A LoCAtIoN VALUe reSpoNSe SUrFACe modeL For mASS ApprAISINg: AN “IterAtIVe” LoCAtIoN ADJUSTMENT FAcTOR IN bARI, ITALy

. the work is focused on a new model of mass appraising including location variable. A location adjustment factor derived from a mathematical iteration was compared to the location adjustment factor based on geostatistical techniques. the work compares three different linear MRA models. The first one uses the location blind linear MRA. The second integrates the linear mrA with a location adjustment factor calculated using spatial interpolation. the second alternative is an application of Location Value response surface models (o’connor, 1982). It represents the first application of these models for mass appraising in Italy. The third approach introduces the Iterative Location Adjustment Factor. This is a factor which measure the influence of location derived from a mathematical iteration. Empirical results seem to prove the validity of Iterative Location Adjustment Factors in specific context with few observations.


INtrodUCtIoN
several authors pointed out the role of externalities and location in property values (krantz et al., 1982; hoch and waddell 1993; des rosiers et al., 1996). Previous research focused on the problem of variability of house prices which remains unexplained in multiple regression models (Anselin and can 1986;dubin 1998). the consequences are for example: the presence of excessive multicollinearity among attributes, spatial autocorrelation among residuals; diminuishing the stability of regression coefficients (Dubin 1988;Anselin and Rey 1991;des rosiers and thériault, 1999). for this reason neighbourhood factors should consider submarket specifics (Adair et al., 1996). this problem is particularly relevant in real estate markets with a limited number of observations. this work proposes a different approach to location variable in mass appraising and automated valuation modelling. After the application of a traditional location blind mrA linear model, the works compare it with an application of Location Value response surface analysis in Italy. It is the first application of this kind of model to the Italian context. the third model derive the location factor from a mathematical iteration instead of geostatistic techniques. The empirical findings of the traditional LAf and the new Iterative Location Adjustment factor converge on comparable solutions. the article is organized as follows: the first paragraph will give a brief outline of Location Value response surface models, in the second paragraph will be proposed the application of an Iterative Location Adjustment factor for mass appraising. After a comparison among the automated valuation methods applied final remarks will be offered at the end.

LoCAtIoN VALUe reSpoNSe SUrFACe modeLS
Location Value Response Surface (LVRS) Analysis has been introduced in Us (o'connor, 1982) for the first time for the appraisal of single family houses in Lucas county, and is different approach to fixed neighbourhoods or composite submarkets analysis (ward et al., 2002). the application of this method requires spatial interpolation of property prices or error term. this method has been applied in the U.s. (Eichenbaum, 1989;Eichenbaum, 1995;ward et al., 1999), in England (Gallimore et al., 1996), andNorthern Ireland (mccluskey et al., 2000). the application of LVrs allows the appraiser to analyze the effect of location using Geographical Information systems (GIS). Among different possible classifications it is possible to observe three main approaches to LVRS. A first approach (McCluskey et al., 2000) consists in calculating a location adjustment factor based on the spatial distribution of the selling prices. A price per square metre is obtained dividing the actual price by the gross floor area of the dwelling. A contour plot overlying the area map portrays the peaks and troughs of property values which are also called value influence centres (VICs). In general term the VIC can be defined as point(s), line(s) or area(s) in a contour map where it is possible to observe a relative maximum (positive) or a minimum (negative) location values (errors). As a consequence VIc may affect the value of near properties. therefore the distance from each VIc is calculated for each observation. the selling price per square meter is regressed on coordinates and the distance of each property to each VIc. the predicted price is then divided by the average estimated price. As a consequence will be determined a local adjustment factor having a mean of 1. In particular better locations will have a factor greater than 1, while poorer locations will have a factor less than 1. this local adjustment factor varying from -1 and 1 will become a measure of impact of location in the final regression model whose predictability will be improved. In the case of bari there is one only VIc and the area is quite homogeneous therefore the measure of distance was the physical distance. A second approach is based on the measure of the variance between actual prices and predicted prices using a mrA model without location variable. this model will present greater value of forecasting error in some areas and lower value in other areas generating a contour map of errors instead of value. Using the error ratio related to under valuation or over valuation and the coordinates of each observation. the impact of each VIc on any property is determined using different possible measures of the distance from the property to the VIc (Eckert, 1990;Eckert et al., 1993). the response surface is depending on the VIc positions and the adopted distance measure. the third approach starts creates an interpolation grid, modelled to reflect the influence on each property of the location ratio factors within its proximity. the method has not been applied to residential flats. It has not been applied outside North America, britain or Northern Ireland. this is the first application to Italian real estate market. A prerequisite is having sufficient amount of data in each zone of the area considered in order to produce the spatial interpolation. there are not a minimum number of observations but real estate market, especially in the Italian context presents a scarcity of data. Location Adjustment factor does not indicate the value of a certain location, but only the comparative location values for real property analysed. spatial interpolation require the surface of the z variable (selling price or error term) to be continuous and the data value at any location can be estimated if sufficient information about the surface is given. In addition the z variable (selling price or error term) must be spatially dependent therefore the value at any specific location is related to the values of surrounding locations.

