The purpose of this paper is to test the asymptotic properties and prediction accuracy of two innovative methods proposed along the hedonic debate: the geographically weighted regression (GWR) and the generalized additive model (GAM).

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Spatial effects and non-
linearity in hedonic pricing
Should we reconsider our
assumptions?
Dr. Marcelo Cajias PATRIZIA Immobilien AG
Sebastian Ertl IRE|BS
ERES 2016 –Regensburg - GERMANY
Source: http://www.qed-energy.com/wp-content/uploads/2013/08/Domestic-EPC.jpg
PATRIZIA Immobilien AG in cooperation with:
5
10
30
0,05
0,20
Asking rents across German cities
Berlin
Hamburg
München
Density

2. 2PATRIZIA Immobilien AG| © 2016
Traditional hedonic decomposition of rents
Asking
Rent (y)
Living
area (x)
xa xb xc xd
E(y|x=xa)
E(y|x=xb)
E(y|x=xc)
E(y|x=xd)
α+βx
TRADITIONAL OLS UNDER HOMOSCEDASTICITY
DISTRIBUTION OF THE SAMPLE IS CENTRED AROUND THE ESTIMATOR 𝛽
Askingrent
Living
area

3. 3PATRIZIA Immobilien AG| © 2016
Traditional hedonic decomposition of rents – Motivation
Size categories at 5 m²
5-10m²
>180m²
20
10
Askingrentin€/m²/p.m.
TRADITIONAL OLS UNDER HOMOSCEDASTICITY
DISTRIBUTION OF THE SAMPLE IS CENTRED ABOUT THE ESTIMATORS 𝛽
Is the expected
conditional
variance of the
errors distributed
homoscedastically
across the sample ?

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Methods for modelling real estate rents and paper goal
OLS = Ordinary least squares; GWR = geographically weighted regression; GAM = Generalized additive model
HEDONIC
COMPOSITION
OF RENTS
building
attributes
neighborhood
characteristics
share of directly
available
amenities
demand of
households for
housing
disposable
income
transport
costs
OLS as a baseline model
GWR has established itself as a prevailing method
that expands the restrictive traditional OLS by
considering spatially varying effects
GAM captures spatial effects based on smooth
functions and expand the traditional hedonic model
by identifying latent nonlinear effects
HEDONIC METHODS FOR ESTIMATING RENTS
𝛽 = 𝑿′
𝑿 −1
𝑿′
𝑌
𝛽𝑖 = 𝑿′
𝑾𝑖 𝑿 −1
𝑿′
𝑾𝑖 𝒀
𝛽 = 𝑿′
𝑿 −1
𝑿′
𝑌 + 𝛿𝑖𝑙 𝑑𝑙 𝑥 𝑎
𝑎𝑖(𝑥 𝑏
)
𝐿
𝑙=1
𝐼
𝑖=1
FACTORS AFFECTING RENTS
STUDY GOAL: EVALUATION OF THE ASYMPTOTIC OUT-OF-SAMPLE PROPERTIES OF LINEAR, SPATIAL
AND NON-LINEAR MODELS IN ESTIMATING THE HEDONIC EQUATION FOR ASKING RENTS

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The Geographically Weighted Regression GWR
GWR is based on the fact that the data generating
process is non-stationary over space and expands
the classical linear model by allowing the
coefficients to vary over space.
GWR places a set of windows (or regions) over the
space based on an initial bandwidth and finds the
optimal bandwidth by minimizing an optimization
criterion
𝛽𝑖 = 𝑿′
𝑾𝑖 𝑿 −1
𝑿′
𝑾𝑖 𝒀
𝑾𝑖 =
𝑤𝑖1 0 ⋯ 0
0 𝑤𝑖2 ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ 𝑤𝑖𝑁
𝑤𝑖𝑗 = 1 −
𝑑𝑖𝑘
𝑏
2 2
if 𝑑𝑖𝑘 < 𝑏
0 else
THE COMPUTATIONAL REQUIREMENTS ON A HEDONIC
REGRESSION VIA GWR CAN BE HIGH AND VERY TIME CONSUMING
WEIGHTING FUNCTION IN
DEPENDENCE OF BANDWIDTH

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The Generalized Additive Method GAM
See: Wood S.N. (2006b) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.
RENT
GAM is estimated via a backfitting algorithm and enables the combination of location and metric variables capturing
both spatial variation and nonlinear effects simultaneously.
Many different smoothing functions available: cubic, cyclic cubic, penalized or thin plate splines
Rentin€/m²/p.m.
30
10
Size
in m²
0m²
100m²
300m²

