spb/wp21_uqo

1# RECORD c6b33e6b839caa1157a283c6f9a1e3350edd05582# TITLE: Modeling unobserved heterogeneity in hedonic price models3# YEAR: 2020.0  VENUE: Real Estate Economics  DOI: 10.1111/1540-6229.123204# AUTHORS: Marc K. Francke; A. van de Minne5# PAGES: 26  CANDIDATE TABLE PAGES (1-based): [11, 14, 15, 17]6# 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.789===== PAGE 1 (score 4, numeric-paren 0, table-words 0, var-words 4) =====1011UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)12UvA-DARE (Digital Academic Repository)13Modeling unobserved heterogeneity in hedonic price models14Francke, M.; van de Minne, A.15DOI1610.1111/1540-6229.1232017Publication date18202119Document Version20Final published version21Published in22Real Estate Economics23License24CC BY-NC-ND25Link to publication26Citation for published version (APA):27Francke, M., & van de Minne, A. (2021). Modeling unobserved heterogeneity in hedonic price28models. Real Estate Economics, 49(4), 1315-1339. https://doi.org/10.1111/1540-6229.1232029General rights30It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)31and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open32content license (like Creative Commons).33Disclaimer/Complaints regulations34If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please35let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material36inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter37to: Library of the University of Amsterdam, Secretariat, P.O. Box 19185, 1000 GD Amsterdam, The Netherlands.38You will be contacted as soon as possible.39Download date:21 Oct 2026404142===== PAGE 11 (score 17, numeric-paren 0, table-words 3, var-words 9) =====4344132445FRANCKE AND VAN DE MINNE46TA B L E 147Descriptive statistics48mean49sd50min51max52Los Angeles53Sales price ($)a546,389,494556,460,729561,550,0005764,250,00058Net Operating Income ($)59325,22360353,2706167,200623,600,00063Age (Years)64456521662679768Size (SqFt)6931,7187031,972715,96472271,75773Years between sales744.61752.87760.177712.0078Garden (R)790.8780081182Mid/Highrise830.1384085186Observations872,26388Unique properties891,93690Heemstede91Sales price (€ )a92484,61293189,10694200,000951,195,00096Age (Years)976598229915100106101Size (SqMt)1021511033910482105288106Years between sales1077.071083.671090.4211016.25111Maintenance [bad] (R)1120.1811301141115Maintenance [average]1160.5911701181119Maintenance [good]1200.2412101221123Row house (R)1240.4412501261127Semidetached (1)1280.0312901301131Semidetached (2)1320.2313301341135Corner home1360.2513701381139Detached1400.0514101421143Yard (yes)1440.9414501461147Observations1482,468149Unique properties1502,065151Note. R gives the reference categories in our model. Semidetached (1) are the properties that are connected via a garage, and semidetached152(2) are the properties that are connected wall-to-wall.153aEstimates for Moran’s 𝐼(sales prices) for Los Angeles and Heemstede are, respectively, +0.04 and +0.23.1543155DATA AND DESCRIPTIVE STATISTICS156We use two different data sources, commercial multifamily real estate (income generating properties)157in the city of Los Angeles and single-family housing (owner-occupied) in Heemstede, a city relatively158close to Amsterdam in the Netherlands.159The first database is provided by Real Capital Analytics, and captures approximately 90% of all160commercial property transactions in the United States over $2.5 million. The database contains 2,263161prefiltered transactions, of which 1,936 are unique properties, in the period 2001–2017. The annual162number of transactions is 140 transactions on average. We observe the net operating income (NOI),163property subtype (garden versus mid/highrise), the age and size of the structure (in square