The analysis of a table 6 shows that both our hypothesis have been confirmed. In the first hypothesis we assumed that accounting for a spatial autocorrelation allows to increase predictive quality of the model. As we can see, the average error term in model [2] and [3] is less than in a standard linear regression according to all the criteria, so these models give us more accurate prediction of a housing price. Simultaneous accounting for SA allows to achieve even more qualitative prediction.

The comparison of a prediction error criteria of models [2] and [3] supports our second hypothesis, where we made an assumption that closeness in characteristic space would have greater influence on a housing price than a closeness in geographic space. According to four out of five criteria the model with characteristic closeness has a greater prediction quality, this means that the model [3] is better than a model [2] in our terms. This means that people are more flexible in a geographical location of a residential property object than in a set of its characteristics, such an information might be important for real estate agents and private owners when evaluating a house or a flat, because a set of a “positive” characteristic may overweight a “negative” location of a house and be a reason of a price increase when selling it.

As it was said earlier the parameter of a space shrinkage is unknown for us and we determined it manually, however, the choice of the value equal to Ѕ may be not optimal from the statistical point of view. To solve the problem, we reestimated weighting matrices using different values for the parameter in the interval from 0.1 to 1 with the step equal to 0.1; moreover, we varied the parameters for geographical and characteristic spaces separately, this lets us to test the assumption that in real life this two spaces are constructed in different ways and have different impact to the final price of a real estate objects.

As far as on the previous step we found out that the best model is a model which includes geographical and characteristic spaces simultaneously we reestimated only this specification. We used the same set of variables in all new models, the only difference was the parameter of a space shrinkage.

Most of prediction error criteria introduce an identical result, so in the table 7 we demonstrate RMSE only, the rest criteria are shown in Appendix 2.

The best model that allows to get the smallest deviation of predicted values from the real data is the specification where the shrinkage parameter for geographical space is equal to 1 and for characteristic space equal to 0,2. In this case the prediction quality improved on 3% in comparison with the previous stage of the research.

Table 7. RMSE of new models specifications with a given parameter of a space shrinkage*

*Note: a basic model specification is identical to all the previous models, the difference refers to a space shrinkage parameter in a weighting matrix

Conclusion

In the project we tested the effect of closeness to other objects on a residential property price. The special feature of the market is that all the real estate objects have geographical location and a set of characteristics, so we can define the closeness in different ways. All the previous works were concentrated on a studying of a geographical distance and the spatial effects related to it. In this research we developed an approach to construct a characteristic space and to measure distance between objects there. This work has a great practical significance, because new approach reflects the unique features of the market, gives new insights about the behavior of buyers and sellers on it and allows to predict a housing price more accurately.

To account a peculiarity of a residential property market that when evaluating a housing price sellers base not only on absolute values of its characteristics but also compare it to the alternatives and adjust the price to the current market situation, we extended a spatial lag model to be able to include a characteristic space into the model.

A new model of a residential property price gave a new understanding of how the market works. For example, taking into account for spatial interactions between objects has shown that characteristics are more important for people and has a greater influence on a price than geographical location. This information may let house sellers to set more optimal price, because if the flat has a “good” set of characteristics or differs from the other objects in a positive way it increases the market price of the flat. In our case significant variables in characteristic space are house floor and total square, so we may assume that this are the most important parameters which have the largest influence on a price and when selling a real estate people pay most attention to it.

Of course, this research suffers from some drawbacks. First of all, our database contains relatively short list of variables, so we could not include some housing parameters to estimate the price more accurately. Secondly, we could not estimate the effect of some variables in characteristics space; the structure of some variables would cause the problem of multicollinearity in a constructed space, so we had to exclude them to avoid more serious problems. Thirdly, a difficulty of model estimation and technical limitations of statistical package forced us to use just a quarter of an initial sample to train models, so the database may contain some random effects that lead to prediction error increase. Moreover, in the database we observed an asking price only, but it may differ from a real sale price; a seller can set a low or a high price in the announcement just to draw an attention of other people, so the introduced dependent variable might be not fully representative. Also we do not have any kind of a house ID, so if a seller put some price on his flat, but changed it in some time, we consider it as two different objects; this fact reduces a model quality and increases forecast error.

However, in general we can consider the project as a successful one. In the research we achieved the goal of the work, found the answer to the research question, proved the existence of spatial autocorrelation on the market and got a new information about it. An introduced approach allows to improve prediction quality by ten per cent, this means that a designed model is more applicable for practical purposes.

There are several directions for further research and model improvement. The developed methodology is not limited by the given set of variables, and if in the future some instruments to collect other characteristics will be developed the model quality can be improved. To get the information about some currently unobservable parameters (e.g., view from the window, materials of decoration, age of the building) the cooperation with the real estate selling portals is possible.

To solve the problem of limited number of observations in a training sample, alternative software and more powerful computers might be used; if this approach will be applied by real estate agents to evaluate the housing price in some kind of a program, the price can be calculated distantly on a server, so this problem is not a significant from the practical point of view. Also some current Machine Learning methods can be useful to overcome the difficulty, for example, bootstrap procedure allows to estimate the same models on different random samples and the to aggregate the coefficients to get consistent result and a qualitative forecast.

Also in the research we paid no attention to the statistical outliers trying to predict the price of all residential property objects by one model. However, it is more conscious to predict the price of an “average” through the information about other “average” objects, because they are closer substitutes, so all the estimators will be more accurate. To evaluate the prices of outliers, we can use different approaches: if the number of such observations is relatively large, we can use another model for them; if we have not enough observations to estimate the model, we can estimate it using both “average” objects and outliers, but to forecast only prices of the second ones. All in all, an additional classification of the objects on the initial stage of price evaluation may improve a predictive power of the model.

shopping housing autocorrelation

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Appendix 1

SE of different specifications with a given parameter of a space shrinkage

MSE of different specifications with a given parameter of a space shrinkage

MAE of different specifications with a given parameter of a space shrinkage

MAPE of different specifications with a given parameter of a space shrinkage

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