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a)Institute of Financial Services Zug (IFZ),

Lucerne University of Applied Sciences

and Arts

Suurstoffi 1

CH-6343 Rotkreuz

Jakob A. Dambon (corresponding author)

Mail: jakob.dambon@hslu.ch

orcID: 0000-0001-5855-2017

Fabio Sigrist

Mail: fabio.sigrist@hslu.ch

orcID: 0000-0002-3994-2244

b)Department of Mathematics,

University of Zurich

Winterthurerstrasse 190

CH-8057 Zurich

c)Fahrländer Partner

Raumentwicklung

Seebahnstrasse 89

CH-8003 Zurich

Stefan S. Fahrländer: sf@fpre.ch

Saira Karlen: ska@fpre.ch

Manuel Lehner: ml@fpre.ch

Jaron Schlesinger: js@fpre.ch

Anna Zimmermann: azi@fpre.ch

Examining the Vintage Effect in Hedonic Pricing using 1

Spatially Varying Coefficients Models: 2

A Case Study of Single-Family Houses in the Canton 3

of Zurich 4

Jakob A. Dambona, b, Stefan S. Fahrländerc, Saira Karlenc, Manuel Lehnerc, Jaron Schlesingerc, Fabio 5

Sigrista, Anna Zimmermannc

6

Abstract: This article examines the spatially varying effect of age on single-family house (SFH) prices. 7

Age has been shown to be a key driver for house depreciation and is usually associated with a negative 8

price effect. In practice, however, there exist deviations from this behavior which are referenced to as 9

vintage effects. We estimate a spatially varying coefficients (SVC) model to investigate the spatial 10

structures of vintage effects on SFH pricing. For SFHs in the Canton of Zurich, Switzerland, we find 11

substantial spatial variation in the age effect. In particular, we find a strong vintage effect in the best 12

urban locations compared to pure depreciative age effects in rural locations. Using cross validation, 13

we assess the potential improvement in predictive performance by incorporating additional spatially 14

varying vintage effects in hedonic models. For out-of-sample observations, we find no considerable 15

difference in predictive performance between a classical spatial hedonic and an SVC hedonic model. 16

2

JEL Classifications: C31, C53, R31, R32 17

Keywords: Gaussian process, spatial statistics, real estate, mass appraisal 18

1 Introduction 19

Hedonic real estate models contain several predictor variables, and age is a key explanatory variable. 20

The marginal effect of the building age on house prices has been well-studied. It has been found that 21

the age effect is nonlinear (Goodman & Thibodeau, 1995; Clapp & Giaccotto, 1998; ). In particular, 22

Case, Clapp, Dubin, and Rodriguez (2004) report a “plausible quadratic form” for the building age. This 23

behavior is a result of two main features of the age as an independent variable: i) In general, older 24

buildings depreciate due to deterioration; ii) “however, beyond some point, only those houses with 25

the best locations and the highest construction quality survive.” (Case, Clapp, Dubin, & Rodriguez, 26

2004, p. 171). The paraboloid appearance of the age effect has also been observed by Fahrländer 27

(2006) and linked to the building material and architectural style. Studies investigating this particular 28

type of behavior, i.e., a deviation from a pure depreciative effect once a particular age has been 29

reached, are referencing to it as a vintage effect (Goodman & Thibodeau, 1995; Clapp & Giaccotto, 30

1998; Rubin, 1993). 31

Over the last two decades, there emerged a special focus on location specific effects due to newly 32

available modeling methodologies. There are numerous publications which show a clear indication of 33

spatially varying covariate effects within hedonic pricing models. For instance, when applying additive 34

mixed regression models on rents in Vienna (Austria), Brunauer, Lang, Wechselberger, and Bienert 35

(2010) find “substantial spatial variation” of covariate effects between the districts of Vienna. 36

Existing methods to model such spatially varying coefficients (SVC) are Bayesian processes (Gelfand, 37

Kim, Sirmans, & Banerjee, 2003) and geographically weighted regression (Fotheringham, Brunsdon, & 38