feet), latitude164and longitude, and the transaction price. The upper panel of Table 1 provides some descriptive statistics.165166167===== PAGE 12 (score 11, numeric-paren 0, table-words 2, var-words 7) =====168169FRANCKE AND VAN DE MINNE1701325171TA B L E 2172Overview of model specifications173Model174Specification175Spatial effect (𝜽𝒑)176Standard177Equation (1): 𝑦𝑝= 𝑥∗178𝑝𝛽+ 𝜖𝑝179Hybrid180Equation (6):181(182𝑦𝑆183𝑝184Δ𝑦𝑅185𝑝186)187=188(189𝑥∗𝑆190𝑝191Δ𝑥∗𝑅192𝑝193)194𝛽+195(196𝜖𝑆197𝑝198Δ𝜖𝑅199𝑝200)201, ∀𝑝∶202(𝑛𝑝= 1203𝑛𝑝> 1204)205RE206Equation (2): 𝑦𝑝= 𝑥∗207𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝜖𝑝208Besag(Hybrid)209(210𝑦𝑆211𝑝212Δ𝑦𝑅213𝑝214)215=216(217𝑥𝑆218𝑝219Δ𝑥𝑅220𝑝221)222𝛽+223(𝜃𝑝2240225)226+227(228𝜖𝑆229𝑝230Δ𝜖𝑅231𝑝232)233, ∀𝑝∶234(𝑛𝑝= 1235𝑛𝑝> 1236)237Equation (9)238Besag(RE)239Equation (7): 𝑦𝑝= 𝑥𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝑗𝑛𝑝𝜃𝑝+ 𝜖𝑝240Equation (9)241SRW(Hybrid)242(243𝑦𝑆244𝑝245Δ𝑦𝑅246𝑝247)248=249(250𝑥𝑆251𝑝252Δ𝑥𝑅253𝑝254)255𝛽+256(𝜃𝑝2570258)259+260(261𝜖𝑆262𝑝263Δ𝜖𝑅264𝑝265)266, ∀𝑝∶267(𝑛𝑝= 1268𝑛𝑝> 1269)270Equation (10)271SRW (RE)272Equation (7): 𝑦𝑝= 𝑥𝑝𝛽+ 𝑗𝑛𝑝𝜙𝑝+ 𝑗𝑛𝑝𝜃𝑝+ 𝜖𝑝273Equation (10)274Note. The models including property random effects—RE, Besag(RE) and SRW(RE)—all have the same property random effect speci-275fication given by 𝜙∼(0, 𝜎2276𝜙𝐼𝑃), see Equation (3).277The spatial effect from the Besag model is provided by 𝜃𝑝|𝜃−𝑝∼278( ∑279𝑞∈𝜕𝑝𝑤𝑝,𝑞𝜃𝑞280𝑑+𝑚𝑝281,282𝜎2283𝜃284𝑑+𝑚𝑝285)286, see Equation (9).287The spatial effect from the spatial random walk model is provided by 𝜃(𝑝) ∼(𝜃(𝑝−1), 𝜎2288𝜃), see Equation (10).289The matrix 𝑥consists of property characteristics and time fixed effects.290The models Standard, Hybrid, and RE have also location fixed effects. All other models do not. The location fixed effects have been291added to 𝑥, denoted by 𝑥∗.292In the models Besag(Hybrid) and SRW(Hybrid), spatial random effects have been added to the one only sales in the hybrid specification293(Equation (6)).294The average transaction price is about $6.4 million, the average size is about 32,000 square feet, and295the average age is 45 years. Most properties are designated garden.296The second database is provided by the Dutch Association of Real Estate Brokers and Real Estate297Experts (NVM), the largest brokers organization in the Netherlands. About 70% of all real estate bro-298kers in the Netherlands are affiliated to the NVM. The database contains residential real estate 2,262299transactions, of which 2,065 are unique properties, in the period 2001 to 2017 for the Dutch city of300Heemstede. The majority of the transactions is one only sales (69%). The annual number of trans-301actions is approximately 145 transactions on average. We observe the property subtype (row houses,302corner house, two types of semidetached homes and detached), the age and size of the structure, the303maintenance level (three groups from bad to good), the presence of a yard, latitude and longitude, and304the transaction price. The lower panel of Table 1 provides some descriptive statistics. The average305transaction price is about €485,000, the average size 151 m2 (1,625 square feet), and the average age306is 65 years. The largest numbers of properties are row houses (44%). The NVM distinguishes between307two types of semidetached homes: (a) two properties are connected via a garage and (b) two properties308are connected wall-to-wall. Most of the semidetached properties fall in the second category, 24% of the309observations. More than half of the properties have an average maintenance level at the time of listing,310compared to 18% badly maintained and 23% well maintained.6 Almost all properties have a yard, and311in only 6% of the transactions, this is not the case.3126See Francke and van de Minne (2017b) for