1# RECORD doi_10_1002_isaf_2872# TITLE: Hedonic price models and indices based on boosting applied to the Dutch housing market3# YEAR: 2007.0 VENUE: Intelligent Systems in Accounting, Finance and Management DOI: 10.1002/isaf.2874# AUTHORS: Martijn Kagie; Michiel Van Wezel5# PAGES: 45 CANDIDATE TABLE PAGES (1-based): [11, 14, 15, 16, 25]6# ABSTRACT: Abstract We create a hedonic price model for house prices for six geographical submarkets in the Netherlands. Our model is based on a recent data‐mining technique called boosting. Boosting is an ensemble technique that combines multiple models, in our case decision trees, into a combined prediction. Boosting enables capturing of complex nonlinear relationships and interaction effects between input variables. We report mean relative errors and mean absolute error for all regions and compare our models with a standard linear regression approach. Our model improves prediction performance by up to 39% compared with linear regression and by up to 20% compared with a log‐linear regression model. Next, we interpret the boosted models: we determine the most influential characteristics and graphically depict the relationship between the most important input variables and the house price. We find the size of the house to be the most important input for all but one region, and find some interesting nonlinear relationships between inputs and price. Finally, we construct hedonic price indices and compare these with the mean and median index and find that these indices differ notably in the urban regions of Amsterdam and Rotterdam. Copyright © 2008 John Wiley & Sons, Ltd.789===== PAGE 1 (score 2, numeric-paren 0, table-words 0, var-words 2) =====1011Hedonic Price Models and Indices based on Boosting applied12to the Dutch Housing Market13Martijn Kagie∗14Michiel van Wezel15Econometric Institute, Erasmus School of Economics and Business Economics16Erasmus University17P.O. Box 1738, 3000 DR, Rotterdam, The Netherlands18Econometric Institute Report EI 2006-1719April 5, 200620Abstract21We create a hedonic price model for house prices for six geographical submarkets in the22Netherlands. Our model is based on a recent data mining technique called boosting. Boosting23is an ensemble technique that combines multiple models, in our case decision trees, into24a combined prediction. Boosting enables capturing of complex nonlinear relationships and25interaction effects between input variables.26We report mean relative errors and mean absolute error for all regions and compare our27models with a standard linear regression approach. Our model improves prediction perfor-28mance with up to 40% compared with Linear Regression. Next, we interpret the boosted29models: we determine the most influential characteristics and graphically depict the relation-30ship between the most important input variables and the house price. We find the size of31the house to be the most important input for all but one region, and find some interesting32nonlinear relationships between inputs and price.33Finally, we construct hedonic price indices and compare these to the mean and median34index and find that these indices differ notably in the urban regions of Amsterdam and35Rotterdam.36137Introduction38Hedonic pricing theory hypothesizes that the price p of a product is determined by a function39p = F ∗(x), where x is a bundle of characteristics that define the product. Hedonic pricing theory40is generally attributed to Court (1939), Lancaster (1966), Griliches (1971b, 1971a) and Rosen41(1974).42In practice the hedonic function F ∗(x) is estimated by a model F(x) which is fitted on a43historical dataset {pi, xi}N441 . Traditionally, these models are Linear Regression or Box-Cox type45models.46In the context of housing and real estate hedonic models are useful in three ways. In the first47place a hedonic model is a very suitable way to predict house prices. House price prediction can48be used for bulk appraisal for property tax, but can also help real estate brokers by determining49the asking price for a house. However, it is also possible to use a hedonic method for a website50∗We thank Jaap Darwinkel and Dree Op ’t Veld for their helpful comments and suggestions. Corresponding51author: E-mail: kagie@few.eur.nl. Phone: +31 10 4088940. Fax: +31 10 4089031.521535455===== PAGE 11 (score 22, numeric-paren 0, table-words 3, var-words 12) =====5657groningen58apeldoorn59eindhoven60amsterdam61rotterdam62zeeland63Size Dataset64338765240266373067849068751069221670NVM Market Share (2003)7177%7291%7351%7476%7556%7633%77Housing Stock789903479763308016834281430733824184678315737784Percentage