Spatial Analysis of Tokyo Apartment Market

Spatial Analysis of Tokyo Apartment Market Morito Tsutsumi 1, Yasushi Yoshida, Hajime Seya 3, Yuichiro Kawaguchi 4 1 Department of Policy and Planning Sciences, University of Tsukuba 1-1-1 Tennodai, Tsukuba-city, 305-8573, Japan (tsutsumi@sk.tsukuba.ac.jp) Chiba University of Commerce 1-3-1 Konodai, Ichikawa-city, 7-851, Japan (yyoshida@mug.biglobe.ne.jp) 3 Graduate School of Systems and Information Engineering, University of Tsukuba 1-1-1 Tennodai, Tsukuba-city, 305-8573, Japan (seya0@sk.tsukuba.ac.jp) 4 Graduate School of Finance, Accounting and Law, Waseda University 1-4-1 Nihonbashi, Chuo-ku, Tokyo, 103-007, Japan (ykawaguchi@waseda.jp) Abstract. This study deals with apartment rent data which has been actually observed in Tokyo apartment market. It makes spatial analysis applying hedonic approach and discusses the spatial characteristics of the market. It also shows the significance of considering spatial autocorrelation of the errors in spatial hedonic model. Keywords: real estate, apartment rent, spatial autocorrelation, kriging, spatial autoregressive error model 1 Introduction Empirical researches on real estate data using spatial econometrics and spatial statistics approaches are developing. However, there are still quite a limited number of researches on Japanese market. Furthermore, most of them use socalled officially assessed land price data, which are provided by the Ministry of Land Infrastructure and Transport. The problem with their works is that the land price data are characterized by data distortions. More specifically, the data reflect ministry s opinion about land market and also contain appraisal smoothing bias. One of the advantages of this study is that it deals with the rent data which has been actually observed in Tokyo housing market. It employs hedonic approach for rent analysis. Although hedonic price or rent regression for real estate has been an indispensable tool in regional and transport analyses, not enough attention had been paid to its estimation in empirical applications except to multicollinearity of the variables. Spatial correlation and heteroscedasticity of error terms that violate the independence assumption on which the statistical analysis is based are often encountered and affect the parameter estimation, but little attention had been paid to them. In the next chapter, after brief description of data set, this study makes preliminary analysis and identifies the overall market characteristics. Then, it examines the residual spatial autocorrelation in a conventional regression model. It interprets the geographic distribution of the residuals from the view point of Tokyo s geological background. As a countermeasure for residual spatial autocorrelation, introducing sophisticated approaches developed in such as quantitative geography, spatial econometrics and spatial statistics have been used recently. However, introducing representative variable, such as dummy variable for zone, is a very simple idea and still often applied in practice. With regard to the latter, the study examines the effect of dummy variables specific to wards. Chapter 4 gives a brief overview of the methodology this study employs. It describes the models, more specifically, regression kriging developed in geostatistics and so-called spatial autoregressive error model in spatial econometrics which the study uses to consider residual spatial autocorrelation. In Chapter 4, the study estimates the models and compares those results and examines their performances. Furthermore, several parameter estimation methods are employed and their performances are compared. The results indicate the qualitative market characteristics. Then, the significance of spatial econometrics and spatial statistics approaches for the analysis of real estate data considering spatial dependence is discussed. Chapter 5 concludes the study. Preliminary Analysis based on Conventional Hedonic Model.1 Data Set and Overall Market Characteristics For empirical analysis, the data set about apartment rent and various kinds of attributes is provided by At Home Co., a provider of real estate market information and relevant support service agency in Japan. They have constructed a network for real estate information where about 56,000 real estate agencies are affiliates. The data set includes the data on the following attributes: position coordinate (latitude and longitude), rent, age, floor area, structure such as reinforced concrete, room type and number of rooms, sewage, gas, parking lot, air conditioner, bathroom, and so on. The present study uses a sample of 150 observations from the data. All of them are for January through May 006.