Charlton, 2002). Applications of these methods consistently show the existence of non-stationary 39

coefficients, e.g., Baton Rouge (LA, United States) (Gelfand, Kim, Sirmans, & Banerjee, 2003), in 40

3

Toronto (ON, Canada) (Wheeler, Páez, Spinney, & Waller, 2014), Singapore (Cao, Diao, & Wu, 2019; 41

van Eggermond, Lehner, & Erath, 2011), and Shenzhen (China) (Geng, Cao, Yu, & Tang, 2011). 42

The goal of this paper is to unite the two frameworks investigating a possible spatially varying vintage 43

effect and to possibly enhance prediction performance of hedonic models using this feature. One of 44

the first observations of spatial differences in the age effects can be found in Malpezzi, Ozanne, and 45

Thibodeau (1987). They compared individual hedonic models for 59 metropolitan areas in the United 46

States and concluded that “[s]everal metropolitan areas exhibited significant deviations from the 47

average depreciation patterns.” (Malpezzi, Ozanne, & Thibodeau, 1987, p. 382). More recent evidence 48

for such behavior are presented in Brunauer, Lang, Wechselberger, and Bienert (2010) as well as 49

Dambon, Sigrist, and Furrer (2020), who found pronounced spatially varying effects on the rents and 50

the prices of apartments, respectively. 51

In this paper, we will model spatially varying vintage effects for single-family houses (SFH) in the Canton 52

of Zurich (Switzerland). Our working hypothesis is that, on average, age has a negative effect on the 53

house prices. However, as indicated above, we expect spatial deviations as we assume that there exist 54

municipalities or city districts where a vintage effect is present. An important question that arises from 55

this is whether, given the existence of such a vintage effect, it can be used to improve predictive 56

performance of hedonic models. 57

To verify our hypothesis on spatial varying vintage effects, we will use a new methodology introduced 58

by Dambon, Sigrist, and Furrer (2020) to model SVCs using Gaussian processes (GP). One of the 59

difficulties of classical GP based SVC models is that they do not scale to large data sets. The novel 60

methodology, in particular, allows for applying model-based SVC models to large spatial data sets. In 61

the next section, we first introduce and then extend the definition on SVC models and, in particular, 62

GP-based SVC models. In Section 3, we present the real estate data and justify the model. The results 63

are discussed in Section 4. In Section 5 we turn to predictive performance before discussing our results 64

in Section 6. 65

4

2 SVC Models 66

SVC models are a generalization of classical linear regression models, where we allow the regression 67

coefficients to vary over space. That is, the effect of a covariate () denoted by the coefficient can 68

depend on a geographic location , which we assume to be two-dimensional. SVC models can be 69

applied to spatial points data sets, where for each of the observations of the response variable 70

(,…,) and covariates ()

(),…,

(),= 1, … , , every observation has 71

an associated location . In summary, SVC models are defined as 72

=()

()

++()

()

+, (1)

where = 1, … , indexes the observations with their corresponding locations and is a classical 73

(0, ) iid error term with > 0. 74

If one assumes that not all coefficients should contain spatial structures, one can define mixed SVC 75

models. Let with 1 be the number of covariates for which we want to model SVCs. Without 76

loss of generality, we define the mixed SVC model as 77

=()

()

++()

()

+

()

++

()

+.

(2)

From now on, we assume that the first coefficient = 1 always models an intercept. In the special case 78

of = 1, we have the classical geostatistical model that is also used in most hedonic models. The exact 79

assumptions for the coefficients (), = 1, … , , and how they are estimated, have yet to be 80

defined. The literature on how to do so for both the classical geostatistical and SVC models is extensive. 81

For geostatistical models, see Cressie (2011) and Heaton et al. (2019) for an overview. For SVC models, 82

see Dambon, Sigrist, and Furrer (2020), Wheeler and Calder (2007), and Wheeler and Waller (2009) for 83

comparisons. 84

2.1 Gaussian Process-based SVC Models 85

We specify the SVC model such that each coefficient is defined by a Gaussian process. Gaussian 86

processes are well-studied (Rasmussen & Williams, 2006) and widely used tools to model dependency 87

structures with applications including – but not limited to – spatial statistics (Gelfand & Schliep, 2016; 88

