Hideo Kozumi

古澄 英男

古澄, 英男

Hideo, Kozumi

Institutional Repositories DataBase (IRDB)

ラプラス確率的フロンティアモデルのベイズ推定

Kwansei Gakuin University Repository

????????????????????????Bayesian Analysisof Laplace Stochastic Frontier Models???????¤ ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????This paper considers Laplace stochastic frontier models from aBayesian point of view. For posterior inference, we develop Gibbs samplingmethods using the mixture representations of the Lapalce distribution.The proposed methods are illustrated by both simulated and real data.Hideo Kozumi?? JEL?C11, C15, D24????????????????????????????????????????Keywords?Gibbs sampling, Laplace distribution, mixture representation,stochastic frontier model1 ????Aigner et al. (1977)?Meeusen and van den Broeck (1977)??????????????????????stochastic frontier model????????????????????????????????????????????????????????Kumbhakar and Lovell (2000)?Parmeter* ?????JSPS ??? 25245035 ?????????????|197|?????? 71 ?? 2 ?and Kumbhakar (2014)?????????????????????????????????????????????????????????????????????? 2??????????????????2???????????????????????????????????????????????????????Stevenson (1980)???????Meeusen and van den Broeck (1977)????????Beckers and Hammond (1987)?Greene (1990)?????????Tsionas (2007)??????????????????????????????????????????????????????Nguyen (2010)? Horrace and Parmeter (2015)??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Nguyen (2010)???????????????????Dempster et al. (1977)????????? EM???????Expectation-Maximization algorithm???????????????????Horrace and Parmeter (2015)????????di®erential evolution??????????????Storn and Price (1997)????????????????????????????????Nguyen (2010)???????????????????????????????????????????????Horrace and Parmeter (2015)????????????????????????????????????????????????????????????????????????|198|?????????????????????????????????????????????????????????????????????????????????????????Markov chainMonte Carlo method: MCMC??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? 2?????????????????????????????????????????Gibbs sampling??????????????????????????????????? 2??????????????????????????????????????????????????? 2???????????????????????????????????????????????????????? 3??????????????????????????????????????????????????????????????????????????????? 4??????????????????????????????? 5?????????????????2 ????????????????2.1 ????????Aigner et al. (1977)?Meeusen and van den Broeck (1977)???????????????????????yi = x0i¯ + vi ¡ ui; (i = 1; : : : ; n) (1)??????????????yi???????????????xi=(xi1; : : : ; xik)0? k £ 1??????????¯ = (¯1; : : : ; ¯k)0 ? k £ 1??????????vi ????????(1)????????????????????????????????????? ui ????????? ui ??????????????????|199|?????? 71 ?? 2 ?-6 -4 -2 0 2 4 60.00.10.20.30.40.5? 1: ?????? La(0; 1)????????? N(0; 1)??????????????????????????????????????????????????ui ???????one-sided error????????????????????? vi ???????two-sided error?????Nguyen (2010)? Horrace and Parmeter (2015)?????????????????????????????? vi ?????????????????????vi » La(0; ¾v)????????????????? La(¹; ¾)?????????f(x) =12¾exp¡jx¡ ¹j¾(¡1 < x <1)??????????? ¹???? 2¾2????? 