Morito Tsutsumi, Yasushi Yoshida, Hajime Seya, Yuichiro Kawaguchi Figure 1 shows the study area, 3 wards of Tokyo including three central wards of Chiyoda, Chuo and Minato. Chiyoda ward is the center of the city and in many ways the center of all Japan which has the Imperial Palace and the Diet, Tokyo Central Station and buildings of ministries and many large corporate headquarters. Chuo is historically the main commercial center of Tokyo, especially before World War II. In Minato ward, many embassies and many highclass apartments which are rented mainly by foreign executives of foreign firms are situated. Figure shows the location of observation properties and their rent values in terms of yen per square meter per month (ca. 1 US dollar = 10 Japanese yen, in May 007). The sample having the highest rent per square meter per month is located in Minato ward samples having high rent are located southwestward. In the Tokyo area, rents in the southern and western areas is said to be generally high. On the other hand, rent in the northern and eastern areas is said to be generally low. Tokyo s central three wards Chiyoda Chuo Minato Figure 1. Tokyo's 3 wards including central three wards Figure. Study area and location of observation points. Conventional Hedonic Model Let the following standard multiple linear regression model be as a basic model: y = X β + u (1) where y is an n 1 vector of log apartment rent; X is an n k matrix of the apartment attributes; β is an unknown parameter vector; and u is an n 1 vector of residuals. The standard assumptions we make about the residuals u in eq. (1) are E u = and Var( u) = σ I, where I is a unit matrix. As is well known, eq. (1) is referred to as hedonic model. ( ) 0 Table 1. Parameter estimates for the basic hedonic model Variable Coef. std dev. t constant 9.5581 0.88 33.80 train -0.7674 0.0914-8.40 BoT, Tokyo * -0.0337 0.0454-0.74 Odakyu * 0.0698 0.1909 0.37 walk -0.138 0.0487 -.54 bus -0.0359 0.0107-3.35 floor area (log.) -0.517 0.1484-1.70 age (sqr.) -0.044 0.0136-3.5 reinforced concrete * 0.018 0.0488 0.45 nos. of rooms -0.0136 0.034-0.40 one-room type * 0.108 0.066 1.55 1K-type * 0.1015 0.0601 1.69 parking lot * 0.497 0.0904.76 self-locking * 0.0470 0.0373 1.6 variance of error 0.0339 - - R-squared adjusted R-squared 0.577 0.537

Spatial Analysis of Tokyo Apartment Market 3 The explained variable is the logarithm of apartment rent [yen/ m per month]. The explanatory variables chosen after trial and error are: constant (intercept), time distance from the nearest station to central Tokyo by train or subway [minutes] (taking the average between the time required to Shinjuku station and the time to Tokyo or Otemachi station), Toei subway, which is operated by Bureau of Tokyo Metropolitan Government [dummy], Odakyu Electric Railway [dummy], time distance from the apartment to the nearest station by walk or bus [minutes], the logarithm of floor area [m ], the root square of age [year], reinforced concrete structure [dummy], the numbers of rooms, so-called one-room type where a very small kitchen is equipped [dummy], 1K type which is one room apartment with separate kitchen [dummy], parking lot [dummy], self-locking [dummy]. Parameter estimates for the conventional hedonic model given by Ordinary Least Squares (OLS) method are presented in Table 1..3 Detecting Spatial Pattern of the Residuals Figure 3 illustrates the spatial distribution of the residuals in the conventional hedonic model described in Section.. It indicates the existence of the residual spatial autocorrelation. More specifically, positive residuals are found southwest, especially high positive residuals in Minato ward. Negative residuals are found northeast. Figure 3. Spatial distribution of the residuals in conventional hedonic model Tests for the presence of spatial autocorrelation by Moran's statistic are carried out. The weighting system is defined as the functions of the physical distances between points. As the values of Moran's statistic depend on the assumed structure of W, in this study, the following five types of function are considered. i j j Table. Type of weight matrix component I II III IV V w = c d w = c d w = c exp( 0. 5 d ) w c exp d w = c exp d i = j w = 0 j j = ( ) Table 3 presents the result of Moran's tests for the residuals. In all cases, null hypothesis of no spatial dependence is rejected at the significance level of 1 %. Table 3. Test for spatial autocorrelation in the residuals of conventional hedonic model ii j j wight function I II III IV V moran's I 0.0 0.440 0.436 0.079 0.34 Z 11.6 8.3 10.3 19.8 14.4 Probabilities for normal distribution to exceed the value of Z 0.00 0.00 0.00 0.00 0.00

