Empirical Results
5.3 Grandiose Clusters: The Space Distribution
One of the few advantages of the hedonic linear regression model estimated via the OLS method is, in our frameworks, the assumption that the residuals pro-vide the comprehensive assessment and evaluation of the unobserved location effects. This framework is wholly inspired by the study of Lipán (2016).
Using the hedonic real estate model described in 2.1, we assume that hedonic models summarized in the table 5.1, properly model termS and N (the struc-tural characteristics and the neighborhood amenities respectively). However, the term Lwhich denotes the factors and effects of location fades into the error term. In other words, we assume that ε∼L+N(0, σIII).
To provide and analyze the "grandiose clusters" which summarise and measure the unobserved effect of the location on the price, we decided to measure the effect on the price in the percentage scale, which shall provide a largely nice overview of the spatial tendencies and trends in the housing market in the entire Czech republic. Firstly, however, we need to be aware of our modeled functional form described in the 3.1. We utilized the log transformation and thus, before calculating the percentage difference of each estate between its actual and fitted value, the proper transformation, that takes the log form into account, needs to be used. The following transformation, which is closely described in Wooldridge (2010), is used:
yˆ =αˆ0exp(logyˆ ),
where the method of moment estimator, which does not rely on the assumption of normality, is αˆ =0 n−1∑︁ni=1exp(uˆi), where uˆi is the residuum of the i-th observation1.
Once the proper transformation of the fitted value is obtained, the percentage error between the actual and fitted value is computed. In the next step, we merge all of the percentage differences from all fourteen regions into one large set, where one spatial distribution and spatial correlation form is assumed.
Then, the proper variogram is modeled.
1For larger overview reference in the Wooldridge (2010).
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The Estimated Variogram function and the estimated variogram models are plotted in the figure 5.4. Is clearly visible that the variogram function is mod-eled the best by the Gaussianmodel.
Spherical
Figure 5.4: Variogram Models
This can be confirmed when inspecting the results present in Table 5.6. Where we report the natural model metrics of the prediction accuracy (in-sample fit).
Confirming that the Gaussian model seems to be the most suitable model to use for the statistical inference. Since the variogram function is selected, the kriging interpolation follows.
Table 5.6: Variograms accuracy Variogram MSE RMSE MAE R2pse n Exponencial 0.004 6.589 0.473 0.875 15
Gaussian 0.001 3.460 0.252 0.959 15 Spherical 0.002 4.866 0.367 0.927 15
The figures 5.5 and 5.6 show the kriging predictions and the variance of the predictions of the "grandiose clusters". The kriging framework allows for the best linear unbiased prediction of the variable of interest, assuming that the proper variogram function is used. The percentage scales are related to the level prices in CZK.
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Legend
(−100%; −25%]
(−25%;0%]
(0%;+5%]
(+5%;+15%]
(+15%;+25%]
(+25%;+50%]
(+50%;+75%]
(+75%;+100%]
(+100%;+inf)
Figure 5.5: Grandiose Clusters - Prediction Mean
Legend (0; 0.25%]
(0.25%;0.50%]
(0.50%; 1%]
(1%; 5%)
Figure 5.6: Grandiose Clusters - Prediction Variance
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In addition, certain and unique distribution patterns of the "grandiose clusters"
are present. First and foremost the capital city of Prague is not largely deter-mined as the "grandiose clusters". This, by all means, does not indicate that price levels in Prague are ”cheap”. But rather indicates that the real estate market in Prague tends to be quite ”deterministic”, which means that once the detailed information about one particular (flat) estate is obtained the pre-dicted price will be within the real estate market perceived price level. On the other hand, one extensively large "grandiose cluster" in Prague is determined.
It turns out that this cluster is within the space of the very historical part of the old town part of the city, which disposes of immense historical value and thus allows for extensively high price levels of the estates within this cluster.
This concept holds the same for other large cities over the Czech Republic.
The (unobserved) effect of "grandiose clusters" is present in most of the cities e.g. in Karlovy Vary, Pardubice, Liberec, Plzeň, České Budějovice and many others.
Another interesting observation to make is for the Central Bohemian Region.
An entire area of Central Bohemian, surrounding the capital city, is identified as the "Grandiose Clusters". There is quite a simple reasoning for it. Given the price levels in Prague, which as discussed above tend to be mostly deterministic, meaning that the effect of location is not extensively largely dictating the price level, many dwellers prefer to seek housing right outside Prague, where many new estate complexes are being built broadly. These complexes very close to Prague, yet outside the capital city, still provide extensively nice traffic options.
Therefore, the dwellers are able to live outside the capital city and yet commute on daily basis fairly easily. As a result of this migrating process, the housing market tends to be adjusting accordingly, and thus these locations, closely outside Prague, are transforming into the "grandiose clusters" also.
Naturally, it is expected that these complexes will reach higher price levels in the future, especially now when multiple freeways and new subway links, which are supposed to provide an additional level of comfort and decrease the commuting time to the capital city.
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Empirical Results 5.4 Geographically Weighted Regression: Empirical Results