# RECORD c6b33e6b839caa1157a283c6f9a1e3350edd0558
# TITLE: Modeling unobserved heterogeneity in hedonic price models
# YEAR: 2020.0  VENUE: Real Estate Economics  DOI: 10.1111/1540-6229.12320
# AUTHORS: Marc K. Francke; A. van de Minne
# PAGES: 26  CANDIDATE TABLE PAGES (1-based): [11, 14, 15, 17]
# ABSTRACT: This paper studies unobserved heterogeneity in hedonic price models, arising from missing property and locational characteristics. Specifically, commercial real estate is very heterogeneous, and data on detailed property characteristics are often lacking. We show that adding mutually independent property random effects to a hedonic price model results in more precise out-of-sample price predictions, both for commercial multifamily housing in Los Angeles and owner-occupied single-family housing in Heemstede, the Netherlands. The standard hedonic price model does not take advantage of the fact that some properties sell more than once. We subsequently show that adding spatial random effects leads to an additional increase in prediction accuracy. The increase is highest for properties without prior sales.


===== PAGE 1 (score 4, numeric-paren 0, table-words 0, var-words 4) =====

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)
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Modeling unobserved heterogeneity in hedonic price models
Francke, M.; van de Minne, A.
DOI
10.1111/1540-6229.12320
Publication date
2021
Document Version
Final published version
Published in
Real Estate Economics
License
CC BY-NC-ND
Link to publication
Citation for published version (APA):
Francke, M., & van de Minne, A. (2021). Modeling unobserved heterogeneity in hedonic price
models. Real Estate Economics, 49(4), 1315-1339. https://doi.org/10.1111/1540-6229.12320
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Download date:21 Oct 2026


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1324
FRANCKE AND VAN DE MINNE
TA B L E 1
Descriptive statistics
mean
sd
min
max
Los Angeles
Sales price ($)a
6,389,494
6,460,729
1,550,000
64,250,000
Net Operating Income ($)
325,223
353,270
67,200
3,600,000
Age (Years)
45
21
2
97
Size (SqFt)
31,718
31,972
5,964
271,757
Years between sales
4.61
2.87
0.17
12.00
Garden (R)
0.87
0
1
Mid/Highrise
0.13
0
1
Observations
2,263
Unique properties
1,936
Heemstede
Sales price (€ )a
484,612
189,106
200,000
1,195,000
Age (Years)
65
22
15
106
Size (SqMt)
151
39
82
288
Years between sales
7.07
3.67
0.42
16.25
Maintenance [bad] (R)
0.18
0
1
Maintenance [average]
0.59
0
1
Maintenance [good]
0.24
0
1
Row house (R)
0.44
0
1
Semidetached (1)
0.03
0
1
Semidetached (2)
0.23
0
1
Corner home
0.25
0
1
Detached
0.05
0
1
Yard (yes)
0.94
0
1
Observations
2,468
Unique properties
2,065
Note. R gives the reference categories in our model. Semidetached (1) are the properties that are connected via a garage, and semidetached
(2) are the properties that are connected wall-to-wall.
aEstimates for Moran’s 𝐼(sales prices) for Los Angeles and Heemstede are, respectively, +0.04 and +0.23.
3
DATA AND DESCRIPTIVE STATISTICS
We use two different data sources, commercial multifamily real estate (income generating properties)
in the city of Los Angeles and single-family housing (owner-occupied) in Heemstede, a city relatively
close to Amsterdam in the Netherlands.
The first database is provided by Real Capital Analytics, and captures approximately 90% of all
commercial property transactions in the United States over $2.5 million. The database contains 2,263
prefiltered transactions, of which 1,936 are unique properties, in the period 2001–2017. The annual
number of transactions is 140 transactions on average. We observe the net operating income (NOI),
property subtype (garden versus mid/highrise), the age and size of the structure (in square feet), latitude
and longitude, and the transaction price. The upper panel of Table 1 provides some descriptive statistics.