5

Banerjee, Gelfand, Finley, & Sang, 2008; Datta, Banerjee, Finley, & Gelfand, 2016), econometrics (Wu, 89

Hernández-Lobato, & Ghahramani, 2014), and time series modeling (Roberts, et al., 2013). They are 90

infinite dimensional stochastic processes that are defined similarly to a finite-dimensional normal 91

distribution. We assume the GP to be jointly independent as well as independent of the error term 92

(, … , )(,). For observations =(, … , ), they are given by 93

(

)

,

()

,

(3)

for = 1, … , . We assume a constant mean and a covariance matrix (), which is defined by a 94

covariance functions () and the corresponding observation locations . The observation locations are 95

being used to model the dependency between observations by computing the distances. In spatial 96

statistics, one usually assumes that closer observations share higher dependency than observations 97

which are far apart1. We use the Euclidean distance denoted by which yields pair-wise distances 98

between all observations, 1,. Here, we assume to have exponential 99

covariance functions ()()= exp

,0, parametrized by variances 0 and 100

ranges > 0. The former parameter defines the extent of variation within an SVC () and the latter 101

defines the decay of spatial dependency with distance. The covariance function is then applied to the 102

distances, which yields the following corresponding covariance matrix 103

(): = ()()=exp

. 104

2.1.1 Example of two sampled Gaussian Processes 105

In this section, we illustrate the interpretation of the parameters for a GP with the help of two samples 106

of a GP. Both are defined by their corresponding parameters given in Table 1. Under the assumption 107

of an exponential covariance function, these parameters, more specifically, the ranges and variances, 108

define the covariance functions given in Figure 1. With the given covariance functions as well as the 109

1 This statement is also referred to as the first law of geography according to Waldo R. Tobler: “Everything is

related to everything else, but near things are more related than distant things.” (Tobler, 1970).

6

mean parameters, we sample the GPs on a regular 101 ×101 from the unit square. The sampled GPs 110

are given in Figure 2. 111

Table 1: Parametrizations of two GPs.

Mean

Range

Variance

Parametrization 1

2

0.25

1

Parametrization 2

-1

0.10

2

112

Figure 1: Covariance functions. Two exponential covariance functions which are depending on a

distance d and parametrized as given in Table 1.

One can clearly see that the variance in

Parametrization 1 is lower than in Parametrization 2. On the other hand, Parametrization 1 has a

greater range which leads to slower decay of the covariance function over distance.

113

7

Figure 2: Visualization of two sampled Gaussian processes. The parametrization and covariance

functions are given in Table 1 and Figure 1, respectively. The GPs values are given at the respected

coordinates x and y in the unit square. The sampled value is given via the color scale.

The influence of each of the corresponding 3 parameters, i.e., the mean , the range , and the 114

variance , can be directly seen from the individual visualized samples in Figure 2. First, we note that 115

the values of each parametrization are scattered around their individual means. The greater range of 116

parametrization 1 relative to parametrization 2 expresses itself by larger color patches in Figure 2. The 117

greater variance of parametrization 2 leads to a wider range of values in the simulation which 118

manifests itself by a wider color range in the visualization. 119

2.2 Maximum Likelihood Estimation of GP-based SVC Models 120

We give a brief summary of a maximum likelihood estimation (MLE) approach for SVC models as 121

introduced in Dambon, Sigrist, and Furrer (2020). Additionally, we extend the framework such that not 122

only full GP-based SVC models as given in (1), but also mixed GP-based SVC models as given in (2) can 123

be estimated. 124

8

With a data matrix , where the entry (): = () is the -th observation of the -th covariate, a 125

mean vector ,…,, and using the independence assumption from above, the 126

distribution of the response is given by 127

, ()

()()+ . (4)

The differences between the response’s distribution as above and as given in Dambon, Sigrist, and 128

