Bootstrap and Regression analysis in Real Estate Appraisal

In real estate valuation, the Multiple Regression Analysis (ARM) finds concretely many applications. This method is coherent with the forecasted, descriptive and explanatory aims of the valuation inquiry, it reduces greatly the existing subjectivity in assessment problems or, at least, it yields to isolate and point it out. However, into practice, even in that cases where theoretical suppositions based on ARM are finally verified, the use of regression is subjected to few obstacles of working nature. Between those, the most obvious are: a) the need to concentrate on statistical measurements having properties which can be analysed with mathematical justifications; b) a limited amount of real data sets referred to prices and to real estates peculiarities. Under those circumstances it becomes lower the chance to verify to what extent is the statistical inference accurate; although the level of likeness in evaluation is an essential requirement of appraisal. Actually, operating in a probabilistic context, the measurement of variability of any statistics pertinent to the case under study gives useful information to verify the accuracy of results, so to apply correctly and efficiently the model to the valuation needs. The improvement of accuracy of statistical inference can be obtained, as it is obvious, having several data sets to use for evaluating. But under real circumstances it is possible to collect data from only one sample which has usually small size. But even having many samples; for most of statistical estimators does not exist a concise formula that allows the variability computation. The same standard error –which is used to find the simplest measure of accuracy of an appraisal– respond to a close mathematical form only if is calculated on the base of arithmetic mean. The bootstrap procedure exposed in the paper allows to overcome the limits shown by the ARM. Conceive about thirty years ago, bootstrap is a procedure that makes possible to find, with the use of a personal computer; the statistical accuracy of an appraisal, just starting from a single real sample data. In other words, the bootstrap is able to simulate the process of random draw of many samples starting from the data of the only sample available. The distribution of estimated statistics for the single samples of bootstrap can so be handled as a distribution obtained using real samples and it is possible to calculate its variability. If we want to work out a statistical likewise of cloning, for example, it is as in the only sample at available for the analysis it is encoded the entire “genetic code” of the population under study, the bootstrap worked “cloning” several times this sample (even endlessly) until it is able to restore the whole population. Even though it looks paradoxical to a first glance; theoretical studies show that for a large amount of statistical properties the measure of accuracy obtained with the bootstrap is reliable in probabilistic terms. The interval of accuracy in an assessment found with the bootstrap using only one sample has an amplitude about the same of the one found with the statistical distribution of many real samples, obviously when those samples are concretely available. The use of personal computers enable to look on a numerical base into variability of appraisals; also in cases where it is not possible to analytically find it in an easy way by using a concise mathematical formula. This paper is divided into two section. In the first, the ARM is applied in a “traditional” form to a sample of residential real properties located in an Italian city so to build a function for price evaluation. The computation of correlation coefficient and the testing of statistical accuracy of regression coefficients yields to verify to what extent the model is fitted to respond to the forecasted aims. In the second part of the paper, it is taken into notice and analysed the variability of statistical estimators which is representative of the goodness of a regression model obtained trough the use of bootstrap. We analyse the theoretical hypotheses & the methodological criteria; as well as how the bootstrap can be used in general problems. On the bootstrap samples –which were found using the real sample– we have computed the variability measurements of correlation coefficient & regression coefficients; so to improve the performance of the model both in an explanatory & forecasting terms.