Bayesian., 2016, 31(2): : (heterogeneity) Bayesian. . Gibbs : O212.8 : A : (2016)

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1 2016, 31(2): Bayesian 1, 2 (1., ; 2., ) : (heterogeney).,. Gibbs Bayesian.,.. : ; Bayesian ; ; Gibbs ; Metropolis-Hastings : O212.8 : A : (2016) Aigner (1977) [1] Meeusen Broeck(1977) [2],.,,., [1], [2], [3], Gamma [4] Gamma [5].,,,,,.. (heterogeney),. [6]. [7], copula. [8],, Bayesian MCMC. [9],,. [10] Gamma Gamma Bayesian, : : : (13ZR ); (14YZ115)

2 [11],., GARCH(p, q).,,. p = q = 0,.,,. Gibbs Bayesian.,, Gibbs. λ α,, Griddy-Gibbs. Nakatsuma( [12]) GARCH, ARMA,, Metropolis-Hastings λ α.. 2,. y = x β + v u, i = 1, 2,, N, t = 1,, T. y i t, x k 1, i t, β k 1. v, v i t v N ( 0, σ 2). u,. v u, u log u = z γ + ε, t = 1, 2,, T, (1) ε = p h ξ, h = λ 0 + λ j ε 2 i,t j + α j h i,t j. i, t ξ N(0, 1), z m 1, γ m 1. z γ, N(0, h ). h GARCH(p, q), λ 1 = = λ p = α 1 = = α q = 0, h λ 0, u, h, (1).,, [8] [11].,.,,. λ = [λ 0, λ 1,..., λ p ], α = [α 1,..., α q ], u = [u, i = 1, 2,, N, t = 1,, T ], θ = [β, γ, α, λ, σ 2 ] u. θ p(θ), Bayesian, p(y x, u, β, σ 2 ), p(u z, γ, α, λ) p(θ), x = [x, i = 1, 2,, N, t = 1,, T ], y = [y, i = 1, 2,, N, t = 1,, T ]. y u p (y, u x, z, θ) =p(y x, u, β, σ 2 )p(u z, γ, α, λ)

3 : Bayesian 129 [ ] =(2πσ 2 ) (NT/2) exp 1 2σ 2 (y + u x β) 2 N T [ (h ) 1/2 exp 1 ] (log u z 2h γ) 2 log u 3 Bayesian Bayesian Gibbs.,,,. β, γ, α, λ σ 2, β, γ, α, λ p (β, γ, α, λ) = p (β) p (γ) p (λ, α).,., β, γ, α, λ., h, λ α λ i 0, α i 0 p λ i + α i < 1., β, γ, α, λ p p q p (β, γ, α, λ) I( λ i + α i < 1) I(λ i 0) I(α i 0)., I( ). i=0 σ 2 Gamma ( [13]),, p 0 > 0 q 0 > 0. p ( σ 2) σ p0 1 exp ( q 0 /σ 2) RG ((p 0 1)/2, q 0 ). Bayesian,. θ u p (θ, u x, z, y) p (y, u x, z, θ) p (θ), N [ T σ (NT +p0+1) (h ) 1/2 exp + N i=0 1 2σ 2 ( N ( 1 2h (log u z γ) 2 log u ) ] I( (y + u x β) 2 + 2q 0 ) p λ i + α i < 1) p q I(λ i 0) I(α i 0). (2), Bayesian. MCMC Monte Carlo,,. MCMC Markov Markov,,. Gibbs.

4 MCMC, Gibbs. MCMC Markov (2), Gibbs. Gibbs Markov {β (i), σ 2(i), γ (1), λ (i), α (i), u (i), i = 1,..., M}., Gibbs, E[ ˆf(θ)] 1 M = f(θ (i) ). M m i=m+1 Gibbs : {β (1), σ 2(1), γ (1), λ (1), α (1), u (1) }, i = 2,..., M, i : 1 p(β (t) σ 2(t 1), γ (t 1), λ (t 1), α (t 1), u (t 1), y, x) β (t), 2 p(σ 2(t) β (t), γ (t 1), λ (t 1), α (t 1), u (t 1), y, x) σ 2(t), 3 p(u (t) β (t), σ 2(t), γ (t 1), λ (t 1), α (t 1), y, x) u (t), 4 p(γ (t) β (t), σ 2(t), λ (t 1), α (t 1), u (t), y, x) γ (t) 5 p(λ (t) β (t), σ 2(t), γ (t), α (t 1), u (t), y, x) λ (t), 6 p(α (t) β (t), σ 2(t), γ (t), λ (t), u (t), y, x) α (t), 7 h = [h, i = 1, 2,, N, t = 1,, T ].. (2). p (β u, γ, λ, α, y, x) exp{ 1 2σ 2 (y + u x β) 2 } N((x x) 1 x (y + u), σ 2 (x x) 1 ), p ( σ 2 y, x, u, β, γ, λ, α ) σ (NT +p0+1) exp{ 1 2σ 2 [ N (y + u x β) 2 + 2q 0 ]} ( (y + u x β) (y + u x β) + 2q 0 σ 2 ) NT +p exp{ (y + u x β) (y + u x β) + 2q 0 2σ 2 }. (3), x (N T ) k, y (N T ) 1. (3), ((y + u x β) (y + u x β) + 2q 0 )/σ 2 Ga ((NT + p 0 + 3)/2, 1/2). = χ 2 NT +p 0+3., χ 2 NT +p NT + p χ 2. (log u z p(γ u, z, λ, α) exp{ γ)2 } N(µ γ, Σ γ ). 2h t=2 γ, µ γ = Σ γ ( z h 1 log u ), Σ γ = ( z h 1 z ) 1. β, γ σ 2,.

