1# RECORD 3372ad7c12532642d33524d3fe7c38467c7847962# TITLE: Estimation of Hedonic Models Using a Multilevel Approach: An Application for the Swiss Rental Market3# YEAR: 2008.0 VENUE: nan DOI: 10.1007/bf033992714# AUTHORS: D. Djurdjević; Christine Eugster; R. Haase5# PAGES: 23 CANDIDATE TABLE PAGES (1-based): [13, 14, 17]6# 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.789===== PAGE 1 (score 2, numeric-paren 0, table-words 0, var-words 2) =====1011© Swiss Journal of Economics and Statistics 122008, Vol. 144 (4) 679–70113a 14Wuest & Partner, Rue de la Rôtisserie 1, CH-1204 Geneva, Switzerland; phone: ++41 (0) 22 15319 30 00; fax: ++41 (0) 22 319 30 01; e-mail: djurdjevic@wuestundpartner.com; web page: 16www.wuestundpartner.com17b 18Wuest & Partner, Gotthardstrasse 6, CH-8002 Zurich, Switzerland; phone: ++41 (0) 44 19289 90 00; fax: ++41 (0) 44 289 90 01; e-mail: eugster@wuestundpartner.com and haase@20wuestundpartner.com; web page: www.wuestundpartner.com21 22Acknowledgement: We are grateful to the participants of the Swiss hedonic day, 27 June 2007, 23Geneva. We also thank Philippe Thalmann and Andrea Baranzini for their helpful comments. 24All remaining errors are ours.25Estimation of Hedonic Models Using a Multilevel 26Approach: An Application for the Swiss Rental Market27Dragana Djurdjevica, Christine Eugsterb and Ronny Haaseb28JEL- Classification: C14, J16, R3129Keywords: hedonic model, rental market, multilevel modelling301. Introduction31Hedonic regression models have traditionally been used to model housing prices. 32Typically, the price of the house is related to the implicit prices of the housing 33attributes. The hedonic model is often estimated by OLS, leading to regression 34coefficients which represent the implicit prices. Most hedonic models include 35characteristics of the location to measure the influence of neighbourhood on 36housing prices, thereby ignoring the inherent hierarchy in the housing decision. 37However, as pointed out by Quigley (1985), a household begins its search proc-38ess by choosing a town to live in, followed by the neighbourhood within the town 39and finally the house to inhabit within this neighbourhood and town. The hierar-40chical nature of the housing decision cannot be ignored and use of the traditional 41OLS estimation technique can lead to unreliable and biased estimates, especially 42where the latter is performed without consideration for the presence of segmented 43markets. With multilevel modelling, a hierarchical structure is recognised when 44it is present in the data. In addition, the parameters of the model are allowed to 45vary across space. Thus, both spatial correlation and heterogeneity are taken into 46account. More detailed information on multilevel modelling can be found in the 47books by Goldstein (1995) and Bryk and Raudenbush (2002).484950===== PAGE 13 (score 20, numeric-paren 3, table-words 6, var-words 5) =====5152Estimation of Hedonic Models Using a Multilevel Approach 5369154Table 3: Estimated Parameters in the Random Intercept Model55Parameter56Coefficient (std. err.)57Fixed58Intercept59 609.555 61(0.013)62Random63Level 2 (municipality level)64σ265u066 670.047 68(0.004)69Level 1 (apartment level)70σ271e72 730.117 74(0.001)75Intra-class correlation760.2977Note : bold significant at 5% level.78After controlling for housing attributes, the intra-class correlation coefficient 79decreases to 24%. This small reduction strongly suggests that rents vary across 80municipalities and provides evidence for the existence of segmented markets.814.2 Random Slope Model82In this section, the coefficient for dwelling area is treated as a random effect. 83The paper focuses on a simple hedonic model that can illustrate the value of the 84multilevel approach in allowing for segmented markets. To this end, we make 85the restrictive assumption that only the rate of increase of rents associated with 86an additional square footage of the dwelling area can vary across municipalities. 87The other explanatory variables are assumed to have a fixed effect. This latter 88assumption can be relaxed and the question of the specification of a complete 89model to explain rents is left for future research.90Table 4 shows that, after controlling for the explained variables, there is a 91significant unexplained variance in rents that remains at municipality level. 