Morito Tsutsumi, Yasushi Yoshida, Hajime Seya, Yuichiro Kawaguchi.4 Introducing Dummy Variables against Residual Spatial Autocorrelation As a countermeasure for residual spatial autocorrelation, introducing representative variable, such as dummy variable for zone, is a very easy to handle and still often applied in practice although more sophisticated approaches have been used recently. Thus, the study examines the effect of dummy variables specific to zones. Chiyoda ward is chosen for the base to which no dummy variable is assigned. Consequently, dummy variables are added to the model. The result of parameter estimation is presented in Table 4. According to the t-values, 16 out of newly added dummy variables are estimated to be significant at the level of 5 %. Only those for Shibuya and Minato wards are positive and others negative. Table 4. Parameter estimates for the hedonic model having dummy variables for wards Dummy variables for wards Variable Coef. std. dev. t constant 9.6975 0.451 39.57 train -0.3138 0.1346 -.33 BoT, Tokyo * -0.038 0.0361-1.06 Odakyu * -0.1471 0.147-1.00 walk -0.1073 0.039 -.74 bus -0.049 0.0081-3.07 floor area (log) -0.5684 0.1187-4.79 age (sqr) -0.0516 0.0104-4.95 reinforced concrete * 0.0101 0.0385 0.6 nos. of rooms 0.0069 0.070 0.6 one-room type * -0.0010 0.0519-0.0 1K-type * 0.0186 0.0461 0.40 parking lot * 0.179 0.074.39 self-locking * 0.0341 0.075 1.4 Chuo * -0.14 0.0763-1.60 Minato * 0.1606 0.0759.1 Shinjuku * -0.0963 0.0703-1.37 Bunkyo * -0.313 0.1058 -.19 Taito * -0.175 0.0961-1.8 Sumida * -0.89 0.0760-3.80 Koutou * -0.841 0.0846-3.36 Shinagawa * -0.1093 0.0851-1.8 Meguro * -0.011 0.0887-0.14 Ota * -0.91 0.117 -.03 Setagaya* -0.0838 0.1053-0.80 Shibuya * 0.4397 0.0990 4.44 Nakano * -0.690 0.1068 -.5 Suginami * -0.1835 0.0896 -.05 Toshima * -0.1661 0.0775 -.14 Kita * -0.3006 0.0851-3.53 Arakawa * -0.3098 0.0815-3.80 Itabashi * -0.4174 0.108-3.86 Nerima * -0.507 0.0995 -.5 Adachi * -0.3891 0.1044-3.73 Katsushika * -0.4053 0.109-3.71 Edogawa * -0.3143 0.0958-3.8 variance of error 0.0168 R-squared adjusted R-squared 0.85 0.771 Figure 4 and 5 illustrates the geographical distribution of the coefficients and t-values of dummy variables for wards. It apparently shows that the dummy variable for Shibuya ward, which is located west of Minato ward, is most significant. Shibuya is known as one of the fashion centers of Japan, particularly for young people. In addition, it includes well-known commercial and residential districts. On the other hand, significantly negative values are found north and east. These results correspond to those in the previous section and indicate the market structure.

Shibuya Minato Figure 4. Visualizing the coefficients of dummy variables for wards Figure 5. Visualizing the t-values for dummy variables Figure 6 illustrates the spatial distribution of the residuals in the present model. Compared with Figure 3, it does not indicate the existence of the residual spatial autocorrelation at a glance. Figure 6. Spatial distribution of the residuals after introducing dummy variables for wards Table 5 presents the result of Moran's tests for the residuals. In case of using type I and II of weight function, null hypothesis of no spatial dependence is again rejected at the significance level of 5 %. However, in case of using exponential types of weight function (type III to V), null hypothesis of no spatial dependence is not rejected. Although this result does not advocate dismantlement of spatial dependence in the residuals, it implies that introduction of such representative variables as widely used in practice let a model satisfy the assumption for OLS. However, this approach may increase the number of parameters, which lead to generalization problem, i.e., worsen the prediction ability. Table 5. Detection of spatial autocorrelation in the hedonic model with dummy variables for wards Weight Matrix I II III IV V moran's I 0.03 0.079-0.08-0.018 0.000 Z.185 1.613-0.945-0.74 0.094 p 0.01 0.05 0.83 0.61 0.46 Thus, the study makes models without using dummy variables for wards in the following chapters.