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FRANCKE AND VAN DE MINNE
1325
TA B L E 2
Overview of model specifications
Model
Specification
Spatial effect (𝜽𝒑)
Standard
Equation (1): 𝑦𝑝= 𝑥∗
𝑝𝛽+ 𝜖𝑝
Hybrid
Equation (6):
(
𝑦𝑆
𝑝
Δ𝑦𝑅
𝑝
)
=
(
𝑥∗𝑆
𝑝
Δ𝑥∗𝑅
𝑝
)
𝛽+
(
𝜖𝑆
𝑝
Δ𝜖𝑅
𝑝
)
, ∀𝑝∶
(𝑛𝑝= 1
𝑛𝑝> 1
)
RE
Equation (2): 𝑦𝑝= 𝑥∗
𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝜖𝑝
Besag(Hybrid)
(
𝑦𝑆
𝑝
Δ𝑦𝑅
𝑝
)
=
(
𝑥𝑆
𝑝
Δ𝑥𝑅
𝑝
)
𝛽+
(𝜃𝑝
0
)
+
(
𝜖𝑆
𝑝
Δ𝜖𝑅
𝑝
)
, ∀𝑝∶
(𝑛𝑝= 1
𝑛𝑝> 1
)
Equation (9)
Besag(RE)
Equation (7): 𝑦𝑝= 𝑥𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝑗𝑛𝑝𝜃𝑝+ 𝜖𝑝
Equation (9)
SRW(Hybrid)
(
𝑦𝑆
𝑝
Δ𝑦𝑅
𝑝
)
=
(
𝑥𝑆
𝑝
Δ𝑥𝑅
𝑝
)
𝛽+
(𝜃𝑝
0
)
+
(
𝜖𝑆
𝑝
Δ𝜖𝑅
𝑝
)
, ∀𝑝∶
(𝑛𝑝= 1
𝑛𝑝> 1
)
Equation (10)
SRW (RE)
Equation (7): 𝑦𝑝= 𝑥𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝑗𝑛𝑝𝜃𝑝+ 𝜖𝑝
Equation (10)
Note. The models including property random effects—RE, Besag(RE) and SRW(RE)—all have the same property random effect speci-
fication given by 𝜙∼(0, 𝜎2
𝜙𝐼𝑃), see Equation (3).
The spatial effect from the Besag model is provided by 𝜃𝑝|𝜃−𝑝∼
( ∑
𝑞∈𝜕𝑝𝑤𝑝,𝑞𝜃𝑞
𝑑+𝑚𝑝
,
𝜎2
𝜃
𝑑+𝑚𝑝
)
, see Equation (9).
The spatial effect from the spatial random walk model is provided by 𝜃(𝑝) ∼(𝜃(𝑝−1), 𝜎2
𝜃), see Equation (10).
The matrix 𝑥consists of property characteristics and time fixed effects.
The models Standard, Hybrid, and RE have also location fixed effects. All other models do not. The location fixed effects have been
added to 𝑥, denoted by 𝑥∗.
In the models Besag(Hybrid) and SRW(Hybrid), spatial random effects have been added to the one only sales in the hybrid specification
(Equation (6)).
The average transaction price is about $6.4 million, the average size is about 32,000 square feet, and
the average age is 45 years. Most properties are designated garden.
The second database is provided by the Dutch Association of Real Estate Brokers and Real Estate
Experts (NVM), the largest brokers organization in the Netherlands. About 70% of all real estate bro-
kers in the Netherlands are affiliated to the NVM. The database contains residential real estate 2,262
transactions, of which 2,065 are unique properties, in the period 2001 to 2017 for the Dutch city of
Heemstede. The majority of the transactions is one only sales (69%). The annual number of trans-
actions is approximately 145 transactions on average. We observe the property subtype (row houses,
corner house, two types of semidetached homes and detached), the age and size of the structure, the
maintenance level (three groups from bad to good), the presence of a yard, latitude and longitude, and
the transaction price. The lower panel of Table 1 provides some descriptive statistics. The average
transaction price is about €485,000, the average size 151 m2 (1,625 square feet), and the average age
is 65 years. The largest numbers of properties are row houses (44%). The NVM distinguishes between
two types of semidetached homes: (a) two properties are connected via a garage and (b) two properties
are connected wall-to-wall. Most of the semidetached properties fall in the second category, 24% of the
observations. More than half of the properties have an average maintenance level at the time of listing,
compared to 18% badly maintained and 23% well maintained.6 Almost all properties have a yard, and
in only 6% of the transactions, this is not the case.
6See Francke and van de Minne (2017b) for a discussion on how the maintenance data in the NVM data are compiled.


