カルマンフィルターによるベータ推定（ ）

Transcription

1 β TOPIX 1 22 β β smoothness priors (the Capital Asset Pricing Model, CAPM) CAPM 1 β β β β smoothness priors :,,. [email protected]., 104

2 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

3 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

4 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 TOPIX TOPIX LIBOR 3 LIBOR Campbell, Lo, and MacKinlay (1996) pp

5 ( ) / /12 TOPIX 1998/ /12 LIBOR / /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 β Quantile-Quantile Quantile-Quantile 2 2 code optim β V (0 0) β CEO β 2 3 β β β 4 NTT IT β IT IT 5 β IT 2000 β IT σ 2 2 β TOPIX 4 nlm

6 Code AIC σ τ NTT SONY KDDI : 109

7 s Time 1: s Time 2: β 3.3 ν i (t) ɛ i (t) 110

8 s Time 3: s 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

9 s Time 5: Bowman-Shenton β 4 1. β 2. β 3. β β β i Kitagawa (1993, 1996) Gordon (1993) 112

10 Bowman-Shenton Ljung-Box Bowman-Shenton Ljung-Box (p ) (p ) (p ) (p ) NTT *** *** SONY ** ** *** *** *** *** *** *** *** *** *** *** ** *** *** *** *** *** *** KDDI *** *** *** *** *** *** *** *** *** *** : 113

11 [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, [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, [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, [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), [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, [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, , Honolulu, Hawaii, January [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, [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 [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, [16] Markowitz, H. M., (1952), Portfolio Selection, Journal of Finance, 7, no. 1, [17] Roll, R., (1977), A Critique of the Asset Pricing Theory s Tests: Part I, Journal of Financial Economics, 4, [18] Sharpe, W. F., (1964), Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk, Journal of Finance, 19, [19] Shiller, R., (1973), A Distributed Lag Estimator Derived from Smoothness Priors, Econometrica, vol. 41, No.4, pp [20] Zalewska, A., (2004), Evolving character of the CAPM beta of the case of the telecom industry, an Indepen working paper. 114

12 urns and Returns s Measurement Equation System Equation : NTT urns and Returns s Measurement Equation System Equation : SONY 115

13 urns and Returns s Measurement Equation System Equation : urns and Returns s Measurement Equation System Equation : 116

14 urns and Returns s Measurement Equation System Equation : urns and Returns s Measurement Equation System Equation : 117

15 urns and Returns s : urns and Returns s : 118

16 urns and Returns s : urns and Returns s : 119

17 urns and Returns s : urns and Returns s : 120

18 urns and Returns s : urns and Returns s : 121

19 urns and Returns s : urns and Returns s : KDDI 122

20 urns and Returns s : urns and Returns s : 123

21 urns and Returns s : urns and Returns s : 124

22 urns and Returns s : urns and Returns s : 125