# RECORD 3372ad7c12532642d33524d3fe7c38467c784796
# TITLE: Estimation of Hedonic Models Using a Multilevel Approach: An Application for the Swiss Rental Market
# YEAR: 2008.0  VENUE: nan  DOI: 10.1007/bf03399271
# AUTHORS: D. Djurdjević; Christine Eugster; R. Haase
# PAGES: 23  CANDIDATE TABLE PAGES (1-based): [13, 14, 17]
# ABSTRACT: SummaryThis paper is an empirical application of a hedonic model to determine rents in Switzerland. Unlike traditional hedonic models, we take account of the fact that data are clustered at different levels. The resulting spatial autocorrelation and heterogeneity permits a consideration of different submarkets within Switzerland. For the empirical analysis, we consider the special case of the clustering of apartments at municipality level. Our results show that even after controlling for housing and locational attributes, a significant part of rental differences are still attributable to municipality differences.


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© Swiss Journal of Economics and Statistics 
2008, Vol. 144 (4) 679–701
a 
Wuest & Partner, Rue de la Rôtisserie 1, CH-1204 Geneva, Switzerland; phone: ++41 (0) 22 
319 30 00; fax: ++41 (0) 22 319 30 01; e-mail: djurdjevic@wuestundpartner.com; web page: 
www.wuestundpartner.com
b 
Wuest & Partner, Gotthardstrasse 6, CH-8002 Zurich, Switzerland; phone: ++41 (0) 44 
289 90 00; fax: ++41 (0) 44 289 90 01; e-mail: eugster@wuestundpartner.com and haase@
wuestundpartner.com; web page: www.wuestundpartner.com
 
Acknowledgement: We are grateful to the participants of the Swiss hedonic day, 27 June 2007, 
Geneva. We also thank Philippe Thalmann and Andrea Baranzini for their helpful comments. 
All remaining errors are ours.
Estimation of Hedonic Models Using a Multilevel 
Approach: An Application for the Swiss Rental Market
Dragana Djurdjevica, Christine Eugsterb and Ronny Haaseb
JEL- Classification: C14, J16, R31
Keywords: hedonic model, rental market, multilevel modelling
1. Introduction
Hedonic regression models have traditionally been used to model housing prices. 
Typically, the price of the house is related to the implicit prices of the housing 
attributes. The hedonic model is often estimated by OLS, leading to regression 
coefficients which represent the implicit prices. Most hedonic models include 
characteristics of the location to measure the influence of neighbourhood on 
housing prices, thereby ignoring the inherent hierarchy in the housing decision. 
However, as pointed out by Quigley (1985), a household begins its search proc-
ess by choosing a town to live in, followed by the neighbourhood within the town 
and finally the house to inhabit within this neighbourhood and town. The hierar-
chical nature of the housing decision cannot be ignored and use of the traditional 
OLS estimation technique can lead to unreliable and biased estimates, especially 
where the latter is performed without consideration for the presence of segmented 
markets. With multilevel modelling, a hierarchical structure is recognised when 
it is present in the data. In addition, the parameters of the model are allowed to 
vary across space. Thus, both spatial correlation and heterogeneity are taken into 
account. More detailed information on multilevel modelling can be found in the 
books by Goldstein (1995) and Bryk and Raudenbush (2002).


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Estimation of Hedonic Models Using a Multilevel Approach 
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Table 3: Estimated Parameters in the Random Intercept Model
Parameter
Coefficient (std. err.)
Fixed
Intercept
 
9.555 
(0.013)
Random
Level 2 (municipality level)
σ2
u0
 
0.047 
(0.004)
Level 1 (apartment level)
σ2
e
 
0.117 
(0.001)
Intra-class correlation
0.29
Note : bold significant at 5% level.
After controlling for housing attributes, the intra-class correlation coefficient 
decreases to 24%. This small reduction strongly suggests that rents vary across 
municipalities and provides evidence for the existence of segmented markets.
4.2 Random Slope Model
In this section, the coefficient for dwelling area is treated as a random effect. 
The paper focuses on a simple hedonic model that can illustrate the value of the 
multilevel approach in allowing for segmented markets. To this end, we make 
the restrictive assumption that only the rate of increase of rents associated with 
an additional square footage of the dwelling area can vary across municipalities. 
The other explanatory variables are assumed to have a fixed effect. This latter 
assumption can be relaxed and the question of the specification of a complete 
model to explain rents is left for future research.
Table 4 shows that, after controlling for the explained variables, there is a 
significant unexplained variance in rents that remains at municipality level. 
Upon interpretation of the other effects, we find the expected relationships. For 
instance, a positive effect is found for an additional m2 dwelling area, an addi-
tional room or a better quality score, whereas an apartment located in an older 
building has a lower rent. When looking at the effect of the dummy for mixed-use 
and residential buildings, there is no preconception about the sign of this effect. 
Here, a positive effect is obtained meaning that apartments located in mixed-use 
buildings have higher rents than apartments located in residential buildings.