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FRANCKE AND VAN DE MINNE
1327
TA B L E 3
Posterior means and in-sample fit statistics for Los Angeles
Besag
SRW
Standard
Hybrid
RE
Hybrid
RE
Hybrid
RE
(Intercept)
3.940***
4.091***
4.053***
4.135***
4.096***
4.029***
4.033***
ln Size
0.932***
0.924***
0.928***
0.945***
0.952***
0.956***
0.961***
ln ( 𝑁𝑂𝐼
Size )
0.735***
0.708***
0.700***
0.613***
0.609***
0.603***
0.602***
Age
−0.001***
−0.001***
−0.001***
−0.003***
−0.004***
−0.004***
−0.004***
Age2
0.000***
0.000
0.000
0.000
0.000*
0.000
0.000***
Mid/Highrise
0.018
0.014
0.019
−0.003***
−0.002***
−0.004***
−0.005***
2002
0.060
0.014
0.032
0.054
0.048
0.089*
0.094**
2003
0.170***
0.153*
0.140*
0.192***
0.184***
0.216***
0.202***
2004
0.275***
0.272***
0.264***
0.308***
0.293***
0.334***
0.326***
2005
0.318***
0.328***
0.336***
0.409***
0.397***
0.439***
0.437***
2006
0.344***
0.350***
0.362***
0.433***
0.421***
0.464***
0.461***
2007
0.309***
0.321***
0.333***
0.416***
0.398***
0.446***
0.437***
2008
0.318***
0.315***
0.329***
0.410***
0.402***
0.444***
0.440***
2009
0.217***
0.220***
0.213***
0.287***
0.264***
0.312***
0.297***
2010
0.192***
0.188***
0.203***
0.281***
0.273***
0.311***
0.309***
2011
0.253***
0.247***
0.258***
0.324***
0.317***
0.365***
0.359***
2012
0.287***
0.292***
0.294***
0.364***
0.356***
0.398***
0.392***
2013
0.320***
0.324***
0.330***
0.454***
0.450***
0.478***
0.475***
2014
0.411***
0.428***
0.439***
0.558***
0.547***
0.592***
0.585***
2015
0.528***
0.545***
0.554***
0.664***
0.656***
0.693***
0.691***
2016
0.628***
0.621***
0.641***
0.762***
0.762***
0.791***
0.796***
2017
0.622***
0.639***
0.648***
0.775***
0.767***
0.814***
0.811***
Location
FE
FE
FE
𝜎𝜖
0.190
0.184
0.124
0.126
0.123
0.137
0.120
𝜎𝜙
0.146
0.149
0.040
𝜎𝜃
0.033
0.029
0.011
0.064
𝜎𝜃∕√𝑑+ ̄𝑤𝑝+
0.010
0.008
Moran 𝐼
0.112
0.079
0.084
0.000
0.003
−0.003
−0.002
DIC
−1,072.1
−919.8
−1,850.6
−2,132.5
−2,412.0
−2,035.4
−2,502.4
WAIC
−1,070.6
−1,031.9
−1,891.2
−2,220.0
−2,443.5
−2,074.4
−2,459.2
Note. The model specifications are provided in Table 2. The omitted dummy variable is garden apartment (for property subtype) and
2001 (for time of sale). NOI stands for net operating income.
Moran’s 𝐼is a measure for spatial autocorrelation. DIC denotes deviance information criterion, and WAIC Watanabe information
criterion.
***means the parameter is statistically significantly different from 0 at the 1% level, **at the 5% level, and *at the 10% level.
estate studies find this “law of diminishing returns” (Bokhari & Geltner, 2018). The coefficient for
Mid/Highrise properties in Los Angeles is positive but insignificant for the standard, hybrid, and RE
model. For the Besag and SRW models, the coefficient becomes statistically significant and negative,
which might indicate an interaction between property type and location, which the location fixed effects
in the standard, hybrid, and RE model do not pick up. Also, ceteris paribus, one would expect that
lowrise housing would be more popular compared to highrise housing. Age has a negative coefficient


