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(3a) (1) 1 G-R收敛诊断 (3b) (2) 1 G-R收敛诊断 (3c) (2) 2 G-R收敛诊断 1000030000500001.01.52.02.5last iteration in chainshrink factormedian97.5%1000030000500001.01.52.02.5last iteration in chainshrink factormedian97.5%1000030000500001.02.03.0last iteration in chainshrink factormedian97.5%10000300005000012345last iteration in chainshrink factormedian97.5%1000030000500001.01.52.02.5last iteration in chainshrink factormedian97.5%1000030000500001.01.52.02.5last iteration in chainshrink factormedian97.5%1000030000500001.02.03.0last iteration in chainshrink factormedian97.5%10000300005000012345last iteration in chainshrink 10000200003000040000500001.01.11.21.31.4last iteration in chainshrink 10000200003000040000500001.01.41.82.2last iteration chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink 100002000030000400005000012345last iteration chainshrink factormedian97.5% (3d) (2) G-R收敛诊断 (3e)(2)G-R收敛诊断 (3f) (3) 230 1 G-R收敛诊断 10000200003000040000500001.01.11.21.31.4last iteration in chainshrink factormedian97.5%10000200003000040000500001.01.41.82.2last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5%10000200003000040000500001.01.11.21.31.4last iteration in chainshrink factormedian97.5%10000200003000040000500001.01.41.82.2last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5%10000200003000040000500001.01.11.21.31.4last iteration in chainshrink factormedian97.5%10000200003000040000500001.01.41.82.2last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5%100002000030000400005000012345last iteration in chainshrink factormedian97.5% (3g)(3)2G-R收敛诊断 (3h)(3)G-R收敛诊断 (3i)(3)G-R收敛诊断 图3参数的G-R收敛诊断图 Fig.3 G-R convergence diagnostic for parameters 由图3可知：给定不同的初始值，各参数的G-R检验统计量随着迭代次数增加逐渐趋235 近于1，表明各个参数的Markov链达到了平稳状态。图1-3说明MCMC抽样算法是有效的，各参数的后验分布达到了稳定状态；图4给出了模型参数边缘后验分布核密度。 10000200003000040000500000.050.100.15IterationsTrace of var10.050.100.150.2005101520density.default(x = y, width = width)N = 15000 Bandwidth = 0.00276Density of var110000200003000040000500000.00240.00260.00280.00300.0032IterationsTrace of var20.00240.00260.00280.00300.00320500150025003500density.default(x = y, width = width)N = 15000 Bandwidth = 1.669e-05Density of var2-0.6-0.4-0.20.00.20.01.02.03.0density.default(x = a)N = 15000 Bandwidth = 0.01473Density1000030000500000.20.40.60.8IterationsTrace of var10.20.40.60.81.0012345density.default(x = y, width = width)N = 15000 Bandwidth = 0.01197Density of var1 (4a)(1)1的后验分布密度图 (4b)(2)1的后验分布图 (4c)(2)2的后验分布图 1000020000300004000050000103050IterationsTrace of var11020304050600.000.030.06density.default(x = y, width = width)N = 20000 Bandwidth = 0.9068Density of var110000200003000040000500000.000.15IterationsTrace of var10.000.050.100.150.200.2504812density.default(x = y, width = width)N = 20000 Bandwidth = 0.004591Density of var110000200003000040000500000.00.10.20.30.4IterationsTrace of var10.00.10.20.30.402468density.default(x = y, width = width)N = 15000 Bandwidth = 