a discussion on how the maintenance data in the NVM data are compiled.313314315===== PAGE 14 (score 169, numeric-paren 142, table-words 4, var-words 2) =====316317FRANCKE AND VAN DE MINNE3181327319TA B L E 3320Posterior means and in-sample fit statistics for Los Angeles321Besag322SRW323Standard324Hybrid325RE326Hybrid327RE328Hybrid329RE330(Intercept)3313.940***3324.091***3334.053***3344.135***3354.096***3364.029***3374.033***338ln Size3390.932***3400.924***3410.928***3420.945***3430.952***3440.956***3450.961***346ln ( 𝑁𝑂𝐼347Size )3480.735***3490.708***3500.700***3510.613***3520.609***3530.603***3540.602***355Age356−0.001***357−0.001***358−0.001***359−0.003***360−0.004***361−0.004***362−0.004***363Age23640.000***3650.0003660.0003670.0003680.000*3690.0003700.000***371Mid/Highrise3720.0183730.0143740.019375−0.003***376−0.002***377−0.004***378−0.005***37920023800.0603810.0143820.0323830.0543840.0483850.089*3860.094**38720033880.170***3890.153*3900.140*3910.192***3920.184***3930.216***3940.202***39520043960.275***3970.272***3980.264***3990.308***4000.293***4010.334***4020.326***40320054040.318***4050.328***4060.336***4070.409***4080.397***4090.439***4100.437***41120064120.344***4130.350***4140.362***4150.433***4160.421***4170.464***4180.461***41920074200.309***4210.321***4220.333***4230.416***4240.398***4250.446***4260.437***42720084280.318***4290.315***4300.329***4310.410***4320.402***4330.444***4340.440***43520094360.217***4370.220***4380.213***4390.287***4400.264***4410.312***4420.297***44320104440.192***4450.188***4460.203***4470.281***4480.273***4490.311***4500.309***45120114520.253***4530.247***4540.258***4550.324***4560.317***4570.365***4580.359***45920124600.287***4610.292***4620.294***4630.364***4640.356***4650.398***4660.392***46720134680.320***4690.324***4700.330***4710.454***4720.450***4730.478***4740.475***47520144760.411***4770.428***4780.439***4790.558***4800.547***4810.592***4820.585***48320154840.528***4850.545***4860.554***4870.664***4880.656***4890.693***4900.691***49120164920.628***4930.621***4940.641***4950.762***4960.762***4970.791***4980.796***49920175000.622***5010.639***5020.648***5030.775***5040.767***5050.814***5060.811***507Location508FE509FE510FE511𝜎𝜖5120.1905130.1845140.1245150.1265160.1235170.1375180.120519𝜎𝜙5200.1465210.1495220.040523𝜎𝜃5240.0335250.0295260.0115270.064528𝜎𝜃∕√𝑑+ ̄𝑤𝑝+5290.0105300.008531Moran 𝐼5320.1125330.0795340.0845350.0005360.003537−0.003538−0.002539DIC540−1,072.1541−919.8542−1,850.6543−2,132.5544−2,412.0545−2,035.4546−2,502.4547WAIC548−1,070.6549−1,031.9550−1,891.2551−2,220.0552−2,443.5553−2,074.4554−2,459.2555Note. The model specifications are provided in Table 2. The omitted dummy variable is garden apartment (for property subtype) and5562001 (for time of sale). NOI stands for net operating income.557Moran’s 𝐼is a measure for spatial autocorrelation. DIC denotes deviance information criterion, and WAIC Watanabe information558criterion.559***means the parameter is statistically significantly different from 0 at the 1% level, **at the 5% level, and *at the 10% level.560estate studies find this “law of diminishing returns” (Bokhari & Geltner, 2018). The coefficient for561Mid/Highrise properties in Los Angeles is positive but insignificant for the standard, hybrid, and RE562model. For the Besag and SRW models, the coefficient becomes statistically significant and negative,563which might indicate an interaction between property type and location, which the location fixed effects564in the standard, hybrid, and RE model do not pick up. Also, ceteris paribus, one would expect that565lowrise housing would be more popular compared to highrise housing. Age has a negative coefficient566567568===== PAGE 15 (score 205, numeric-paren 178, table-words 1, var-words 