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: October 7, 2011/November 21, 2011, Accepted: November 22, 2011 Abstract: 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, price and rate of return, time series model, empirical analysis, financial engineering 1. 2008 1 Graduate School of International Accounting, Chuo University, Shinjuku, Tokyo 162 8478, Japan 2 College of Arts and Sciences, the University of Tokyo, Meguro, Tokyo 153 8902, Japan 3 Nomura Research Institute, Ltd., Chiyoda, Tokyo 100 0005, Japan a) hiroshi.ishijima@gmail.com [7] c 2012 Information Processing Society of Japan 74
[8] Google Earth/Google Maps [7], [8] / 1 [7], [8] 2 [7], [8] [7] [8] [7] [8] 3 4 5 2. 2.1 [7] 2 ( ) = ( k ) ( k ) k (1) Lancaster [9] Rosen [13] hedonic model 1 [7] 2 2.2 [7] ( ) c 2012 Information Processing Society of Japan 75
=( ) +( ) +( ) (2) H ij,t = { H λ ij,t 1 λ log H ij,t (λ 0 ) (λ =0 ) (5) Case and Shiller [3] weighted repeated sales index 2.3 N H N n i N i=1 n i = N H K t i j H ij,t k x (k) ij,t [8] 2 K Hij,t = α t + β (k) t x (k) ij,t + ε ij,t (3) k=1 K Hij,t = α t + k=1 ( β (k) t ) + ν (k) i,t x (k) ij,t + ε ij,t (4) 2 i =1,...,N; j =1,...,n i 3 4 1 i ii linear pricing Luenberger [11] 1 3 4 2 1 1 1 Box-Cox Box and Cox [1] [8] i j H ij,t Box-Cox λ =1 λ 1 [14] Box-Cox λ =0 Box-Cox [8] λ =1 λ =0 H ij,t 5 Box-Cox 3 4 2 1 2 1 α t N x (l) ij,t (l =1,...,N) K α t := β (l) t x (l) ij,t (6) l=1 x (l) ij,t =1(l = i ) x(l) ij,t = 0(l i ) (6) 1 α t 2 k β (k) t i ν (k) i,t 4 ε ij,t 0 N H ε ij,t ( ) ν i,t := ν (1) i,t...ν(k) i,t...ν(k) i,t 0 K G 4 c 2012 Information Processing Society of Japan 76
Hsiao [6] Fitzmaurice et al. [4] McCulloch et al. [12] SAS 9.1.3 MIXED Littell et al. [10]3 REML; Restricted Maximum Likelihood BLUP Best Linear Unbiased Prediction 4 G t (t =1,...,T) 3 4 5 Box-Cox Ĥ t (t =1,...,T) Box-Cox ˆλ ( ) 1/ˆλ 1+ˆλ Ĥ t = Ĥ t (ˆλ 0 ) ) (t=1,...,t) exp (Ĥ t (ˆλ=0 ) (7) 2.4 i j t H ij,t 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 t 1 (x ij,t ) t 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 ) t i j 2 R ij,t 2 (8) R ij,t = m t + μ i,t + η ij,t (9) (i =1,...,N; j =1,...,n i ) 2 92 *1 m t μ i,t 0 N H η ij,t 0 N H 3. 3.1 2006 2 2011 1 20 5 6 6 1 AGE WALK 1 1 2006 2 2007 1 2008 1 2008 2009 1 5 2 *1 9 2 1 3 4 c 2012 Information Processing Society of Japan 77
1 N Table 1 Quarterly reported the number of observations (N) and average prices (in ten thousand yen) for apartment data. 2 Table 2 Averages of floor space (square meters), age of apartment (years) and walking distance from nearest subway/railway station (minutes) for apartment data. 1 5 2 2 1 1 2 2 5 3 AIC Table 3 Comparison of AICs when apartment prices are estimated quarterly by mixed and fixed effect models, and the distortion coefficients estimated by the mixed effect model. 3.2 1 2006 2 2011 1 3 4 3 AIC 4 Gurka et al. [5] λ λ 1 λ 0 λ 0.06 0.29 c 2012 Information Processing Society of Japan 78
Table 4 4 Quarterly reported the averages and standard deviations for pseudo return of apartment prices. 1 2 5 4 1 [8] 3.3 1 4 2.4 2006 2 2011 1 1 2006 3 2011 1 19 4 3.1 2006 2007 1 2008 1 2009 1 5 5 3.4 9 9 m t μ i,t η ij,t 9 SAS 9.1.3 MIXED 5 4 9 5 m t 4 29 m t c 2012 Information Processing Society of Japan 79
5 Table 5 Decomposition of pseudo return of apartment prices. 1 Fig. 1 Google Maps An example display of return index on Google Maps. [8] 1 [8] 1 c 2012 Information Processing Society of Japan 80
