IPSJ SIG Technical Report Vol.2011-MPS-85 No /9/ : Time Series Modeling of Real Estate Prices and Its Application Hiroshi Ishijima, 1 A
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1 1 2 3 : Time Series Modeling of Real Estate Prices and Its Application Hiroshi Ishijima, 1 Akira Maeda 2 and Tomohiko Taniyama 3 As real estate and financial asset markets are merging in these days, there is a strong need for us to have a theoretical foundation for analysis of real estate investments in conjunction with both domestic and international financial investments. The purpose of this paper is to present a dynamic equilibrium model to evaluate prices of not only financial assets but also pieces of real estate. In particular, we extend our previous model to a sophisticated one that allows us to create pseudo returns on real estate and to estimate risks and returns on real estate investments. The results of our theory and statistical analysis here highlight the role of real estate investments, contrasting to that of financial ones. Keywords: real estate, time series model, price, rate of return, empirical analysis, financial engineering ) 8) Google Earth/Google Maps 7),8) 7),8) 1 Graduate School of International Accounting, Chuo University. 2 College of Arts and Sciences, University of Tokyo. 3 Nomura Research Institute, Ltd. 1 c 2011 Information Processing Society of Japan
2 7),8) 7) 8) 7) 8) ) 2 ( ) = ( k ) ( k ) (1) k Lancaster 9) Rosen 13) hedonic model 1 7) ) () = ( ) + ( ) + ( ) (2) Case and Shiller 3) weighted repeated sales index 2.3 M N n i N i=1 n i = M K t i j H ij,t k x (k) ij,t 8) 2 2 c 2011 Information Processing Society of Japan
3 H ij,t = α t + H ij,t = α t + K k=1 K k=1 β (k) t x (k) ij,t + ε ij,t (3) ( β (k) t ) + ν (k) i,t x (k) ij,t + ε ij,t (4) 2 i = 1,..., N; j = 1,..., n i i ii linear pricing Luenberger 11) Box-Cox Box and Cox 1) 8) i j H ij,t Box-Cox H ij,t = { H λ ij,t 1 λ (λ 0 ) log H ij,t (λ = 0) λ = 1 λ 1 8) H ij,t 5 Box-Cox 3 4 (5) α t N x (l) ij,t (l = 1,..., N) α t := K l=1 β (l) t x (l) ij,t (6) x (l) ij,t = 1 (l = i ) x(l) ij,t = 0 (l i ) (6) 1 α t 2 k β (k) t ν (k) i,t i 4 ε ij,t 0 M ( ) ε ij,t ν i,t := ν (1) i,t... ν(k) i,t... ν(k) i,t 0 K G 4 Hsiao 6), Fitzmaurice et al. 4), McCulloch et al. 12) SAS MIXED Littell et al. 10) 3 REML; Restricted Maximum Likelihood BLUP Best Linear Unbiased Prediction 4 G t (t = 1,..., T ) c 2011 Information Processing Society of Japan
