Temple (1999) Barro and Sala-i-Martin (1992) β σ β σ Togo (2002) Quah (1993) 1) 2 Kakamu and Fukushige (2006) Barro and Sala-i-Martin (1

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1 41, 1, Regional Income Distribution and Convergence Hypothesis Kazuhiko Kakamu, Hajime Wago and Yoshihiro Ohtsuka σ This paper estimates the regional income distribution from a Bayesian point of view and examines the convergence hypothesis. We utilize the bivariate normal mixture models to examine whether the regional income distribution is unimodal or bimodal. In addition, we introduce the spatial econometric models to examine the spatial correlations in the regional income distribution. From the empirical results using the data from 1986 to 2004, we confirm that the regional income distribution is unimodal and that there exists strong spatial heterogeneity. Moreover, we cannot find the evidence of σ-convergence in this period. : 1. 1 Barro and Sala-i- Martin (1992) 2 1 β 1 σ β 1 1 σ 1 1 : ( kakamu@le.chiba-u.ac.jp). : ( wago@ism.ac.jp). : ( ed094001@g.hit-u.ac.jp).

2 Temple (1999) Barro and Sala-i-Martin (1992) β σ β σ Togo (2002) Quah (1993) 1) 2 Kakamu and Fukushige (2006) Barro and Sala-i-Martin (1992) (2005) (2005) Pen World Table σ 47 Kakamu and Wago (2008) Kakamu et al. (2008) 1) β σ 2

3 183 Hashiguchi (2010) Kakamu et al. (2011) Seya et al. (2011) 1 (2005) 2 2) 3) σ Barro and Sala-i-Martin (1992) σ σ ) 2 (2005) 3)

4 i t 1 y it y it σ y it = µ t + ɛ it, ɛ it N (0, σ 2 t ). (2.1) (2.1) exp(y it ) Nishino and Kakamu (2009) (2.1) L 1 (Y µ, σ 2 ) = T n t=1 1 2πσ 2 t exp { (y it µ t ) 2 2σ 2 t }, (2.2) y t = (y 1t,, y nt ) Y = (y 1,, y T ) µ = (µ 1,, µ T ) σ 2 = (σ1, 2, σt 2 ) µ t σt 2 t σt 2 σ s it y it = µ sit,t + ɛ it ɛ it N (0 σ 2 s it,t). (2.3) s it = j (2.3) L 2 (Y µ, σ 2, w, S) = T n t=1 w sit,t exp 2πσs 2 it,t { (y it µ sit,t) 2 2σ 2 s it,t }, (2.4) w jt t j s t = (s 1t,, s nt ) S = (s 1,, s T ) µ = {{µ jt } T t=1} 2 j=1 σ2 = {{σ 2 jt }T t=1} 2 j=1 w = {{w jt} T t=1} 2 j=1 (2.3) y it = µ jt + ɛ it, ɛ it N (0, σ 2 jt), Pr(s it = j s i,t 1 = k) = p jk. (2.5)

5 185 µ 1t < µ 2t Π = p 11 p 12 = p 11 1 p 22 w = (w1, w 2 ) = p 21 p 22 1 p 11 p 22 (w 1, 1 w 1 ) i s i1 w si1 σ (2.5) L 3 (Y µ, σ 2, w, Π, S) = { n w si1 T T t=2 n t=1 p sit,s i,t 1 } 1 2πσs 2 it,t exp { (y it µ sit,t) 2 2σ 2 s it,t }, (2.6) (2.1) (2.5) σ 4) 2.3 (2.1) y it y it = µ t + ρ t n l=1 d il y lt + ɛ it, ɛ it N ( 0, σ 2 t ), (2.7) d il i, l i l d ii = 0 Anselin (1988) ρ t l i 5) (2.7) T n ( ) L 4 (Y µ, σ 2 1, ρ, D) = I ρ t D exp e2 it 2πσ 2 t 2σt 2, (2.8) t=1 D = {d il } ρ = (ρ 1,, ρ T ) I n n e it ρ t n l=1 d ily lt µ t = y it 4) Rostow (1960) take-off s it s it 5) LeSage and Pace (2009)