Furrer (2020) are twofold. The first entries of the mean vector are the means of the GP as defined 129

in (3), while the further entries are the coefficients , … , . For simplicity, we identify them with 130

, … , , respectively. The second difference is the sum building the covariance matrix. Since only 131

covariates = 1, … , are defined to have SVCs, only covariance matrices and the respective 132

covariates enter the sum. 133

The model is thus fully parametrized by the covariance parameters ,

, … , ,

,134

and the mean parameters . We define (,) as our parameter of interest 135

which we estimate by maximizing the log-likelihood of (4). Since there exists no analytical solution, we 136

have to turn to numeric optimization. Once the estimate

is found, one can use it to predict the SVCs 137

for (new) locations using the conditional distribution, i.e., one obtains

(), = 1, … , . The 138

estimator and predictor is implemented in the statistical software R (R Core Team, 2020) and can be 139

used via the package varycoef (Dambon, Sigrist, & Furrer, 2019). 140

3 Data & Model 141

3.1 Data 142

The analysis is based mainly on transaction data and to a small extent on offer data for SFH in the 143

Canton of Zurich including a margin of 10 km width to account for margin effects when modeling the 144

data on the border of the Canton of Zurich. The data is provided by Fahrländer Partner 145

Raumentwicklung (FPRE), Zurich (Switzerland) and was collected (except for the offer data) by Swiss 146

9

banks and insurance companies in their day-to-day business2. It covers a time span of 6 consecutive 147

quarters ranging from the last quarter of 2018 to the first quarter of 2020 and consists of 2904 148

observations. For locations with limited data availability, namely for those locations with less than 3 149

observations, we used carefully selected offer data to enhance the available transaction data. By doing 150

so we obtained a data set consisting of about 20% offer data (583 observations) and 80% transaction 151

data (2321 observations). The data set contains approximately 45% of the transactions in the Canton 152

of Zurich for the given time period. An overview of the data alongside with some summary statistics is 153

given in Table 2 (a) and (b). 154

Due to Swiss banking secrecy, the exact geographic locations of the SFH cannot be disclosed. Here, 155

FPRE works with a fine grid of cells. For each observation we know the corresponding cell it falls in and 156

the location is given by a representative centroid of that cell, c.f. Table 2 (c) and Figure 3. The centroid’s 157

location is provided in the LV03 coordinate reference system (Federal Offce of Topography swisstopo, 158

1900). The cell’s resolution is higher in densely populated areas. The median cell size is 3.658 , 159

with the total range of areas extending from 0.147 to 36.316 . In total, we observe data at 160

618 distinct cells, of which 337 (equal to 1678 observations) lie in the Canton of Zurich. Additionally, 161

each cell is labeled with a location type, see Table 2 (c), which will turn out helpful when analyzing our 162

findings in Section 4. 163

2 A total of three observations were removed from the data set, for which real estate experts from Fahrländer

Partner Raumentwicklung (FPRE) assume that they were incorrectly classified as arm’s length transactions.

10

Table 2: Description and summary statistics of underlying data set.

(a) Continuous variables

Variable

Description

Min.

Median

Max.

Std. Deviation

price

Adjusted transaction price

in Swiss Francs

excluding parking and

special factors

120’000

1’260’000

10’900’000

917’821

yoc

Year of construction

1920

1995

2020

26

volume

Building volume in (SIA

Zürich, 2003)

290

803

3134

300

plot size

Plot size in

100

489

3232

333

renov

Need for renovation

(difference between actual

and theoretical building

condition, higher meaning

better; h.m.b.)

0.00

0.00

4.00

0.83

standard

Standard; h.m.b.

2.00

3.00

5.00

0.65

micro

Micro-location ; h.m.b.

2.00

3.50

5.00

0.64

11

(b) Categorical variables, reference level in italics.

