Download Rent Analysis Beyond Conditional Expectation – How to Estimate

rent analysis beyond conditional expectation
– how to estimate the spatial distribution of
quoted rents by using geographically
weighted regression
Anne Seehase, M. Sc., Amt für Statistik, Landeshauptstadt Magdeburg,
Germany
July 1, 2016
Scorus-Conference 2016 – Lisbon
content
1. Motivation
2. Alternative Data Sources
3. Methodology
4. Empirical Application
5. Conclusion
Anne Seehase
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
1 / 25
motivation
motivation
∙ 57 percent of all households in Germany are living in a rented
accommodation (Statistisches Bundesamt, Wiesbaden 2015,
effective 2013).
∙ Most areas with a high population density are embossed by tight
and dynamically developing rental markets.
∙ It is expected that the increasing number of households caused by
the demographic transition and urbanization leads to increasing
rents also in cities like Magdeburg, which right now are
characterized by quite a relaxed rental market.
∙ Monitoring of the developments at the rental markets are relevant
for urban planning and social welfare.
Anne Seehase
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
3 / 25
alternative data sources
alternative data sources
∙ Traditional data collection to rental analysis focus asset rents:
→ information of the common (average) level of residental cost over a longer
time-period
→ insufficient conclusions for the market price by re-letting in very dynamic
markets
∙ Alternative data sources make it possible to focus actual rent prices by re-letting:
→ information about the current market situation and development
Advantages
Disadvantages
∙ In-time-data-collection of actual offers
∙ Easy availability of a high number of rental
property offers, e.g. via online platforms or the
advertisement section in local newspapers
∙ Easy and cheap data-collection by the dint of
∙ Web-crawling-algorithms
∙ Web-APIs
∙ Database of commercial service providers
Anne Seehase
∙ Rental offers prices may differ from the real
price by re-letting.
∙ Data are biased by the interest of supplier.
∙ The researcher has no bearing on the
characteristics covered in the rental property
offer.
∙ Very good housing supplies are likely to be
re-hired without a publication on an online
platform.
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
5 / 25
methodology
1500
hedonic price regression
∙ Background: The rental price of a property is
determined by the individual characteristics of
the flat and the characteristics of the
surrounding neighborhood.
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Cold Rent in €
∙ Method of hedonic modeling: Estimation of the
prices of the individual characteristic by multiple
regression-models
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∙ It will be possible to estimate the expected price
of a hypothetical average flat and to compare
this price depending on the location of the flat
or the date of the offer.
Anne Seehase
●
●
●
0
∙ Enables the estimation of the conditional
expectation by a special constellation of
characteristics.
500
●●
40
60
80
100
120
140
Living area in square meters
© Department of Statistics, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
7 / 25
1500
hedonic price regression
∙ Background: The rental price of a property is
determined by the individual characteristics of
the flat and the characteristics of the
surrounding neighborhood.
●
1000
●
●
●
●
●
●
●
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●
●
●
●
●
● ●
●
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●
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●
●
●
Cold Rent in €
∙ Method of hedonic modeling: Estimation of the
prices of the individual characteristic by multiple
regression-models
●
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●
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∙ It will be possible to estimate the expected price
of a hypothetical average flat and to compare
this price depending on the location of the flat
or the date of the offer.
Anne Seehase
●
●
●
0
∙ Enables the estimation of the conditional
expectation by a special constellation of
characteristics.
500
●●
40
60
80
100
120
140
Living area in square meters
© Department of Statistics, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
7 / 25
1500
quantile regression (qr)
∙ The quantile regression enables the analysis of
the conditional quantile to the quantile value
τ ∈ (0, 1) of a dependent variable in conjunction
with a set of explanatory variables (Koenker and
Basset (1978)).
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Quantile Regression Model for τ ∈ (0, 1)
Cold Rent in €
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=β0 (τ ) + β1 (τ )X1i + . . . + β(p−1) (τ )X(p−1)i + Ui (τ )
T
=Xi β(τ ) + Ui (τ )
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Yi
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i = 1, . . . , n
Assumption: FUi (τ ) (0) = τ ⇒ QY (τ |Xi = xi ) = xTi β(τ ),
Ui (τ ) for i = 1, . . . , n are independent.