1????????? La(0; 1)????????????????????????????????????????????????????????????????? ui ??????Nguyen (2010)??????ui » Exp(¾¡1u ) (2)| 200|????????????????????????????????????????Exp(°¡1)???? ° ?????????????????????? µ = (¯0; ¾v; ¾u)0 ???????????????????????????f(yjµ) =nYi=1Z 1012¾vexp¡jyi ¡ x0i¯ + uij¾v¼(uij¾u)dui (3)??????????????¼(uij¾u) ? (2) ??????? ui ???????¼(uij¾u) = 1¾uexp¡ ui¾u????(3)???????????????????????????????????????????????????????????????????????????????????????????????????????????????? ¯; ¾v; ¾u ?????????????????????¯; ¾v; ¾u ????????????¯ » N(¯0;B0); ¾v » IG(n0; s0); ¾u » IG(m0; d0)??????????¯0, B0?n0?s0?m0?d0 ???????????????????????IG(®; ¯)????????????????????f(x) =¯®¡(®)x¡(®+1) exp¡¯x???????¡(®)???????????2.2 ???????? I?????????????????????????Kotz et al. (2001)????????²i » N(0; 1); zi » Exp(1=2)???²i ? zi ??????????????????vi » La(0; ¾v)??vi = ¾vpzi²i (4)| 201|?????? 71 ?? 2 ???????????Park and Casella (2008)???????????????????????????? I?????????(4)????? zi ????????vi j zi » N(0; ¾2vzi)??????????????y = (y1; : : : ; yn)0?z = (z1; : : : ; zn)0?u =(u1; : : : ; un)0 ?????z? u??????????????????????f(yjµ; z;u) =nYi=11p2¼¾2vziexp¡ (yi ¡ x0i¯ + ui)22¾2vzi(5)?????????????? II??????????????????? I????????????????²i ??? (¡1; 1)???????? U(¡1; 1)?zi ?????? Ga(2; 1)????????:²i » U(¡1; 1); zi » Ga(2; 1)????????? Ga(®; ¯)????????f(x) =¯®¡(®)x®¡1 exp (¡¯x)??????????????? La(0; ¾v)????? vi ??²i ? zi ????vi = ¾vzi²i?????????Mallick and Yi (2014)????????????????II???????????????? zi ????????vi j zi » U(¡¾vzi; ¾vzi)?????????? I?II??????????????????????????????????????z? u???????????????????|202|?????????????????????????f(yjµ; z;u) =nYi=112¾vziI jyi ¡ x0i¯ + uij < ¾vzi (6)??????????I(¢)?????????3 ??????????(5)?? (6)??????????????????????????????????????????????(5)?? (6)??? z? u????????????????????????????????MCMC????????????????3.1 ?????????? I????????? I?????? (5)??????????¯?¾v?¾u?z?u????????????????????????¾v ???????????????(3)??? z????????????????¾v j y;¯;u » IGn0 + n; s0 +nXi=1jyi ¡ x0i¯ + uij???????¾u ???????????????ui » Exp(¾¡1u )?????? ¾u ????????¼(¾ujy;¯; ¾v; z;u) /nYi=1¾¡1u exp¡ ui¾u£ ¾¡(m0+1)u exp¡ d0¾u????????????????¾u j y;u » IGm0 + n; d0 +nXi=1ui??????????????????????????¯ ??????????¯ j y; ¾v; z;u » N(^¯ ; B^)????????B^ = nXi=1xix0i¾2vzi+B¡10!¡1; ^¯ = B^(nXi=1xi(yi + ui)¾2vzi+B¡10 ¯0)|203|?????? 71 ?? 2 ?????????? ui (i = 1; : : : ; n)???????????¼(uijy;¯; ¾v; ¾u; z) / exp¡ (yi ¡ x0i¯ + ui)22¾2vzi£ exp¡ ui¾u??????????????(yi ¡ x0i¯ + ui)22¾2vzi+ui¾u=nui ¡x0i¯ ¡ yi ¡ ¾2vzi¾uo2+ (yi ¡ x0i¯)22¾2vzi????????????????ui j y;¯; ¾v; ¾u; z » Nx0i¯ ¡ yi ¡ ¾2vzi¾u; ¾2vziI(ui > 0)????????ui ????????????? (0;1)???????????????????????????? zi (i = 1; : : : ; n) ???????????????(5) ?? zi »Exp(1=2)????????zi ???????????¼(zijy;¯; ¾v; ¾u;u) /1zi1=2exp¡ (yi ¡ x0i¯ + ui)22¾2vzi£ exp¡zi2?????????????±2i =(yi ¡ x0i¯ + ui)2¾2v?????¼(zijy;¯; ¾v; ¾u;u) /1zi1=2exp¡12±2izi+ zi????????°i = 1=zi ?????¼(°ijy;¯; ¾v; ¾u;u) / °1=2i exp¡12±2i °i +1°i£ °¡2i/ °¡3=2i exp¡ (°i ¡ 1=±i)22°i=±i????????????????°i ??????? IGauss(1=±i; 1)????????????????1zij y;¯; ¾v;u » IGauss1±i; 1?????????????? IGauss(¹; ¸)?????????f(x) =¸2¼x31=2exp¡¸(x¡ ¹)22¹2x???????|204|?????????????????????????3.2 ?????????? II??????? II?????? (6)?????????????¯ ?¾v?¾u?z?u????????????????????????????????¾v ? ¾u ?????????????? I???????????????¾v j y;¯;u » IGn0 + n; s0 +nXi=1jyi ¡ x0i¯ + uij¾u j y;u » IGm0 + n; d0 +nXi=1ui???????