Houses in Dataset853.42%863.15%872.22%881.97%891.79%901.41%91Total Area Size (m2)92288.1093498.3894308.9895303.3396448.83972689.3898Size Land (m2)99272.45100496.17101304.20102239.02103337.751041645.30105Percentage Land10694.57%10799.56%10898.45%10978.80%11075.25%11161.18%112Population Size113217593114189309115392773116855094117899781118349683119Population Density120799121382122129112335771242664125213126Apartments12745.73%12822.31%12924.21%13079.41%13161.58%13210.47%133Terraced13434.96%13535.64%13672.65%13718.20%13831.80%13948.29%140Detached1418.92%14225.60%1439.92%1440.94%1452.30%14625.36%147Semi Detached14810.39%14916.44%15017.35%1511.45%1524.31%15315.88%154Table 2: Some characteristics of the 6 regions.1553.1156Regions157Figure 4 shows where the regions are located in the Netherlands and Table 2 summaries some158characteristics of the 6 regions. A brief description of the six regions is given below1:159City of Groningen and environs: The city of Groningen is located in the province Groningen160in the north east of the Netherlands. The region contains the municipalities of Groningen161(179185 inhabitants, 76 km2), Haren (19048, 46 km2), Eelde (6920, 13 km2), Norg (7160,162110 km2) and Peize (5280, 28 km2). Groningen is the largest city in the north west of the163Netherlands and the capital of the province with the same name. It is surrounded by small164rural villages. With 77% the NVM has a relatively high market share in this region.165Apeldoorn and environs: Apeldoorn is located in the province Gelderland in the east of the166Netherlands. The region contains the municipalities of Apeldoorn (156000 inhabitants, 340167km2) and Epe (33309, 156 km2). Like Groningen Apeldoorn is mainly surrounded by small168villages. This region is interesting, because all the house types are almost equally present in169this region and the NVM has a very high market share of 91% in this region.170Eindhoven and environs: Eindhoven is located in the province Noord-Brabant in the south east171of the Netherlands. The region contains the municipalities of Eindhoven (207870 inhabitants,17288 km2), Nuenen (23367, 34 km2), Son en Bruegel (15070, 26 km2), Valkenswaard (31091,17355 km2), Veldhoven (42545, 32 km2), Waalre (16502, 22 km2) and Geldrop (27670, 13 km2).174Eindhoven is the largest city in the south east of the Netherlands and has became this mainly,175because of the factories and head quarters of DAF Trucks and Philips, that were located in176this city. Eindhoven itself is a typical Dutch city with a large amount of terraced houses.177The NVM has a market share of 51% in this region.178Amsterdam: Amsterdam is located in the province Noord-Holland in the west of the Nether-179lands. The region contains the municipalities of Amsterdam (739104 inhabitants, 161 km2),180Amstelveen (78886, 44 km2), Diemen (24049, 12 km2) and Ouder-Amstel (13055, 24 km2).181Amsterdam is the largest city in the Netherlands and the capital of the country. The city182is situated on the borders of the river IJ. Amstelveen, Diemen and Duivendrecht (Ouder-183Amstel) are suburbs of Amsterdam. The NVM has a market share of 76% in this region.184Rotterdam: Rotterdam is located in the province Zuid-Holland in the west of the Netherlands.185The region contains the municipalities of Rotterdam (598923 inhabitants, 206 km2), Capelle186aan den IJssel (65354, 14 km2), Krimpen aan den IJssel (29046, 8 km2),Nieuwekerk aan187den IJssel (22334,17 km2), Vlaardingen (74058,24 km2), Schiedam (75619, 18 km2), Al-188brandswaard (19607, 22 km2) and Nederlek (14831, 28 km2). Rotterdam, the second largest189city in the Netherlands, is one of the largest sea ports in the world and located in delta of1901All statistics that are used in the descriptions and Table 2 are derived from the online StatLine database of the191CBS (de Jonge, Reedijk, and Sluis 2005), except the NVM market shares.19211193194195===== PAGE 12 (score 16, numeric-paren 0, table-words 3, var-words 10) =====196197the rivers Rhine, Maas, Lek and IJssel and near the North Sea coast. The market share of198the NVM real estate brokers in this region is 56%.199Zeeland: The region Zeeland contains almost the complete province of Zeeland, located in the200south west of the Netherlands. The region contains the municipalities of Borsele (22318201inhabitants, 142 km2), Goes (36591 inhabitants, 93 km2), Hulst (27887, 201 km2), Kapelle202(11627, 37 km2), Middelburg (46350, 49 km2), Reimerswaal (20996, 102 km2), Terneuzen203(55412, 251 km2), Veere (22130, 133 km2), Vlissingen (45236, 34 km2), Schouwen-Duiveland204(34416, 231 km2), Noord-Beveland (7124, 86 km2) and Sluis (24596, 280 km2). Zeeland exist205of a number of islands and peninsulas (Schouwen-Duivenland, Noord- and Zuid-Beveland,206Walcheren and Zeeuws-Vlaanderen) and because of that 40% of the total area of Zeeland is207water. Zeeland is one of the less dense populated regions in the Netherlands and also the208NVM has a relative small market share (33%) in this region. Besides the large size of this209region these two reasons would make that Zeeland is expected to be the region that is the210hardest to predict.2113.2212Variables213For these 6 regions all transactions of existing houses (no newly built houses) that were sold by214NVM brokers in the year 2004 are used to create the datasets. The transaction price in euros215is the target variable or response. There are 83 explanatory variables, of which 12 variables are216continuous, 12 variables are categorical and the other 59 are binary. Both the continuous and217the categorical variables have some missing values. Not all categorical variables are compulsory218to fill in by the broker who sold the house and continuous variables contain some odd values like2199999 and 0’s for number of total rooms and 1’s for house sizes. These odd values are replaced by220missing values. Binary values measure whether something is present or not, so missing values are221replaced by 0 in the dataset. A description of the variables for the 6 regions is given in Table 9222up to 14 in the Appendix. The floor specific variables (except rooms per floor) are summarized in223Table 15 for all regions together. When a variable has the same value for all houses in a dataset224the variable is omitted in the table. For the districts only the 5 biggest districts are given in the225tables, but all districts are used in the computations.2263.3227Demographic Data228According to many real estate brokers location is one of the most important characteristics that229determine the house price. In the dataset location is included in the terms of a district code. In230the different regions 3 levels of geographic positioning can be used: On municipality, district and231neighborhood level. Of course we want to include a positioning at the lowest possible level. How-232ever, the disadvantage of the use of neighborhood coding, is its number of possibilities. Because233there are many different neighborhoods and some of them are very small, there are neighborhoods234where during the time of one year only one or two houses are sold and often even none. The use235of neighborhoods coding will thus definitely lead to classification problems (because one tries to236estimate the price of a house positioned in a neighborhood unknown to the model) and overfitting237of the dataset (because many neighborhoods are represented in the dataset by just a couple of238houses). On the other hand the use of district coding has the disadvantage that, especially in239big cities as Amsterdam and Rotterdam, the defined districts are quite large, with up to 80, 000240inhabitants and a surface of up to 4, 000 hectares.241Due to these considerations we use another way represent location, namely with the use of242demographic data. Statistics Netherlands (Centraal Bureau voor Statistiek, CBS) publishes de-243mographic information for all the municipalities, districts and neighborhoods in the Netherlands.244This information is free accessible on the internet via StatLine (de Jonge, Reedijk, and Sluis 2005).245The neighborhood codes of the houses available in the dataset are matched with the CBS neigh-246borhood information. When a characteristic is missing for some neighborhood, this characteristic247is replaced by the value for the district the neighborhood belongs to. When this information is24812249250251===== PAGE 14 (score 18, numeric-paren 0, table-words 5, var-words 3) =====252253Region254Best Model255MAE256Improvement257MRE258Improvement259groningen260LSBoost (5)261€20110.4026226.47%26311.32%26432.01%265apeldoorn266LSBoost (5)267€26643.7426824.38%2699.31%27032.39%271eindhoven272LSBoost (5)273€21856.9527426.71%2758.06%27631.98%277amsterdam278LADBoost (5)279€29320.4728032.84%2819.38%28244.00%283rotterdam284LADBoost (5)285€22637.8828634.60%28710.91%28840.15%289zeeland290LSBoost (5)291€28914.2929231.16%29314.83%29438.03%295Table 3: Out-of-sample performance of the best models per region. Improvement is the improve-296ment with respect to the Linear Regression model. The number between the parentheses represents297the tree depth of the base learner. See text for more