5 : Bayesian 131 u, i t u p(u y, x, z, β, γ, h) 1 exp{ 1 u 2σ 2 (y + u x β) 2 1 (log u z 2h γ) 2 }.,,., u 1. 1( u ) 1 U[0, f 1 (u (t 1) )] ω 1. 2 U[0, f 2 (u (t 1) )] ω 2. 3 U A u (t). U A A, A = {x; f i (x) ω i, i = 1, 2}. 1 f 1 f 2 u, p(u y, x, z, β, γ, h) = f 1 (u )f 2 (u ). f 1 (u ) = exp{ (y + u x β) 2 /(2σ 2 )}, f 2 (u ) = exp{ 1 2h (log u z γ) 2 log(u )}. λ α N T p(λ, α u, γ) (h ) 1 2 exp{ h = λ 0 + p λ j ε 2 i,t j + (log u z γ)2 2h }. α j h i,t j., u, γ, λ α,. Nakatsuma( [12]) GARCH, Metropolis-Hastings. h = λ 0 + w = ε 2 h, p λ j ε 2 i,t j + ε 2, GARCH : l ε 2 = λ 0 + (λ j + α j ) ε 2 i,t j + w l = max{p, q}, α j h i,t j α j w i,t j, w N ( 0, 2h 2 ). E(w ε 2, h, λ, α) = 0, Var(w ε 2, h, λ, α) = 2h 2. λ, Metropolis-Hastings : 2( λ) 1 π(λ) λ. 2 U(0, 1) e.

6 e α(λ (t 1), λ ), λ (t) = λ ;, λ (t) = λ (t 1) ; { α(λ (t 1), λ ) = min 1, p(λ β, σ 2, α, u, y, x)π(λ (t 1) } ). p(λ (t 1) β, σ 2, α, u, y, x)π(λ ) 2 λ, GARCH :,, λ, π ( ε 2 λ, α, u, γ ) N T N 1 exp{ w2 h 4h 2 } T 1 exp{ (ε2 ζ λ ) 2 } h N(ˆµ λ, ˆΣ λ ). ζ = [ ι, ε 2 i,t 1,..., ε 2 i,t p], ι = 1 + 4h 2 α j ι i,t j, ε 2 = ε 2 + λ α, u, γ N(ˆµ λ, ˆΣ λ ) ˆµ λ = ˆΣ λ N ε 2 ζ 2h 2 p I(λ i 0). i=0, ˆΣ λ = ( ζ ζ 2h 2 ) 1. α j ε 2 i,t j. α, w (α) α Taylor, GARCH π (α λ, u, γ) N T N 1 exp{ w2 h 4h 2 } T 1 exp{ (w (α ) ζ (α α ) ) 2 } h N(ˆµ α, ˆΣ α ). 4h 2 α α, α w (α) = arg min 2h 2, (0 < α j < 1, j = 1,..., q). ζ = [ζ 1,..., ζ q ], ζ j w αj. dw (α j ) dw i,t s = h i,t j + α s, dα j dα j s=1 ζ j = dw (α j ) dα j = h i,t 1 (α ) + αsζ i,t s,j. s=1

7 : Bayesian 133, α ˆµ α = ˆΣ α N α λ, γ, u N(ˆµ α, ˆΣ α ) q I(α i 0). Y α = w (α ) + ζ (α ), X α = ζ, t=2 Y α Xα 2h 2, ˆΣ α = ( t=2 4 (X α) (X α) 2h 2 ) 1. Gibbs,., p = q = 1, : y = x β + v u, i = 1, 2, t = 1,, T, log u = z γ + ε, ε = h ξ, h = λ 0 + λ 1 ε 2 i,t 1 + α 1 h i,t 1. k = 3, m = 3, x, z 1,, x, z k 1, 0.5, [14]. β [1, 1, 1], γ [ 0.3, 1, 1]. σ 2 = 0.05, λ 0 = 0.05, λ 1 = 0.2, α 1 = 0.7. p 0 = 4, q 0 = , , T=100 T=500 T=1000 sd sd sd β β β γ γ γ σ λ λ α ,, β, z. λ 0, λ 1, α 1 σ 2,,., λ 0, λ 1 α 1,,., Gibbs Bayesian.,, λ 0, h,., λ 0 Metropolis-Hasting., λ 0, λ 0, Metropolis-Hasting.