0.007091Density of var11000020000300004000050000-0.4-0.20.00.2IterationsTrace of var2-0.4-0.20.00.2012345density.default(x = y, width = width)N = 15000 Bandwidth = 0.0127Density of var2 240 (4d)(2)的后验分布密度图 (4e)(2)的后验分布图 (4f)(3)1的后验分布图 10000200003000040000500000.00.10.20.30.4IterationsTrace of var10.00.10.20.30.402468density.default(x = y, width = width)N = 15000 Bandwidth = 0.007091Density of var11000020000300004000050000-0.20.00.2IterationsTrace of var2-0.4-0.20.00.2012345density.default(x = y, width = width)N = 15000 Bandwidth = 0.0127Density of var210000200003000040000500001020304050IterationsTrace of var1010203040500.000.020.04density.default(x = y, width = width)N = 15000 Bandwidth = 1.159Density of var110000200003000040000500000.00.10.20.3IterationsTrace of var20.00.10.20.3024681012density.default(x = y, width = width)N = 15000 Bandwidth = 0.004599Density of var210000200003000040000500001020304050IterationsTrace of var1010203040500.000.020.04density.default(x = y, width = width)N = 15000 Bandwidth = 1.159Density of var110000200003000040000500000.00.10.20.3IterationsTrace of var20.00.10.20.3024681012density.default(x = y, width = width)N = 15000 Bandwidth = 0.004599Density of var2 (4g)(3)2的后验分布密度图 (4h)(3)的后验分布图 (4i)(3)的后验分布图 图4参数的后验分布密度曲线 Fig.4 Posterior distribution of parameters 245 从图4可知，模型1-3中各参数的边缘后验分布核密度估计的曲线平滑，有明显的单峰 对称特征，说明参数贝叶斯估计值的误差非常小；表1给出了参数的MCMC估计结果。 表1 贝叶斯PSTR模型的MCMC估计结果 Tab.1 MCMC estimation for Bayesian PSTR models 参数 估计值 标准差 MC误差 95%置信区间 AIC BIC 模型一 (1)1 0.1213 0.0177 0.0002 （0.0865，0.1561） -194.9392 -160.5287 模型2 (2)1 -0.1873 0.1269 0.0032 （-0.4360,0.0614） -456.6780 -322.2675 (2)2 0.5263 0.0786 0.0026 （0.3722,0.6804） (2) 37.1815 5.8181 0.0026 (-0.3139, 0.0073) (2) 0.1176 0.3066 0.2714 （-0.4833,0.7185） 模型3 (3)1 0.1973 0.0478 0.0006 (0.1035, 0.2911) -336.9561 -202.5456 (3)2 -0.1533 0.0820 0.0016 (-0.3139, 0.0073) (3) 23.5137 7.6086 0.2326 (8.6008, 38.4266) (3) 0.0348 0.0303 0.0012 (-0.0247, 0.0943) 根据表1的估计结果，可以作如下解读： 250 （1）相比于固定效应面板数据模型1，分别以各省实际GDP增长率和经济规模为控制变量的两机制面板平滑转换模型具有更小的AIC、BIC值，表明两机制面板平滑转移模型的拟合度更好，itI和itS具有非线性关系。 （2）以实际GDP增长率为控制变量的模型2中，FH系数在-0.1873与0.339之间变化。当实际GDP增长率超过10.16%时，FH系数为正值。0(2)2，表明经济增长越快的地区，255 FH系数越大，该地区的资本流动性越小，收入的增加促进本地投资。如天津、上海、江苏、广东、重庆等地区，自身经济发达，投资机会多，吸引本地投资；值得注意的是西藏、青海等经济欠发达而经济增长率高的地区，储蓄率与本地投资率具有高相关性。另一方面，可以从时间维度进行分析：在某一地区经济发展速度越快的阶段，它的FH系数就越大，地区的资本流动性越小，此时本地投资与储蓄的相关性越强，收入的增加也促进本地投资。 260 （3）以各地区经济规模为控制变量的模型3中，FH系数的变化范围为0.044至0.1973。由于0(3)2，表明经济发达（规模越大）的地区，FH系数小，资本流动性强。如广东、江苏、浙江等经济大省，资金富足，可以在其它经济发展速度快的地区寻求投资机会。而西藏、青海、宁夏、海南等地区，经济滞后（经济规模小），FH系数反而比较大，储蓄率与投资率呈正相关关系，收入的增加促进本地投资。 265 （4）由表2中可知，平滑转换斜率)3()2(，表明相对于经济规模，FH系数对实际GDP增长率变化的敏感度要高。 3 结论 本文针对非线性OLS法估计面板平滑转换模型参数时算法难以收敛的问题，构造贝叶斯面板平滑转换模型，设计了MH-Gibss混合抽样算法估计模型参数。利用中国各地区投资270 与储蓄面板数据进行实证研究表明，面板平滑转换模型各参数的迭代轨迹是收敛的，参数估计结果的MC误差均比较小，且参数的后验密度曲线呈钟形，说明MH-Gibbs混合抽样算法能够有效地模拟了参数的边缘后验分布。相比于非线性OLS法，贝叶斯面板平滑转换模型 利用MCMC算法估计模型参数，简化了计算的复杂度，是一种有效的研究工具。 [参考文献] (References)275 [1] Hubbard R G. 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