comments.298MAE299MRE300Region301LSBoost302LADBoost303Impr. LADB304LSBoost305LADBoost306Impr. LADB307groningen308€20110.40309€20141.72310-0.16%31111.32%31211.62%313-2.65%314apeldoorn315€26643.74316€27824.49317-4.43%3189.31%3199.86%320-5.91%321eindhoven322€21856.95323€21822.293240.16%3258.06%3268.09%327-0.37%328amsterdam329€29283.76330€29320.47331-0.13%3329.55%3339.38%3341.78%335rotterdam336€22067.10337€22637.88338-2.59%33910.96%34010.91%3410.46%342Table 4: Comparison of the results of LSBoost and LADBoost343of the optimal parameters is a time-consuming task that we considered to be beyond the scope of344our research.3454.2346Experiment 1: Data with Districtcoding347This subsection gives the results of the experiments using districtcoding but without demographic348variables. In Table 3 a summary of the best results is given. A complete overview of all results349can be found in Appendix D, Table 17.350On each dataset 10 boosting experiments are performed (5 LSBoost and 5 LADBoost, with351depths 1 up to 5), except on the Amsterdam and Rotterdam regions, where some less interesting352experiments were skipped, because of their long computation times. All these boosting experiments353outperformed the LR model on their dataset. Adding an extra layer always reduced both absolute354and relative error, but the effect of every extra layer is less than the previous one. The improvement355of a Boosting model with depth 5 with respect to a depth 4 model in terms of absolute error is356around €200. It is therefore expected that the effect of adding a 6th layer is negligible. There357are two possible reasons why extra layers increase the prediction performance. In the first place358because extra layers allow for extra interaction and in the second place, since the number of models359is constant, extra layers just can make more splits on the data and therefore predict more accurate.360The boosting models with depth 5 reduce the absolute error with around 25% – 35% and361relative error with around 30% – 45% with respect to the LR model.362But which of the two363boosting algorithms, LSBoost and LADBoost, performed best differed by dataset.364When we365look at the relative error, LADBoost is the best algorithm in the large cities of Amsterdam and366Rotterdam, where LSBoost performed best in the other 4 regions. However, the difference in367performance between the two algorithms is very small. Only in Apeldoorn and environs LSBoost368is really better than LADBoost. Since LADBoost tries to ignore outliers, it is possible that the369worse performance in Apeldoorn is explained by the bad prediction of expensive houses. In Table3704 a comparison is made between the results of both the boosting algorithms.3714.2.1372Submarkets373It is possible that prediction accuracy is very different among different types of houses or market374segments. In real estate literature market segments are often called submarkets. We analyze the375predictive performance of our models after a very simple market segmentation, in which each of376the six possible house types (detached, apartments etc.) represents a segment.377The prediction error (MAE or MRE) of a submarket is the weighted mean test error of that378submarket, i.e. in every test set the houses belonging to the submarket are collected and the37914380381382===== PAGE 15 (score 19, numeric-paren 0, table-words 5, var-words 7) =====383384Region385Best386Size387LSBoost388LADBoost389Worst390Size391LSBoost392LADBoost393groningen394mid-terraced3959013969.67%3979.24%398detached39930240024.30%40124.98%402apeldoorn403mid-terraced4045484055.08%4065.15%407detached40861540914.39%41014.82%411eindhoven412apartment4139034146.92%4156.93%416detached41737041813.11%41912.26%420amsterdam421apartment42267424239.06%4248.92%425detached4268042723.68%42821.90%429rotterdam430stepped431614328.00%4338.04%434detached43517343619.66%43718.57%438zeeland439stepped440344419.58%4429.83%443detached44456244520.29%44620.82%447Table 5: Best and worst predicted house type per region448mean is taken and multiplied by the number of houses belonging to the submarket in that test449set. Afterward the sum of all these means is taken and divided by the sum of the house counts.450Table 5 gives the best and worst predicted house types per region. More detailed results can451be found in Appendix D, Table 18.452The terraced houses, especially the mid-terraced house, are predicted relatively well in all453regions. There are many terraced houses and when looking at the variables included, we may454conclude that these datasets are especially created for terraced houses (and other single house-455hold residences). Many of the variables are uninteresting for apartments (gardens, floors etc.).456Apartments are also predicted well, except in Zeeland. Detached houses seem to be the biggest457problem, also when there are relative many houses of this type in the region, e.g. in Apeldoorn458and Zeeland. Even then detached houses are quite complex to predict. Detached houses are often459quite unique and besides the included variables also the architectonic style of the house may have a460high influence. Also by the other house types we see when a house becomes more exotic prediction461performance decreases. There are two approaches that may improve the prediction of these exotic462houses, but these approaches create also new problems. Both approaches are based on extending463the datasets. First we can include houses sold in other years in the dataset, but then we have464to correct the prices to compensate for market dynamics. The second approach is to combine465multiple regions. Disadvantage is we get an enormous number of districts, but maybe the use of466demographic data (and preferable other environmental variables) can replace the districts.467It was expected that LADBoost would predict ordinary houses better and LSBoost reached468better results on the larger dwellings.469The submarket results don’t confirm this expectation.470Sometimes LADBoost predicts detached dwellings even better than LSBoost. Overall there does471not seem to be a pattern in which algorithm predicts best on the different house types.4724.3473Experiment 2: Data with Demographic Variables474In Section 3.3 datasets were created where the district codes were replaced by a number of demo-475graphic variables on neighborhood level. The LSBoost and LADBoost, with depth 5, were also476trained on these datasets. In Table 6 the best models per region are listed. A complete overview477of the results is given in Table 19 in the Appendix.478The results of all the boosting models in all 6 regions are better on these new datasets compared479with the results on the old ones, discussed in Section 4.2. The size of the performance increase480differs among regions and models. Overall, the adaptation of the datasets has more influence481on the LSBoost models than on the LADBoost models. In the region of Amsterdam prediction482performance of the models has increased by about 10%, but in the region of Rotterdam this is483only 3% to 4%. It does not come as a surprise that Amsterdam is the region where performance484gain is highest, since the districts in Amsterdam are extremely large. With the use of demographic485information the model can discover a better separation between different locations.486We can conclude that LSBoost or LADBoost models with depth 5 and built with the use of487demographic data are the best choices among the models evaluated so far. These models improve488the Mean Relative Error with 35% up to 49% compared with the original Linear Regression model.48915490491492===== PAGE 16 (score 13, numeric-paren 0, table-words 2, var-words 7) =====493494Region495Best Model496MAE497Improvement498MRE499Improvement500groningen501LSBoost (5)502€18892.045036.06%50410.41%5058.04%506apeldoorn507LSBoost (5)508€25497.435094.30%5108.64%5117.20%512eindhoven513LSBoost (5)514€20748.225155.07%5167.71%5174.34%518amsterdam519LADBoost (5)520€26373.3652110.05%5228.51%5239.28%524rotterdam525LADBoost (5)526€21936.935273.10%52810.42%5294.49%530zeeland531LSBoost (5)532€27189.765335.96%53413.90%5356.27%536Table 6: Results of the best models per region with the use of demographic data. Improvement is537the improvement with respect to the model trained on the normal dataset.5385539Interpretation540Parametric techniques often have the advantage that a useful interpretation can be given to the541model parameters, e.g., in linear regression the model parameters can be interpreted as the partial542prices of the product characteristics. Although not parametric, regression trees are also highly543interpretable and can be written as an equivalent set of if-then rules. Boosted trees lack both544these appealing properties. Not all is lost, however, because two useful tools exist that can be545used for the interpretation of these ensemble models:546Relative importance plots, that visualize how important the various independent variables are547relative to one another in predicting the dependent variable. In regression trees, the relative548importance of a variable is measured examining the effect of each split on that variable on the549model outcome. Roughly speaking, this effect is high if the split results in a large difference550in model outcome for the right and the left subtree (with preferably