●
Xi =
. . .dependent variable
(1, X1i , . . . , X(p−1)i )
. . .independent variables
(β0 (τ ), . . . , β(p−1) (τ ))
. . .parameters depending on τ
0
●
●
Yi
40
60
80
100
120
140
Living area in square meters
β(τ ) =
Ui (τ )
Anne Seehase
© Department of Statistics, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
. . .perturbation depending on τ
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
8 / 25
1500
quantile regression (qr)
∙ The quantile regression enables the analysis of
the conditional quantile to the quantile value
τ ∈ (0, 1) of a dependent variable in conjunction
with a set of explanatory variables (Koenker and
Basset (1978)).
●
1000
●
●
●
●
●
Quantile Regression Model for τ ∈ (0, 1)
Cold Rent in €
●
●
●
=β0 (τ ) + β1 (τ )X1i + . . . + β(p−1) (τ )X(p−1)i + Ui (τ )
T
=Xi β(τ ) + Ui (τ )
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500
Yi
●
●
●
●
●
●
●
●
●
●
●
i = 1, . . . , n
Assumption: FUi (τ ) (0) = τ ⇒ QY (τ |Xi = xi ) = xTi β(τ ),
Ui (τ ) for i = 1, . . . , n are independent.
●
●
●
Xi =
. . .dependent variable
(1, X1i , . . . , X(p−1)i )
. . .independent variables
(β0 (τ ), . . . , β(p−1) (τ ))
. . .parameters depending on τ
conditional exspectation
conditional quantile
0
Yi
40
60
80
100
120
140
Living area in square meters
β(τ ) =
Ui (τ )
Anne Seehase
© Department of Statistics, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
. . .perturbation depending on τ
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
8 / 25
1500
quantile regression (qr)
∙ The quantile regression enables the analysis of
the conditional quantile to the quantile value
τ ∈ (0, 1) of a dependent variable in conjunction
with a set of explanatory variables (Koenker and
Basset (1978)).
●
1000
●
●
●
●
●
Quantile Regression Model for τ ∈ (0, 1)
Cold Rent in €
●
●
●
=β0 (τ ) + β1 (τ )X1i + . . . + β(p−1) (τ )X(p−1)i + Ui (τ )
T
=Xi β(τ ) + Ui (τ )
●
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●
●
●●
500
Yi
●
●
●
●
●
●
●
●
●
●
●
i = 1, . . . , n
Assumption: FUi (τ ) (0) = τ ⇒ QY (τ |Xi = xi ) = xTi β(τ ),
Ui (τ ) for i = 1, . . . , n are independent.
●
●
●
Xi =
. . .dependent variable
(1, X1i , . . . , X(p−1)i )
. . .independent variables
(β0 (τ ), . . . , β(p−1) (τ ))
. . .parameters depending on τ
conditional exspectation
conditional quantile
0
Yi
40
60
80
100
120
140
Living area in square meters
β(τ ) =
Ui (τ )
Anne Seehase
© Department of Statistics, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
. . .perturbation depending on τ
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
8 / 25
principal of the geographically weighted regression
∙ Toblers first law of geographic: “Everything is related to everything else, but near
things are more related than distant things.” (Tobler (1970))
∙ The principal of the geographically weighted regression (GWR) is based on the
ideas of Fotheringham et al. (2002). The geografical weighted extension of
QR-Modell is based on the description from Chen et al. (2014) and McMillen (2013).
∙ The basic idea of the geographical weighting is a local estimation of the model at
special target-points which are denoted by their geographical position
si = (lati , loni ).
Model of the Geografically Weighted Quantile Regression (GWQR)
Yi = XTi β(τ ; si ) + Ui (τ ; si ),
i = 1, . . . , n
Assumption: FUi (τ ;si ) (0) = τ ⇒ QY (τ ; si |Xi = xi ) = xTi β(τ ; si ), Ui (τ ; si ) is independent.