6 (2.1) ɛ it y it = µ t + u it, (2.9) u it = ρ t n l=1 d il u lt + ɛ it, ɛ it N ( 0, σ 2 t ), ρ t (2.10) T n ( ) L 5 (Y µ, σ 2 1, ρ, D) = I ρ t D exp e2 it 2πσ 2 t 2σt 2, (2.10) t=1 e it = y it µ t ρ t n l=1 d il(y lt µ t ) (2.1) (2.7) (2.10) 2.5 (2.5) (2.7) y it = µ sit,t + ρ t n l=1 d il y lt + ɛ it, ɛ it N (0, σ 2 s it,t), Pr(s it = j s i,t 1 = k) = p jk. (2.11) (2.11) L 6 (Y µ, σ 2, ρ, w, Π, S, D) = { n t=1 w si1 T t=2 p sit,s i,t 1 } T n I ρ t D 1 2πσs 2 it,t { exp e2 it 2σ 2 s it,t }, (2.12) n e it = y it ρ t l=1 d ily lt µ sit,t (2.5) (2.10)

7 187 y it = µ sit,t + u it, u it = ρ t n l=1 d il u lt + ɛ it, ɛ it N ( 0, σ 2 s it,t), (2.13) Pr(s it = j s i,t 1 = k) = p jk. (2.13) L 7 (Y µ, σ 2, ρ, w, Π, S, D) = { n t=1 w si1 T t=2 p sit,s i,t 1 } T n I ρ t D 1 2πσs 2 it,t { exp e2 it 2σ 2 s it,t }, (2.14) n e it = y it µ sit,t ρ t l=1 d il(y lt µ slt,t) ) χ 2 Hansen (1992) Garcia (1998) 6) (2001) (2005) (2007)

8 (2.5) (2.10) 2 T π( µ, σ 2, ρ, w, Π) = π(w 1 ) π(p jj ) π(µ jt )π(σjt)π(ρ 2 t ), (3.1) (2.14) (3.1) j=1 π( µ, σ 2, ρ, w, Π, S Y, D) π( µ, σ 2, ρ, w, Π)π(S w, Π)L 7 (Y µ, σ 2, ρ, w, Π, S, D), (3.2) µ jt σ 2 jt t = 1,, T, j = 1, 2 ρ t t = 1,, T w 1 p jj (µ 1t, µ 2t ) N (µ 0 1 2, τ 2 0 I 2 )I(µ 1t < µ 2t ), σ 2 jt IG(ν 0 /2, λ 0 /2), ρ t U( 1, 1), w 1 BE(a 0, b 0 ), p jj BE(a 0, b 0 ), (3.3) IG( ) U( ) BE( ) I(µ 1t < µ 2t ) µ jt µ 1t < µ 2t n l=1 d il = 1 ρ t ( 1, 1) Sun et al. (1999) 3.2 (3.2) w 1 p jj j = 1, 2 t=1 w 1 BE(â, ˆb), p jj BE(â j, ˆb j ), â = n 11 + a 0, ˆb = n n11 + b 0, â j = n jj + a 0, ˆbj = n kj + b 0,

9 189 n 11 1 s i1 = 1 (i = 1,..., n) n kj = T t=2 n 1(s it = j, s i,t 1 = k) 1(s it = j, s i,t 1 = k) 1 w 1 w 1 w 1 p jj µ jt σjt 2 t = 1,, T j = 1, 2 µ 1t N (ˆµ 1t, ˆτ 2 1t)I [,µ2t ), µ 2t N (ˆµ 2t, ˆτ 2 2t)I (µ1t, ], σ 2 jt IG(ˆν jt /2, ˆλ jt /2) (j = 1, 2), ˆτ jt 2 = σ 2 jt ˆµ jt = ˆτ 2 jt ( ) n n 2 1(s it = j) ρ t d il 1(s lt = j) + τ0 2 σ 2 jt { l=1 ( n n 1(s it = j) ρ t d il 1(s lt = j) y it 1(s it = j) ρ t ˆν jt = n j + ν 0, ˆλjt = n l=1 l=1 d il y lt 1(s it = j) {( n y it µ jt ρ t n l=1 } ) 2 1 ] + τ0 2 µ 0, d il (y lt µ slt,t), ) 1(s it = j)} 2 + λ 0, n j = n 1(s it = j) µ jt (j = 1, 2) µ 1t < µ 2t ρ t t = 1,, T π(ρ t µ, σ 2, ρ t, S, Y, D) ( I ρ t D exp n ) n (y it µ sit,t ρ t l=1 d il(y lt µ slt,t)) 2, (3.4) 2σ 2 s it,t ρ t = (ρ 1,, ρ t 1, ρ t+1,, ρ T ) (Metropolis- Hastings MH) Ohtsuka and Kakamu (2010) HM MH Ohtsuka and Kakamu (2009)

