1# RECORD 1cd473df8f393883694de4530b0d1c667382890c2# TITLE: Exploring a multilevel approach with spatial effects to model housing price in San José, Costa Rica3# YEAR: 2021.0 VENUE: Environment and Planning B Urban Analytics and City Science DOI: 10.1177/239980832110411224# AUTHORS: E. Pérez-Molina5# PAGES: 18 CANDIDATE TABLE PAGES (1-based): [9, 10]6# ABSTRACT: A multilevel model of the housing market for San José Metropolitan Region (Costa Rica) was developed, including spatial effects. The model is used to explore two main questions: the extent to which contextual (of the surroundings) and compositional (of the property itself) effects explain variation of housing prices and how does the relation between price and key covariates change with the introduction of multilevel effects. Hierarchical relations (lower level units nested into higher level) were modeled by specifying multilevel models with random intercepts and a conditional autoregressive term to include spatial effects from neighboring units at the higher level (districts). The random intercepts and conditional autoregressive models presented the best fit to the data. Variation at the higher level accounted for 16% of variance in the random intercepts model and 28% in the conditional autoregressive model. The sign and magnitude of regression coefficients proved remarkably stable across model specifications. Travel time to the city center, which presented a non-linear relation to price, was found to be the most important determinant. Multilevel and conditional autoregressive models constituted important improvements in modeling housing price, despite most of the variation still occurring at the lower level, by improving the overall model fit. They were capable of representing the regional structure and of reducing sampling bias in the data. However, the conditional autoregressive specification only represented a limited advance over the random intercepts formulation.789===== PAGE 1 (score 4, numeric-paren 0, table-words 0, var-words 4) =====1011Exploring a Multilevel12Approach with Spatial13Effects to Model14Housing Price in San15Jos´e, Costa Rica16Journal Title17XX(X):1–1818©The Author(s) 202019Reprints and permission:20sagepub.co.uk/journalsPermissions.nav21DOI: 10.1177/ToBeAssigned22www.sagepub.com/23SAGE24Eduardo P´erez-Molina125Abstract26A multilevel model of the housing market for San Jos´e Metropolitan Region (Costa27Rica) was developed, including spatial effects. Hierarchical relations (lower level units28nested into higher level) were modeled by specifying multilevel models with random29intercepts and a conditional autoregressive (CAR) term to include spatial effects from30neighboring units at the higher level (districts). The random intercepts and conditional31autoregressive models presented the best fit to the data. Variation at the higher level32accounted for 16% of variance in the random intercepts model and 28% in the CAR33model. The sign and magnitude of regression coefficients proved remarkably stable34across model specifications. Multilevel and CAR models represented an important35improvement in modeling housing price, despite most of the variation still occurring36at the lower level, by improving the overall model fit and expanding the interpretation37of model results. However, the CAR specification only represented a limited advance38over the random intercepts formulation.39Keywords40Multilevel, Conditional autoregressive model, Housing prices, San Jos´e-Costa Rica411Programa de Investigaci´on en Desarrollo Urbano Sostenible, Universidad de Costa Rica42Corresponding author:43Eduardo P´erez Molina, Universidad de Costa Rica Programa de Investigcio´on en Desarrollo Urbano44Sostenible, Bo Los Profesores, Calle B, No. 13, Mercedes, Montes de Oca, 115103, Costa Rica.45Email: eduardo.perezmolina@ucr.ac.cr46Prepared using sagej.cls [Version: 2017/01/17 v1.20]474849===== PAGE 9 (score 12, numeric-paren 0, table-words 3, var-words 4) =====5051P´erez-Molina52953Table 2. Model comparison criteria.54GaLM55MLRE56CARM57VPM58Mean Deviance599.363 · 10160−7.692 · 10161−5.542 · 101629.04563Deviance std. error642.002 · 10−2658.721 · 10−2667.37 · 10−1671.140 · 10−168pD691.002 · 101707.194 · 101715.683 · 101726.179 · 10173DIC741.037 · 10275−4.987761.415777.083 · 10178Mean VPC79–801.664 · 10−1812.838 · 10−1829.127 · 10−283VPC std. error84–851.827 · 10−4862.745 · 10−4872.832 · 10−488The VPC estimate for the VPM model refers only to the district level variation (for comparison purposes);89mean VPC for both levels in this model is 2.031 · 10−1 with a standard error of 2.145 · 10−4.904, that formalizes the municipal level variance) were also modeled with hyperpriors τu91equal to 1/σ292u and τs equal to 1/σ293s that follow uninformative Gamma distributions.94Models were estimated using WinBUGS (Spiegelhalter et al. 2003), called through95the R2BUGS package (Sturtz et al. 2005). Three chains of 400000 iterations each were96simulated with the first 25000 iterations discarded as burn-in. The resulting chains were97thinned by selecting one out of every 25 iterations to reduce autocorrelation problems.98The final posterior samples consisted of three chains of 15000 simulated instances each.99Convergence was evaluated through visual inspection of the MCMC trace plots for the100model parameters, autocorrelation plots, and the Gelman and Rubin diagnostic (these101plots are annexed in the supplemental material; for models with multilevel effects, six102randomly selected parameters are reported as illustration, although all parameters were103visually inspected for convergence and can be consulted in the full model output files104available online, see supplemental material).105Results106Model Estimates107The results for the estimated models are presented in tables 2 (model fit) and 3 (parameter108means and 95% posterior intervals), and figure 2 (patterns for systematic variation at the109higher level units).110The DIC of the four models is shown in figure 2. Two models (MLRE and CARM)111have substantially lower DIC values than the other two (GaLM and VPM). The two best112models do not differ markedly: in terms of fit alone, the higher order CAR structure did113not improve the model fit. Additionally, the two worst models (GaLM and VPM) in terms114of DIC also are broadly similar, which suggests the spatial multilevel structure may to a115large extent control for the spatial effects determined by the covariates.116In table 2, the variance partitioning coefficient (VPC) is also reported. This is the117fraction of variance ascribed to the higher (σ2118u, district) or lower (σ2119e) level units;120specifically, the VPC is defined as the sum of σ2121u and σ2122e into the higher level variance,123σ2124u. As can be seen in figure 2, the GaML model includes by construction all of the125variance in the lower level, because of the absence of a σ2126u term. More interestingly, the127Prepared using sagej.cls128129130===== PAGE 10 (score 24, numeric-paren 0, table-words 3, var-words 7) =====13113210133Journal Title XX(X)134Table 3. Regression model results for natural logarithm of housing price per m2 of living area135(mean, 2.5% and 97.5% percentiles).136Mean1372.5% perc.13897.5% perc.139Mean1402.5% perc.14197.5% perc.142GaML143MLRE144β11457.85 · 10−21466.33 · 10−21479.36 · 10−21487.20 · 10−21495.59 · 10−21508.80 · 10−2151β21526.13 · 10−11534.26 · 10−11548.01 · 10−11555.06 · 10−11562.64 · 10−11577.56 · 10−1158β3159−1.13 · 10−1160−1.48 · 10−1161−7.79 · 10−2162−9.45 · 10−2163−1.42 · 10−1164−4.91 · 10−2165β4166−1.11 · 10−2167−2.07 · 10−2168−1.47 · 10−3169−1.20 · 10−2170−2.24 · 10−2171−1.65 · 10−3172β51731.12 · 10−21744.63 · 10−31751.78 · 10−21768.40 · 10−31771.83 · 10−31781.50 · 10−2179γ11805.711815.431826.001835.931845.531856.34186γ21874.14 · 10−4188−3.36 · 10−31894.19 · 10−3190−6.36 · 10−5191−6.42 · 10−31926.41 · 10−3193γ31941.88 · 10−21959.34 · 10−31962.83 · 10−21972.03 · 10−21984.06 · 10−31993.67 · 10−2200γ4201−1.95 · 10−2202−3.46 · 10−2203−4.27 · 10−3204−2.59 · 10−2205−5.42 · 10−22062.14 · 10−3207σ2208u209–210–211–2121.11 · 10−22136.25 · 10−32141.76 · 10−2215σ2216e2176.31 · 10−22185.83 · 10−22196.81 · 10−22205.51 · 10−22215.07 · 10−22225.98 · 10−2223CARM224VPM225β0226–227–228–2296.922306.882316.96232β12336.96 · 10−22345.38 · 10−22358.54 · 10−2236–237–238–239β22403.83 · 10−12411.18 · 10−12426.47 · 10−1243–244–245–246β3247−7.23 · 10−2248−1.25 · 10−1249−1.98 · 10−2250–251–252–253β4254−1.13 · 10−2255−2.18 · 10−2256−7.23 · 10−4257–258–259–260β52618.06 · 10−32621.60 · 10−32631.45 · 10−2264–265–266–267γ12686.072695.672706.48271–272–273–274γ22758.59 · 10−4276−4.55 · 10−32776.22 · 10−3278–279–280–281γ32821.08 · 10−2283−4.37 · 10−32842.61 · 10−2285–286–287–288γ4289−1.17 · 10−2290−3.49 · 10−22911.14 · 10−2292–293–294–295σ2296s297–298–299–3008.438 · 10−33012.163 · 10−33021.837 · 10−2303σ2304u3052.28 · 10−23061.11 · 10−23074.03 · 10−23086.798 · 10−33091.863 · 10−33101.430 · 10−2311σ2312e3135.60 · 10−23145.16 · 10−23156.08 · 10−23165.899 · 10−23175.427 · 10−23186.410 · 10−2319variance of the random intercept models (with and without CAR) is relatively small but320not unimportant, with a VPC for MLRE of 0.16 and a larger value of 0.28 for CARM321(almost twice as much); if only the district levels are included, as in the VPM model,3220.09 of the total variance can be explained –a figure that doubles when considering also323a second, higher level of municipalities.324The VPC estimates are in general lower than the equivalents estimated by Bivand et al.325(2017) when replicating models by Osland et al. (2016) (a VPC in the order of 0.58 for326Stavanger, Norway, although