Yi
. . . dependent variable
Xi
= (1, X1i , . . . , X(p−1)i )
. . . independent variables
Ui (τ, si )
. . . perturbation depending on τ
β(τ ; si )
= (β0 (τ, si ), . . . , β(p−1) (τ ; si ))
. . . parameters depending on τ
Anne Seehase
at location si
Rent Analysis by Using Geographically Weighted Quantile Regression
at location si
July 1, 2016
9 / 25
principal of the geographically weighted regression
Basic Principle
Anne Seehase
∙ Nearby observations will be integrated with a higher weight.
d
∙ The weights are denoted by a kernel function K( hi0 )|i=1,...,n of the scaled distance di0 of the
observation i to the target-point by the bandwidth h.
∙ Fixed kernel weighting routine
∙ Adaptive kernel weighting routine
Gaussian Kernel Function
1.0
Tri−Cube−Kernel Function
●●
●
0.35
0.8
●
●
−1
0
1
0.25
kernal weight
−2
●
●
0.05
●● ●
●
0.15
0.6
0.4
0.2
0.0
h0
B1
kernal weight
●
E0 d10
●
2
scaled distance
Rent Analysis by Using Geographically Weighted Quantile Regression
●● ●
−2
−1
0
●
1
●
2
scaled distance
July 1, 2016
10 / 25
empirical application
empirical application
Data
∙ Rental offers from the empirica systema AG database
∙ Published between January 1, 2012 and the March 31, 2016
∙ Subset of the row data by
∙ geocoded offers with a location precision fewer than 10 meters
∙ accommodations with a living area of 30 to 250 squares meters
∙ 18442 offers are included into the research
Methodical Specification
Anne Seehase
∙ Semi-log model; dependent variable: log of price per square meter
∙ First step: Estimation a global global OLS-model and global QR-models by the
quantile values τ ∈ {0.1, 0.25, 0.5, 0.75, 0.9}
∙ Seccond step: Estimation the local GWR-models and GWQR-models by the quantile
values τ ∈ {0.1, 0.25, 0.5, 0.75, 0.9}
∙ Spatial weighting by a gaussian kernel-function and an adaptive bandwith
∙ target-points: location of each observation
∙ mapping: target points are the center points of a grid over the town area
which covered with buildings (adaptive bandwidth have to be under 1 km)
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
12 / 25
semi-log qr/gwqr-model for τ = 0.5
GWQR
τ = 0.5
mean
min
1st quantile
median
3rd quantile
Intercept
1.69
1.55
1.66
1.68
1.72
log(living area)
-0.03
-0.07
-0.03
-0.02
-0.02
elevator
0.02
-0.05
-0.00
0.01
0.04
balcony/terrace
0.03
-0.02
0.02
0.03
0.04
parking
0.06
0.02
0.05
0.06
0.08
kitchen
0.04
-0.01
0.03
0.04
0.05
social housing
-0.04
-0.10
-0.04
-0.04
-0.03
good condition
-0.00
-0.03
-0.00
0.00
0.01
bad condition
-0.08
-0.17
-0.11
-0.08
-0.05
time
0.02
0.01
0.02
0.02
0.02
GWR
Intercept
1.63
1.49
1.58
1.61
1.65
log(living area)
-0.01
-0.07
-0.02
-0.01
-0.00
elevator
0.04
-0.01
0.02
0.03
0.06
balcony/terrace
0.03
-0.01
0.01
0.03
0.04
parking
0.06
0.03
0.05
0.06
0.07
kitchen
0.05
0.01
0.04
0.05
0.06
social housing
-0.06
-0.11
-0.07
-0.06
-0.05
good condition
0.01
-0.02
0.01
0.01
0.01
bad condition
-0.07
-0.14
-0.1
-0.07
-0.05
time
0.02
0.02
0.02
0.02
0.02
Significance levels [0,001]’***’;(0.001,0.01]’**’;(0.01;0.05]’*’;(0.5,0.1]’°’,(0.1,1]’ ’
Anne Seehase
Rent Analysis by Using Geographically Weighted Quantile Regression
max
1.91
0.004
0.11
0.05
0.11
0.07
-0.00
0.01
0.02
0.03
1.90
0.02
0.11
0.06
0.11
0.08
-0.01
0.02
0.03
0.03
QR
Std.