Variable

Description

Levels

Observations

transaction

quarter

Transaction quarter

20184: 4th Quarter of 2018

20191: 1st Quarter of 2019

20192: 2nd Quarter of 2019

20193: 3rd Quarter of 2019

20194: 4th Quarter of 2019

20201: 1st Quarter of 2020

430

562

571

579

389

373

energy

Energy standard

1: Insulated shell

2: Enhanced energy efficiency

2791

113

SFH type

Type of SFH

1: detached

2: semi-detached

3: row house

1680

846

378

(c) Observation locations

Coordinates

Description

Range

Easting 03

Coordinates in the LV03 coordinate

reference system (Federal Office of

Topography swisstopo, 1900) in .

200 ×10800 ×10

Northing 03

100 ×10400 ×10

Variable

Description

Levels

Observations

FPRE type

Type of location

1: Top-Locations

2: Urban Agglomerations

3: Other Agglomerations

4: Rural Areas

439

898

885

682

164

12

Figure 3: Spatial distribution of observations and classification of locations within the Canton of

Zurich. The Canton of Zurich is divided into 563 cells which are classified into types of location. The

number of observations within such cells is depicted with the color-coded observation count. The

type of cell and the representative location are depicted by the orange symbols.

3.2 Model 165

The model has the natural logarithm of the transaction price as the response variable. Further, we 166

standardize the year of construction (yoc) using the following transformation 167

.=

2000

20 .

The advantage of working with . rather than the actual year of construction or age is a numerical 168

stable optimization process of the MLE. As mentioned above, we expect a quadratic effect (Case, 169

13

Clapp, Dubin, & Rodriguez, 2004; Clapp & Giaccotto, 1998; Goodman & Thibodeau, 1995; Fahrländer, 170

2006), which is why we also include the covariate .. As we expect spatial variation in these 171

coefficients, we use SVCs for these variables, c.f. first line in (5). The plot size as well as the volume 172

enter the model under a natural logarithm transformation. The rest of the continuous covariates 173

, and are included without further transformation. Thus, all continuous 174

covariates have approximately the same standard deviations which results in a well-behaved numeric 175

optimization procedure for estimating the model. 176

The categorical variables , and and the error term complete 177

our model which can be formulated as: 178

= log =

(

)

+

(

)

.+

(

)

.

+log +log

+++

+ + +

+ .

(5)

Comparing the general mixed SVC model (2) and our explicit hedonic model (5) we note that we have 179

= 3 and =16 including the intercept and all factor levels deviating from the reference levels. The 180

model is therefore fully parametrized by 181

= (,)=(

,

,

,

,

,

,,

, … ,

).

We will use a numeric optimization over the profile likelihood. Thus, we must optimize over the 182

covariance parameters and the mean parameters are determined implicitly by calculating the 183

generalized least square estimate. 184

3.3 Observation Locations 185

As the LV03 coordinates for the centroid’s locations =(03,03) cover a fairly large range, 186

we standardized them to kilometers using the following formula: 187

14

.3

.3=1003

03600000

200000

Again, this ensures a well-behaved numeric optimization while remaining interpretable as the ranges 188

now act as a scaling factor on the kilometer distances. 189

4 Results 190

4.1 Parameter Estimates 191

We first take a look on the ML estimates

, which are given in Table 3. Here, we find that the 192

mean estimates match our expectations. In particular, the vintage-related covariates, i.e., . and 193

., show the following: 194

1) Reminding ourselves that . is derived from the year of construction (yoc) and not the 195

building age, the positive mean effect is equivalent to a depreciation of the building price with 196

increasing age. 197

2) Further, the quadratic effect . is very close to 0, but positive. A larger quadratic mean 198

effect would correspond to an emphasized vintage effect. 199

All other mean effects have plausible signs, too. That is, all other coefficients of continuous covariates 200

are positive and therefore in line with our expectations. As for the categorical covariates, we observe 201

some temporal price volatility for the transaction quarter, a premium for stand-alone, detached SFH 202

compared to other SFH for the type of SFH and a premium for houses with enhanced energy efficiency 203

for the energy standard. 204

The estimates for ranges show that the range for the intercept is considerably larger than those for 205

. and .. This will be expressed by larger spatial structures for the SVC modeling the 206

intercept compared to the other two SVCs, see Table 3. The small range of . and . indicate 207

that the corresponding SVCs will behave much more selective in their deviations from the mean. 208

Finally, the variances suggest higher variations in the first SVC, which can also be interpreted as a mean 209

pricing level for the respective location. 210

15

211

212

213

214

215

Table 3: Mean and covariance estimates

of SVC model (5).