4 {Ĥ1,..., Ĥt,..., ĤT } 2.4 i j t H ij,t (7) t 1 H ij,t 1 t 1 t t R R ij,t := (H ij,t H ij,t 1)/ H ij,t 1 t t 1 t x ij,t t 1 H ij,t 1 = H t 1 (x ij,t) 7 t 1 Ĥt 1 Ĥt 1 (x ij,t ) R pseudo return R R ij,t H ij,t Ĥt 1 (x ij,t ) Ĥ t 1 (x ij,t) 2 3 t i j 3 R ij,t = m t + µ i,t + η ij,t (9) (i = 1,..., N; j = 1,..., n i ) m t µ i,t 0 N H η ij,t 0 M (8) AGE WALK AIC 4 4 c 2011 Information Processing Society of Japan
5 Gurka et al. 5) λ λ 1 λ 0 λ ) m t µ i,t η ij,t m t 4 9 8) 1 8) c 2011 Information Processing Society of Japan
6 ) , , MSCI Japan Net,, BPI,, MSCI Kokusai Net Index,, WGBI Non JPY HFRX 6 7, 6. x y 2 Markowitz MV Luenberger 11) 2 1: 2: MV MVP 2 MVP MVP MVP 90% 6 c 2011 Information Processing Society of Japan
7 6 MVP 6 100% ) Box, G.E.P. and Cox, D.R.: An Analysis of Transformations (with Discussion), Journal of the Royal Statistical Society: Series B, Vol.26, pp (1964). 2) Campbell, J. Y. and Viceira, L. M.: Strategic Asset Allocation: Portfolio Choice for Long-Term Investors, Oxford University Press (2002). See also its appendix which can be found at campbell/papers.html (accessed ) 3) Case, K.E. and Shiller, R.J.: The Efficiency of the Market for Single-Family Homes, American Economic Review, Vol.79, No.1, pp (1989). 4) Fitzmaurice, G.M., Laird, N.M. and Ware, J.H.: Applied Longitudinal Analysis, John Wiley & Sons, Inc (2004). 5) Gurka, M.J., Edwards, L.J., Muller, K.E. and Kupper, L.L.: Extending the Box- Cox Transformation to the Linear Mixed Model, Journal of Royal Statistical Society A, Vol.169, No.2, pp (2006). 6) Hsiao, C.: Analysis of Panel Data: Second Edition, Cambridge University Press (2003). 7) Vol.8 No.2 pp ) Vol.4 No.2 pp ) Lancaster, K.: A New Approach to Consumer Theory, Journal of Political Economy, Vol.74, pp (1966). 10) Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D. and Schabenberber, O.: SAS for Mixed Models: Second Edition, SAS Publishing (2006). 11) Luenberger, D.G.: Investment Science, Oxford University Press (1997) : (2002). 12) McCulloch, C.E., Searle, S.R. and Neuhaus, J.M.: Generalized, Linear, and Mixed Models: Second Edition, John Wiley & Sons (2008). 13) Rosen, S.: Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition, Journal of Political Economy, Vol.82, pp (1974). 7) 2 7 c 2011 Information Processing Society of Japan
8 ) 1 P j,t = E t [( Pj,t+1 + D P j,t+1) M C t+1 ] (j = 1,..., N P ) (10) = L i,t D + E t [ Hi,t+1 M C t+1 ] (i = 1,..., N H ) (11) D = b i,t M Z t (i = 1,..., N H ) (12) P j,t t N P j D P j,t t j M C t+1 := δ u(c t+1, Z t+1)/ C t+1 / u(c t, Z t)/ C t u δ C t t Z t := (Z 1,t... Z k,t... Z K,t ) t t M i D t i L i,t t i 1 b i,t := (b i1,t... b ik,t... b ik,t ) t i K M Z k,t := u(c t, Z t)/ Z k,t / u(c t, Z t)/ C t (k = 1,..., K) M Z t := ( M Z 1,t... M Z k,t... M Z K,t) t M C t+1 t + 1 M C t+1 = δ δ ( ) LiDi H 1 ( ) D P 1 j = 1 + i, j. (13) P j H i = = τ=0 [ ] E t δ τ L i,t+τ b i,t+τ Mt+τ Z M Z t+τ := u (C t+τ, Z t+τ ) / Z t+τ / u (C t+τ, Z t+τ ) / C t 1 (14) b i,t = b i i, t (15) 14 [ ] = b ie t δ τ L i,t+τ Mt+τ Z τ=0 [ ] +1 = b i δ 1 δ τ+1 E t+1 Li,t+τ+1 Mt+τ+1 Z = δ 1 b i = δ 1 b i τ=0 k=1 [ ] δ k E t+1 Li,t+k Mt+k Z (16) [ [ ] δ k E t+1 Li,t+k Mt+k] Z δ 1 b i E t+1 Li,t Mt Z. (17) k=0 t E t [+1 ] = δ 1 b i [ δ k E t Li,t+k Mt+k] Z δ 1 b i L i,t Mt Z = δ 1 δ 1 b i L i,t Mt Z. k=0 (18) 8 c 2011 Information Processing Society of Japan
9 E t [+1] = ( δ 1 1 ) δ 1 b il i,tm Z t. (19) i j j m r m,t := D P m,t/p m,t 1 t i L i,t D / = b i L i,t M Z t / M Z t = M Z t b i L i,t M Z t = r m,t 1 + r m,t δσ iε i,t. (20) σ iε i,t r m,t/(1 + r m,t) i ε i,t N (0, 1) r m,t ε m,t N (0, 1) r m,t /(1 + r m,t ) r m,t := ˆµ m,t 1 δσ m,t 1 ε m,t (21) 1 + r m,t [ ] [ ( ) ] 2 r ˆµ m,t 1 := E m,t t 1 1+r m,t, σm,t 1 2 := δ 2 r E m,t t 1 1+r m,t ˆµ m,t 1 ε i,t ε m,t E t [+1] = µ m,t 1 + σ m,t 1 ε m,t + σ i ε i,t. (22) µ m,t 1 := ( δ 1 1 ) δ 1 ˆµ m,t 1 η i,t+1 N (0, 1) / := (+1 ) / 23 i 1 2 ii 3 iii t 4 t Case and Shiller 3) Weighted Repeated Sales Index Case and Shiller 3), p.126 l.6 Case and Shiller 3) ˇP i,t = Čt + Ȟi,t + Ňi,t. (24) ˇP i,t t i Čt t Ȟi,t Čt Ȟi,t σ 2 h Ňi,t Čt Ȟi,t 2) 1 = µ m,t 1 + σ m,t 1 ε m,t + σ i ε i,t + σ η i,t η i,t+1. (23) ( σ η i,t) 2 := Et [ ( Hi,t E t [ Hi,t ]) 2 ] η i,t+1 ε m,t ε i,t 9 c 2011 Information Processing Society of Japan
10 1 N Table 1 As for the data used in the apartment price analysis, the number of observations (N) and average price (in ten thousand yen) for entire data and for each of real estate area classes are shown quarterly. 全体札幌市都心 5 区都区部名古屋市大阪市福岡市 N 平均 N 平均 N 平均 N 平均 N 平均 N 平均 N 平均 2006_2 1, _3 1, _4 1, _1 2, , _2 3, , _3 3, , _4 3, , _1 3, , _2 3, , _3 3, , _4 3, , _1 3, , _2 3, , _3 3, , _4 3, , _1 3, , _2 3, , _3 3, , _4 2, , _1 2, , Table 2 As for the data used in the apartment price analysis, we report averages of floor space (square meters), age of apartment (years) and walking distance from nearest subway/railway station (minutes). These are shown for entire data and for each of real estate area classes. 属性札幌市都心 5 区都区部名古屋市大阪市福岡市全体平均面積 ( 平米 ) 平均築年数 ( 年 ) 平均駅徒歩 ( 分 ) AIC Table 3 Comparison of AICs when apartment prices are estimated quarterly by mixed and fixed effect models, respectively. Also the distortion coefficients estimated by the mixed effect model are shown. 固定効果 AIC 混合効果混合効果の推定 λ 2006_2 12, , _3 12, , _4 12, , _1 16, , _2 27, , _3 26, , _4 27, , _1 27, , _2 28, , _3 27, , _4 27, , _1 28, , _2 30, , _3 31, , _4 31, , _1 32, , _2 32, , _3 27, , _4 23, , _1 21, , c 2011 Information Processing Society of Japan