10 ) (3.4) ρ t m ρ t [ 1, 1] 8) 1 = ρ t1 < ρ t2 < < ρ tm < ρ t,m+1 = 1. ρ k t (k {1,..., m}) [ρ tk, ρ t,k+1 ] [ρ tk, ρ t,k+1 ] ω tk = π(ρ t µ, σ 2, ρ t, S, Y, D) m h=1 π(ρh t µ, σ 2, ρ t, S, Y, D), [ρ tk, ρ t,k+1 ] ρ k t π(ρ k t ) = ω tk(ρ t,k+1 ρ tk ) m j=1 ω j(ρ t,j+1 ρ tj ), ρ k t [ρ tk, ρ t,k+1 ] ρ t Ohtsuka and Kakamu (2010) s it S sit S s it S sit s it Pr{s it = j µ, σ 2, w, Π, Y, D, S it } = w jit n k=1 2 n h=1 w hit k=1 q 1 2πσs 2 kt,t q 1 2πσs 2 kt,t { } exp e kt(s it=j) 2 2σs 2 kt,t { exp e kt(s it =h) 2 2σs 2 kt,t }, (3.5) p si2,jw j, (t = 1) wjit = p si,t+1,jp j,si,t 1 (2 t T 1) p j,st 1 (t = T ) n e kt (s it = h) = y kt µ skt,t ρ t d kl (y lt µ slt,t) ρ t d ki (y it µ h,t ), l=1,l i (3.6) 7) Ritter and Taner (1992) (2001) Bauwens and Lubrano (1998) Wago (2004) Ohtsuka and Kakamu (2009) 8) Ohtsuka and Kakamu (2009)

11 191 (3.6) s it s i,t 1 s i,t+1 S sit p si,t+1,j(j = 1, 2) s it Hamilton (1989) s it (3.5) 9) D Kakamu et al. (2008) 10) a 0 = 1.0, b 0 = 1.0, µ 0 = 0.0, τ 0 = 10, ν 0 = 3.0, λ 0 = , 000 MCMC 15, 000 5, 000 Ox version 6.2 (Doornik (2006) ) Geweke (1999) (2009) 11) 1 6 9) (2005) 10) 4 4

12 ) (2009) Chib(1995) 12) Togo (2002) Kakamu and Fukushige (2006) 2 Barro and Sala-i-Matin (1992) Togo (2002) 2 Togo (2002) Fujita and Tabuchi (1997) 2 ( ) 3 ( ) 11) π(y ) θ S (2009) π(y ) =»» g(θ) 1 E π(y S, θ)π(θ) " # 1 1 MX g(θ (i) ) M π(y S (i), θ (i) )π(θ (i). ) M θ g(θ) = ξ 1 (2π) d/2 V θ 1/2 exp[ 0.5(θ θ) V 1 θ (θ θ)] I{(θ θ) V 1 θ (θ θ) F 1 (ξ)}. θ V θ d F χ 2 d (ξ) d χ 2 ξ = 0.95 µ jt π(µ) µ 0 τ 2 0 µ 1t µ 2t 10, , ) ξ = 0.95, 0.9 ξ = 0.95 χ 2 d

13 (186) (163) (205) (187) (105) Ergodic t (t = 1,, T 1) 7.75, , t , , Ergodic Kawagoe (1999) < < 3 5 {µ t } T t=1 {σ2 t } T t=1 {ρ t} T t=1 95% Geweke (1992) p 13) 3 5 CD p % µ µ t σ µ t 13) (Inefficiency Factor: IF) Chib (2001) X IF = ˆρ l. (4.1) l=1 ˆρ l l l

14 µ t 2.5%CI 97.5%CI CD IF µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ ) CI CD IF Geweke (1992) p Chib (2001) % 0 3 σt 2 σ % σ

15 195 4 σ t 2.5%CI 97.5%CI CD IF σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ ) CI CD IF Geweke (1992) p Chib (2001) σ Braun and Kubota (2000) 5. 2 σ Barro and Sala-i-Martin (1992)

16 ρ t 2.5%CI 97.5%CI CD IF ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ) CI CD IF Geweke (1992) p Chib (2001) σ 1 Paap and van Djik (1996) 2

17 µ t ( : 95% : ) ρ t ( : 95% : )

18 σt 2 ( : 95% : ) 0.5 Kakamu et al. (2010) Kakamu and Wago (2010) Regional Convergence in Japan: A Bayesian Spatial Econometric Perspective 19 th International Conference on Computational Statistics ( ) ( ) Anselin, L. (1988). Spatial Econometrics: Methods and Models, Dordrecht, Kluwer. Barro, R. J. and Sala-i-Martin, X. (1992). Regional growth and migration: A US-Japan comparison, J. Jpn. Int. Econ. 6(4), Bauwens, L and Lubrano, M. (1998). Bayesian inference on GARCH models using the Gibbs sampler, Econometrics Journal, 1, c23 c46.