4) =====5695701328571FRANCKE AND VAN DE MINNE572TA B L E 4573Posterior means and in-sample fit statistics for Heemstede574Besag575SRW576Standard577Hybrid578RE579Hybrid580RE581Hybrid582RE583(Intercept)5848.010***5858.060***5868.135***5878.836***5888.891***5899.127***5909.095***591ln Size5920.898***5930.889***5940.876***5950.723***5960.714***5970.685***5980.694***599Age6000.004***6010.002**6020.003***6030.003***6040.003***6050.0016060.001607Age2608−0.000***609−0.000***610−0.000***611−0.000***612−0.000***613−0.000***614−0.000***615Semidetached (1)6160.148***6170.162***6180.137***6190.083***6200.074***6210.088***6220.073***623Semidetached (2)6240.199***6250.214***6260.205***6270.167***6280.152***6290.174***6300.163***631Corner Home6320.094***6330.117***6340.104***6350.100***6360.087***6370.109***6380.094***639Detached6400.337***6410.357***6420.347***6430.307***6440.292***6450.301***6460.286***647Maintenance [average]6480.128***6490.127***6500.133***6510.129***6520.130***6530.123***6540.128***655Maintenance [good]6560.216***6570.209***6580.208***6590.204***6600.208***6610.206***6620.207***663Yard6640.0256650.031*6660.0256670.0136680.0146690.0156700.01667120026720.047**6730.041*6740.036*6750.031*6760.034**6770.032**6780.031**67920036800.039*6810.0366820.0276830.0266840.029*6850.038*6860.038**68720046880.078***6890.084***6900.068***6910.071***6920.071***6930.075***6940.068***69520056960.141***6970.133***6980.134***6990.128***7000.130***7010.140***7020.139***70320067040.171***7050.180***7060.170***7070.168***7080.171***7090.185***7100.181***71120077120.261***7130.261***7140.252***7150.247***7160.253***7170.270***7180.272***71920087200.280***7210.290***7220.278***7230.284***7240.283***7250.295***7260.286***72720097280.244***7290.232***7300.232***7310.225***7320.227***7330.239***7340.241***73520107360.224***7370.213***7380.216***7390.204***7400.213***7410.228***7420.228***74320117440.224***7450.212***7460.223***7470.216***7480.224***7490.228***7500.238***75120127520.135***7530.132***7540.133***7550.115***7560.116***7570.132***7580.133***75920137600.135***7610.136***7620.135***7630.118***7640.113***7650.123***7660.121***76720147680.176***7690.173***7700.185***7710.158***7720.161***7730.177***7740.184***77520157760.216***7770.207***7780.230***7790.221***7800.229***7810.232***7820.241***78320167840.344***7850.340***7860.354***7870.334***7880.343***7890.358***7900.365***79120177920.420***7930.415***7940.434***7950.403***7960.408***7970.420***7980.427***799Location800FE801FE802FE803𝜎𝜖8040.1718050.1648060.1028070.0968080.0958090.1168100.098811𝜎𝜙8120.1398130.0118140.070815𝜎𝜃8160.1198170.1188180.0578190.053820𝜎𝜃∕√𝑑+ ̄𝑤𝑝+8210.0498220.048823Moran’s 𝐼8240.0898250.0378260.0198270.005828−0.021829−0.017830−0.011831DIC832−1,671.4833−1,517.1834−2,835.4835−2,581.9836−3,538.2837−2,813.2838−3,395.0839WAIC840−1,670.3841−1,664.6842−2,925.9843−2,700.3844−3,569.0845−2,834.2846−3,384.3847Note. The model specifications are provided in Table 2. The omitted dummy variables are row house (for property subtype), maintenance848[bad], and having no yard. Semidetached (1) are houses connected by a garage, and semidetached (2) are houses that are connected wall-849to-wall and 2001 (for time of sale).850Moran’s 𝐼is a measure for spatial autocorrelation. DIC denotes deviance information criterion, and WAIC Watanabe information851criterion.852***means the parameter is statistically significantly different from 0 at the 1% level, **at the 5% level, and *at the 10% level.853854855===== PAGE 16 (score 23, numeric-paren 0, table-words 8, var-words 7) =====856857FRANCKE AND VAN DE MINNE8581329859and the square of age a positive coefficient. This confirms expectations, as depreciation is fastest when860a property is young, see Bokhari and Geltner (2018). Here, the