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Djurdjevic / Eugster / Haase
Table 4: Estimated Parameters in the Random Coefficient Model
Parameters
Coefficient (std. err.)
Fixed effects
Intercept (overall mean effect)
 
9.549 
(0.019)
Dwelling area (overall mean effect)
 
0.603 
(0.017)
Macro-location (in log)
 
0.789 
(0.024)
Age in years
 
−0.007 
(0.000)
Age squared
 
0.006 
(0.000)
Number of rooms
 
0.052 
(0.005)
Number of apartments
 
−0.0002 
(0.000)
Floor level
 
0.009 
(0.001)
Condition
 
0.065 
(0.003)
Standard
 
0.024 
(0.004)
Micro-location
 
0.009 
(0.003)
Mixed building
 
0.038 
(0.005)
Dhj_2
 
0.007 
(0.003) 
Dhj_3 
 
0.036 
(0.004)
Random effects
Level 2 (municipality level)
σ2
u0 (intercept)
 
0.022 
(0.004)
σu01 (covariance)
 
0.020 
(0.004)
σ2
u1 (dwelling area)
 
0.036 
(0.006)
Level 1 (apartment level)
σ2
e
 
0.017 
(0.000)
Note : bold significant at 5% level.
In the paper, the rate of increase in rents for additional square footage is allowed 
to vary across municipalities, between municipalities with higher average rents 
and between municipalities with lower average rents. A correlation term that is 
not significantly different from zero implies that the rate of increase in rental 
prices for additional square footage is similar across municipalities regardless 
of the municipality’s mean rent. This correlation coefficient can be calculated 


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Estimation of Hedonic Models Using a Multilevel Approach 
693
as σu01 / σu0σu1 and is equal to 0.71 in our random coefficient model. This has 
some interesting implications. It suggests that there is a relatively strong associa-
tion between average municipality rents and the municipality effects of square 
footage – indicating that municipalities with higher average rents tend to have 
a greater rate of increase in rental prices associated with increases in dwelling 
area.
Figure 3 provides an illustration of the variability of the effect of dwelling area 
at municipality level. The upper part of Figure 3 shows how the observations 
are distributed according to dwelling area. Two municipalities have been repre-
sented: one with circles and another with squares. We can draw a black line for 
the average relationship between rents and dwelling area. The circles are signifi-
cantly above the average line whereas the squares are significantly below it. As a 
consequence, Figure 3 shows that different intercepts have to be considered. The 
Figure 3: Relationship between Dwelling Area and Rents (in log)
10.7
10.0
9.3
8.6
7.9
log (price)
2.3 
3.1 
3.9 
4.7 
5.5
log (area)
11.0
10,2
9.4
8.6
7.8
log (price)
2.3 
3.1 
3.9 
4.7 
5.5
log (area)