92Upon interpretation of the other effects, we find the expected relationships. For 93instance, a positive effect is found for an additional m2 dwelling area, an addi-94tional room or a better quality score, whereas an apartment located in an older 95building has a lower rent. When looking at the effect of the dummy for mixed-use 96and residential buildings, there is no preconception about the sign of this effect. 97Here, a positive effect is obtained meaning that apartments located in mixed-use 98buildings have higher rents than apartments located in residential buildings.99100101===== PAGE 14 (score 38, numeric-paren 18, table-words 5, var-words 7) =====102103692 104Djurdjevic / Eugster / Haase105Table 4: Estimated Parameters in the Random Coefficient Model106Parameters107Coefficient (std. err.)108Fixed effects109Intercept (overall mean effect)110 1119.549 112(0.019)113Dwelling area (overall mean effect)114 1150.603 116(0.017)117Macro-location (in log)118 1190.789 120(0.024)121Age in years122 123−0.007 124(0.000)125Age squared126 1270.006 128(0.000)129Number of rooms130 1310.052 132(0.005)133Number of apartments134 135−0.0002 136(0.000)137Floor level138 1390.009 140(0.001)141Condition142 1430.065 144(0.003)145Standard146 1470.024 148(0.004)149Micro-location150 1510.009 152(0.003)153Mixed building154 1550.038 156(0.005)157Dhj_2158 1590.007 160(0.003) 161Dhj_3 162 1630.036 164(0.004)165Random effects166Level 2 (municipality level)167σ2168u0 (intercept)169 1700.022 171(0.004)172σu01 (covariance)173 1740.020 175(0.004)176σ2177u1 (dwelling area)178 1790.036 180(0.006)181Level 1 (apartment level)182σ2183e184 1850.017 186(0.000)187Note : bold significant at 5% level.188In the paper, the rate of increase in rents for additional square footage is allowed 189to vary across municipalities, between municipalities with higher average rents 190and between municipalities with lower average rents. A correlation term that is 191not significantly different from zero implies that the rate of increase in rental 192prices for additional square footage is similar across municipalities regardless 193of the municipality’s mean rent. This correlation coefficient can be calculated 194195196===== PAGE 15 (score 8, numeric-paren 0, table-words 2, var-words 4) =====197198Estimation of Hedonic Models Using a Multilevel Approach 199693200as σu01 / σu0σu1 and is equal to 0.71 in our random coefficient model. This has 201some interesting implications. It suggests that there is a relatively strong associa-202tion between average municipality rents and the municipality effects of square 203footage – indicating that municipalities with higher average rents tend to have 204a greater rate of increase in rental prices associated with increases in dwelling 205area.206Figure 3 provides an illustration of the variability of the effect of dwelling area 207at municipality level. The upper part of Figure 3 shows how the observations 208are distributed according to dwelling area. Two municipalities have been repre-209sented: one with circles and another with squares. We can draw a black line for 210the average relationship between rents and dwelling area. The circles are signifi-211cantly above the average line whereas the squares are significantly below it. As a 212consequence, Figure 3 shows that different intercepts have to be considered. The 213Figure 3: Relationship between Dwelling Area and Rents (in log)21410.721510.02169.32178.62187.9219log (price)2202.3 2213.1 2223.9 2234.7 2245.5225log (area)22611.022710,22289.42298.62307.8231log (price)2322.3 2333.1 2343.9 2354.7 2365.5237log (area)238239240===== PAGE 17 (score 96, numeric-paren 66, table-words 5, var-words 6) =====241242Estimation of Hedonic Models Using a Multilevel Approach 243695244Table 5: Comparison between the Different Models245Parameters246OLS 1247OLS 2248ML249Coeff. (std. err.)250Coeff. (std. err.)251Coeff. (std. err.)252Fixed effects253Intercept (overall mean effect)254 9.512 (0.037)255 9.513 (0.037)256 9.549 (0.019)257Dwelling area (overall mean effect)258 0.614 (0.019)259 0.630 (0.011)260 0.603 (0.017)261Macro-location (in log)262 0.766 (0.025)263 0.761 (0.022)264 0.789 (0.024)265Age in years266 −0.007 (0.000)267 −0.007 (0.000)268 −0.007 (0.000)269Age squared270 0.005 (0.000)271 0.005 (0.000)272 0.006 (0.000)273Number of rooms274 0.053 (0.005)275 0.054 (0.005)276 0.052 (0.005)277Number of apartments278 −0.0001 (0.000)279 −0.0001 (0.000)280 −0.0002 (0.000)281Floor level282 0.006 (0.001)283 0.007 (0.001)284 0.009 (0.001)285Condition286 0.063 (0.003)287 0.063 (0.003)288 0.065 (0.003)289Standard290 0.018 (0.004)291 0.019 (0.004)292 0.024 (0.004)293Micro-location294 0.007 (0.003)295 0.007 (0.003)296 0.009 (0.003)297Mixed building298 0.033 (0.005)299 0.032 (0.005)300 0.038 (0.005)301Dhj_2302 0.006 (0.003)303 0.005 (0.003)304 0.007 (0.003)305Dhj_3 306 0.037 (0.004)307 0.037 (0.004)308 0.036 (0.004)309Cantonal dummies310For canton of Zurich311 0.047 (0.004)312 0.048 (0.004)313For canton of Bern314 0.052 (0.006)315 0.052 (0.006)316For canton of Zug317 0.029 (0.010)318 0.034 (0.010)319For canton of Solothurn320 0.040 (0.010)321 0.038 (0.010)322For canton of Basel City323 0.085 (0.008)324 0.092 (0.007)325For canton of Basel Land326 0.078 (0.007)327 0.083 (0.007)328For canton of St.Gallen329 0.036 (0.007)330 0.037 (0.007)331For canton of Aargau332 0.056 (0.005)333 0.054 (0.005)334Cantonal interaction terms 335For canton of Zurich336 −0.048 (0.009)337For canton of Bern338 −0.030 (0.016)339For canton of Zug340 −0.053 (0.025)341For canton of Solothurn342 −0.063 (0.024)343For canton of Basel City344 0.036 (0.015)345For canton of Basel Land346 0.031 (0.025)347For canton of St.Gallen348 −0.049 (0.018)349For canton of Aargau350 −0.006 (0.671)351R 23520.8583530.8593540.877355Log-likelihood3565493.043575512.953586277.26359SSE360245.85361244.98362185.61363Notes: the reference canton is composed of other cantons; bold significant at 5% level; italics at 36410% level; 11 913 observations.365366367===== PAGE 18 (score 14, numeric-paren 0, table-words 4, var-words 6) =====368369696 370Djurdjevic / Eugster / Haase37111 If we consider the simplest version of the multilevel model (model with a random intercept 372only), the true values of u0 j (level 2 residuals which represent departures from the overall mean 373β0) are unknown, but we can obtain estimates of them given the observed data and the esti-374mated parameters of the model. These estimated or predicted residuals are often referred to 375As regards the fixed effects, the different regressions yield similar results. It is 376common to many hedonic pricing models that not all variables exhibit the proper 377sign when estimated with OLS. This is not the case in this application: there is 378no variable that is treated differently according to the type of estimation. How-379ever, the estimators differ in their estimates of the magnitude of the fixed effects. 380For example, the ML estimate for the number of apartments in the building is 381100% greater (in absolute term) than the corresponding OLS estimates. This is 382also the case for the effect of floor level (+ 50%), of standard (+ 33%) and to 383a lesser extent for the effect of micro-location (+ 20%). A particular attention 384should be devoted to the effect of the macro-location, which is closer to one in 385the ML specification. In the hedonic model, we expect the coefficient for the 386macro-situation to be close to one since this variable reflects the relative attrac-387tiveness of municipality A with respect to municipality B. If the generalised m2 388asked prices in municipality A are 1% higher than in municipality B, then the 389yearly rental price of a specific apartment should be 1% higher in municipality A 390than in municipality B. However, as advertisements offer insufficient information 391on the characteristics of apartments, the m2 asked price for one municipality is 392computed without correcting for the structural attributes (except for floor area). 393As a consequence, the contextual effects of the macro-location in the hedonic 394model are potentially confounded with the rent impact of apartment attributes. 395That is why the coefficient for the macro-location in the hedonic model is not 396equal to one.397The locational attributes (cantonal dummies and interaction terms) included 398in specifications OLS 1 and OLS 2 measure large-scale effects. However, the very 399local influences may prove to be more important in the prediction of rental prices. 400If we use the sample R2 and the sum of squared errors SSE to measure goodness 401of fit, the estimated SSE falls by 33% relative to the OLS results whereas the R2 402rises from 0.858 to 0.877, which is a small, though significant amount. Use of a 403likelihood ratio statistic to check whether the different specifications are signifi-404cantly different yields the result that the OLS specifications are rejected against 405the ML specification.406This section also provides some information on the precision of models in 407terms of rental prediction. The following figure represents the absolute differences 408between actual and predicted rents (as % of actual rents).11 The class 0%–5% 409