and measures the HPIs with the baselineof the HPI in period t. The overlapping-period hedonic model, proposed by Shimizu et al.
(2010), captures the successive changes in the regression coefficients with a certain period
length, which is called a window. Our proposed hedonic models also assume that themarket structure changes each period, and moreover one of the models assumes that the
regression coefficients have a stochastic nature.
Assuming a semi-log for the hedonic function, we describe the restricted hedonic model:

ln pit =KX

βk xkit +TXδs ds + εit

(1)

s=1

k=1

where pit is the transaction price for housing unit i in period t, βk are the coefficients in kth
housing characteristic, xkit is the housing attribute in kth housing characteristics and ith
housing unit in period t, δs denotes the time dummy coefficients in period s, ds is an index
that takes the constant value of 1 when s = 1, while for 2 5 s 5 T , it is a time dummy
variable, taking the value of 1 when s = t and the value of 0 otherwise, and εit is theidiosyncratic error in housing unit i in period t. This model assumes that the regression
coefficient βk is constant through the periods. Using ordinary least squares, pˆt is estimatedas follows.
lnˆpt =KX

βˆk xk + δˆ1 + δˆt

(2)

k=1

lnˆp1 =KX

βˆk xk + δˆ1

(3)

k=1

where βˆk , δˆ1 , and δˆt are the estimated parameters. Subtracting equation 3 from equation 2,
the HPI in period t is calculated by:

ln(

pˆt
) = δˆtpˆ1(4)

where the HPI in period t = 1 is used as the reference. In the same way, the change in the

6HPI from t − 1 to t is calculated by:

ln(

pˆt
ˆ .
) = δˆt − δt−1
pt−1
ˆ

(5)

This indicates that the HPI in period t is normalized by the HPI in period t − 1, so that
equation (5) enables us to compare the HPIs between periods t and t − 1. However, therestricted hedonic model has several problems. First, the HPIs are estimated using only thetime dummy variables without any housing attributes. This causes an estimation bias if the
housing characteristics have been changing during the periods. Second, autocorrelation in
the time series data should be considered. For instance, seasonality and its changes overtime may affect the HPIs.
Assuming structural changes in the market, the overlapping-period hedonic model
defined by Shimizu et al. (2010) is given as follows:

ln pit =KX

βk xkit +τXδs ds + εit

(6)

s=1

k=1

where t = 1, 2, ..., τ and τ indicates a specific period length (time window) satisfying 1 5
τ 5 T.
The HPIs are estimated with successive time windows [1, τ ], [2, τ + 1], ..., [r, τ + r −1], ..., [T − τ + 1, T ]. This successive manner of estimating the HPIs can take into accountchanges in the parameters. With the overlapping-period hedonic model, the period lengthτ is the length of the overlapping estimate period, so that the model is a type of restricted
hedonic model with respect to a certain period length τ . The overlapping-period hedonic
models for all periods are estimated successively by shifting the period length τ one by
one.

3.2

Proposed models

We now extend the hedonic model using time-varying coefficients. Here, we modify theoverlapping-period hedonic model with a time window size [t−τ, t+τ ], which means that7

the period length is 2τ + 1. We regress the housing prices within the time window:

ln pit =KX

βk xkit + δt + εit , f or t ∈ [t − τ, t + τ ].

(7)

k=1

The estimated HPIs can be considered as smoothing values in the designated time window. When τ = 0, we separately estimate the HPIs for each t = 1, 2, ..., T . We call thisthe ”Separate Hedonic Model” and our first proposed model as a benchmark of the HPI
values.
We then designate the time window size as a fixed value. Ideally, a time window size
should be determined by some criteria such as AIC, BIC, and so forth. However, since thisanalysis is the starting point for the time-varying coefficients approach, we set the window
size to be τ = 1. We call this model the ”Rolling Hedonic Model” and our second proposedmodel to compare with the separate hedonic model.
Although both the separate hedonic model and the rolling hedonic model deal withthe coefficients of housing attributes as non-stochastic parameters, the coefficients can be
redefined as stochastic parameters. The reason is that even though we consider the chronological changes in the HPIs, we still need to predict the coefficients βˆk .
To smooth the coefficients, we employ another approach, where we deal with the dynamic changes in the coefficients. We specify a hedonic function with a random walkmodel. The hedonic function is slightly modified:ln pit =KX

βkt xkit + δt + εit .