with accessibility to the CBD being a higher level covariate)327and Dong et al. (2015) (a VPC of 0.38 for Beijing), but the difference is reasonable and328should be ascribed to the complexity of a polycentric region and to a very local effect of329spatial dependence, likely present in San Jos´e and not incorporated into the models.330Estimated coefficients for the four proposed models are shown in table 3. Overall,331travel time to San Jos´e is the most important determinant (about an order of magnitude332greater than other coefficients); the 95% posterior intervals of both β2 and β3 exclude3330.00 for all models and are consistent in their signs (positive for β2 and negative for β3,334see table 3). Taken jointly, these two coefficients suggest the relation between housing335price and accessibility to the CBD follows a concave curve: it rises near the CBD, where336Prepared using sagej.cls337338339===== PAGE 11 (score 11, numeric-paren 0, table-words 1, var-words 9) =====340341P´erez-Molina34211343the urban environment is physically deteriorated –particularly in the municipality of San344Jos´e proper– and at some point, the slope changes downward, reflecting the Alonso-345Mills-Muth model’s trade-off between accessibility and land rent.346In general, determinants with variation at the lower level (βk) are more significant347and larger in magnitude than variables with variation at the higher level (γp), as can be348seen in table 3. For models GaLM, MLRE, and CARM, no βk coefficient has 0.00 within349its 95% posterior interval whereas half the γp coefficients do –and the γk coefficients350for the MLRE and CARM models, which are a better fit for the data, are as a rule351not significant. However, it is important to note that accessibility, the most important352set of determinants in the estimated models, presents variation at the lower (individual353property) level. Thus, this result coincides with previous findings by Osland et al.354(2016), who found socioeconomic variables to be relatively weak but accessibility to355become stronger explanations of housing price variance when accounting for spatial356dependency (albeit their accessibility measure was at the zonal, not individual level).357A proper accounting of hierarchical structure (and spatial dependence) clarifies the role358of theoretically key covariates, regardless of their level in the hierarchy.359When comparing coefficients across model specifications, magnitudes remain360remarkably stable with three important exceptions: travel time to CBD (β2 and β3),361distance to nearest bus stop (β5), and number of jobs in the district (γ3). The first two362determinants decrease in magnitude as more spatial effects are considered: they are, in363absolute terms, largest for the GaLM model and smallest for the CARM model, with364the coefficients of the MLRE model being intermediate. Number of jobs has larger365coefficients for the GaLM and MLRE models, decreasing for the CARM model. This366characteristic suggests accessibility was serving to control for certain urban patterns,367particularly for socioeconomic self-sorting: it is very likely the best districts (higher368level units) in terms of accessibility attract both firms (and jobs) and higher income369households, which in turn would result in a more attractive spatial context.370Finally, with regard to the VPM model, one should note σ2371s (municipal level variance)372and σ2373u (district level variance) present similar values. The multilevel variance in the374higher levels should be, therefore, apportioned between these two levels. An evident375extension of the MLRE and CARM models will be to include municipal level random376effects and covariates, although the main contribution of these is likely to explanation of377the housing price rather than prediction, since most of this variations is already controlled378for in the district level random effects (crucially, land use regulation is a municipal level379covariate not currently included in the models).380The estimated mean posterior of η for the MLRE and CARM models are shown in381figure 2. In addition, the random effect deviations from the overall mean, at municipal382and district level, of the VPM model are also included. The larger values are shown383in dark red and the lower values, in dark blue, with intermediate values in white. Non-384significant parameters (with 0.00 in the 95% posterior interval) are shown in yellow. Only385districts and municipalities with data are represented in figure 2.386The η represent the total systematic variation, caused by the higher level covariates387and the random effect. Additionally, they represent the variation unexplained by the388lower level determinants (which is to say, accessibility and house size). They are broadly389Prepared using sagej.cls390