¯j (j = 1; : : : ; k)???????????(6)???¼(¯j j y; ¾v; z;u) /nYi=1I(jyi ¡ x0i¯ + uij < ¾vzi)£ ¼(¯j j ¯¡j)??????????????¼(¯j j ¯¡j)? ¯¡j= (¯1; : : : ; ¯j¡1; ¯j+1; : : : ; ¯k)0?????? ¯j ????????????¸ij =Pr 6=j xir¯r ???xij 6= 0????bLij = minyi ¡ ¸ij + ui ¡ ¾vzixij;yi ¡ ¸ij + ui + ¾vzixijbUij = maxyi ¡ ¸ij + ui ¡ ¾vzixij;yi ¡ ¸ij + ui + ¾vzixij??????????????¯j j y;¯¡j ; ¾v; z;u » ¼(¯j j ¯¡j)Imaxi:xij 6=0bLij < ¯j < mini:xij 6=0bUij(7)????????¯j ?????????????????? maxi:xij 6=0 bLij ;mini:xij 6=0 bUij????????????????????????????ui (i = 1; : : : ; n)??????????¼(uijy;¯; ¾2v; ¾u; z) / I jyi ¡ x0i¯ + uij < ¾vzi£ exp¡ ui¾u??????????? (x0i¯ ¡ yi ¡ ¾vzi;x0i¯ ¡ yi + ¾vzi)?????????? Exp(¾¡1u )?????ui j y;¯; ¾2v; ¾u; z » Exp(¾¡1u )I jyi ¡ x0i¯ + uij < ¾vzi|205|?????? 71 ?? 2 ???????????????zi (i = 1; : : : ; n)?????????????????????¼(zi j y;¯; ¾v; ¾u;u) / I jyi ¡ x0i¯ + uij < ¾vzi£ 1zi£ z2¡1i exp(¡zi)= exp(¡zi)I jyi ¡ x0i¯ + uij < ¾vzi?????????zi ????????????? (x0i¯ ¡ yi ¡ ¾vzi; x0i¯ ¡yi + ¾vzi)??????????zi j y;¯; ¾v; ¾u;u » Exp(1)I jyi ¡ x0i¯ + uij < ¾vzi???????? I?II????????????????????????????????????????????????MCMC?? 1???????????????Gelfand and Smith (1990)??????????????????????????????????????????????? I?????????????????????????? I????? II???????????????????????? II??????????????????????????????????????? I, II1. ¾v ? ¼(¾vjy;¯;u)???????????2. ¾u ? ¼(¾ujy;u)???????????3. zi (i = 1; : : : ; n)? ¼(zijy;¯; ¾v; ¾u;u)???????????4. ?????????? I?¯ ? ¼(¯jy; ¾v; z;u)????????????????????? II?¯j (j = 1; : : : ; k)? ¼(¯j jy;¯¡j ; ¾v; z;u)???????????5. ui (i = 1; : : : ; n)? ¼(uijy;¯; ¾v; ¾u; z)???????????|206|??????????????????????????????????? I? II?????¯ ??????????????????????????????????? I?? ¯ ???????????????????????????? II??? ¯j ????????????????????????????????????????????????????????????????????¾v ? ¾u ?????????? z????????????????????????? 2?????????????????? 3??????????????????¾v?¾u?z????????????????????¾v ? ¾u ?????????????????z???????????????????????4 ???4.1 ??????????????????????????????????????????????????????????????????????????????????????????????????yi = ¯1 + ¯1xi1 + ¢ ¢ ¢+ ¯kxik + vi ¡ ui; (i = 1; : : : ; n)xij » N(0; 1); vi » La(0; ¾v); ui » Exp(¾¡1u )????????????????¯j = 1 (j = 1; : : : ; k)?¾v = ¾u = 0:1????????????????????n = 100; 300?k = 2; 5; 10????????????¯ ? ¾v ????????????????????????¯ = 0; B0 = 100I; n0 = 1; s0 = 0:1???¾u ???????????van den Broeck et al. (1994)??????¾u » IG(1;¡ log r0); r0 = 0:875????????????????????? 30,000?????????????????????????????????????????????|207|?????? 71 ?? 2 ?????????? 10,000??????????????burn-in period?????????? 20,000?????????????? 1?? 2??????????????????????????????????????????????????????????????????????????????????????????2???????????????????????????????????????????????????????IF???????????????????????????????????? I????????????? II????????????????????????????????????????????????????????????????????????? 2????????????????????????????????????????????? II??????????????????????????????????????????????????????????????????????????????????? II??????????????? I?????????????? ¯j ??????????????????????????????????????????????(7)??????????