β TOPIX 1 22 β β smoothness priors (the Capital Asset Pricing Model, CAPM) CAPM 1 β β β β smoothness priors :,,. E-mail: koiti@ism.ac.jp., 104
1 TOPIX β Z i = β i Z m + α i (1) Z i Z m α i α i β i (the Capital Asset Pricing Model CAPM) CAPM 1 β i Campbell, Lo, and MacKinlay (1996) CAPM TOPIX CAPM Roll Roll (1977) (2003) CAPM β i β i β i 1 TOPIX 1 22 β i β i CAPM CAPM Sharpe (1964) Lintner (1965) CAPM E[R i R f ]=β i (E[R m R f ]) β i = Cov[R i,r m ] Var[R m ] (2) R i i R m R f β i R m R f R i R f CAPM CAPM 1 β i β i Campbell, Lo, and MacKinlay (1996) β i CAPM Roll TOPIX CAPM β i β i Blume (1971, 1975) Clarkson and Thompson (1990) β 1 β i 1 1 1 105
β i smoothness priors Smoothness priors β i β i Kalman (1960) (2) β i β i Kalman (1960) β i β i Zalewska (2004) β Zalewska (2004) AIC (Akaike Information Criterion) 2 i Z i (t) =β i Z m (t)+α i (3) Z i (t) E[R i R f ] Z m (t) E[R m R f ] α i i CAPM β i β i β i β i (t) β i (t) ν i (t) =β i (t) β i (t 1), (4) ν i (t) N(0,τ 2 i ) τ i i smoothness priors (4) Kitagawa and Gersch (1985) smootness priors smoothness priors (3) (4) β i (t) =β i (t 1) + ν i (t) (5) Z i (t) =Z m (t)β i (t)+ɛ i (t) (6) ɛ i (t) ɛ i (t) N(0,σ 2 i ) σ i i (5) (6) β i (t) 2 Kalman (1960) β i (t) β i (t) x(t t j), [j =0, 1, ] β i (t t 1) = β i (t 1 t 1), V (t t 1) = V (t 1 t 1) + τi 2, (7) 2 (1993) Durbin and Koopman (2001) Hamilton (1994) 106
K(t) =V (t t 1)Z m (t){z m (t)v (t t 1)Z m (t)+σi 2 } 1, β i (t t) =β i (t t 1) + K(t){Z i (t) Z m (t)β i (t t 1)}, V (t t) ={1 K(t)Z m (t)}v (t t 1), (8) V (t t j) =E[{β i (t) β i (t t j)} 2 ] β(0 0) V (0 0) β i (t) Akaike (1973) AIC (Akaike Information Criterion) AIC AIC(i) = 2l( ˆθ i )+2p (9) l(θ i ) p ˆθ i AIC i {Z i (1),,Z i (T )} f(z i (1),,Z i (T ) θ i )= T f(z i (t) Z i (1),,Z i (t); θ i ), (10) t=1 θ i =(σ i,τ i ) f(z i (t) Z i (1),,Z i (t); θ i ) f(z i (t) Z i (1),,Z i (t); θ i ) 1 [ = 2πv2 (t) exp 1 2v 2 (t) (Z i(t) Z m (t)β i (t t 1)) 2] (11) l(θ i )= 1 2 { T log(2π)+ + T t=1 T log v 2 (t) t=1 1 v 2 (t) (Z i(t) Z m (t)β i (t t 1)) 2}, (12) v 2 (t) =Z m (t)v (t t 1)Z m (t)+σi 2 R optim Nelder-Mead 3 3.1 1 22 TOPIX TOPIX 1 1998 1 2003 12 6 72 71 3 1998 2003-0.8-1.6 LIBOR 3 LIBOR 3 1 1 3 5 Campbell, Lo, and MacKinlay (1996) pp. 184 107
( ) 25 1998/01-2003/12 TOPIX 1998/01-2003/12 LIBOR 3 1998/01-2003/12 1: R i = log P t P t 1, (13) P t TOPIX (12) R optim β i (t) (5) (6) τ i, σ i, AIC optim 0 4 (Bowman-Shenton (Jarque-Bera) ) Ljung-Box 3.2 6-27 β Quantile-Quantile Quantile-Quantile 2 2 code optim β V (0 0) β 1 1999 6 2000 6 2001 6 CEO β 2 3 β 1.0 1.2 β 0.2 0.4 2004 2 16 β 4 NTT 1998 2003 IT β IT IT 5 β 0.5 0.2 IT 2000 β IT σ 2 2 β TOPIX 4 nlm 1000 200 108
Code AIC σ τ NTT 0-149.3359 0.07534621 0.05882475 SONY 0-64.40624 0.1435226 0.03899718 0-170.9791 0.0643951 0.06440982 0-119.4261 0.09381163 0.004122669 0-105.2728 0.1038671 0.03153384 0-152.0062 0.0730043 0.02377539 4502 0-158.2433 0.0709385 0.02374597 6752 0-142.9274 0.07851107 0.02417267 6753 0-119.2637 0.09456484 0.02551953 6902 0-175.7362 0.06173184 0.02517676 7201 0-90.45424 0.1159477 0.03341797 7751 0-146.1517 0.07626406 0.02624219 8058 0-142.2841 0.07883462 0.02532104 9020 0-170.8066 0.06367031 0.02346094 9022 0-201.2063 0.05186767 0.02268473 KDDI 9433 0-72.4038 0.1305883 0.03283984 2914 0-140.844 0.08041787 0.02553174 4452 0-180.9324 0.05882841 0.02141521 6701 0-115.7291 0.09526309 0.02877051 6971 0-89.49805 0.1159477 0.03341797 7974 0-108.1091 0.1029107 0.03000534 3102 0-93.36037 0.1159477 0.03341797 2: 109