Morito Tsutsumi, Yasushi Yoshida, Hajime Seya, Yuichiro Kawaguchi 3 Spatial Hedonic Modeling 3.1 Overview of Modeling Approach Spatial model is an essential tool for both practitioners and academics to analyze real estate data where spatial association or correlation is a key concept. Therefore, modeling techniques have been introduced from the fields of geography, geostatistics, spatial econometrics and so on (e.g. Benirschka (1994), Can (199), Dubin (1988), Velante et al. (005)). However, empirical analyses on real estate data are often faced with data problems. This study makes models based on both geostatistics and spatial econometrics. 3. Geostatistics Approach An alternative to consider the correlation among the residuals is the following regression kriging model: () I y = X β + ε, Σ Var ε = σ H ( φ) + τ, () where Σ is the n n covariance matrix of the errors, σ is so-called partial-sill variance, and τ is nugget variance. H (φ) is an n n matrix having ( i, j) th entry exp( si s j φ), where s i denotes the location coordinates of i th observation t and φ is a parameter. Let θ = ( β, σ, τ, φ). Velante et al. (005) uses the same type of model and demonstrates its significance by analyzing the asking rent data in US. They separate the results into three groups by the relative success of the spatial model over the conventional regression model. Kim et al. (003) uses kriging to estimate the values of unobserved explanatory variables for spatial hedonic model based on conventional spatial econometrics approach. On the contrary, this study uses regression kriging to estimate the rent values themselves similar to Valente et al (005). Knight et al. (1998) discusses a variety of data problems that confront real estate researches such as missing data, measurement error and censored data. They demonstrate impressive gains by using the Gibbs sampler to deal with missing data in a hedonic house price model. However, the deficiency with Kight et al. s work is that their models lack empirical spatial considerations. Tsutsumi et al. (006) employs Bayesian spatial modeling and improves upon the results without spatial consideration significantly. The present study employs the following four methods for parameter estimation. (i) Generalized least squares method and weighted least squares method (GLS-WLS): The method of weighted least squares is applied for fitting variogram model. Then the covariance matrix Σ is calculated based on the results and the trend parameter is estimated by generalized least squares method. (ii) Maximum likelihood method (ML): It assumes that the error obeys normal distribution: y = X β + ε, ε~ N( 0, σ H ( φ) + τ I), (3) then, maximize the log likelihood, L ( β,, τ y, X ) φ. (iii) Restricted maximum likelihood method (RML): According to Kitanidis and Lane (1985), maximum likelihood method often yields biased estimates especially unless the number of observation is large. Restricted maximum likelihood method is a counter measure for the problem and this method is also used in the present study. For more detail, see Kitanidis and Lane (1985). (iv) Markov Chain Monte Carlo (MCMC): We assume the prior distribution of θ, p(θ), can be formulated as p( θ ) = p( β ) p( σ ) p( τ ) p( φ ). Eq.() can be recast as a hierarchical model y θ, w ~ N( Xβ + w, τ I), where w σ, φ ~ N( 0, σ H ( φ)). The priors chosen for the present study are β ~ N( c, T ), σ ~ IG( aσ, bσ ), τ ~ IG( aτ, bτ ), and p ( φ) 1 φ, where IG(, ) denotes Inverse Gamma distribution. In this paper, we abbreviate to show full conditional distributions for sampling under these specifications due to space limitation. See Banerjee et al. (003) for more detail. As sampling methods, Gibbs sampler is used for ( β, w, τ, σ ) and Metropolis-Hastings algorithm for (φ).