Covariates

Mean

Range

Variance

and

Intercept

8.951622

28.439374

0.118325

.

0.107510

2.041053

0.001574

.

0.005834

2.197881

0.000313

log()

0.478684

log( )

0.189723

0.032742

0.095569

0.030550

factor( )20191

-0.002353

factor( )20192

0.012518

factor( )20193

0.009341

factor( )20194

0.016490

factor( )20201

0.012574

factor( )2

-0.017312

factor( )3

-0.042931

factor()2

0.028165

nugget

0.025688

16

Table 4: Summary Statistics of SVCs

a) estimated

Intercept

()

.

(

)

.

(

)

Minimum

8.922

-0.003295

-0.026603

Mean

9.389

0.100902

0.006979

Maximum

10.088

0.149384

0.056561

b) spatially predicted

Intercept

(

)

.

() .

()

Minimum

8.883

0.035420

-0.010333

Mean

9.333

0.103460

0.006225

Maximum

10.079

0.137420

0.035978

216

4.2 Visualization and Interpretation of SVCs 217

In Figure 4 we visualize fitted and predicted SVCs. Specifically, the figure shows the estimated SVCs for 218

the observation locations the model has been trained on as well as for the spatial predictions for all 219

other cell’s centroid where we did not have any observations. The quality of these coincides with the 220

previous parameter estimates interpretations from Section 4.1 and real estate experts’ knowledge. 221

For the intercept’s SVC, c.f. Figure 4 (a), which also can be interpreted as a mean price level, we can 222

see that the highest values are achieved close to the city or Zurich and the Lake Zurich, with a local 223

peak at the city of Winterthur. As expected, the lowest values can be found towards the North and 224

East, which are more rural areas. 225

17

Figure 4: Estimated and spatially predicted SVCs of model (5). Note that the coefficients’ value spans

are according to their variance estimates , i.e., descending from (a) to (c).

As for the . and . SVCs, c.f. Figure 4 (b) and (c), one can see small scale local deviations. 226

Examples are a very small value of the . SVC to the West of Zurich (in Table 4 one can verify that 227

the negative SVC effect is almost identical to 0) and a relatively high value for the . SVC close to 228

the city center. There is also some clustering of below average values (recall . 0.11) in Zurich, 229

adjacent to Lake Zurich and the city of Winterthur. However, an individual interpretation of both 230

panels is cumbersome and inadequate as the fitted SVCs originate from the same covariate. As we are 231

simultaneously modeling a linear and quadratic effect, one could therefore interpret the results as 232

18

spatially varying quadratic effects. Using the SVCs

() and

() for all observation locations 233

within the training data and the Canton of Zurich, we back-transform the estimated effects to receive 234

the marginal effect (,) for the year of construction [1920,2020]: 235

(,)()

2000

20 +()

2000

20

. (6)

236

Figure 5: Back-transformed, aggregated effect of year of construction. The grey curves correspond

to the marginal effects as defined in (6) and the red lines are the aggregated marginal effects as

defined in (7). The most extreme are displayed with their respective names, i.e., Fluntern (a district

of the city of Zurich), Bonstetten (a suburb to the West of the city of Zurich), and Guggenbühl (a

district of the city of Winterthur).