estimated coefficient is also lower861compared to other studies because of the inclusion of NOI (and age squared is not always significantly862different from zero). It is well known that depreciation results in lower NOI, and not so much in higher863cap rates (Bokhari & Geltner, 2018; Geltner & van de Minne, 2017). As such, most depreciation is864“captured” by the NOI variable.865The year 2001 is the omitted category, and therefore, the reference year. The estimates of the year866fixed effects for the standard, hybrid, and RE models are always smaller than that of the other models867(especially compared to the SRW models). This is explained by the differences in the estimate for NOI868per square foot, which considerably differ between models. Given that NOI also “captures” changes in869the macroeconomic environment, this is expected. In other words, models with a high estimate for NOI870per square foot (like the standard model) will result in less variation in the time fixed effects and vice871versa. Note that the crisis and subsequent recovery are still clearly visible in all models. However, the872timing differs. The trough of the time fixed effects is in 2010 for the standard, hybrid, RE, and Besag873models. The trough is a full year earlier for the other models.9874The residual standard error is highest in the standard model, 𝜎𝜖= 0.19. In other words, the model-fit875is quite low. The standard error reduces to 0.18 in the Hybrid model, and is around 0.13 for the other876models. The standard error of the property random effects 𝜎𝜙is 0.15 in the RE and Besag models,877and it reduces to 0.04 in the SRW models. The standard errors of the spatial property effects 𝜎𝜃in878the Besag and SRW are difficult to compare, because the underlying models are different. Note that in879the SRW models, 𝜎𝜃is much lower in the RE model than in the Hybrid model. The Moran 𝐼statistic880suggests that there is some spatial autocorrelation left in the residuals only for the nonspatial models:881The standard, hybrid, and RE model. The WAIC10 of the Hybrid model is actually higher compared to882the standard model, meaning worse model fit. The WAIC for the RE does improve considerably over883the standard model with 820 points. The models including both property and spatial random have the884lowest WAIC. The best performing model is SRW(RE).8854.1.2886Heemstede887Compared to row houses, detached houses are valued the highest, followed by semidetached and corner888houses. Compared to poorly maintained houses, average and good maintained houses sell at a premium889of 14% and 23%, respectively. The estimated premium for a yard sits at 2% on average, however, is890statistically insignificant different from zero for most of our models. The coefficient for size varies891between .69 and .90, depending on the specification, indicating that prices increase less than propor-892tional to property size. Age has a positive coefficient and the square of age a negative coefficient. In893other words, older houses have higher values. An 80 years old house—built in the thirties—has a 16%894premium compared to a new house. This has most likely to do with vintage effects (see, e.g., Coulson895& McMillen, 2008; Francke & van de Minne, 2017b; Wilhelmsson, 2008), combined with the fact that896we hold constant for physical deterioration by controlling for maintenance. Interestingly, the effect of897age on house prices is statistically insignificant for the SRW models (however, the age squared term is898significant).8999Although it should be noted that the different index levels do not differ from each other significantly between 2009 and 2010.900This is not shown here, but is available upon request.90110The Watanabe–Akaike or widely applicable