3.3 Spatial Econometrics Approach Spatial Analysis of Tokyo Apartment Market 7 Another alternative to consider the correlation among the residuals is the following regression model called spatial autoregressive error model: y = X β + ε, ε = λ Wε + u, (4) where W = { w } is called spatial weight matrix, which denotes the effect of each zone, λ is a parameter, u is an error vector. In order to determine the effects of spatial autocorrelation, we must design the spatial weight matrix W. The weighting system is often defined as the functions of the physical distances between points such as shown in Table, where c j is a constant which leads w. (5) i =1 This study employs five types of weight matrix shown in Table. The study assumes that the error vector, u, obeys normal distribution u~n( 0, σ u I) and applies maximum likelihood (ML) method to estimate the parameters of the model. Kriging, explained in the previous section, assumes spatial stationarity which enables us to predict the value at arbitrary point or site. However, since spatial econometrics approach does not assume spatial stationarity, it cannot be used for predicting the value of the data which does not enter into parameter estimation. Haining (1990) calls the approach which develops a description for the observed data first and uses this to predict the values of unobserved data "sequential approach". Predicting based on sequential approach violates the assumptions on spatial weight matrix, so it is not suitable for the models based on spatial econometrics approach. An alternative is called "simultaneous approach" where unobserved value is regarded as missing value and both parameter values and missing values are estimated simultaneously. Referring to Martin (1984) and Haining (1990), suppose we have n samples and values on rent are missing from h of these. Let y = y o y p where y o denotes the ( n h) dimensional vector of observed rent values and y p denotes the 1 h dimensional vector of unknown rent values. Herewith, the covariance matrix of ε ( = ( I λ W ) u), Var ( ε), can be formulated as Var() ε V oo Vop (6) = Vpo Vpp so that V is the covariance matrix for the sub-vectors of y and y above mentioned. Maximizing log likelihood qr ( ) function, L β, σ, λ, y y, leads to the estimator of y, ŷ : u p o 1 ( Vˆ ) ( y X ˆ β ) p y ˆ = X ˆ β + Vˆ. (7) p p po oo o o p For more detail, see Martin (1984) or Haining (1990). To examine the prediction ability of the model, this study applies simultaneous approach and presents the results in the next chapter. q r 4 Empirical Results 4.1 Comparison among the Models and the Estimation Methods The estimation results of the basic regression model and the present spatial models, i.e., regression kriging and spatial autoregressive error model, are shown in Table 6 and 7. The number of iterations for MCMC is 10, 000 and the discarded (i.e. burn-in) is 1,000. As mentioned above, the present study tests five types of weight matrix for spatial autoregressive error model. These tables show significant success of the spatial models over the conventional regression model. It is interesting that kriging works better that spatial autoregressive error model despite of its unrealistic assumption, stationarity.

Morito Tsutsumi, Yasushi Yoshida, Hajime Seya, Yuichiro Kawaguchi Table 6. Comparative reuslt of parameter estimates for basic regression model and univesal kriging models Basic Regression Model Regression Kriging Estimation Method OLS WLS-GLS ML REML MCMC Variable Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t constant 9.5581 0.88 33.80 9.5358 0.775 34.45 9.38 0.7 34.46 9.3330 0.3007 31.04 9.577 0.373 40.34 train -0.7674 0.0914-8.40-0.440 0.141-5.40-0.3541 0.1341 -.64-0.395 0.147 -.31-0.4663 0.1143-4.08 BoT, Tokyo * -0.0337 0.0454-0.74-0.0398 0.0379-0.89-0.0454 0.0339-1.34-0.0466 0.0354-1.3-0.0403 0.0381-1.06 Odakyu * 0.0698 0.1909 0.37-0.1016 0.1730 0.40-0.1313 0.1517-0.87-0.1385 0.1584-0.87-0.0971 0.1740-0.56 walk -0.138 0.0487 -.54-0.1100 0.0373-3.3-0.1078 0.0338-3.19-0.1071 0.0353-3.03-0.1119 0.0380 -.94 bus -0.0359 0.0107-3.35-0.0301 0.0097-3.7-0.074 0.0087-3.16-0.063 0.0091 -.88-0.0307 0.0099-3.10 floor area (log) -0.517 0.1484-1.70-0.491 0.1103 -.8-0.4785 0.1011-4.73-0.4757 0.1058-4.50-0.494 0.110-4.40 age (sqr) -0.044 0.0136-3.5-0.047 0.0099-4.49-0.0480 0.0090-5.3-0.0483 0.0094-5.1-0.0468 0.0099-4.73 reinforced