===== PAGE 15 (score 205, numeric-paren 178, table-words 1, var-words 4) =====

1328
FRANCKE AND VAN DE MINNE
TA B L E 4
Posterior means and in-sample fit statistics for Heemstede
Besag
SRW
Standard
Hybrid
RE
Hybrid
RE
Hybrid
RE
(Intercept)
8.010***
8.060***
8.135***
8.836***
8.891***
9.127***
9.095***
ln Size
0.898***
0.889***
0.876***
0.723***
0.714***
0.685***
0.694***
Age
0.004***
0.002**
0.003***
0.003***
0.003***
0.001
0.001
Age2
−0.000***
−0.000***
−0.000***
−0.000***
−0.000***
−0.000***
−0.000***
Semidetached (1)
0.148***
0.162***
0.137***
0.083***
0.074***
0.088***
0.073***
Semidetached (2)
0.199***
0.214***
0.205***
0.167***
0.152***
0.174***
0.163***
Corner Home
0.094***
0.117***
0.104***
0.100***
0.087***
0.109***
0.094***
Detached
0.337***
0.357***
0.347***
0.307***
0.292***
0.301***
0.286***
Maintenance [average]
0.128***
0.127***
0.133***
0.129***
0.130***
0.123***
0.128***
Maintenance [good]
0.216***
0.209***
0.208***
0.204***
0.208***
0.206***
0.207***
Yard
0.025
0.031*
0.025
0.013
0.014
0.015
0.016
2002
0.047**
0.041*
0.036*
0.031*
0.034**
0.032**
0.031**
2003
0.039*
0.036
0.027
0.026
0.029*
0.038*
0.038**
2004
0.078***
0.084***
0.068***
0.071***
0.071***
0.075***
0.068***
2005
0.141***
0.133***
0.134***
0.128***
0.130***
0.140***
0.139***
2006
0.171***
0.180***
0.170***
0.168***
0.171***
0.185***
0.181***
2007
0.261***
0.261***
0.252***
0.247***
0.253***
0.270***
0.272***
2008
0.280***
0.290***
0.278***
0.284***
0.283***
0.295***
0.286***
2009
0.244***
0.232***
0.232***
0.225***
0.227***
0.239***
0.241***
2010
0.224***
0.213***
0.216***
0.204***
0.213***
0.228***
0.228***
2011
0.224***
0.212***
0.223***
0.216***
0.224***
0.228***
0.238***
2012
0.135***
0.132***
0.133***
0.115***
0.116***
0.132***
0.133***
2013
0.135***
0.136***
0.135***
0.118***
0.113***
0.123***
0.121***
2014
0.176***
0.173***
0.185***
0.158***
0.161***
0.177***
0.184***
2015
0.216***
0.207***
0.230***
0.221***
0.229***
0.232***
0.241***
2016
0.344***
0.340***
0.354***
0.334***
0.343***
0.358***
0.365***
2017
0.420***
0.415***
0.434***
0.403***
0.408***
0.420***
0.427***
Location
FE
FE
FE
𝜎𝜖
0.171
0.164
0.102
0.096
0.095
0.116
0.098
𝜎𝜙
0.139
0.011
0.070
𝜎𝜃
0.119
0.118
0.057
0.053
𝜎𝜃∕√𝑑+ ̄𝑤𝑝+
0.049
0.048
Moran’s 𝐼
0.089
0.037
0.019
0.005
−0.021
−0.017
−0.011
DIC
−1,671.4
−1,517.1
−2,835.4
−2,581.9
−3,538.2
−2,813.2
−3,395.0
WAIC
−1,670.3
−1,664.6
−2,925.9
−2,700.3
−3,569.0
−2,834.2
−3,384.3
Note. The model specifications are provided in Table 2. The omitted dummy variables are row house (for property subtype), maintenance
[bad], and having no yard. Semidetached (1) are houses connected by a garage, and semidetached (2) are houses that are connected wall-
to-wall and 2001 (for time of sale).
Moran’s 𝐼is a measure for spatial autocorrelation. DIC denotes deviance information criterion, and WAIC Watanabe information
criterion.
***means the parameter is statistically significantly different from 0 at the 1% level, **at the 5% level, and *at the 10% level.