1.658
-0.02
0.01
0.02
0.07
0.05
-0.03
0.00
-0.10
0.02
OLS
1.67
-0.00
0.03
0.02
0.06
0.06
-0.05
0.01
-0.08
-0.02
July 1, 2016
Sign.
***
***
*
***
***
***
***
***
***
***
***
***
***
***
***
***
***
***
13 / 25
spatial heterogeneity of the time effect (magdeburg)
Anne Seehase
tau=0.1
tau=0.9
no evaluation
0.0111 to 0.0131
0.0131 to 0.0152
0.0152 to 0.0172
0.0172 to 0.0192
0.0192 to 0.0213
0.0213 to 0.0233
0.0233 to 0.0253
0.0253 to 0.0274
0.0274 to 0.0294
0.0294 to 0.0314
0.0314 to 0.0335
0.0335 to 0.0355
0.0355 to 0.0375
0.0375 to 0.0396
0.0396 to 0.0416
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
14 / 25
prediction of the cold rent per square meter (in €)
Anne Seehase
GWQR−model (tau=0.1)
GWQR−model (tau=0.9)
no evaluation
3.82 to 4
4 to 4.18
4.18 to 4.36
4.36 to 4.54
4.54 to 4.72
4.72 to 4.9
4.9 to 5.08
5.08 to 5.26
5.26 to 5.44
5.44 to 5.62
5.62 to 5.8
5.8 to 5.98
5.98 to 6.16
6.16 to 6.34
6.34 to 6.52
Flat characteristics: 61.8 square meters, balcony, good condition effective July 1,2012
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
15 / 25
prediction of the cold rent per square meter (in €)
Anne Seehase
GWQR−model (tau=0.1)
GWQR−model (tau=0.9)
no evaluation
3.82 to 4
4 to 4.18
4.18 to 4.36
4.36 to 4.54
4.54 to 4.72
4.72 to 4.9
4.9 to 5.08
5.08 to 5.26
5.26 to 5.44
5.44 to 5.62
5.62 to 5.8
5.8 to 5.98
5.98 to 6.16
6.16 to 6.34
6.34 to 6.52
Flat characteristics: 61.8 square meters, balcony, good condition effective July 1,2015
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
16 / 25
prediction of the cold rent per square meter (in €)
Anne Seehase
GWQR−model (tau=0.5)
GWR−model
no evaluation
3.82 to 4
4 to 4.18
4.18 to 4.36
4.36 to 4.54
4.54 to 4.72
4.72 to 4.9
4.9 to 5.08
5.08 to 5.26
5.26 to 5.44
5.44 to 5.62
5.62 to 5.8
5.8 to 5.98
5.98 to 6.16
6.16 to 6.34
6.34 to 6.52
Flat characteristics: 61.8 square meters, balcony, good condition effective July 1,2012
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
17 / 25
prediction of the cold rent per square meter (in €)
Anne Seehase
GWQR−model (tau=0.5)
GWR−model
no evaluation
3.82 to 4
4 to 4.18
4.18 to 4.36
4.36 to 4.54
4.54 to 4.72
4.72 to 4.9
4.9 to 5.08
5.08 to 5.26
5.26 to 5.44
5.44 to 5.62
5.62 to 5.8
5.8 to 5.98
5.98 to 6.16
6.16 to 6.34
6.34 to 6.52
Flat characteristics: 61.8 square meters, balcony, good condition effective July 1,2015
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
18 / 25
prediction of the cold rent per square meter (in €)
●
●
●
●
conditional median 2012
conditional median 2015
conditional mean 2012
conditional mean 2015
Milchweg 2012
Milchweg 2015
Hasselbachplatzviertel 2012
Hasselbachplatzviertel 2015
Fliedergrund 2012
Fliedergrund 2015
0.6
0.4
0.2
0.0
density
0.8
1.0
●
Milchweg
Hasselbachplatzviertel
Fliedergrund
Anne Seehase
●
3
4
●
●●
5
●
●
6
Price per square meter
7
8
© Amt für Statistik, Landeshauptstadt Magdeburg
Data: Empirica−Systeme
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
19 / 25
conclusion
conclusion
Anne Seehase
∙ Geocoded online-data are a substantial data-source for the analysis of
rental data.