concrete * 0.018 0.0488 0.45-0.009 0.0371 0.59-0.013 0.0341-0.36-0.0135 0.0358-0.38-0.0097 0.0381-0.5 nos. of rooms -0.0136 0.034-0.40 0.0010 0.053-0.54 0.008 0.031 0.1 0.008 0.04 0.1 0.001 0.057 0.05 one-room type * 0.108 0.066 1.55 0.0551 0.0464.1 0.0659 0.048 1.54 0.0678 0.0448 1.51 0.0539 0.0493 1.09 1K-type * 0.1015 0.0601 1.69 0.0383 0.044.30 0.0407 0.0407 1.00 0.0414 0.046 0.97 0.0387 0.043 0.90 parking lot * 0.497 0.0904.76 0.09 0.075 3.45 0.1974 0.064 3.07 0.196 0.0669.93 0.079 0.0716.90 self-locking * 0.0470 0.0373 1.6 0.030 0.063 1.79 0.065 0.04 1.10 0.059 0.053 1.0 0.0309 0.088 1.07 variance of error 0.0339 nugget (tau^) 0.0091 0.058 0.0107 0.0088 partial-sill (sigma^) 0.047 0.0096 0.0379 0.04 range (phi) 3.4015 7.395 10.6900 3.0705 R-squared 0.577 0.944 0.916 0.91 0.944 adjusted R-squared 0.537 0.938 0.908 0.904 0.939 * dummy variable Table 7. Comparative reuslt of parameter estimates for spatial autoregressive error model Weight Matrix I II III IV V Variable Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t constant 9.6993 0.3 41.77 9.6860 0.3804 5.46 9.537 0.978 3.01 9.8819 0.39 41.31 9.6936 0.474 39.18 train -0.6688 0.1058-6.3-0.6737 0.0907-7.43-0.5097 0.1157-4.41-0.6991 0.118-5.91-0.5403 0.171-4.5 BoT, Tokyo * -0.011 0.0385-0.9-0.0106 0.0378-0.8-0.0439 0.0353-1.5-0.0317 0.0411-0.77-0.0395 0.0371-1.07 Odakyu * -0.005 0.1399-0.15-0.0078 0.1494-0.05-0.1395 0.1364-1.0-0.0810 0.1389-0.58-0.155 0.137-0.95 walk -0.1088 0.0363 -.99-0.113 0.0390 -.88-0.103 0.0349-3.45-0.16 0.034-3.59-0.1177 0.033-3.54 bus -0.0309 0.0078-3.95-0.034 0.0084-3.87-0.083 0.0077-3.67-0.0316 0.0078-4.03-0.091 0.0075-3.89 floor area (log) -0.4161 0.1085-3.83-0.3875 0.1184-3.7-0.4649 0.1073-4.33-0.4917 0.107-4.59-0.514 0.103-4.98 age (sqr) -0.0500 0.0096-5.18-0.0496 0.0106-4.70-0.0507 0.0097-5. -0.051 0.0097-5.40-0.0514 0.0093-5.53 reinforced concrete * 0.0007 0.0358 0.0 0.0109 0.0387 0.8-0.0106 0.0368-0.9-0.0087 0.0354-0.4-0.0131 0.0349-0.38 nos. of rooms -0.0036 0.050-0.14 0.0001 0.069 0.00 0.0107 0.044 0.44 0.0078 0.044 0.3 0.0136 0.034 0.58 one-room type * 0.0697 0.0449 1.55 0.0899 0.0509 1.77 0.0785 0.0464 1.69 0.0559 0.0440 1.7 0.0616 0.0434 1.4 1K-type * 0.0699 0.0409 1.71 0.0751 0.0460 1.63 0.0473 0.048 1.10 0.047 0.04 1.1 0.0403 0.0409 0.99 parking lot * 0.305 0.0850.71 0.3311 0.0886 3.74 0.170 0.0670 3.4 0.1767 0.0768.30 0.1874 0.069.71 self-locking * 0.030 0.055 1.6 0.0339 0.085 1.19 0.078 0.061 1.06 0.0397 0.053 1.57 0.0333 0.046 1.35 variance of error 0.0183 0.009 0.0180 0.0183 0.0169 lambda 0.6887 0.9577 0.979 0.5898 0.7771 R-squared 0.748 0.71 0.753 0.748 0.768 adjusted R-squared 0.74 0.685 0.79 0.73 0.746 loglikelihood 77.44 7.93 8.81 76.05 83.60 4. Further Discussion In Table 6, estimated nugget plus partial sill by those methods except for REML equals c.a. 0.03, which corresponds with the variance of error estimated by OLS. This could imply the regression kriging model is identified properly. The values of estimated range which represents the extent of existing spatial correlation are shown in terms of kilometer. Thus, the estimated range is about 3 km to 10 km. Figure 7 illustrated the plot of semi-variance values and best-fitted semivariogram for WLS-GLS and ML. As is often pointed out, WLS-GLS leads to robust estimation, and consequently gives rather small range estimates in this case. From the view point range estimation, MCMC gives similar result. To identify the market characteristics and consider business strategy, these results are quite interesting. In order to examine the prediction ability of the models, we use 50 observations for validation apart from the 150 observations shown in Figure 1 and used for estimates, and calculate root mean square error (RMSE). The conventional regression model estimated by OLS and regression kriging models use the parameters presented in Table 6 to predict the rent values of 50 points. As mentioned in Section 3.5, spatial econometrics models require simultaneous approach to predict the values of the data which are not used for parameter estimation. The results based on the approach are presented in Table 8. Some estimates such as for the variable of one-room type drastically changes from those in Table 7. This study also applies simultaneous approach for kriging employing maximum likelihood method and presents the result in Table 9, which is similar to that in Table 6.