===== PAGE 16 (score 23, numeric-paren 0, table-words 8, var-words 7) =====

FRANCKE AND VAN DE MINNE
1329
and the square of age a positive coefficient. This confirms expectations, as depreciation is fastest when
a property is young, see Bokhari and Geltner (2018). Here, the estimated coefficient is also lower
compared to other studies because of the inclusion of NOI (and age squared is not always significantly
different from zero). It is well known that depreciation results in lower NOI, and not so much in higher
cap rates (Bokhari & Geltner, 2018; Geltner & van de Minne, 2017). As such, most depreciation is
“captured” by the NOI variable.
The year 2001 is the omitted category, and therefore, the reference year. The estimates of the year
fixed effects for the standard, hybrid, and RE models are always smaller than that of the other models
(especially compared to the SRW models). This is explained by the differences in the estimate for NOI
per square foot, which considerably differ between models. Given that NOI also “captures” changes in
the macroeconomic environment, this is expected. In other words, models with a high estimate for NOI
per square foot (like the standard model) will result in less variation in the time fixed effects and vice
versa. Note that the crisis and subsequent recovery are still clearly visible in all models. However, the
timing differs. The trough of the time fixed effects is in 2010 for the standard, hybrid, RE, and Besag
models. The trough is a full year earlier for the other models.9
The residual standard error is highest in the standard model, 𝜎𝜖= 0.19. In other words, the model-fit
is quite low. The standard error reduces to 0.18 in the Hybrid model, and is around 0.13 for the other
models. The standard error of the property random effects 𝜎𝜙is 0.15 in the RE and Besag models,
and it reduces to 0.04 in the SRW models. The standard errors of the spatial property effects 𝜎𝜃in
the Besag and SRW are difficult to compare, because the underlying models are different. Note that in
the SRW models, 𝜎𝜃is much lower in the RE model than in the Hybrid model. The Moran 𝐼statistic
suggests that there is some spatial autocorrelation left in the residuals only for the nonspatial models:
The standard, hybrid, and RE model. The WAIC10 of the Hybrid model is actually higher compared to
the standard model, meaning worse model fit. The WAIC for the RE does improve considerably over
the standard model with 820 points. The models including both property and spatial random have the
lowest WAIC. The best performing model is SRW(RE).
4.1.2
Heemstede
Compared to row houses, detached houses are valued the highest, followed by semidetached and corner
houses. Compared to poorly maintained houses, average and good maintained houses sell at a premium
of 14% and 23%, respectively. The estimated premium for a yard sits at 2% on average, however, is
statistically insignificant different from zero for most of our models. The coefficient for size varies
between .69 and .90, depending on the specification, indicating that prices increase less than propor-
tional to property size. Age has a positive coefficient and the square of age a negative coefficient. In
other words, older houses have higher values. An 80 years old house—built in the thirties—has a 16%
premium compared to a new house. This has most likely to do with vintage effects (see, e.g., Coulson
& McMillen, 2008; Francke & van de Minne, 2017b; Wilhelmsson, 2008), combined with the fact that
we hold constant for physical deterioration by controlling for maintenance. Interestingly, the effect of
age on house prices is statistically insignificant for the SRW models (however, the age squared term is
significant).
9Although it should be noted that the different index levels do not differ from each other significantly between 2009 and 2010.
This is not shown here, but is available upon request.
10The Watanabe–Akaike or widely applicable information criterion (WAIC, Watanabe, 2010) is based on the series expansion
of LOO cross-validation. WAIC can be viewed as an improvement of the deviance information criterion (DIC, Spiegelhalter,
Best, Carlin, & van der Linde, 2002). Lower values indicate a better model-fit.


===== PAGE 17 (score 15, numeric-paren 0, table-words 3, var-words 2) =====

1330
FRANCKE AND VAN DE MINNE
TA B L E 5
LOO cross-validation
Besag
SRW
Standard
Hybrid
RE
Hybrid
RE
Hybrid
RE
Los Angeles
Mean
0.000
−0.005
0.001
−0.001
0.000
−0.001
0.000
|Mean|
0.146
0.146
0.140
0.111
0.108
0.111
0.109
Standard deviation
0.191
0.190
0.184
0.150
0.147
0.148
0.146
Minimum
−0.777
−0.769
−0.759
−0.814
−0.830
−0.850
−0.846
Maximum
1.022
1.005
1.003
0.747
0.755
0.772
0.775
Heemstede
Mean
0.000
0.004
−0.001
0.001
0.001
0.004
−0.001
|Mean|
0.137
0.137
0.133
0.108
0.105
0.109
0.106
Standard deviation
0.172
0.172
0.164
0.173
0.130
0.136
0.133
Minimum
−0.656
−0.639
−0.663
−0.719
−0.547
−0.634
−0.653
Maximum
0.463
0.462
0.467
0.491
0.457
0.418
0.426
Note. The table provides summary statistics of the leave-one-out residuals. The model specifications are provided in Table 2 .
Since we have no variables that move with the economic cycle (like NOI) in Heemstede, the coef-
ficients of the time dummy variables can be interpreted as a log price index. Between 2001 and 2008,
prices increased by 32%–34%. Subsequently, prices dropped between 2008 and 2013 by 13%–16%.
Note that the crisis took relatively long in the Netherlands. From 2013 to 2017, prices increased by
33%–36%. The difference between the models is negligible.
The Moran 𝐼statistic suggests that there is some spatial autocorrelation left in the residuals only for
the standard model. The WAIC of the hybrid model is almost similar to the standard model. The RE
model performs better, the WAIC of the RE model is 1,261 points lower compared to the hybrid model.
The models including both property random and spatial effects have the lowest WAIC. The WAIC of
the best performing model, Besag(RE), is 643 points lower compared to the RE model.
4.2
Leave-one-out cross-validation
Table 5 provides the results of the LOO cross-validation, the upper part for Los Angeles, and the
lower part for Heemstede. In general, the out-of-sample model fit is slightly better for single-family in
Heemstede compared to multifamily housing in Los Angeles.
In Los Angeles, the standard deviation of the LOO residuals is similar for the standard and hybrid
models, about 0.190. In the RE model, the standard deviation is 0.184, which is still not a big improve-
ment over the standard model. The main reason for this small reduction is the relative small portion
of repeat sales. Adding spatial random effects reduces the standard deviation considerably though, to
0.146 in the best performing model SRW(RE), a reduction of 24% (23%) compared to the standard
(hybrid) model. The spatial models including property random effects perform better than the hybrid
spatial models, although the differences seem small.
In Heemstede, the standard deviation of the LOO residuals is similar for the standard and hybrid
model, both 0.172. In the RE model, the standard deviation is 0.164, a small reduction of 4.7% com-
pared to the standard model. Adding spatial random effects reduces the standard deviation even more,
to 0.130 in the best performing model Besag(RE), a reduction of 24% compared to the standard and
hybrid model. The spatial models including property random effects perform better than the hybrid
spatial models.