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Sample description
0'
20'
40'
60'
80'
4 €
6 €
8 €
10 €
12 €
14 €
16 €
Zwickau
Gelsenkirchen
Chemnitz
MärkischerKreis
Unna
Duisburg
Magdeburg
Recklinghausen
Meißen
Wuppertal
Ennepe-Ruhr-Kreis
Leipzig
Mönchengladbach
Dortmund
Bochum
Wesel
Essen
Bielefeld
Dresden
Lübeck
RegionHannover
Kiel
Main-Kinzig-Kreis
Mettmann
Rhein-Sieg-Kreis
StädteregionAachen
Rhein.-Berg.Kreis
Erftkreis
Bremen
Rhein-Neckar-Kreis
Neuss
Nürnberg
Offenbach
Potsdam
Ludwigsburg
Berlin
Esslingen
Böblingen
Bonn
Wiesbaden
Düsseldorf
Köln
Hamburg
Stuttgart
FrankfurtamMain
München
Rentin€/m²/p.m.
SamplesizeacrossNUTS3-areas
Rent mean
Rent median
Rent 25%|75%-quantile
Number of observations
SAMPLE DESCRIPTION
 46 markets across Germany with a
total of 573,272 observations of
internet offers of rental flats from
2013-Q1 until 2015-Q2 from the
empirica system databank.
 16 relevant hedonic
characteristics:
 Area in m², Age, Number of rooms,
 With bathtub, With built-in-kitchen,
With balcony, With park slot, With
balcony and terrace, With terrace,
With elevator, Brand new dwelling,
Refurbished dwelling and As-good-
as-new dwelling.
 2 socioeconomic variables on a
ZIP-code level and yearly basis
from the GfK-databank :
 Purchasing power per household,
 Number of inhabitants per
household

8. 8PATRIZIA Immobilien AG| © 2016
In-sample estimation and out-of-sample forecasting approach
Estimate predicted
fuctional form for
each market and
model
2013-Q1 2015-Q22013-Q2
Predict the vector of asking rents in 𝑡 + 1
with the predicted functional form from 𝑡
Calculate forecast evaluation indicators:
Save results and go to next period.
𝐸𝑉𝑗,𝑡+1
𝑝
=
1
𝑁𝑗,𝑡+1
𝑒𝑟𝑟𝑜𝑟𝑗,𝑡+1
𝑝
− 𝑀𝐸𝑗,𝑡+1
𝑝
2
𝑀𝑆𝐸𝑗,𝑡+1
𝑝
=
1
𝑁𝑗,𝑡+1
(𝑒𝑟𝑟𝑜𝑟𝑗,𝑡+1
𝑝
)2
𝑀𝑆𝑃𝐸𝑗,𝑡+1
𝑝
=
1
𝑁𝑗,𝑡+1
𝑦𝑗,𝑡+1
𝑦𝑗,𝑡+1
𝑝 − 1
2
𝑀𝐸𝑗,𝑡+1
𝑝
=
1
𝑁𝑗,𝑡+1
(𝑒𝑟𝑟𝑜𝑟𝑗,𝑡+1
𝑝
)
2013-Q3

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Out-of-sample results:
aggregated quantile distribution of forecasting indicators
Mean error
[€/m²/p.a.]
0.0
0.1
0.2
0.3
10 20 30 40 50 60 70 80 90
Error variance
[€/m²/p.a.]
0
1
2
3
4
10 20 30 40 50 60 70 80 90
Mean squared error
[€/m²/p.a.]
0
1
2
3
4
10 20 30 40 50 60 70 80 90
Quantile distribution
Mean squared percentage error
[%]
1.5
2.0
2.5
3.0
3.5
4.0
10 20 30 40 50 60 70 80 90
Quantile distribution
OLS GAM GWR
 A relative dominance of the GAM model
in comparison to the OLS and the GWR.
 OLS shows remarkable results with an
acceptable forecasting inaccuracy
 Strong discrimination of the GWR model
in contrast to the linear and semi-
parametric model.
 The GAM and the OLS model remain
almost equivalent up to the 70%
quantile for both the MSE and MSPE
 OLS dominates when forecasting
extreme values as the MSE and MSPE
are relative lower in comparison to the
GAM for the quantiles up to 70%.

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Thank you very much!
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Germany
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AUTHORS
DR. MARCELO CAJIAS PATRIZIA IMMOBILIEN AG
SEBASTIAN ERTL IRE|BS
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