Empirical Results
5.2 Spatial Modeling and Spatial Stability
In this park of empirical results, the spatial model described in the methodol-ogy section 4.3. Before however, any statistical inference based on the spatial models is performed, the proper form of the spatial weight matrix WWW must be specified.
After examining multiple types and constraints for the correctly specified form of WWW, using various criteria such as the AIC, BIC, and the Log-likelihood we concluded that the most suitable type of spatial weight matrix, for each region, is constructed using the k−nearest neighbors method. To determine the ”optimal” number of k, for the various number of k, starting from k = 2 all the way to the k = 100, using a step of size 2, the spatial weight matrix was constructed (once again for each region) and spatial models were estimated while preserving all models criteria.
This step was performed for both spatial lag and spatial error models separately not to rely on the strict assumption that the optimal form of the spatial matrix is the same for both models. Finding the numberk for which theAICandBIC reaches its minimums, respectively, where the log-likelihood function reaches its maximum yields the value of kwhich should be selected for the final models for the inference purposes.
The evaluation of various parameter k for both spatial lag model and spatial error model can be analyzed in the figure 5.2 and 5.3 respectively. The esti-mated results of the spatial lag model are presented in table 5.4 and results of the spatial error model are presented in table 5.5.
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Vysocina (26) Zlín (14)
Pilsner (52) South Bohemian (64) South Moravian (76)
Moravian-Silesian (28) Olomouc (94) Pardubice (16)
Central Bohemian (36) Hradec Králové (16) Liberec (12)
Aussig (14) Capital city Prague (44) Carlsbad (16)
0 25 50 75 100 0 25 50 75 100
Num. of Neirest neighbour units
Criteria AIC BIC logLik
Spatial Lag model
Models stability evaluation for different W matrices
Figure 5.2: Models Stability Evaluation - Spatial Lag Model (different k used for the WWW matrix)
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Vysocina (10) Zlín (14)
Pilsner (52) South Bohemian (8) South Moravian (66)
Moravian-Silesian (28) Olomouc (10) Pardubice (16)
Central Bohemian (34) Hradec Králové (16) Liberec (12)
Aussig (14) Capital city Prague (18) Carlsbad (14)
0 25 50 75 100 0 25 50 75 100
Num. of Neirest neighbour units
Criteria AIC BIC logLik
Figure 5.3: Models Stability Evaluation - Spatial Error Model (differ-ent k used for the WWW matrix)
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Empirical Results 5.2 Spatial Modeling and Spatial Stability