information criterion (WAIC, Watanabe, 2010) is based on the series expansion902of LOO cross-validation. WAIC can be viewed as an improvement of the deviance information criterion (DIC, Spiegelhalter,903Best, Carlin, & van der Linde, 2002). Lower values indicate a better model-fit.904905906===== PAGE 17 (score 15, numeric-paren 0, table-words 3, var-words 2) =====9079081330909FRANCKE AND VAN DE MINNE910TA B L E 5911LOO cross-validation912Besag913SRW914Standard915Hybrid916RE917Hybrid918RE919Hybrid920RE921Los Angeles922Mean9230.000924−0.0059250.001926−0.0019270.000928−0.0019290.000930|Mean|9310.1469320.1469330.1409340.1119350.1089360.1119370.109938Standard deviation9390.1919400.1909410.1849420.1509430.1479440.1489450.146946Minimum947−0.777948−0.769949−0.759950−0.814951−0.830952−0.850953−0.846954Maximum9551.0229561.0059571.0039580.7479590.7559600.7729610.775962Heemstede963Mean9640.0009650.004966−0.0019670.0019680.0019690.004970−0.001971|Mean|9720.1379730.1379740.1339750.1089760.1059770.1099780.106979Standard deviation9800.1729810.1729820.1649830.1739840.1309850.1369860.133987Minimum988−0.656989−0.639990−0.663991−0.719992−0.547993−0.634994−0.653995Maximum9960.4639970.4629980.4679990.49110000.45710010.41810020.4261003Note. The table provides summary statistics of the leave-one-out residuals. The model specifications are provided in Table 2 .1004Since we have no variables that move with the economic cycle (like NOI) in Heemstede, the coef-1005ficients of the time dummy variables can be interpreted as a log price index. Between 2001 and 2008,1006prices increased by 32%–34%. Subsequently, prices dropped between 2008 and 2013 by 13%–16%.1007Note that the crisis took relatively long in the Netherlands. From 2013 to 2017, prices increased by100833%–36%. The difference between the models is negligible.1009The Moran 𝐼statistic suggests that there is some spatial autocorrelation left in the residuals only for1010the standard model. The WAIC of the hybrid model is almost similar to the standard model. The RE1011model performs better, the WAIC of the RE model is 1,261 points lower compared to the hybrid model.1012The models including both property random and spatial effects have the lowest WAIC. The WAIC of1013the best performing model, Besag(RE), is 643 points lower compared to the RE model.10144.21015Leave-one-out cross-validation1016Table 5 provides the results of the LOO cross-validation, the upper part for Los Angeles, and the1017lower part for Heemstede. In general, the out-of-sample model fit is slightly better for single-family in1018Heemstede compared to multifamily housing in Los Angeles.1019In Los Angeles, the standard deviation of the LOO residuals is similar for the standard and hybrid1020models, about 0.190. In the RE model, the standard deviation is 0.184, which is still not a big improve-1021ment over the standard model. The main reason for this small reduction is the relative small portion1022of repeat sales. Adding spatial random effects reduces the standard deviation considerably though, to10230.146 in the best performing model SRW(RE), a reduction of 24% (23%) compared to the standard1024(hybrid) model. The spatial models including property random effects perform better than the hybrid1025spatial models, although the differences seem small.1026In Heemstede, the standard deviation of the LOO residuals is similar for the standard and hybrid1027model, both 0.172. In the RE model, the standard deviation is 0.164, a small reduction of 4.7% com-1028pared to the standard model. Adding spatial random effects reduces the standard deviation even more,1029to 0.130 in the best performing model Besag(RE), a reduction of 24% compared to the standard and1030hybrid model. The spatial models including property random