the AppLICAtIoN oF IterAtIVe LOcATION ADJUSTMENT FAcTOR IN bArI, ItALY. dAtA ANd methodS
In the residential real estate market of bari the location factor have been built avoiding the construction of contour maps. In fact in some institutional context it may be not easy to collect data for several problems. In italian context neither real price nor characteristics are always clearly indicated in the transaction and the data are often incomplete. there are few organized databank of real transactions. developing a real Estate market observatory in order to test and apply mass appraisalautomated valuation models it is not a easy task in Italy. real Estate market observatory founded in 1998 collects real transactions from several sources. It has groups of real estate transactions in several parts of the city of bari in the south east of Italy. this kind of sample are often recurring in real estate markets without an institutional organization of property data. Although the number of observations are poor this works tries to explore the power of mathematical criterion of minimum square least of representing real estate market contexts like Italy with few data (kauko and d'Amato, 2008). the work uses a sample of 20 observations in the administrative area of Carrassi Poggiofranco in bari. these observations are related to residential dwellings in a semicentral location in the urban area of bari. the list of 20 real observations is indicated in the paragraph 1.1 of the Appendix of this work.
In this work the sample has been analysed considering the following variables indicated in the table 1. the paragraph 1.2 of Appendix shows the output of linear regression model. It is possible to observe a good r 2 equal to 0.89 an acceptable test f of fisher, a good performing tstudent Gossett test except for bALcoNY variable. the output shows no presence of collinearity. the mean absolute percentage error whose formula is indicated in the formula (2) was calculated in order to test this first regression model.
In the formula (2) PS means predicted selling price while AS indicated actual selling price, n is the number of observations. the proposed linear regression model has a mAPE of 15,261%. In order to improve the predictability of the model a location adjustment factor was considered in the model. the location of 20 observations in term of longitude and latitude in the area of bari is indicated in the table 2.
the geographic distribution of 20 observations in the urban context of bari is indicated in the figure 1. In the middle of area it is possible to observe the urban park "Largo 2 Giugno". spatial correlation among the 20 observations was preliminary detected using moran's I (moran, 1948; moran, 1950) test. this index measures autocorrelation between values of the x vector. It ranges from -1 to +1 and each observation is only compared with its relevant neighbourhood. Positive moran's I indicates positive autocorrelation which means that high values for x or (market basket value or price per square meters) should be located near other high values while lower market basket values should be located near other lower market basket values.
where: x is the variable (the market basket value), and w ij represents the set of neighbours j for observation i.
In this case, as in previous examples in literature, inverse squared distance among the observations has been considered (des rosiers and Thériault, 1999) The final result showed positive autocorrelation assuming a value of 0,7954. A market basket value (say price per unit) has been calculated in order to produce a contour map. contour map is a map created joining all the points having similar measure (similar price per square meter). the market basket value has been obtained dividing the actual property price by the square meters. In the following figure 2 is indicated the contour map.
starting from the spatial distribution of the market basket value it has been possible to  observe the relationship between the price per unit of observations and their location through a linear semivariogram. the surface obtained allowed the application of an universal kriging to generate a surface in order to model location variable in this residential property market. kriging is a spatial interpolation technique which relies on analysis of the spatial variance of a phenomenon. spatial variability is used to build experimental variogram and observe means differentials between values. In this application the "regional" variable is the price per square meter (cressie, 1993). Variograms are then formally approximated with a formal function. In this case the theoretical function is linear to obtain the best adjustment for value variations resulting from proximity. the universal kriging was carried out using sUrfEr 8. therefore a second mrA has been runned considering the value influence center clearly individuated in the kriging whose coordinates are indicated in the table 3. the output of this regression model is indicated in the paragraph 1.4 of Appendix. the r 2 is 0.93, the f di fisher test and the t-test of student Gossett are both satisfying. the mean absolute percentage error is 11.08 with a significative improvement compared to the first MRA model presented in the formula (1). this work proposed the research of a location adjustment factor without using geostatistical tehnique. for this reason a third linear regression model has been applied to the same sample of 20 observations selected in this work. the mrA model is indicated in the following formula (5).