11 4 Table 4 For generated pseudo return of apartment prices, we quarterly report the average and standard deviation for entire data and for each of real estate area classes. 全体札幌市都心 5 区都区部名古屋市大阪市 µ σ µ σ µ σ µ σ µ σ µ σ µ σ 2006_3-0.28% 5.26% 2.55% 9.35% -3.39% 2.33% 2.83% 2.23% -7.10% 3.22% -3.78% 6.13% -1.89% 2.60% 2006_4 5.14% 8.03% 9.12% 1.98% -3.43% 7.64% 5.34% 2.92% 3.23% 7.24% 5.64% 5.75% 17.99% 16.34% 2007_1 4.06% 7.14% -7.62% 3.42% 8.93% 6.56% 4.76% 6.39% 5.93% 4.39% 0.63% 4.19% 2.73% 9.97% 2007_2 1.22% 5.72% 1.03% 3.00% 1.68% 4.64% -1.27% 2.88% 11.47% 8.51% 6.77% 0.90% -4.45% 2.20% 2007_3 0.87% 5.60% -0.21% 1.74% 4.54% 3.08% 1.74% 6.12% 2.37% 7.38% -3.65% 2.12% -2.78% 2.50% 2007_4 0.78% 5.32% -2.41% 3.00% -3.26% 5.10% 1.76% 3.13% -2.81% 3.16% 6.34% 2.84% -0.08% 11.51% 2008_1 0.73% 7.15% 3.97% 4.33% -2.10% 12.40% 1.63% 3.19% 2.44% 7.71% -1.91% 7.79% 0.92% 6.96% 2008_2-2.10% 4.74% 0.33% 1.12% 1.52% 2.58% -4.29% 1.11% -3.40% 4.16% -2.94% 5.60% 3.54% 9.57% 2008_3 0.15% 2.79% -1.31% 0.55% 0.27% 2.90% -0.77% 1.35% 5.41% 4.94% -0.50% 1.43% 2.70% 1.18% 2008_4-4.10% 3.41% -5.20% 4.25% -5.07% 3.07% -3.81% 0.41% -4.50% 5.93% -1.59% 2.85% -6.77% 6.07% 2009_1-1.25% 4.30% 5.50% 5.75% -2.27% 2.57% -2.11% 1.16% 1.81% 3.81% -3.53% 1.56% -0.20% 9.70% 2009_2 0.01% 4.03% -2.59% 3.90% 0.15% 2.35% 0.35% 1.78% 0.13% 5.42% 0.01% 4.55% 0.98% 9.50% 2009_3 1.68% 3.87% 2.03% 5.88% 1.09% 1.06% 1.38% 2.39% 2.37% 2.80% 2.36% 5.02% 2.57% 8.72% 2009_4 1.04% 3.00% -1.80% 2.12% 0.73% 2.63% 2.25% 2.07% -1.57% 3.39% -0.91% 2.55% 4.16% 3.29% 2010_1 3.59% 3.59% 2.93% 1.67% 4.05% 5.15% 2.95% 2.46% 6.55% 5.80% 4.62% 3.40% 2.35% 3.17% 2010_2-0.47% 3.77% -3.48% 2.60% 3.81% 3.95% 0.00% 1.70% 0.08% 4.44% -3.23% 1.09% -3.26% 6.88% 2010_3 1.08% 4.23% 4.21% 3.25% -2.57% 1.10% 1.78% 2.32% 2.81% 8.67% -2.66% 2.71% 5.00% 1.77% 2010_4 1.90% 4.42% -2.81% 3.31% 3.65% 2.87% 2.46% 3.83% 2.92% 2.97% 4.35% 1.77% -5.99% 4.54% 2011_1 0.52% 5.52% 10.68% 7.56% 1.03% 1.38% -2.69% 3.63% 5.39% 3.87% 3.89% 2.36% 6.34% 10.15% 福岡市 5 Table 5 Decomposition of generated pseudo return of apartment prices: Quarterly generated pseudo return of apartment prices is decomposed into the entire market factor and into each factor of real estate area classes. 