This is what we visualize in Figure 5 in averaged form for 4 different location types. The grey lines are 237

the marginal effects (,), grouped in panels by FPRE type of location and filtered such that 238

i) there are at least 5 observations per location and ii) cropped to the span of observed years of 239

construction at corresponding location. This is to ensure that we have sufficient data backing up the 240

results and that we do not extrapolate to unobserved years of construction. We define these set of 241

observation locations as ,{1, 2,3,4}. The red line is obtained by aggregating all marginal effects 242

by type of location, i.e., 243

19

,()=

1

||(,)

(7)

for year of construction [1920,2020]. In all panels in Figure 5 a depreciation part is present for all 244

years of construction >1975. This holds not only for all ,, but also for most of individual 245

(,). A first hint of a vintage effect is observable for , with = 1, i.e., top locations, 246

and = 3, i.e., other agglomerations. The estimated effects have pronounced curvatures which are 247

going back to a substantial quadratic component, i.e.,

(). The other two effects with = 2, i.e., 248

urban agglomerations, and = 4, i.e., rural areas, appear almost linear which means that the age of a 249

building is exclusively associated with a price depreciation. 250

Looking at the individual marginal effects (,), one can observe some variety within each 251

location type . However, the conveyed message stays the same. At top locations (= 1) a vintage 252

effect is present where some of the oldest SFHs have the same marginal effect for year of construction 253

as just recently built SFHs. Here, one location (Fluntern, a district of Zurich) stands out as it gives a 254

premium of approximately 0.5 log-points for an SFH built in the 1920s compared to an SFH built in the 255

1960s and 1970s, ceteris paribus. At other types of locations, the individual marginal effects are of a 256

purely depreciating nature, with exception to 3 locations at other agglomerations (= 3); for instance 257

Guggenbühl (a district of Winterthur). At this location, we again note that the observed real estate 258

objects are rather old, but, that the range of the marginal effect is quite small. These two locations 259

therefore represent two urban districts with rather old real estate objects but two very different 260

marginal effects of age, ceteris paribus. While the prices of SFHs in Guggenbühl seem rather in-261

sensitive to age, very old SFHs are sought after in Fluntern. Here, as well as in other top locations, the 262

vintage effect is most pronounced. In any case, both manifestations of the marginal age effects are 263

deviations from the usual depreciative nature of age with respect to SFH pricing. Finally, we address 264

the municipality of Bonstetten and its marginal effect displayed in Figure 5. It suggests a relatively 265

steep depreciation with age. The strongly negative age effect in Bonstetten is mainly driven by two 266

observations with year of construction 1920, being sold at relatively low transaction prices. As we 267

20

observe no transactions with year of construction between 1920 and 1979, the two observations 268

mentioned have a high leverage and the marginal effect is extrapolated. Therefore, the depicted 269

age effect for Bonstetten should be treated with caution. 270

We conclude this section by noting that we observe spatially varying age effects which clearly deviate 271

from a pure depreciation. These effects are locally pronounced and mostly appear at top locations, 272

which backs our hypothesis of spatially varying vintage effects and is in line with both initial citations 273

taken from Case, Clapp, Dubin, and Rodriguez (2004) and Malpezzi, Ozanne, and Thibodeau (1987). 274

5 Predictive Performance 275

In this section, we will assess the implications for predictive performance of our findings from above. 276

As suggested in Section 4, there appears to be a spatially varying age effect that deviates from a linear 277

depreciation with age. Now, we investigate if one can exploit this to enhance classical hedonic models 278

to increase predictive performance. 279

To examine this, we validate and compare our findings to a classical hedonic model with only the mean 280

price, i.e., the intercept depending on spatial location . Thus, we use a geostatistical model similarly 281

defined as the SVC model in (5) but with = 1: 282

= log

=

(

)+

.

+

.

+log +log

+++

+ + +

+ .

(8)

Using the same data (2904 observations) as before, the two models (5) and (8) were trained on two 283

folds in a very similar manner as actual hedonic pricing is conducted. That is, the first 5 quarters of 284

transaction data was exclusively used as training data. For the validation data, observations from the 285

last observed quarter, i.e., the 1st quarter of 2020, and from within the Canton of Zurich were selected 286

and randomly divided into two folds of 113 observations each. The corresponding rest is being used 287

21

for the training data. Therefore, for both folds the training data consists of 2791 observations and 113 288

testing locations, respectively. The split of the data into training and validation sets is illustrated in 289

Table 5. 290

Table 5: Layout of validation.