(8)

k=1

In the random walk model, since the coefficients in period t fluctuate from the previous
period, we have

βdt = βdt−1 + υdt(9)where βdt is the dth value of the coefficient and υdt is a stochastic innovation term that
follows a Gaussian distribution.
We then estimate the models with the Maximum Likelihood method in the frameworkof Kalman filtering. This estimation procedure optimizes coefficient-wise the variance8

Table 1: Sample size per year
Year

Sample Size

2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018

1731
10,52820,2617485
15,81819,74428,32623,453
11,055
9178
8163
4500

* Outliers are excluded(σd )2 , which is the variance of the noise υdt , which implies that the strength of the smoothing is automatically determined. Therefore, we can distinguish the dynamic coefficientsand the nearly non-dynamic coefficients. We call this the ”Dynamic Hedonic Model”,
which is our third proposed model, and considers the stochastic variation between periods.

4

The data

To compare these methods, we use the housing sales data in Dubai from 2007 to 2018.
Although the sample size fluctuates each year, every year has more than 1,000 samples.
Therefore, we think that the estimation result obtained from the data can be stable evenafter the exclusion of all the outliers (Table 1).
This data includes building coordinates, floor area space, and buyers’ information suchas nationality, gender, and age. Therefore, building fixed effects are controlled though thecoordinates of the building. Taking advantage of the POIs from OpenStreetMap 4 , we havemeasured the distances from each building to the nearest grocery store, restaurant, park,
police station, mosque, university, school, hospital, and coastal line. Since the shape of theroads in Dubai is complex and measures the distances in a small scale, we calculated the
distances via the road network. Moreover, we assume that land use restrictions affect the
housing prices, so that a residential land use dummy is included in our models.
4

https://www.openstreetmap.org/

9

Table 2: Housing attributes
Variablegen dummy
AGE AT CONTRACT TIME
UNIT SIZE SQM
RES
grocery dist st
restaurant dist stpark dist st
police dist st
mosque dist stuniv dist st
school dist st
hospital dist st
coastal dist st

Description
Gender dummy of buyer
Buyer’s age at contract time
Floor space (m2 )Land use regulation dummy (residential use)
Street distance to the nearest grocery store
Street distance to the nearest restaurant
Street distance to the nearest park
Street distance to the nearest police station
Street distance to the nearest mosque
Street distance to the nearest university
Street distance to the nearest school
Street distance to the nearest hospitalStreet distance to the nearest coastal line

The housing attributes used in this paper are shown in Table 2. For the preprocessingof the data, some variables have been trimmed; AGE AT CONTRACT TIME lower than 18
and higher than 100, and UNIT SIZE SQM lower than 20 and higher than 400 are omitted,
respectively. Moreover, all the variables are standardized. We then conducted a list-wisedeletion of the data, which means the data for analysis does not include either NA or
NULL values.

5

Result

The estimation results are illustrated in Figure 2. For simplicity, we only show the results
for const, UNIT SIZE SQM, gen dummy, and grocery dist st. The coefficients of const can beinterpreted as annual standard prices. UNIT SIZE SQM is the most influential variable forprices, having the largest coefficient. gen dummy and grocery dist st are taken as examplesof dummy and continuous variables, respectively.
Direct inspection of figure 2 indicates that there are distinctive trends for all the variables. Observing UNIT SIZE SQM, we find a gradual increase in the coefficient, whichmeans that the floor space of the properties becomes more important as time passes. Focusing on one of the property owners’ characteristics, gen dummy, we see it decreases according to the time periods. This presumably means that gender makes no difference inthe recent Dubai’s real estate market of Dubai.