Table5.4:SpatialLagModel SpatialLagModel DependentVariable:LogofPrice (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) Meters0.011∗∗∗0.010∗∗∗0.008∗∗∗0.010∗∗∗0.005∗∗∗0.006∗∗∗0.008∗∗∗0.009∗∗∗0.003∗∗∗0.007∗∗∗0.007∗∗∗0.007∗∗∗0.008∗∗∗0.009∗∗∗ (0.0003)(0.0003)(0.0004)(0.0001)(0.0003)(0.0004)(0.0004)(0.0003)(0.0003)(0.0004)(0.0002)(0.0002)(0.0005)(0.0004) Room−0.0010.00040.001−0.002∗∗∗0.005∗∗∗0.0002−0.005∗∗∗0.007∗∗∗0.003∗−0.005∗∗∗0.006∗∗∗0.002∗∗0.0010.003 (0.001)(0.001)(0.001)(0.0004)(0.001)(0.001)(0.0005)(0.002)(0.001)(0.001)(0.001)(0.001)(0.001)(0.003) Floor−0.008∗∗0.0090.013∗∗0.0030.014∗∗0.017∗∗0.0001−0.0030.014∗0.012∗0.017∗∗∗0.013∗∗∗0.023∗∗∗−0.001 (0.003)(0.006)(0.005)(0.002)(0.006)(0.008)(0.006)(0.003)(0.007)(0.006)(0.004)(0.004)(0.009)(0.005) Floorzero−0.061∗∗∗−0.0230.016−0.079∗∗∗0.023−0.020−0.012−0.066∗∗∗−0.082∗∗−0.057∗−0.0020.009−0.057∗−0.092∗∗∗ (0.020)(0.026)(0.025)(0.012)(0.034)(0.033)(0.029)(0.018)(0.032)(0.030)(0.019)(0.016)(0.031)(0.024) FloorTop−0.0003−0.060∗∗∗−0.008−0.029∗∗∗−0.005−0.058∗∗0.051∗−0.045∗∗∗−0.174∗∗∗−0.066∗∗−0.040∗∗−0.040∗∗∗−0.072∗∗−0.026 (0.018)(0.022)(0.022)(0.011)(0.027)(0.029)(0.031)(0.017)(0.029)(0.026)(0.016)(0.014)(0.029)(0.021) Buildingtype:Concrete−0.028∗−0.042∗0.017−0.097∗∗∗−0.042−0.037−0.0200.016−0.081∗∗∗0.003−0.010−0.030∗−0.027−0.0003 (0.016)(0.022)(0.024)(0.012)(0.027)(0.033)(0.028)(0.016)(0.031)(0.028)(0.020)(0.016)(0.030)(0.023) Condition:Verygood0.140∗∗∗0.147∗∗∗0.139∗∗∗0.0080.118∗∗∗0.107∗∗∗0.084∗∗∗0.141∗∗∗0.173∗∗∗0.099∗∗∗0.067∗∗∗0.069∗∗∗0.116∗∗∗0.099∗∗∗ (0.014)(0.020)(0.022)(0.010)(0.026)(0.028)(0.025)(0.015)(0.028)(0.025)(0.017)(0.015)(0.026)(0.022) Condition:Afterreconstruction0.199∗∗∗0.204∗∗∗0.182∗∗∗0.046∗∗∗0.175∗∗∗0.097∗∗0.155∗∗∗0.229∗∗∗0.207∗∗∗0.084∗∗0.096∗∗∗0.065∗∗∗0.157∗∗∗0.186∗∗∗ (0.018)(0.025)(0.029)(0.013)(0.037)(0.039)(0.033)(0.019)(0.037)(0.033)(0.020)(0.019)(0.035)(0.028) Private0.162∗∗∗0.163∗∗0.262∗∗∗0.193∗∗∗0.0390.086∗∗0.103∗∗∗0.113∗∗∗0.0510.114∗∗∗0.0400.056∗∗∗0.0440.098∗∗∗ (0.014)(0.063)(0.047)(0.013)(0.029)(0.038)(0.029)(0.013)(0.034)(0.037)(0.026)(0.021)(0.037)(0.036) Kitchenette−0.0040.087∗∗∗0.145∗∗∗0.018∗0.069∗∗∗0.129∗∗∗0.0290.085∗∗∗0.076∗∗0.086∗∗∗0.094∗∗∗0.072∗∗∗0.0380.009 (0.017)(0.021)(0.023)(0.010)(0.026)(0.026)(0.025)(0.016)(0.032)(0.023)(0.017)(0.014)(0.028)(0.024) Balcony/Terrace0.055∗∗∗0.168∗∗∗0.104∗∗∗0.060∗∗∗0.077∗∗∗0.111∗∗∗0.046∗0.064∗∗∗0.054∗∗0.061∗∗0.085∗∗∗0.090∗∗∗0.093∗∗∗0.105∗∗∗ (0.017)(0.019)(0.022)(0.009)(0.022)(0.025)(0.026)(0.013)(0.025)(0.024)(0.014)(0.012)(0.023)(0.019) Garage−0.048∗−0.020−0.0410.033∗∗∗0.0320.0260.0230.075∗∗∗0.042−0.0130.106∗∗∗0.0100.0120.020 (0.027)(0.026)(0.025)(0.010)(0.033)(0.035)(0.030)(0.024)(0.035)(0.034)(0.017)(0.015)(0.032)(0.029) Concrete×NewEstate0.1490.3220.127∗∗∗0.1030.0520.203∗∗∗−0.398∗∗∗0.352∗0.196∗∗∗0.057−0.2110.339∗∗0.191∗∗∗0.077 (0.166)(0.295)(0.031)(0.080)(0.312)(0.041)(0.111)(0.189)(0.043)(0.250)(0.238)(0.137)(0.036)(0.211) Brick×NewEstate0.336∗∗∗0.334∗∗∗−0.0030.304∗∗∗0.149∗∗∗0.272∗∗∗0.0440.041∗∗0.101∗∗∗0.244∗∗∗ (0.077)(0.036)(0.012)(0.044)(0.043)(0.033)(0.038)(0.021)(0.017)(0.033) Constant2.542∗∗∗4.034∗∗∗1.783∗∗∗6.197∗∗∗1.681∗∗∗2.386∗∗∗4.204∗∗∗3.394∗∗∗3.572∗∗∗2.616∗∗∗1.046∗∗∗3.271∗∗∗3.541∗∗∗6.488∗∗∗ (0.187)(0.248)(0.442)(0.246)(0.425)(0.318)(0.376)(0.237)(0.539)(0.415)(0.290)(0.268)(0.600)(0.427) ρ0.757∗∗∗0.648∗∗∗0.809∗∗∗0.55∗∗∗0.844∗∗∗0.791∗∗∗0.671∗∗∗0.708∗∗∗0.727∗∗∗0.777∗∗∗0.881∗∗∗0.736∗∗∗0.709∗∗∗0.508∗∗∗ (0.013)(0.017)(0.03)(0.016)(0.029)(0.022)(0.026)(0.017)(0.037)(0.028)(0.019)(0.018)(0.041)(0.029) Log-like-414.037-264.013-52.694267.727-223.369-209.098-113.342-207.359-214.123-28.96925.76533.0259.67989.677 AIC.862.074562.025137.389-501.455480.738450.196260.684448.717460.24591.938-17.53-32.05112.642-145.353 BIC.958.013649.72213.369-397.628561.9523.397338.696543.839535.516166.45871.83961.66479.323-68.78 R2 pse.0.7240.8150.6190.7860.5330.6750.710.7340.6140.6270.7230.6760.6190.757 Observations2,0871,2858533,3198757177271,9898165921,4181,831477668 Note:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01
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Table5.5:SpatialErrorModel SpatialErrorModel DependentVariable:LogofPrice (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) Meters0.012∗∗∗0.010∗∗∗0.008∗∗∗0.010∗∗∗0.005∗∗∗0.007∗∗∗0.008∗∗∗0.010∗∗∗0.003∗∗∗0.007∗∗∗0.007∗∗∗0.007∗∗∗0.008∗∗∗0.009∗∗∗ (0.0003)(0.0003)(0.0004)(0.0001)(0.0003)(0.0004)(0.0004)(0.0003)(0.0003)(0.0004)(0.0002)(0.0002)(0.0004)(0.0003) Room−0.0010.00010.001−0.001∗∗∗0.004∗∗∗−0.0001−0.005∗∗∗0.006∗∗∗0.003∗∗∗−0.005∗∗∗0.006∗∗∗0.002∗∗∗0.00040.002 (0.001)(0.001)(0.001)(0.0003)(0.001)(0.001)(0.0004)(0.002)(0.001)(0.001)(0.001)(0.001)(0.001)(0.003) Floor−0.006∗0.0040.012∗∗0.0010.0040.010−0.001−0.006∗∗∗0.0070.0050.009∗∗0.009∗∗0.015∗−0.003 (0.003)(0.006)(0.005)(0.002)(0.005)(0.008)(0.005)(0.002)(0.007)(0.006)(0.004)(0.004)(0.008)(0.004) Floorzero−0.054∗∗∗−0.0360.018−0.076∗∗∗0.003−0.0350.012−0.075∗∗∗−0.049∗−0.050∗−0.0160.002−0.056∗∗−0.071∗∗∗ (0.019)(0.025)(0.025)(0.011)(0.028)(0.033)(0.027)(0.017)(0.030)(0.029)(0.018)(0.016)(0.027)(0.022) FloorTop−0.011−0.071∗∗∗−0.006−0.0100.022−0.049∗0.036−0.028∗−0.101∗∗∗−0.037−0.022−0.035∗∗∗−0.037−0.023 (0.017)(0.022)(0.022)(0.011)(0.023)(0.029)(0.028)(0.016)(0.028)(0.026)(0.015)(0.013)(0.025)(0.020) Buildingtype:Concrete−0.089∗∗∗−0.0330.026−0.107∗∗∗−0.050∗−0.097∗∗∗−0.024−0.021−0.108∗∗∗−0.014−0.012−0.037∗∗−0.006−0.001 (0.019)(0.024)(0.026)(0.014)(0.026)(0.034)(0.028)(0.016)(0.032)(0.029)(0.021)(0.017)(0.029)(0.023) Condition:Verygood0.130∗∗∗0.141∗∗∗0.131∗∗∗0.0140.126∗∗∗0.111∗∗∗0.078∗∗∗0.116∗∗∗0.164∗∗∗0.082∗∗∗0.076∗∗∗0.070∗∗∗0.104∗∗∗0.099∗∗∗ (0.014)(0.018)(0.022)(0.010)(0.022)(0.028)(0.023)(0.014)(0.026)(0.025)(0.017)(0.015)(0.024)(0.020) Condition:Afterreconstruction0.192∗∗∗0.191∗∗∗0.175∗∗∗0.044∗∗∗0.138∗∗∗0.147∗∗∗0.149∗∗∗0.192∗∗∗0.166∗∗∗0.070∗∗0.094∗∗∗0.085∗∗∗0.143∗∗∗0.180∗∗∗ (0.017)(0.024)(0.029)(0.013)(0.031)(0.037)(0.030)(0.018)(0.036)(0.032)(0.020)(0.019)(0.032)(0.027) Private0.193∗∗∗0.099∗0.238∗∗∗0.177∗∗∗0.085∗∗∗0.097∗∗0.115∗∗∗0.115∗∗∗0.0430.089∗∗0.044∗0.108∗∗∗0.0230.089∗∗∗ (0.017)(0.059)(0.047)(0.013)(0.028)(0.038)(0.029)(0.013)(0.033)(0.037)(0.025)(0.023)(0.034)(0.034) Kitchenette−0.0220.074∗∗∗0.133∗∗∗0.018∗0.0130.071∗∗∗0.0170.102∗∗∗−0.0070.082∗∗∗0.080∗∗∗0.076∗∗∗0.083∗∗∗0.035 (0.018)(0.020)(0.024)(0.009)(0.023)(0.027)(0.023)(0.015)(0.032)(0.024)(0.017)(0.013)(0.026)(0.023) Balcony/Terrace0.029∗0.164∗∗∗0.097∗∗∗0.074∗∗∗0.076∗∗∗0.115∗∗∗0.067∗∗∗0.042∗∗∗0.103∗∗∗0.063∗∗∗0.076∗∗∗0.087∗∗∗0.052∗∗0.099∗∗∗ (0.017)(0.018)(0.022)(0.009)(0.019)(0.024)(0.025)(0.013)(0.024)(0.023)(0.013)(0.012)(0.022)(0.018) Garage−0.0380.051∗−0.041∗0.046∗∗∗0.055∗0.0130.092∗∗∗0.109∗∗∗0.071∗∗0.0060.098∗∗∗0.034∗∗−0.004−0.004 (0.026)(0.027)(0.025)(0.010)(0.032)(0.037)(0.031)(0.025)(0.034)(0.035)(0.016)(0.016)(0.031)(0.029) Concrete×NewEstate0.1020.461∗0.150∗∗∗0.1180.1020.225∗∗∗−0.516∗∗∗0.323∗0.315∗∗∗0.085−0.2220.288∗∗0.212∗∗∗0.104 (0.155)(0.277)(0.032)(0.074)(0.258)(0.043)(0.114)(0.175)(0.050)(0.239)(0.225)(0.130)(0.036)(0.197) Brick×NewEstate0.562∗∗∗0.356∗∗∗0.0160.251∗∗∗0.344∗∗∗0.391∗∗∗0.0210.078∗∗∗0.104∗∗∗0.279∗∗∗ (0.121)(0.037)(0.013)(0.042)(0.057)(0.038)(0.038)(0.021)(0.018)(0.033) Constant13.104∗∗∗13.486∗∗∗13.585∗∗∗14.858∗∗∗14.221∗∗∗14.106∗∗∗14.118∗∗∗13.552∗∗∗14.324∗∗∗14.132∗∗∗14.099∗∗∗14.228∗∗∗13.980∗∗∗14.004∗∗∗ (0.048)(0.088)(0.110)(0.026)(0.057)(0.103)(0.070)(0.055)(0.070)(0.090)(0.159)(0.057)(0.067)(0.058) λ0.836∗∗∗0.864∗∗∗0.906∗∗∗0.79∗∗∗0.776∗∗∗0.868∗∗∗0.816∗∗∗0.885∗∗∗0.794∗∗∗0.86∗∗∗0.962∗∗∗0.889∗∗∗0.791∗∗∗0.779∗∗∗ (0.014)(0.015)(0.026)(0.016)(0.022)(0.019)(0.023)(0.014)(0.023)(0.025)(0.014)(0.015)(0.03)(0.03) Log-like-346.849-241.211-54.509407.054-120.551-198.209-70.222-103.59-190.605-25.06983.72976.20854.41112.973 AIC.727.698516.422141.018-780.108275.102428.418174.444241.179413.21184.139-133.458-118.417-76.819-191.945 BIC.823.637604.116216.998-676.282356.263501.619252.456336.301488.481158.659-44.089-24.702-10.139-115.372 R2 pse.0.7490.8340.6230.8120.6670.6980.7550.7680.6670.6450.7490.70.7050.785 Observations2,0871,2858533,3198757177271,9898165921,4181,831477668 Note:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01
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Empirical Results 5.2 Spatial Modeling and Spatial Stability
In the case of the spatial lag model 5.4, we can observe that all spatial lag models dispose of considerably better in-sample fit metrics such as AIC, BIC, and the log-likelihood. In the case of all regions, the spatial lag models have greatly higher values of loglikelihood functions and lower values of both AIC and BIC compared to the OLS models. As far as the strength of spatial de-pendence goes, we can observe significant, and extensively strong parameters ρ in the case of all regions. Interestingly enough, the spatial dependence among the dependent variable, denoted by the parameter ρ, seems to be the lowest, yet quite strong, in the case of the capital city Prague and the Zlin region.
These results quite match the results found in the table 5.3, where both of these Regions are also, in terms of the magnitudes of spatial dependence, on the lowest positions.