===== PAGE 18 (score 11, numeric-paren 0, table-words 2, var-words 1) =====

FRANCKE AND VAN DE MINNE
1331
TA B L E 6
Absolute mean of LOO residuals as a function of the number of sales per property
# Sales
Besag
SRW
per property
Standard
Hybrid
RE
Hybrid
RE
Hybrid
RE
Prop.
Los Angeles
1
0.148
0.149
0.149
0.115
0.112
0.115
0.114
1,643
2
0.138
0.141
0.115
0.098
0.096
0.098
0.095
261
3
0.139
0.121
0.112
0.112
0.101
0.110
0.102
30
Total
0.146
0.146
0.140
0.111
0.108
0.111
0.109
2,263
Heemstede
1
0.139
0.139
0.139
0.112
0.108
0.112
0.110
1,703
2
0.129
0.130
0.102
0.093
0.090
0.094
0.091
644
3
0.120
0.100
0.091
0.088
0.087
0.088
0.083
117
Total
0.137
0.137
0.133
0.108
0.105
0.109
0.106
2,468
Note. The model specifications are provided in Table 2.
The best performing models measured by the standard deviation of the LOO residuals coincide with
the best ones measured by the WAIC criterion. Unlike LOO statistics, one cannot compare WAICs over
different data sets.
Table 6 provides the absolute mean of the LOO residuals as a function of the number of sales per
property (the first column). The final column gives the corresponding number of properties. Note that
when the number of sales per property is 𝑛, in the leave-on-out analysis 𝑛−1 sales of the property
have been used to estimate the model.
In the standard HPM, the-out-of-sample model fit increases when the number of sales per property
increases, although the gain is relatively small. In Los Angeles, it goes down from 0.148, when having
only one sale, to 0.139, when having three sales per property (−6.1%), and in Heemstede from 0.139,
when having one sale, to 0.120, when having three sales per property (−13.8%).
Note that the reduction is much higher for RE models. In Los Angeles, it goes down from 0.149,
when having only one sale, to 0.112, when having three sales per property (−24.8%), and in Heemstede
from 0.139, when having one sale, to 0.091, when having three sales per property (−35.0%). The prop-
erty random effects HPM clearly takes advantage of the fact that some properties transact more than
once.
Note that the hybrid model performs less than the RE model, in particular when the number of sales
per property is two. In fact, the hybrid model performs equal to the standard model with just two sales.
This is expected, given that the hybrid model can only get property-level estimates if the number of
sales is three or more (because one observation is lost in the LOO analysis).
When having only one sale per property (zero during the LOO analysis), the property random effects
HPM including spatial effects performs better than the model excluding the spatial effects. The dif-
ference in performance becomes smaller when the number of sales per property increases. Then the
property random effects pick up most of the unobserved heterogeneity, and there is almost no additional
gain from the spatial random effects.
4.3
Spatial effects
Figure 1 gives the TSP routes for both Los Angeles and Heemstede. The (random) starting point is
given in the figure as well. In both cases, the TSP route goes clockwise. Figure 2 provides the size
of the spatial effects 𝜃along this route for the Besag and SRW models. Higher (lower) values for the
spatial random effect means that the property has a higher (lower) value than can be explained by just