PRICE CONSTANT X DATE X SUI X BALCONY X ELEVATOR X ILAF the formula (5) has the same variables of formula (4) except for a new variable indicated as ILAf (Iterative Location Adjustment factor) instead of LAf (Location Adjustment factor). this variable is the physical distance in km of the coordinates (longitude and latitude) of each point from a virtual point whose coordinates should be defined after a mathematical non linear iteration in order to reach the highest level of R 2 . In the Appendix paragraph 1.3 is indicated the formula. After several iterations carried out through the command "Excel Solver" it has been possible to define an Iterative Location Adjustment Factor. It is Iterative because it is simply based on non linear iterations. the coordinates of this point (for this study we call it iterative location adjustment point) will varies in a mathematical iteration in order to select the appropriate Iterative Location Adjustment factor. At this stage using solver command of Excel it is assumed the following goal function indicated in formula (6).
where: r 2 is the well known coefficient of determination. the constraints will regard the coordinates of the iterative location adjustment point. It will vary according to these contraints that must be applied to the coordinates. the value of these constraints are indicated in the table 4. In this way the virtual point to be individuated through non linear iterations is inside the area individuated by the coordinates of the points. several iterations were carried out using the simple function solver included in the well known MS Office Excel. The iterations selected an Iterative location Adjustment Point as VIc without using geostatistics techniques. the report of iterations is indicated in the Appendix with the paragraph 1.4. the iterative locatin adjustment factor has the coordinates indicated in the table 5. therefore a third linear regression model was runned considering the same variable of model 4. In this model the term ILAf -Iterative Location Adjustment factor indicates the distance among each point of the sample and the coordinates of the Iterative Value Influence center indicated in the table 5. the formula (7) shows the linear multiple regression model obtained.
the model indicated in the formula 6 is linear having the same characteristics of the model indicated in the formula (4). the variable ILAf has a positive marginal price. the t-student Gossett test of ILAf variable shows a satisfying a 3.429. the iteration indicated an undesired place near a crossroad with problem of traffic, noise and pollution. This is the reason why the marginal is positive. the unpleasent place can be easily individuated in the kriging of market basket value in the figure 3. It is worth to notice the convergence between empirical findings of kriging technique and the iterations proposed. In the Appendix paragraph 1.5 are indicated the statistics of this third regression containing ILAf -Iterative Location Adjustment factor. the r 2 is 94.0 the fisher and the t-student Gossett tests are encouraging. the mean absolute percentage error is equal to 11.07. It presents a small improvement compared to the first model and to the second -Location Value response surface Model. The final Table 6 compares the three mass appraising models. The comparison seems to confirm that the Iterative Location Adjustment factor may represent an interesting tool to develop for implementing mass Appraisal and Automated Valuation systems.

FINAL remArkS ANd FUtUre dIreCtIoNS oF reSeArCh
the works demonstrated that it is possble to produce an Iterative Location Adjustment factor using a mathematical iteration instead of the well known geostatistical techniques. Among three different models the Iterative Location Adjustment factor based on mathematical modelling showed and interesting performance. the iteration were carried out with a quite simple software like MS Office Excel using solver function. more complex analysis with more than one or two VIcs may require the use of mathLab or solver programming offered by frontline. further researches may verify the Iterative Location Adjustment factor in area with more than one VIc and with different formal function from the linear one.
reFereNCeS Adair, A. s., berry, j. N. and mcGreal, s. w. (1996)   1.6. SpSS ouput regression model on 20 observations in the residential real estate market of bari using iterative location adjustment factor