8 , λ 0. λ 0 U[δ, 1], δ, λ [15], ,, log y =β 0 + β 2 log x 1 + β 3 log x β 4 log x 1 log x β 5 log x 2 log x 2 + β 6 log x 1 log x 2 + v u, log u =γ 0 + γ 1 z + ε., y, x 1, x 2 i t,. z i t. [16]., p q 1. Gibbs ,. 2 β 0 β 1 β 2 β 3 β 4 β (0.433) (0.292) (0.244) (0.124) (0.195) (0.153) γ 0 γ 1 σ 2 λ 0 λ 1 α (0.893) (0.209) (0.001) (0.003) (0.136) (0.148) 2, γ,. λ , λ 1 α ,.,. h GARCH(1,1),., [8] [11],. : [1] Aigner D, Knox-Lovell C A, Schmdt P. Formulation and estimation of stochastic frontier production function models [J]. J Econometrics, 1977, 6: [2] Meeusen W, van der Broeck J. Efficiency estimation from Cobb-Douglas production functions wh Composed Error[J]. Internat Econom Rev, 1977, 18: [3] Stevenson R E. Likelihood functions for generalized stochastic frontier model[j]. J Econometrics, 1980, 13: [4] Greene W H. A gamma-distributed stochastic frontier model[j]. J Econometrics, 1994, 61: [5],. Gamma Bayesian [J]., 2013, 28(4): [6] Greene W. Reconsidering heterogeney in panel data estimators of the stochastic frontier model[j]. J Econometrics, 2005, 126:

9 : Bayesian 135 [7] Carta A, Steel M F J. Modelling multi-output stochastic frontiers using copulas[j]. Comput Statist Data Anal, 2012, 56: [8] Tsionas E. Inference in dynamic stochastic frontier models[j]. J Appl Econom, 2006, 21: [9] Chen Yi-Yi, Schmidt P, Wang Hung-Jen. Consistent estimation of the fixed effects stochastic frontier model[j]. J Econometrics, 2014, 181: [10] Griffin J, Steel M. Flexible mixture modelling of stochastic frontiers[j]. J Prod Anal, 2008, 29: [11] Gal n J, Veiga H, Wiper M. Bayesian estimation of inefficiency heterogeney in stochastic frontier models[j]. J Prod Anal, 2014, 42: [12] Nakatsuma T. Bayesian Analysis of ARMA-GARCH Models A Markov chain Sampling Approach[J]. J Econometrics, 2000, 95: [13] Fernandez C, Osiewalski J, Steel M F J. On the use of panel data in stochastic frontier models[j]. J Econometrics, 1997, 79: [14] Wang Hung-Jen, Schmidt P. One step and two step estimation of the effects of exogenous variables on technical efficiency levels[j]. Prod Anal, 2002, 18: [15] Gal n J E, Pollt M. Inefficiency persistence and heterogeney in Colombian electricy utilies[j]. Energ Econ, 2014, 46: [16] Growsch T, Jamasb T, Pollt M. Qualy of service, efficiency and scale in network industries: an analysis of European electricy distribution[j]. Appl Econ, 2005, 41: Bayesian inference for dynamic heterogeney stochastic frontier model CHENG Di 1, ZHANG Shi-bin 2 (1. School of Math. Sci., Inner Mongolian Univ., Hohhot , China; 2. Dept. of Math., Shanghai Marime Univ., Shanghai , China) Abstract: If heterogeney of the inefficiency term is disregarded, will result in the incorrect estimate of this term in the stochastic frontier model. By combining the influence from characteristic differences of individuals wh the time-varying property of variance, a dynamic heterogeney stochastic frontier model is proposed. dynamic heterogeney stochastic frontier model is given. By the Gibbs sampling, the methodology for Bayesian analysis of the For each model parameter, the posterior distribution is derived. A simulation study shows that under the crerion of minimizing the posterior mean square error, the Bayesian estimate is close to s true value for small and medium sized samples. From the Bayesian analysis based on the real electric power company generation data, is evidenced that there exists the time-varying property for the variance of the logarhm inefficiency term. Keywords: stochastic frontier model; Bayesian inference; heterogeney; Gibbs sampling; Metropolis-Hastings sampling MR Subject Classification: 62F15; 65C60

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