s 0.5 1.0 1.5 2.0 1998 1999 2000 2001 2002 2003 Time 1: s 0.8 1.0 1.2 1.4 1.6 1998 1999 2000 2001 2002 2003 Time 2: β 3.3 ν i (t) ɛ i (t) 110
s 0.2 0.4 0.6 0.8 1.0 1998 1999 2000 2001 2002 2003 Time 3: s 0.8 0.9 1.0 1.1 1.2 1.3 1998 1999 2000 2001 2002 2003 Time 4: NTT 1. Quantile-Quantile 2. Bowman-Shenton (Jarque-Bera) 3. Ljung-Box Bowman-Shenton Ljung-Box Bowman-Shenton Ljung-Box 3 p 4 3 *** 1% ** 5% Ljung-Box 12 3 111
s 0.2 0.3 0.4 0.5 1998 1999 2000 2001 2002 2003 Time 5: Bowman-Shenton β 4 1. β 2. β 3. β β β i Kitagawa (1993, 1996) Gordon (1993) 112
Bowman-Shenton Ljung-Box Bowman-Shenton Ljung-Box (p ) (p ) (p ) (p ) NTT 0.8300 0.0065 *** 0.0000 *** 0.1588 SONY 0.4734 0.0951 0.0016 ** 0.6258 0.4734 0.0951 0.0016 ** 0.6258 0.4671 0.5509 0.0000 *** 0.2269 0.0000 *** 0.4553 0.0000 *** 0.9637 0.0000 *** 0.0027 *** 0.0000 *** 0.1711 0.9370 0.1464 0.0000 *** 0.3193 0.8450 0.5210 0.0000 *** 0.6653 0.2481 0.4699 0.0000 *** 0.4783 0.7662 0.3247 0.0000 *** 0.0344 ** 0.0855 0.1385 0.0000 *** 0.3584 0.0583 0.3719 0.0000 *** 0.5045 0.0035 *** 0.3101 0.0000 *** 0.0772 0.7622 0.9843 0.0000 *** 0.5211 0.1577 0.5774 0.0000 *** 0.8257 KDDI 0.1739 0.5256 0.0000 *** 0.0071 *** 0.1515 0.6817 0.0000 *** 0.9975 0.9811 0.6888 0.0000 *** 0.0871 0.8509 0.5885 0.0000 *** 0.6851 0.0000 *** 0.6516 0.0000 *** 0.2925 0.8155 0.5098 0.0000 *** 0.2006 0.0000 *** 0.7953 0.0000 *** 0.7606 3: 113
[1] (1993) FORTRAN 77 [2] (2003) 41 6 [3] Akaike, H., (1973), Information theory and an extension of the maximum likelihood principle, Proc. 2nd International Symposium on Information Theory (B. N. Petrov and F. Csaki eds.) Akademiai Kiado, Budapest, 267-281. [4] Blume, M.E., (1971), On the Assessment of Risk, Journal of Finance, 24, 1-9. [5] Blume, M. E., (1975), s and their Regression Tendencies, Journal of Finance, 30, 785-795. [6] Campbell, Lo, and MacKinlay (1996), The Econometrics of the Financial Markets, Princeton Press. [7] Clarkson, P.M. and R. Thompson, (1990), Empirical Estimates of When Investors Face Estimation Risk, Journal of Finance, 45, 2, 431-453. [8] Durbin, J. and Koopman, S.J., (2001), Time Series Analysis by State Space Methods, Oxford University Press. [9] Gordon, NJ, Salmond, D. J., Smith. A. M., (1993), Novel approach to nonlinear/non-gaussian Bayeaian state estimation, IEEE Proceedings-F, 140 (2), 107-113. [10] Hamilton, J. D., (1994), Time Series Analysis, Princeton University Press. [11] Kalman, R. E., (1960), A New Approach to Linear Filtering and Prediction Problems, Transaction of the ASME-Journal of Basic Engineering, 35-45. [12] Kitagawa, G., (1993), A Moneta Carlo Filtering and smoothing method for non-gaussian nonlinear state space models, Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis, 110-131, Honolulu, Hawaii, January 25-29. [13] Kitagawa, G., (1996), Monte