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Estimation of Hedonic Models Using a Multilevel Approach 
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Table 5: Comparison between the Different Models
Parameters
OLS 1
OLS 2
ML
Coeff. (std. err.)
Coeff. (std. err.)
Coeff. (std. err.)
Fixed effects
Intercept (overall mean effect)
 9.512 (0.037)
 9.513 (0.037)
 9.549 (0.019)
Dwelling area (overall mean effect)
 0.614 (0.019)
 0.630 (0.011)
 0.603 (0.017)
Macro-location (in log)
 0.766 (0.025)
 0.761 (0.022)
 0.789 (0.024)
Age in years
 −0.007 (0.000)
 −0.007 (0.000)
 −0.007 (0.000)
Age squared
 0.005 (0.000)
 0.005 (0.000)
 0.006 (0.000)
Number of rooms
 0.053 (0.005)
 0.054 (0.005)
 0.052 (0.005)
Number of apartments
 −0.0001 (0.000)
 −0.0001 (0.000)
 −0.0002 (0.000)
Floor level
 0.006 (0.001)
 0.007 (0.001)
 0.009 (0.001)
Condition
 0.063 (0.003)
 0.063 (0.003)
 0.065 (0.003)
Standard
 0.018 (0.004)
 0.019 (0.004)
 0.024 (0.004)
Micro-location
 0.007 (0.003)
 0.007 (0.003)
 0.009 (0.003)
Mixed building
 0.033 (0.005)
 0.032 (0.005)
 0.038 (0.005)
Dhj_2
 0.006 (0.003)
 0.005 (0.003)
 0.007 (0.003)
Dhj_3 
 0.037 (0.004)
 0.037 (0.004)
 0.036 (0.004)
Cantonal dummies
For canton of Zurich
 0.047 (0.004)
 0.048 (0.004)
For canton of Bern
 0.052 (0.006)
 0.052 (0.006)
For canton of Zug
 0.029 (0.010)
 0.034 (0.010)
For canton of Solothurn
 0.040 (0.010)
 0.038 (0.010)
For canton of Basel City
 0.085 (0.008)
 0.092 (0.007)
For canton of Basel Land
 0.078 (0.007)
 0.083 (0.007)
For canton of St.Gallen
 0.036 (0.007)
 0.037 (0.007)
For canton of Aargau
 0.056 (0.005)
 0.054 (0.005)
Cantonal interaction terms 
For canton of Zurich
 −0.048 (0.009)
For canton of Bern
 −0.030 (0.016)
For canton of Zug
 −0.053 (0.025)
For canton of Solothurn
 −0.063 (0.024)
For canton of Basel City
 0.036 (0.015)
For canton of Basel Land
 0.031 (0.025)
For canton of St.Gallen
 −0.049 (0.018)
For canton of Aargau
 −0.006 (0.671)
R 2
0.858
0.859
0.877
Log-likelihood
5493.04
5512.95
6277.26
SSE
245.85
244.98
185.61
Notes: the reference canton is composed of other cantons; bold significant at 5% level; italics at 
10% level; 11 913 observations.


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11 If we consider the simplest version of the multilevel model (model with a random intercept 
only), the true values of u0 j (level 2 residuals which represent departures from the overall mean 
β0) are unknown, but we can obtain estimates of them given the observed data and the esti-
mated parameters of the model. These estimated or predicted residuals are often referred to 
As regards the fixed effects, the different regressions yield similar results. It is 
common to many hedonic pricing models that not all variables exhibit the proper 
sign when estimated with OLS. This is not the case in this application: there is 
no variable that is treated differently according to the type of estimation. How-
ever, the estimators differ in their estimates of the magnitude of the fixed effects. 
For example, the ML estimate for the number of apartments in the building is 
100% greater (in absolute term) than the corresponding OLS estimates. This is 
also the case for the effect of floor level (+ 50%), of standard (+ 33%) and to 
a lesser extent for the effect of micro-location (+ 20%). A particular attention 
should be devoted to the effect of the macro-location, which is closer to one in 
the ML specification. In the hedonic model, we expect the coefficient for the 
macro-situation to be close to one since this variable reflects the relative attrac-
tiveness of municipality A with respect to municipality B. If the generalised m2 
asked prices in municipality A are 1% higher than in municipality B, then the 
yearly rental price of a specific apartment should be 1% higher in municipality A 
than in municipality B. However, as advertisements offer insufficient information 
on the characteristics of apartments, the m2 asked price for one municipality is 
computed without correcting for the structural attributes (except for floor area). 
As a consequence, the contextual effects of the macro-location in the hedonic 
model are potentially confounded with the rent impact of apartment attributes. 
That is why the coefficient for the macro-location in the hedonic model is not 
equal to one.
The locational attributes (cantonal dummies and interaction terms) included 
in specifications OLS 1 and OLS 2 measure large-scale effects. However, the very 
local influences may prove to be more important in the prediction of rental prices. 
If we use the sample R2 and the sum of squared errors SSE to measure goodness 
of fit, the estimated SSE falls by 33% relative to the OLS results whereas the R2 
rises from 0.858 to 0.877, which is a small, though significant amount. Use of a 
likelihood ratio statistic to check whether the different specifications are signifi-
cantly different yields the result that the OLS specifications are rejected against 
the ML specification.
This section also provides some information on the precision of models in 
terms of rental prediction. The following figure represents the absolute differences 
between actual and predicted rents (as % of actual rents).11 The class 0%–5%