全体 札幌市 都心 5 区 都区部 名古屋市 大阪市 福岡市 2006_3-1.80% 2.48% -3.39% 2.83% -7.06% -3.78% -1.89% 2006_4 6.31% 9.09% -3.39% 5.34% 3.25% 5.65% 17.90% 2007_1 2.58% -7.46% 8.90% 4.75% 5.90% 0.64% 2.73% 2007_2 2.54% 1.04% 1.68% -1.27% 11.45% 6.77% -4.44% 2007_3 0.34% -0.21% 4.51% 1.74% 2.35% -3.62% -2.74% 2007_4-0.08% -2.40% -3.25% 1.76% -2.80% 6.32% -0.08% 2008_1 0.81% 3.87% -2.05% 1.63% 2.38% -1.86% 0.92% 2008_2-0.88% 0.32% 1.51% -4.28% -3.39% -2.93% 3.52% 2008_3 0.97% -1.30% 0.27% -0.77% 5.40% -0.49% 2.70% 2008_4-4.49% -5.19% -5.07% -3.81% -4.50% -1.61% -6.75% 2009_1-0.13% 5.48% -2.27% -2.11% 1.80% -3.52% -0.20% 2009_2-0.16% -2.53% 0.14% 0.35% 0.12% 0.01% 0.93% 2009_3 1.95% 2.02% 1.16% 1.40% 2.31% 2.34% 2.48% 2009_4 0.48% -1.79% 0.73% 2.25% -1.56% -0.91% 4.14% 2010_1 3.91% 2.95% 4.05% 2.95% 6.51% 4.62% 2.38% 2010_2-1.01% -3.47% 3.80% -0.01% 0.07% -3.22% -3.25% 2010_3 1.43% 4.20% -2.56% 1.78% 2.81% -2.66% 4.99% 2010_4 0.76% -2.80% 3.64% 2.46% 2.91% 4.35% -5.97% 2011_1 4.10% 10.63% 1.04% -2.68% 5.38% 3.89% 6.33% 11 c 2011 Information Processing Society of Japan
12 6 4 3 % % % Table 6 Besides traditional four assets, as for three assets regarded as alternative investments, we report their risk and return which are respectively measured by standard deviation and average in annual percentage. Also the return per unit risk (return-to-risk ratio) is shown. Following the panel reports the correlation coefficient between assets in percentage. 2010_3Q 2010_2Q 国内株式 国内債券 外国株式 外国債券 マンション ヘッジファンド 金 標準偏差 ( 年率 ) 21.52% 2.12% 25.70% 9.39% 4.64% 9.32% 12.48% 期待値 ( 年率 ) -8.28% 2.38% 0.02% -0.59% 3.71% 3.48% 11.97% リターン リスク比 相関係数 国内株式 国内債券 外国株式 外国債券 マンション ヘッジファンド 金 国内株式 % % 87.59% 51.31% 38.93% 85.59% 29.64% 国内債券 % % % % % % % 外国株式 87.59% % % 73.49% 53.47% 88.81% 43.81% 外国債券 51.31% % 73.49% % 39.77% 49.80% 53.64% マンション 38.93% % 53.47% 39.77% % 45.21% 29.25% ヘッジファンド 85.59% % 88.81% 49.80% 45.21% % 43.70% 金 29.64% % 43.81% 53.64% 29.25% 43.70% % 15.00% 金 10.00% 5.00% 0.00% MVP 内債 マンション 外債 ヘッジファンド 外株 2010_4Q 2011_1Q -5.00% % 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% リターン ( 年率 %) リスク ( 年率 %) 内株 1 Google Maps : Fig. 1 An example display of return index on Google Maps: According to return index values, different colors are assigned to the pinned locations. These pins are placed at the representative location, namely the city hall, for each of real estate area classes MVP Markowitz MV Fig. 2 Risk and return profiles of traditional four assets and of three assets regarded as alternative investments. Also the efficient frontier given by mean-variance model of Markowitz is shown in solid line. 12 c 2011 Information Processing Society of Japan
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1,a) 2 3 2011 8 19 2011 10 7 / 2011 11 21 2011 11 22 Time Series Modeling of Real Estate Prices and Its Application Hiroshi Ishijima 1,a) Akira Maeda 2 Tomohiko Taniyama 3 Received: August 19, 2011, Revised:
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