Quarters:

2018Q4

2019Q1

2019Q2

2019Q3

2019Q4

2020Q1

Within Ct. of ZH:

Outside Ct. of ZH:

Legend:

Training

Training

Validation

Fold 1:

Fold 2:

291

The root mean square error (RMSE) was chosen as a measure of comparison and computed for in-292

sample estimates and out-of-sample predictions. The results are given in Table 6. While the SVC model 293

has an advantage in terms of in-sample fit, the RMSE of the two models on the out-of-sample data are 294

virtually identical. This shows that while the SVC models are better at accounting for spatially varying 295

age effects as discussed in the previous section, this feature does not translate to more accurate out-296

of-sample predictions. Probably, the reason behind this is a too heterogeneous set of samples within 297

each location grid cell. As the Swiss banking secrecy does not allow to disclose exact SFH locations, the 298

data set at hand reaches its limits with respect to application of SVC models. A possible vintage effect 299

is present at a much smaller scale, or, even at an object individual level. A possible increase of 300

predictive performance would only be observable if spatially varying age effects, such as vintage 301

effects, would be observable at a greater scale. Thus, with the current resolution of the spatial data 302

set at hand, we cannot use this to our advantage. 303

22

Table 6: Results of the two-fold cross validation between the SVC and geostatistical model. The

best performing method is highlighted using italic font.

In-Sample RMSE

Out-of-Sample RMSE

Model

Fold 1

Fold 2

Fold 1

Fold 2

SVC Model (5)

0.14637

0.13732

0.18888

0.20999

Geostatistical Model (8)

0.16036

0.15980

0.18744

0.19669

6 Conclusion 304

To the best of our knowledge, the presented work is the first of its kind to investigate a spatially varying 305

age effect for SFH. While we find a purely depreciative age effect for some locations in the Canton of 306

Zurich, there appears to be a substantial price premium for older SFHs for other locations. The 307

existence of a not purely depreciative age effect is in line with the scientific literature and the 308

assumptions of real estate experts. In this context, we consider it very likely that age or the year of 309

construction acts as a proxy for unmeasured covariates that directly have an impact on prices, such as 310

quality of built or architectural style (e.g. room height, architectural details) of the object as has been 311

suggested by the existing literature (Case, Clapp, Dubin, & Rodriguez, 2004; Goodman & Thibodeau, 312

1995). 313

Overall, our analysis suggests a spatially varying or at least object specific age effect, which for certain 314

locations manifests as a vintage effect. Further research on the topic based on data from different 315

regions or with much higher resolution would be desirable. 316

317

318

319

320

23

7 List of Abbreviations 321

Ct.

Canton

FPRE

Fahrländer Partner Raumentwicklung

GP(s)

Gaussian process(es)

h.m.b.

higher meaning better

ML(E)

maximum likelihood (estimation)

RMSE

root mean square error

SFH

single-family house

SVC

spatially varying coefficients

yoc

year of construction

ZH

Zurich

322

8 Declarations 323

8.1 Availability of Data and Materials 324

The data used for this analysis is subject to Swiss banking secrecy and can therefore neither be made 325

available publicly nor on request. 326

8.2 Competing Interests 327

The authors declare that they have no competing interests. 328

8.3 Funding 329

This study was jointly funded by Innosuisse (the Swiss Innovation Agency) and Fahrländer Partner 330

Raumentwicklung (FPRE) in the framework of a project on space-time machine learning models for 331

valuation and prediction of real estate objects (Innosuisse project number 28408.1 PFES-ES). The 332

design of this study, the collection, analysis and interpretation of the data and the writing of the 333

manuscript were not influenced by the funding body. 334

24

8.4 Author’s Contributions 335

JD and FS contributed the statistical fundamentals serving as a basis for this paper. The model 336

estimates and other calculations were carried out by SK, with support from JD. An analysis of the data 337

and results was performed by JD and SK. The interpretation of the results was performed to a large 338

extent by JD, with support from ML, AZ and JS. JD was responsible for writing the paper, with selective 339

contributions from SK. FS, ML, AZ and SF have revised the paper and have initiated a number of 340

changes to the paper. 341

8.5 Acknowledgements 342

Not applicable. 343

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