an equal probability to551turn right or left) and/or the split is likely to occur. (The probability that a split occurs552is indicated by the number of patterns that travel through the corresponding node relative553to the number of patterns in the dataset. Generally this probability is higher for nodes554close to the root.) In an ensemble of regression trees such as built by boosting, the relative555importances are simply averaged over all trees in the ensemble.556Partial dependence plots, that visualize the partial dependence of the implemented function557on a subset of the independent variables. For a grid of values for this variable subset, the558‘expected’ output of the model at that point is computed. Ideally, this expectation at a grid559point should be computed with respect to the conditional distribution (given the grid point)560of the variables not in the subset. In practice it is obtained by averaging the model outputs561over all instances in the dataset, keeping the values in the selected variable subset fixed to562the grid point, thus using the dataset to approximate the distribution.563A more detailed description of these plot types is given in Appendix B.564Both relative importance plots and partial dependence plots are made for the six datasets565using an LSBoost model. An LSBoost model, with 300 CART base learners with depth 5 and566learningrate 0.1, is trained on the complete 2004 dataset of a region. First we will use the datasets567with district coding, but in Subsection 5.2 we will also create relative importance plots with the568inclusion of demographic variables and partial dependence plots of some demographic variables.5695.1570Interpretation: Data with Districtcoding571Figure 5 shows the relative importance plots of the 6 regions. Only the 10 most important variables572in a dataset are plotted. Except in Zeeland the most important variable in all regions is a variable573that measures some size: volume, size or lot size. In Zeeland kind is the most important variable,574but kind measures house size implicit. In general there are 6 variables that are in the top tens for575all 6 datasets: volume, size, lot size, kind, construction year and district. It is remarkable that,576although the size of the house is very important in the hedonic model, the number of rooms in a577house hardly has an influence. The total size of these rooms is far more important.578We created partial dependence plots for 5 of the 6 most important variables mentioned in the579previous section. Only for district no plots are created, because of the large number of different580values this variable can have.58116582583584===== PAGE 17 (score 5, numeric-paren 0, table-words 0, var-words 5) =====585586volume587kind588size589district590lot size591constr year592type593rooms GF594garage595garden size596Groningen597059820599406006060180602100603lot size604kind605volume606size607district608type609constr year610garage 2+611maint inside612garden maint613Apeldoorn614061520616406176061880619100620volume621lot size622type623kind624district625c year626size627maint out628bath 2F629blcny 1F630Eindhoven631063220633406346063580636100637size638volume639district640kind641lot size642maint inside643constr year644garden size645garden E−W646heating647Amsterdam648064920650406516065280653100654volume655size656lot size657district658kind659constr year660rooms661maint inside662type663garage664Rotterdam665066620667406686066980670100671kind672district673volume674lot size675c year676size677type678maint in679garage680heating681Zeeland682068320684406856068680687100688Figure 5: Relative Importance of predictors with Districtcoding.68917690691692===== PAGE 25 (score 16, numeric-paren 0, table-words 3, var-words 5) =====693694Lot Size695Volume696Apartments697Region69819856992004700%70119857022004703%70419857052004706%707Groningen708587.39709240.50710-59.06%711356.21712327.93713-7.94%71449.12%71546.06%716-6.22%717Apeldoorn718864.85719654.82720-24.29%721384.59722379.00723-1.45%72413.33%72522.04%72665.32%727Eindhoven728335.85729254.53730-24.21%731425.88732389.39733-8.57%7349.25%73523.48%736153.80%737Amsterdam738170.72739108.14740-36.65%741337.70742269.60743-20.17%74471.24%74579.26%74611.26%747Rotterdam748292.01749138.13750-52.70%751420.17752305.51753-27.29%75449.54%75561.46%75624.06%757Zeeland758675.99759467.29760-30.87%761428.07762377.89763-11.72%76416.20%76510.93%766-32.57%767Table 8: Differences of variable values between 1985 and 2004768index.769In all 6 regions the same pattern is visible only the absolute effects differ. Prices raises in the770first 10 years with 40% in Zeeland up to 