¯j ????????????????????????????????????????????????????????????????????????????????????? II?????????????????????4.2 ???????Christensen and Greene (1976)????????????????????????????????????????????????????????????????????????? Greene (1990)?van denBroeck et al. (1994)?Koop et al. (1995)???????????????????????????????????????¡ log CPF= ¡¯1 ¡ ¯2 logQ¡ ¯3(logQ)2 ¡ ¯4 log PLPF¡ ¯5 log PKPF+ v ¡ u|208|?????????????????????????? 1: ???????????????? (n = 100)?????????? I ?????????? IIk ???? ?????? IF ???? ?????? IF2 ¯1 1.017 0.025 33.815 1.022 0.022 460.344¯2 0.998 0.021 14.587 1.001 0.020 90.988¾u 0.112 0.020 29.436 0.110 0.020 142.585¾v 0.127 0.034 39.526 0.132 0.032 276.5725 ¯1 1.004 0.034 54.468 1.007 0.029 598.534¯2 0.992 0.018 8.662 0.992 0.017 72.706¯3 0.998 0.019 11.152 0.997 0.020 64.818¯4 0.995 0.019 4.448 0.995 0.018 120.064¯5 0.991 0.017 12.524 0.992 0.016 143.955¾u 0.112 0.020 29.197 0.111 0.019 192.582¾u 0.106 0.038 55.852 0.109 0.033 404.34310 ¯1 1.006 0.030 35.773 0.999 0.026 781.836¯2 0.999 0.016 10.489 1.000 0.015 268.690¯3 0.971 0.018 6.927 0.974 0.018 87.592¯4 1.004 0.016 8.197 1.004 0.014 93.547¯5 0.997 0.018 7.730 1.002 0.018 402.423¯6 0.994 0.016 3.531 0.994 0.015 97.016¯7 1.016 0.017 1.909 1.013 0.017 258.013¯8 1.004 0.015 6.008 1.004 0.015 371.971¯9 0.978 0.015 4.249 0.975 0.016 289.292¯10 0.997 0.015 7.654 0.994 0.015 144.891¾u 0.091 0.019 25.468 0.093 0.019 336.216¾u 0.117 0.034 37.858 0.110 0.031 510.145?????????C ????????Q?????PL?PK?PF ???????????????????????????????????????Greene (1990)?????????????????????????????????????30; 000????????????????????? 10; 000?????????????????????? 20; 000 ????????????? 3???????????????????????????????????????V F =Var(u)Var(v) + Var(u)| 209|?????? 71 ?? 2 ?? 2: ???????????????? (n = 300)?????????? I ?????????? IIk ???? ?????? IF ???? ?????? IF2 ¯1 0.997 0.016 53.807 0.993 0.014 994.453¯2 1.009 0.010 7.172 1.007 0.010 528.858¾u 0.098 0.010 25.599 0.100 0.011 381.788¾v 0.114 0.018 47.234 0.110 0.018 668.0995 ¯1 0.994 0.023 93.118 0.993 0.013 862.109¯2 0.992 0.009 5.817 0.994 0.009 97.477¯3 0.992 0.010 7.122 0.991 0.010 275.259¯4 0.997 0.009 16.985 0.998 0.009 191.796¯5 0.983 0.009 11.190 0.985 0.008 279.970¾u 0.112 0.012 46.111 0.113 0.010 225.018¾u 0.084 0.026 105.986 0.084 0.017 570.01610 ¯1 1.006 0.015 40.114 1.005 0.009 563.153¯2 1.012 0.008 11.482 1.011 0.008 612.759¯3 0.996 0.009 10.952 0.994 0.009 753.699¯4 1.002 0.008 10.129 1.001 0.008 689.078¯5 1.007 0.009 8.900 1.010 0.008 571.756¯6 1.004 0.008 11.688 1.005 0.008 567.865¯7 0.999 0.009 11.725 0.996 0.009 568.825¯8 1.013 0.008 10.651 1.012 0.008 387.041¯9 1.004 0.009 11.556 1.004 0.008 222.331¯10 1.004 0.008 8.796 1.003 0.007 481.374¾u 0.079 0.010 31.041 0.079 0.008 130.004¾u 0.112 0.017 40.131 0.112 0.012 262.795??????????? TEf ???????????????uf ????????????????TEf = exp(¡uf )??????uf ??????????¼(uf jy) =Z¼(uf j¾u)¼(¾ujy)d¾u?????????????????????????????????????????????????????????????????????????????????????? II???????????????????? 3???????????????????????????????? I?????????|210|?????????????????????????0 20 40 60 80 1000.00.20.40.60.81.010 20 40 60 80 1000.00.20.40.60.81.010 20 40 60 80 1000.00.20.40.60.81.020 20 40 60 80 1000.00.20.40.60.81.020 20 40 60 80 1000.00.20.40.60.81.0u0 20 40 60 80 1000.00.20.40.60.81.0u0 20 40 60 80 1000.00.20.40.60.81.0v0 20 40 60 80 1000.00.20.40.60.81.0v? 2: ?????????????????????????????????? I ????????????????? II ?????????? II?????????????????????????? 95%??????????2???????????????????????????????????????? II???????????logQ???????????????????? I?????????????????????????logPk=PF????????????????? I?? 95%?????????????????????? II??????????????