10Figure 2: Estimated coefficients with full data
We also find that the estimated coefficients of the dynamic hedonic model are similar
to those of the separate hedonic model. In particular, the coefficients of the influential
variables const and UNIT SIZE SQM are completely overlapped. This indicates that the
dynamic hedonic model weakly smooths the coefficients. We presume that because thesample size each year is large enough to obtain reliable estimated values, the coefficientsare not necessarily smoothed out.
We have also examined the stability of the estimation for a small sample size, using thefollowing procedure. We pick 1% samples each year at random and estimate the coefficients. This procedure is repeated 30 times to test the stability of the estimations.
The results are shown in Figure 3. The estimation of the separate hedonic model is far
more unstable than the others, so that we present the results obtained from both the rolling
and the dynamic hedonic models in Figure 4. These figures show that both the rolling anddynamic hedonic models are more stable than the separate hedonic model. Comparing
the dynamic hedonic model to the rolling hedonic model, the estimated coefficients of theinfluential variables, such as const and UNIT SIZE SQM, are approximately equal. However, the dynamic hedonic model is more stable in the estimation of the less influential
variables, such as gen dummy and grocery dist st.
The possible reasons for the stability of the dynamic hedonic model are as follows.
First, The dynamic hedonic model refers to all the data used to estimate and smooth the11Figure 3: Estimation stability
coefficients. This means that the dynamic hedonic model can refer to a larger number ofsamples than the other methods. Second, the automatic control of strength of the smoothing contributes to the stability. A smoothing with an appropriate strength makes the coefficients temporally stable, while the rolling hedonic model is not as stable as the dynamic
hedonic model due to the fixed length of the time window. Another finding is that temporal changes in the coefficients of gen dummy and grocery dist st’ in the dynamic hedonic
model are small even if only a small number of samples are available.

6

Concluding remarks

Based on time-varying coefficients, we have compared three proposed methods for measuring HPIs.
The results indicate that the dynamic hedonic model shows the highest predictive
power of the three and the HPI roughly follows the other two HPIs. However, since thesample size in this analysis is sufficient, the coefficients only yield sight differences between the three methods. Moreover, since the trends of the coefficients differ for each12

Figure 4: Estimation stability (showing only Rolling and Dynamic Hedonic models)variable, observing them helps identify the factors that are more important for explaining
housing price indexes.
In a stability analysis using a small sample size, we confirmed that the separate hedonic
model is far more unstable than the others. The dynamic hedonic model is the most stable
method of the three, for all the variables. We consider that the dynamic hedonic model isapplicable even to only a small number of samples, because the method refers to all thesamples for smoothing the coefficients.
Although we have only assumed a random walk error in the dynamic hedonic model,
it can be extended to linear (see Appendix) and non-linear trends. Further exploration isneeded for the model structure of time-varying coefficients.

Acknowledgement
The authors are grateful to Dubai Land Department for data provision and helpful comments.

13

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16Appendix
In addition to a random walk coefficients model, we also attempt a linear trend coefficientsmodel. The latter model considers a linear trend among the coefficients, so that the firstdifference of the dth value of a coefficient in period t is equal to the one in period t − 1 withan error:βdt − βdt−1 = βdt−1 − βdt−2 + υdt(10)where υdt is a stochastic innovation term that follows a Gaussian distribution. This estimation procedure also optimizes the coefficient-wise variance (σd )2 of the variance noiseυdt .
Despite the development of the model, we would not highlight it, because the predictive power of the model is not superior to the dynamic hedonic model proposed in
the main text. The figure below illustrates the R-squared between the two models, whichshows the dynamic hedonic model is the better one to employ.

Figure 5: Comparison between random walk and trend models

17