90% in Amsterdam, according to the mean and median771indices. The Fisher hedonic shows raises of 58% in Zeeland up to 106% in Amsterdam, which772means that the price raise was accompanied by a quality decline.773In the next 5 years, 1995 up to 2000 these effects became even stronger. In 2000 houses were774almost 4 times so expensive in Amsterdam than in 1985. Corrected for quality this is even 5 times.775Although both effects are the strongest in Amsterdam this pattern is also visible in the other 5776regions. The quality decline is the largest in big cities, such as Amsterdam and Rotterdam, but777also in Eindhoven. In the 3 more rural regions the quality decline is less impressive. Due to778these differences in quality decline, mean and median indices give a wrong notion about in which779regions prices have increased the most. There is no discussion about the fact that Amsterdam is780the region with the highest price increase and Zeeland the region with the lowest. But for the781other 4 regions the mean and median indices give a deceptive view. The most striking example782is Rotterdam. According to the mean and median indices this region is positioned fith out of the783six regions in terms of price increase, but according to the hedonic indices it is positioned second,784since the quality decline of the dwellings in Rotterdam is relatively high.785Table 8 shows some means of important variables in the years of 1985 and 2004. When these786important characteristics have become of less quality (i.e. the average house volume has become787smaller, the average lot size has decreased or more dwellings are apartments) this would be a788decent explanation of the quality decline the hedonic models found between 1985 and 2004. Both789lot size and volume have decreased during these 20 years in all of the 6 regions. Lot size with 24%790in Eindhoven up to 59% in Groningen and volume with 1% in Apeldoorn up to 27% in Rotterdam.791Also the relative amount of apartments has increased in 4 of the 6 regions. The regions with a large792decline in the average house volume are also the 2 regions with a large quality decline according793to the hedonic price index: Amsterdam and Rotterdam. The quality decline during the 90’s can794therefore mainly be explained by the fact that houses have become smaller during this period.795These hedonic indices give only a general overview of all possible dwellings. Since we can see796in Table 8 that in some regions one reason of the decline in quality is the increase of the portion of797apartments, it is possible that the average qualities of the different house types did not decrease.798When one wants to check whether this is the case one could create hedonic indices per house type.799Although the hedonic indices give a good impression of the quality differences during a period,800they are not 100% reliable. There are three reasons for this.801The first reason is a theoretic one and discussed in the previous section. Both the Paasche and802Laspeyres index do not account for substitution effects. The Fisher’s Ideal index only estimates803these effect by using the geometric mean of the other two indices. However, the Laspeyres and804Paasche index are situated very close to each other and thus the Fisher’s index seems to be a good805estimation.806In the second place the boosting models make prediction errors. Since the data quality and807quantity become less going further back in history, this will be a problem. Especially for the808Paasche-like index, because this approach uses only the base period model.809The third reason has to do with the meaning of the variables. There are two variables which810meanings change over time. The first is construction year. This variable measures in fact two811things: the age of the house and the period it is built in, e.g. the architectonic style and average812quality of houses built in that period. This means a construction year of 1980 has a different81325814815816===== PAGE 26 (score 0, numeric-paren 0, table-words 0, var-words 0) =====81781819858191995820100821200822300823400824500825Groningen826year827index82819858291995830100831200832300833400834500835Apeldoorn836year837index83819858391995840100841200842300843400844500845Eindhoven846year847index84819858491995850100851200852300853400854500855Amsterdam856year857index85819858591995860100861200862300863400864500865Rotterdam866year867index86819858691995870100871200872300873400874500875Zeeland876year877index878Figure 10: Hedonic and normal indices of the six regions: Fisher’s Hedonic Index (solid), Mean879Index (dashed) and Median Index (dotted).88026881