|211|?????? 71 ?? 2 ?? 3: ?????????????????????? I???? ?????? 95%???? IF??? -7.774 0.366 [-8.462, -7.030] 6.339logQ 0.445 0.041 [ 0.359, 0.522] 13.938(logQ)2 0.028 0.003 [ 0.023, 0.034] 13.978log PLPF0.291 0.072 [ 0.150, 0.431] 8.514log PKPF0.041 0.060 [-0.077, 0.160] 12.250¾v 0.084 0.015 [ 0.057, 0.115] 30.402¾u 0.092 0.029 [ 0.032, 0.146] 43.782V F 0.382 0.194 [ 0.044, 0.739] 42.334TEf 0.915 0.084 [ 0.690, 0.998] 5.272?????????? II???? ?????? 95%???? IF??? -7.497 0.049 [-7.591, -7.414] 1366.08logQ 0.416 0.010 [ 0.397, 0.431] 1872.32(logQ)2 0.030 0.001 [ 0.029, 0.031] 1386.70log PLPF0.261 0.014 [ 0.236, 0.281] 1902.61log PKPF0.070 0.032 [ 0.019, 0.143] 1033.51¾v 0.089 0.014 [ 0.061, 0.116] 348.17¾u 0.082 0.026 [ 0.035, 0.139] 588.12V F 0.311 0.171 [ 0.050, 0.697] 592.09TEf 0.924 0.077 [ 0.715, 0.998] 57.38???????????????????????????????????? II????????????????????????????????????????????????5 ????????Nguyen (2010)? Horrace and Parmeter (2015)?????????????????????????????????????????????????????????MCMC?? 1???????????????????????????????????????2???????????????????????????????????????????????????? I???????????????????????|212|?????????????????????????0 20 40 60 80 1000.00.40.810 20 40 60 80 1000.00.40.810 20 40 60 80 1000.00.40.820 20 40 60 80 1000.00.40.820 20 40 60 80 1000.00.40.830 20 40 60 80 1000.00.40.830 20 40 60 80 1000.00.40.840 20 40 60 80 1000.00.40.840 20 40 60 80 1000.00.40.850 20 40 60 80 1000.00.40.850 20 40 60 80 1000.00.40.8u0 20 40 60 80 1000.00.40.8u0 20 40 60 80 1000.00.40.8v0 20 40 60 80 1000.00.40.8v? 3: ?????????????????????????????? I ????????????????? II ??????? II????????????????????????????????????????? II????????????????????????????????????????????????????????????|213|?????? 71 ?? 2 ??????????????????????????????????????????????????????? II????????????????? I???????????????????? II????????????????????????????????????????????????????????? II????????????????????????????????????????????????????Damien et al. (2000)?????(7)?? ¯j??????????????????¯ ????????????????????????????????????????????????????????????????????????????????????????????????????????????????? II????????????????????????????????????????????????????????????????[1] Aigner, D. J., Lovell, C. A. K., and Schmidt, P. (1977). \Formulation andestimation of stochastic frontier production functions," Journal of Econo-metrics 6, 21{37.[2] Beckers, D. and Hammond, C. (1987). \A tractable likelihood functionfor the normal-gamma stochastic frontier model," Economics Letters 24,33{38.[3] Christensen, L. R. and Greene, W. H. (1976). \Economies of scale in U.S.electric power generation," Journal of Political Economy 84, 655{676.[4] Damien, P., Qin, Z. S., and Walker, S. G. (2000). \Uniform scale mixturemodels with applications to variance regression," University of MichiganWorking Paper, No. 00-014.[5] Dempster, A. P., Laird, N. M., and Rubin, D. B. 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ラプラス確率的フロンティアモデルのベイズ推定

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