Carlo filter and smoother for non-gaussian nonlinear state space models, Journal of Computational and Graphical Statistics, Vol.5, No.1, 1-25. [14] Kitagawa, G. and Gersch, W., (1985), A Smoothness Priors Time Varying AR Coefficient Modeling of Nonstationary Covariance Time Series, IEEE Trans. Automat. Control, Vol AC-30, No.1, pp. 48-56. [15] Lintner, J., (1965), The Valuation of Risk Assets and Selection of Risky Investments in Stock Portfolios and Capital Budgets, Review of Economics and Statistics, 47, 13-37. [16] Markowitz, H. M., (1952), Portfolio Selection, Journal of Finance, 7, no. 1, 77-91. [17] Roll, R., (1977), A Critique of the Asset Pricing Theory s Tests: Part I, Journal of Financial Economics, 4, 129-176. [18] Sharpe, W. F., (1964), Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk, Journal of Finance, 19, 425-442. [19] Shiller, R., (1973), A Distributed Lag Estimator Derived from Smoothness Priors, Econometrica, vol. 41, No.4, pp. 775-788. [20] Zalewska, A., (2004), Evolving character of the CAPM beta of the case of the telecom industry, an Indepen working paper. 114
urns and Returns s 0.2 0.0 0.1 0.8 1.0 1.2 Measurement Equation System Equation 0.15 0.00 0.10 0.4 0.2 0.0 0.2 6: NTT urns and Returns s 0.8 0.4 0.0 0.4 1.0 1.2 1.4 1.6 1.8 Measurement Equation System Equation 0.8 0.4 0.0 0.4 0.1 0.0 0.1 0.2 7: SONY 115
urns and Returns s 0.2 0.0 0.2 0.85 0.95 1.05 Measurement Equation System Equation 0.2 0.0 0.2 0.10 0.00 0.10 8: urns and Returns s 0.15 0.00 0.10 0.20 0.2 0.3 0.4 0.5 Measurement Equation System Equation 0.15 0.00 0.10 0.10 0.00 9: 116
urns and Returns s 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Measurement Equation System Equation 0.4 0.2 0.0 0.2 0.1 0.1 0.3 10: urns and Returns s 0.2 0.0 0.1 0.3 0.5 0.7 0.9 Measurement Equation System Equation 0.2 0.0 0.1 0.2 0.2 0.6 11: 117
urns and Returns s 0.20 0.05 0.10 0.4 0.8 1.2 0.2 0.0 0.1 0.6 0.2 0.2 12: urns and Returns s 0.2 0.0 0.1 0.0 0.4 0.8 0.2 0.0 0.1 0.2 0.1 0.1 0.3 13: 118
urns and Returns s 0.3 0.1 0.1 0.6 1.0 1.4 0.3 0.1 0.1 1.0 0.5 0.0 0.5 14: urns and Returns s 0.2 0.0 0.1 0.8 1.0 1.2 1.4 0.15 0.00 0.10 0.2 0.0 0.2 15: 119
urns and Returns s 0.3 0.1 0.1 0.5 1.0 1.5 2.0 0.3 0.1 0.1 0.2 0.0 0.2 0.4 16: urns and Returns s 0.2 0.0 0.2 0.6 0.8 1.0 1.2 0.2 0.0 0.2 0.2 0.0 0.2 17: 120
urns and Returns s 0.2 0.0 0.1 0.2 0.8 1.2 1.6 0.2 0.0 0.2 0.1 0.1 0.3 18: urns and Returns s 0.15 0.00 0.10 0.2 0.3 0.4 0.5 0.6 0.15 0.00 0.10 0.20 0.2 0.0 0.1 19: 121
urns and Returns s 0.15 0.05 0.05 0.15 0.00 0.10 0.20 0.30 0.15 0.05 0.05 0.15 0.2 0.0 0.1 0.2 20: urns and Returns s 0.4 0.0 0.2 0.4 1.0 1.5 2.0 0.2 0.0 0.2 0.4 0.6 0.2 0.2 21: KDDI 122
urns and Returns s 0.2 0.0 0.1 0.0 0.2 0.4 0.6 0.2 0.0 0.2 0.2 0.0 0.2 0.4 22: urns and Returns s 0.15 0.00 0.10 0.2 0.4 0.6 0.8 1.0 0.15 0.00 0.10 0.6 0.2 0.2 23: 123
urns and Returns s 0.4 0.2 0.0 0.2 0.8 1.2 1.6 0.3 0.1 0.1 0.8 0.4 0.0 24: urns and Returns s 0.4 0.0 0.4 0.8 1.0 1.2 1.4 0.4 0.0 0.4 0.05 0.05 0.15 25: 124
urns and Returns s 0.3 0.1 0.1 0.6 1.0 1.4 0.3 0.1 0.1 0.5 0.3 0.1 0.1 26: urns and Returns s 0.2 0.2 0.4 0.8 1.2 1.6 0.2 0.0 0.2 0.4 0.2 0.2 0.6 27: 125