effects perform better than the hybrid1031spatial models.103210331034===== PAGE 18 (score 11, numeric-paren 0, table-words 2, var-words 1) =====10351036FRANCKE AND VAN DE MINNE103713311038TA B L E 61039Absolute mean of LOO residuals as a function of the number of sales per property1040# Sales1041Besag1042SRW1043per property1044Standard1045Hybrid1046RE1047Hybrid1048RE1049Hybrid1050RE1051Prop.1052Los Angeles1053110540.14810550.14910560.14910570.11510580.11210590.11510600.11410611,6431062210630.13810640.14110650.11510660.09810670.09610680.09810690.09510702611071310720.13910730.12110740.11210750.11210760.10110770.11010780.1021079301080Total10810.14610820.14610830.14010840.11110850.10810860.11110870.10910882,2631089Heemstede1090110910.13910920.13910930.13910940.11210950.10810960.11210970.11010981,7031099211000.12911010.13011020.10211030.09311040.09011050.09411060.09111076441108311090.12011100.10011110.09111120.08811130.08711140.08811150.08311161171117Total11180.13711190.13711200.13311210.10811220.10511230.10911240.10611252,4681126Note. The model specifications are provided in Table 2.1127The best performing models measured by the standard deviation of the LOO residuals coincide with1128the best ones measured by the WAIC criterion. Unlike LOO statistics, one cannot compare WAICs over1129different data sets.1130Table 6 provides the absolute mean of the LOO residuals as a function of the number of sales per1131property (the first column). The final column gives the corresponding number of properties. Note that1132when the number of sales per property is 𝑛, in the leave-on-out analysis 𝑛−1 sales of the property1133have been used to estimate the model.1134In the standard HPM, the-out-of-sample model fit increases when the number of sales per property1135increases, although the gain is relatively small. In Los Angeles, it goes down from 0.148, when having1136only one sale, to 0.139, when having three sales per property (−6.1%), and in Heemstede from 0.139,1137when having one sale, to 0.120, when having three sales per property (−13.8%).1138Note that the reduction is much higher for RE models. In Los Angeles, it goes down from 0.149,1139when having only one sale, to 0.112, when having three sales per property (−24.8%), and in Heemstede1140from 0.139, when having one sale, to 0.091, when having three sales per property (−35.0%). The prop-1141erty random effects HPM clearly takes advantage of the fact that some properties transact more than1142once.1143Note that the hybrid model performs less than the RE model, in particular when the number of sales1144per property is two. In fact, the hybrid model performs equal to the standard model with just two sales.1145This is expected, given that the hybrid model can only get property-level estimates if the number of1146sales is three or more (because one observation is lost in the LOO analysis).1147When having only one sale per property (zero during the LOO analysis), the property random effects1148HPM including spatial effects performs better than the model excluding the spatial effects. The dif-1149ference in performance becomes smaller when the number of sales per property increases. Then the1150property random effects pick up most of the unobserved heterogeneity, and there is almost no additional1151gain from the spatial random effects.11524.31153Spatial effects1154Figure 1 gives the TSP routes for both Los Angeles and Heemstede. The (random) starting point is1155given in the figure as well. In both cases, the TSP route goes clockwise. Figure 2 provides the size1156of the spatial effects 𝜃along this route for the Besag and SRW models. Higher (lower) values for the1157spatial random effect means that the property has a higher (lower) value than can be explained by just1158