The coefficients of all of the lag models yield, in the case of all fourteen regions, almost similar results in terms of signs of the coefficients estimates to the results of the OLS models. Some interesting conclusions can be made from the model estimates. For example, we can observe that the effect of the private type of ownership is always positive in the case of all regions. (We are fully aware that, like mentioned in the methodology section 4.3.1, coefficients of the spatial lag models are not the true marginal effect however, the signs of the effects can be interpreted directly). This effect is natural as the other types of ownership usually have certain limitations associated with itself. Another, yet interesting observation about the coefficients of the spatial lag models to make is the fluctuation of the effect of the interactive term Concrete × New estate, which is very similar to the fluctuations estimated in the OLS models. This effect can be, to a certain degree, decomposed into two individual factors. Firstly, the new estates of the concrete type are not being built in all fourteen regions and moreover, certainly, not in the same volumes. Secondly, the effect ofnew estate is mostly independent of the building type. In other words, the effect of the new estate is relatively similar for the brick estate and for the concreteestate.
Lastly, this supports the fact that thebrickis much-preferred building material compared to concrete.
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The similar conclusions can be made from the spatial error models in table 5.5, which seems to be the best model thus for almost every single region. An extensively strong presence of the spatial dependency among the error term is present in all regions. Based on both spatial models thus far, the effect of spatial dependency among the error term is considerably higher compared to the spatial dependency among the dependent variable. This may indicate to us that all of our models, which are operating with more independent variables, still suffer from a certain degree of imperfection, which, in the case of the housing data, can be naturally expected. This may also be due to the fact that the spatial lag model and the spatial error model assume that the effects do not vary in space.
Even though both spatial models estimated thus far do allow to model, in a certain however limited way, the spatial nature of the data, the main drawback are still present. The main drawback of modeling methodology thus far is that the effects of price determinants are not allowed to vary in space. This is, in the case of the housing data, a major drawback and quite an extensive assumption.
To model the spatial heterogeneity and to allow the coefficients to vary in the space, we also estimate the GWR models, as described in section 4.3.3.
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Empirical Results 5.3 Grandiose Clusters: The Space Distribution