Spatial Analysis of Tokyo Apartment Market 9 ML WLS-GLS Figure 7. Plot of semi-variance values and best-fitted semivariogram Table 10 summarizes the results showing the calculated RMSE for these models. Among the four parameter estimation methods for regression kriging, ML gives the best root mean square error (RMSE). Nevertheless, other methods also work out. Spatial autoregressive models using the exponential types of weight function also works out. As is well known, the specification of spatial weight matrix affected the results to a remarkable degree. Table 8. Resutl of parameter estimates for spatial autoregression error model based on simltanoues approach Weight I II III IV V Variable Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t Coef. std. dev. t constant 9.6588 0.384 5.14 9.573 0.167 57.7 9.4545 0.3051 30.99 9.5330 0.1891 50.4 9.7048 0.1749 55.47 train -0.6871 0.0685-10.03-0.6066 0.0815-7.45-0.5088 0.0895-5.69-0.4501 0.099-4.54-0.581 0.0967-6.01 BoT, Tokyo * -0.0116 0.031-0.37-0.0063 0.086-0. -0.0431 0.084-1.5-0.0404 0.083-1.43-0.038 0.0305-1.07 Odakyu * 0.0001 0.09 0.00-0.0307 0.0783-0.39-0.1300 0.0837-1.55-0.104 0.0766-1.57-0.0793 0.0759-1.04 walk -0.1133 0.084-3.98-0.1083 0.04-4.47-0.105 0.054-4.74-0.1103 0.033-4.73-0.1159 0.031-5.0 bus -0.039 0.0064-5.16-0.0335 0.0069-4.83-0.085 0.0063-4.49-0.0304 0.0069-4.41-0.0336 0.0069-4.88 floor area (log) -0.3784 0.0845-4.48-0.3767 0.0711-5.30-0.4467 0.0767-5.83-0.4850 0.0711-6.8-0.4610 0.0713-6.46 age (sqr) -0.0498 0.008-6.09-0.050 0.0067-7.81-0.0511 0.0073-6.97-0.0515 0.0067-7.70-0.050 0.0067-7.75 reinforced concrete * 0.0103 0.060 0.40-0.0018 0.00-0.09-0.011 0.038-0.47-0.0139 0.011-0.66-0.0111 0.005-0.54 nos. of rooms -0.0008 0.001-0.04-0.0168 0.0165-1.0 0.0079 0.0180 0.44 0.0046 0.0163 0.8-0.0047 0.0163-0.9 one-room type * 0.0913 0.0378.4 0.0670 0.030. 0.0806 0.0340.37 0.064 0.0306.04 0.050 0.097 1.75 1K-type * 0.0757 0.0357.1 0.066 0.088.30 0.050 0.034 1.55 0.0440 0.095 1.49 0.0547 0.09 1.87 parking lot * 0.3141 0.0537 5.85 0.6 0.0407 5.55 0.19 0.0441 4.97 0.1886 0.041 4.58 0.1870 0.0417 4.48 self-locking * 0.0341 0.00 1.55 0.03 0.0174 1.8 0.048 0.0199 1.5 0.048 0.0178 1.39 0.081 0.0175 1.60 variance of error 0.016 0.01 0.013 0.011 0.01 lambda 0.973 0.813 0.964 0.875 0.730 loglikelihood 14.59 143.5 141.06 148.68 140.05 Table 9. Resutl of parameter estimates for kriging by maximum likelihood based on simltanoues approach Variable Coef. std. dev. t constant 9.3617 0.7 34.39 train -0.3301 0.1341 -.46 BoT, Tokyo * -0.0453 0.0339-1.34 Odakyu * -0.1380 0.1517-0.91 walk -0.1060 0.0338-3.14 bus -0.065 0.0087-3.06 floor area (log) -0.4885 0.1011-4.83 age (sqr) -0.0477 0.0090-5.9 reinforced concrete * -0.0140 0.0341-0.41 nos. of rooms 0.0017 0.031 0.07 one-room type * 0.0610 0.048 1.4 1K-type * 0.0384 0.0407 0.94 parking lot * 0.1957 0.064 3.05 self-locking * 0.073 0.04 1.13 nugget 0.0063 partial-sill 0.06 range 8.070

Morito Tsutsumi, Yasushi Yoshida, Hajime Seya, Yuichiro Kawaguchi Table 10. Summary of the prediction results for the models Approach sequential simultaneous non-spatial spatial Model Regression Kriging Spatial Autoregressive Error Model OLS Estimation Method WLS ML REML MCMC ML ML Weigh Matrix --- --- --- --- --- --- I II II IV V RMSE 0.769 0.554 0.549 0.550 0.553 0.551 0.635 0.589 0.567 0.563 0.600 5 Concluding Remarks This study analyzed the Tokyo's 3 wards apartment market applying hedonic approach. Preliminary analysis gave the overall market characteristics and suggested the spatial consideration in the error terms. Then, the study applied regression kriging developed in geostatistics and so-called spatial autoregressive error model in