∙ Enables a lot of analysis in detail; high potential for geostatistical
analysis
∙ Data-restriction has to be kept in mind.
∙ Quantile regression allows a detailled view at the conditional distribution
of the variable of interest.
∙ Main advantages: rubustness aigainst outlier
∙ Disadvantages: local estimation could lead to quantile-crossing
∙ The extension of the geographical weighting allows to consider the
location of interest.
∙ Time trends could be analyzed considering the geographical position.
∙ Indicator for the development in special districts (social segmentation
or heterogenity).
∙ GWQR-Model based on local estimation.
∙ computationally intensive
∙ difficults in modell-selection
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
21 / 25
Anne Seehase
Thank you for your attention!
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
22 / 25
references
Anne Seehase
∙ Chen, Vivian Y.; Deng, Wen-Shuenn; Yang, Tse-Chuan; Matthews, Stephan:
Geographically Weighted Quantile Regression (GWQR): An Application to
U.S. Mortality Data. In: Geographical Analysis 44(2) 2012, pp. 134-150
∙ Fotheringham, A. Stewart ; Brunsdon, Chris ; Charlton, Martin:
Geographically Weighted Regression: The Analysis of Spatially Varying
Relationships. Wiley 2002
∙ Koenker, Roger: Quantile regression, Cambridge university press 2005
∙ Koenker, Roger; Basset Jr., Gilbert: Regression quantiles. In: Econometrica:
journal of the Econometric Society, 1978, pp. 33-50
∙ McMillen, Daniel P.: Quantile regression for spatial data. Berlin, Springer,
2013
∙ Tobler, Waldo R.: A Computer Movie Simulating Urban Growth in the
Detroit Region. In: Economic Geography, Vol. 46,1999, pp. 234-240
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
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quantile regression (qr)
Anne Seehase
∙ In case of empirical analysis, the unknown parameter β(τ ) has to be estimated.
∙ Solution of the minimization problem:
n
∑
T
β̂(τ ) = arg min Rτ (β) = arg min
ρτ (yi − xi β)
β∈Rp
β∈Rp
∙ The loos-function ρτ is defined by:
ρτ (u) = (u)(τ − I(u < 0)) =
{
τ ·u
u·u
i=1
,
,
for
for
u≤0
u<0
0.2
0.4
loos
0.6
0.8
1.0
∙ Rτ (β) is not differentiable. → The minimization-problem has to be solved by methods of
linear programming.
0.0
quadratic loos
tau=0.5
tau=0.75
tau=0.25
−1.0
−0.5
0.0
0.5
1.0
u
Rent Analysis by Using Geographically Weighted Quantile Regression
July 1, 2016
24 / 25
principal of the geographically weighted regression
Basic Principle
∙ Nearby observations will be integrated with a higher weight.
d
∙ The weights are denoted by a kernel function K( hi0 )|i=1,...,n of the scaled distance di0 of the
observation i to the target-point by the bandwidth h.
∙ Fixed kernel weighting routine
∙ Adaptive kernel weighting routine
Gaussian Kernel Function
1.0
Tri−Cube−Kernel Function
●●
●
0.35
0.8
●
●
−1
0
1
0.25
kernal weight
−2
●
●
0.05
●● ●
●
0.15
0.6
0.4
0.2
0.0
h0
B1
kernal weight
●
E0 d10
●
2
●● ●
−2
scaled distance
−1
0
●
1
●
2
scaled distance
Minimization Problem
Anne Seehase
β̂(τ ; s0 ) = arg min
β∈Rp
n
∑
T
(ρτ (yi − xi β(s0 )) · K(
i=1
Rent Analysis by Using Geographically Weighted Quantile Regression
di0
))
h
July 1, 2016
25 / 25