9 4. 1 4 1 2008 2 1.93 2009 2011 1 1.75 1 10 2006 3 2011 1 19 MSCI Japan Net BPI MSCI Kokusai Net Index WGBI Non JPY HFRX 1 Bloomberg GOLDS Cmdty 6 7 6 x y 6 % % % Table 6 Risk (standard deviation, in annual %), return (average, in annual %), return per unit risk (return-to-risk ratio) for traditional and alternative assets, and the correlation coefficient (%) between these assets. c 2012 Information Processing Society of Japan 81
2 MVP Markowitz MV Fig. 2 Risk and return profiles of traditional and alternative assets, and the efficient frontier given by mean-variance model of Markowitz (solid line). 2 Markowitz MV Luenberger [11] 2 1: 2: 2 2 1 4 2 1 6 MV 7 1 2 2 2 MVP 2 MVP MVP MVP 90% 76.6% 15.5% 6 MVP 6 100% 2010 1 5. 5 1 2 c 2012 Information Processing Society of Japan 82
3 4 5 [1] 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.211 252 (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 is, available from http://kuznets.fas.harvard.edu/ campbell/papers.html (accessed 2011-08-09). [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.125 137 (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.273 288 (2006). [6] Hsiao, C.: Analysis of Panel Data: Second Edition, Cambridge University Press (2003). [7] Vol.8 No.2 pp.95 98 (2011). [8] Vol.4 No.2 pp.1 12 (2011). [9] Lancaster, K.: A New Approach to Consumer Theory, Journal of Political Economy, Vol.74, pp.132 157 (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.34 55 (1974). [14] Vol.3, pp.30 42 (2003). [7] 2 3 1 2 3 1 [7] 1 P j,t = E t [( Pj,t+1 + D P j,t+1) M C t+1 ] (j =1,...,N P ) (A.1) = L i,t D + E t [ Hi,t+1 M C t+1 ] (i =1,...,N H ) (A.2) D = b i,t M Z t (i =1,...,N H ) (A.3) P j,t t N P j Dj,t P t j M t+1 C := 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 N H i Di,t H t i L i,t t i 1 b i,t := (b i1,t...b ik,t...b ik,t ) t i K M k,t Z := u(c t, Z t )/ Z k,t / u(c t, Z t )/ C t (k = 1,...,K) M t Z := ( M Z 1,t... M k,t Z... M ) K,t Z A.1 A.2 A.3 t c 2012 Information Processing Society of Japan 83
M t+1 C t +1 M t+1 C = δ A.1A.2 δ ( ) Li Di H 1 ( ) 1 D P j =1+ i, j. (A.4) H i P j = 1 + A.2 A.3 [ ] = E t δ τ L i,t+τ b i,t+τ Mt+τ Z (A.5) τ=0 M Z t+τ := u(c t+τ, Z t+τ )/ Z t+τ / u(c t, Z t )/ C t 1 b i,t = b i i, t (A.6) A.5 [ ] = b i E t δ τ L i,t+τ Mt+τ Z (A.7) τ=0 +1 = b i δ 1 τ=0 = δ 1 b i k=1 = δ 1 b i k=0 [ ] δ τ+1 E t+1 Li,t+τ+1 Mt+τ+1 Z [ ] δ k E t+1 Li,t+k Mt+k Z [ ] δ k E t+1 Li,t+k Mt+k Z [ ] δ 1 b i E t+1 Li,t Mt Z. (A.8) 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 k=0 = δ 1 δ 1 b i L i,t M Z t. (A.9) E t [+1 ] = ( δ 1 1 ) δ 1 b i L i,t M Z t. (A.10) A.4 A.4 A.4 i j j m r m,t := Dm,t/P P m,t 1 t i L i,t Di,t H / = b i L i,t Mt Z / Mt Z = M t Z b i L i,t M Z t = r m,t 1+r m,t δσ i ε i,t. (A.11) σ i ε 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 1+r m,t := ˆμ m,t 1 δσ m,t 1 ε m,t (A.12) ˆμ m,t 1 := E t 1 [ rm,t 1+r m,t ], σ 2 m,t 1 := δ 2 E t 1 [ ( rm,t 1+r m,t ˆμ m,t 1 ) 2 ] ε i,t ε m,t A.11 A.12A.10 E t [+1 ] = μ m,t 1 + σ m,t 1 ε m,t + σ i ε i,t. (A.13) μ m,t 1 := ( δ 1 1 ) δ 1 ˆμ m,t 1 η i,t+1 N (0, 1) Δ / := (+1 ) / Δ = μ m,t 1 + σ m,t 1 ε m,t + σ i ε i,t + σ η i,t η i,t+1. ( σ η i,t) 2 := Et [ ( ΔHi,t (A.14) E t [ ΔHi,t ]) 2 ] η i,t+1 ε m,t ε i,t A.14 i 1 2 c 2012 Information Processing Society of Japan 84
ii 3 iii 4 1 3 t 4 t +1 2 3 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. (A.15) ˇP i,t t i Čt t Ȟi,t Č t ΔȞi,t σh 2 Ňi,t Čt Ȟi,t [2] 1 1971 1999 2004 10 2006 5 2007 4 2010 FP SAS / & 2010 JAFEE FP 1963 1990 3 4 1996 6 MS 1999 4 Ph.D. Engineering-Economic Systems and Operations Research Minor: Economics 1999 4 2004 4 2007 4 2011 4 2004 10 2007 4 2010 9 1978 2004 3 4 2010 3 2010 FP c 2012 Information Processing Society of Japan 85