spatial econometrics. Below are main conclusions. The residuals in the conventional hedonic model indicated strong spatial association in the market. As a countermeasure for residual spatial autocorrelation, the study applied the way of introducing dummy variables for wards as representative variables, which is still often used in practice, and verified its effect. The results showed the advantage of regression kriging over conventional regression model in prediction precision although it assumes stationarity across the area, which appears to be unrealistic. The study applied several parameter estimation methods for kriging and presented that the estimates of range particularly differs among the methods. The results based on kriging identified the range which represents the extent of existing spatial correlation. The estimated range is about 3 km to 10 km. The results also showed the advantage of spatial autoregressive error model over conventional regression model. The study presented that the simultaneous approach for spatial autoregressive error model in which the unobserved rent is regarded as missing value and both observed and missing values are contained, made good predictions. The study applied several types of spatial weight matrix for spatial autoregressive error model and compared the results. As is often mentioned, the specification of spatial weight matrix affected the results to a remarkable degree. Acknowledgement. The authors would like to thank At Home Co., Ltd. for their collaboration on providing the apartment rent data. References 1. Anselin, L. (1988) Spatial Econometrics: Methods and Models. Dordrecht : Kluwer Academic.. Banerjee, S., Carlin, B.P. and Gelfand, A.E.(003): Hierarchical Modeling and Analysis for Spatial Data, Chapman & Hall/CRC. 3. Benirschka, M. and Binkley, J. K.(1994) : Land price volatility in a geographically dispersed market, Journal of the American Journal of Agricultural Economics, 76, 185-195. 4. Can, A.(199): Specific and estimation of hedonic housing price models, Regional Science and Urban Economics,, 453-474. 5. Cressie, N.(1985): Fitting variogram models by weighted least squares. Mathematical Geology, 17, 563-586. 6. Dubin, R.A. (1988): Estimation of regression coefficient in the presence of spatially autocorrelated error terms. The Review of Economics and Statistics, 70, 466-474. 7. Haining, R.(1990): Spatial Data Analysis in the Social and Environmental Sciences, Cambridge University Press. 8. Kim C. W., Phipps T. T. and Anselin L. (003): Measuring the benefits of air quality improvement: a spatial hedonic approach, Journal of Environmental Economics and Management, 45 (1), 4-39. 9. Kitanidis,P.K. and Lane,R.W.(1985): Maximum Likelihood Parameter Estimation of Spatial Processes by the Gauss-Newton Method, Journal of Hydrology,79, 53-71. 10. Knight, J. R., Sirmans, C. F., Gelfand, A. E. and Ghosh S. K.(1998): Analyzing Real Estate Data Problems Using the Gibbs Sampler, Real Estate Economics 6 (3), 469-49. 11. Martin, R. J. (1984): Exact maximum likelihood for incomplete data from a correlated gaussian process, Communications in Statistics: Theory and Methods, 13, 176-188. 1. Tsutsumi, M. Yoshida, Y., Seya, H., Kawaguchi, Y.: Bayesian Spatial Modeling for Apartment Rent in Tokyo, the International Symposium on Statistical Analysis of Spatio-Temporal Data, pp.66-69, 006 (presented at the International Symposium on Statistical Analysis of Spatio-Temporal Data, The University of Tokyo, Tokyo, November 13-15, 006.) 13. Valente, J. Wu, S.S. Gelfand, A. and Sirmans, C. F.(005): Apartment rent prediction using spatial modeling, Journal of Real Estate Research, 7, 105-136.