19 199 Braun, A. R. and Kubota, K. (2000). The effect of government capital on labor productivity in Japanese prefecture, mimeo. Chib, S. (1995). Marginal likelihood from the Gibbs output, J. Am. Stat. Assoc., 1, Chib, S. (2001). Markov chain Monte Carlo methods: Computation and inference, Handbook of Econometrics (eds. J. J. Heckman and E. Leaper), Elsevier, pp Doornik, J. A. (2006). Ox: An Object Oriented Matrix Programming Language, Timberlake Consultants Press, London. Fujita, M. and Tabuchi, T. (1997). Regional growth in postwar Japan, Reg. Sci. Urban Econ., 27, Garcia, R. (1998). Asymptotic null distribution of the likelihood ratio test in Markov switching models, Int. Econ. Rev., 39, Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, Bayesian Statistics 4 (eds. J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M, Smith), Oxford University Press, pp Hansen, B. E. (1992). The likelihood ratio test under nonstandard conditions: Testing the Markov switching model of GNP, J. Appl. Econ., 7, S61 S82. Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, Hashiguchi, Y. (2010). Bayesian estimation of spatial externalities using regional production function: The case of China and Japan, Economics Bulletin, 30, Kakamu, K. and Fukushige, M. (2006). Productivity convergence of manufacturing industries in Japanese MEA, Applied Economics Letters, 13, Kakamu, K. and Wago, H. (2008). Small-sample properties of panel spatial autoregressive models: Comparison of the Bayesian and maximum likelihood methods, Spatial Economic Analysis, 3, Kakamu, K., Polasek, W. and Wago, H. (2008). Spatial interaction of crime incidents in Japan, Mathematics and Computers in Simulation 78(2 3), Kakamu, K., Wago, H. and Tanizaki, H. (2010). Estimation of regional business cycle in Japan using Bayesian panel spatial autoregressive approach, Handbook of Regional Economics (ed. T. P. Nolin), Nova Science Publishers, pp Kakamu, K., Polasek, W. and Wago, H. (2011). Production technology and agglomeration for Japanese prefectures during forthcoming in Pap. Reg. Sci. Kawagoe, M. (1999). Regional dynamics in Japan: A reexamination of Barro regressions, J. Jpn. Int. Econ., 13, LeSage, J. and Pace, R. K. (2009). Introduction to Spatial Econometrics, Chapman & Hall/CRC, London. (2005) (2007).. Nishino, H. and Kakamu, K. (2009). Grouped data estimation and testing of Gini coefficient using lognormal distributions, mimeo. Ohtsuka, Y. and Kakamu, K. (2009). Estimation of electric demand in Japan: A Bayesian spatial autoregressive AR(p) approach, Inflation: Causes and Effects (ed. L. V. Schwartz), Nova Science Publishers, pp Ohtsuka, Y. and Kakamu, K. (2010). Comparison of the sampling efficiency in spatial autoregressive model, mimeo. (2001). 31(3), Paap, R. and van Dijk, H. K. (1998). Distribution and mobility of wealth of nations, Eur. Econ. Rev., 42, Quah, D. (1993). Empirical cross-section dynamics in economic growth, Eur. Econ. Rev., 37, Ritter, C. and Tanner, M. A. (1992). Facilitating the Gibbs sampler:the Gibbs stopper and the griddy Gibbs sampler, J. Am. Stat. Assoc., 87,

20 Rostow, W. W. (1960). The Stages of Economic Growth: A Non-Communist Manifesto, Cambridge university press. Seya, H., Tsutsumi, M. and Yamagata, Y. (2011). Income Convergence in Japan: A Bayesian spatial durbin model approach, forthcoming in Econ. Model. Sun, D., Tsutakawa, R. K. and Speckman, P. L. (1999). Posterior distribution of hierarchical models using CAR(1) distributions, Biometrika, 86(2), Temple, J. (1999). The new growth evidence, J. Econ. Lit., 37, Togo, K. (2002). Productivity convergence in Japan s manufacturing industries, Econ. Lett., 75, Wago, H. (2004). Bayesian estimation of smooth transition GARCH model using Gibbs sampling, Mathematics and Computers in Simulation, 64, (2005). -. (2009). 60,

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w