Transcript 连玉君_2013_SFA_随机边界模型

随机边界模型
Stochastic Frontier Models
连玉君
中山大学 岭南学院
[email protected]
2013年12月9日
New Course: http://baoming.pinggu.org/Default.aspx?id=93
提纲
• SFA 简介
• 截面SFA模型
• 面板SFA模型
• 双边SFA模型
I. SFA 简介
y
²ú³ö±ß½ç
x
ʵ¼Ê²ú³ö
×î´ó²ú³ö
SFA 的模型设定思想
TE ( q, z ) 
q
1
f ( z)
(18.1)
q : 实际产出; f ( z ) : 理论产出; z : 要素投入
qi  f ( zi ,  )  TEi
(18.2)
qi  f ( zi ,  )  TEi  exp(vi )
(18.3)
vi ~ N (0,  v2 )
yi  [ f ( xi ,  )exp(vi )]  TEi
(18.4)
Stochastic Frontier , SF
ln( qi )  ln  f ( zi ,  )  vi  ui
where, ui   ln(TEi )  0 (0  TEi  1)
(18.5)
TEi  exp( ui )
(18.6)
SFA 图示
y1
Source: Porcelli(2009)
实证分析中的模型设定
k
ln( qi )   0    j ln( z ji )  vi  ui
(18.7)
yi  xi    i ,  i  vi  ui
(18.8)
j 1
Q: 两个干扰项如何处理？
Normal-half Normal model (hN):
yi  xi   vi  ui ,
vi
iid N (0,  v2 ), ui
iid N  (0,  u2 )
(18.9)
iid N  ( ,  u2 )
(18.10)
iid Exp( u )
(18.11)
Normal-truncated Normal model (tN):
yi  xi   vi  ui ,
vi
iid N (0,  v2 ), ui
Normal-Exponential model (Exp):
yi  xi   vi  ui ,
vi
iid N (0,  v2 ), ui
Note: 假设 v, u 不相关，且二者与 x 也不相关
正态分布和半正态分布的密度函数图
1.0
+
2
ui = |Ui| ~ N (0, u )
2
Ui ~ N(0, u )
u = 0.8
0.8
Density
0.6
0.4
0.2
0.0
-4.0
-3.0
-2.0
-1.0
0.0
x
1.0
2.0
3.0
4.0
指数分布的密度函数图
f (u)
5
 u = 0.2
4
Density
f(u) =  exp( u)
 = 1/ u
3
2
1
 u = 0.5
u=1
0
0.0
0.5
1.0
u
1.5
2.0
半正态分布和指数分布对比
2.0
Density
1.6
1.2
Exponential
Half-Normal
0.8
0.4
0.0
0.0
0.5
1.0
1.5
u
2.0
2.5
3.0
效率的估计
• Jondrow, Lovell, Materov and Schmidt (1982)，JLMS82
TEi  1  E (ui  i )

    i   

E (ui  i )  i + 









i


(18.25)
• Battese and Coelli (1988)，BC88
TEi  E exp  ui   i 
1     i   
1 2


exp



 

 i
2 

 1     i   
(18.26)
II. 面板随机边界模型
Panel SFA
• Review: linear FE v.s. RE)
– FE (Fixed Effect Model)
yit  i  xit    it ,
 it ~ N (0, 2 )
– RE (Random Effect Model)
yit  xit   i  it ,
 it ~ N (0, 2 ), i ~ N (0,  a2 )
– Pooled OLS
yit  0  xit    it ,
 it ~ N (0, 2 )
II. 面板随机边界模型
Panel SFA
• 可能的通用模型：
yit  yit*   it ,
yi*t  i  xi' t 
ai : 公司个体效应, N -1 个公司虚拟变量;
it  i  vit  i  uit
i : 不随时间变化的常规干扰项;
随时间变化的常规干扰项;
+i : 不随时间变化的无效率项 (persistent component)
u+it : 随时间变化的无效率项 (transient component)
vit :
Panel SFA: Pooled SFA model
yit    xit'   vit  uit ,
SF
vit
iid N (0,  v2 ),
uit
iid N  (0,  u2 )
(18.31)
Panel SFA:随机效应模型 (RE-SFA)
效率不随时间变化
• Pitt and Lee (1981), PL81
yit    xit'   vit  ui ,
vit
N (0,  v2 ),
ui
N  (0,  u2 )
(18.31)
Panel SFA:固定效应模型 (FE-SFA)
效率不随时间变化
• Schmidt and Sickles (1984), SS84
yit    xit'   vit  ui ,
(18.31), PL81
yit  i  xit'   vit，
(18.34)， FE
i    ui ,
• TE的估计
ˆ M  max ˆ j  ,
j
(18.36)
uˆi  ˆ M  ˆi
TEi  expuˆi ,
(18.37) JLMS82
Panel SFA: 效率时变模型
• Cornwell, Schmidt and Sickles (1990), CSS90
yit    xit'   vit  uit ,
  uit =it  i  i1t  i 2t 2 ,
(18.38)
• Lee and Schmidt (1993), LS93
yit    x i' t   vit  uit ,
uit  g (t )  ui ,
Note : g (t ) is year dummies
(18.40)
Panel SFA: 效率时变模型
• Battese and Coelli(1992), BC92, 应用非常广泛
yit    x it'   vit  uit ,
uit  g (t )  ui ,
uit
(18.42)
 exp    t  Ti    ui ,
1.0
 = -0.1  decreasing
Inefficiency Effect
0.8
0.6
0.4
0.2
 = 0.1  increasing
0.0
1
2
3
4
5
6
Time Period
7
8
9
10
Panel SFA: True FE SFA
• Greene难题 (Greene Problem)
– True-Model:
yit  i  xit'   vit  uit
SF
(18.43)
inEff
– Estimate-Model:
yit  0  xit'   vit  uit
SF
(18.44)
inEff
– Implications:
• TE 的估计值将是有偏的
• 把那些个体异质性(公司文化, CEO特征等)影响产出的因素都归为“无
效率项”了
Panel SFA: True FE SFA
• Greene(2005), TFE
yit    i  x it'   (vit  uit )
SF
(18.45)
inEff
i ：N  1个公司虚拟变量
vit
N (0,  v2 ),
uit
N  (0,  u2 )
• 估计方法: 蛮力法 (brute force approach)
– 直接估 N 个公司虚拟变量和 k 个  参数即可
– 需要采用一些特殊的数值计算技巧
Panel SFA: True RE SFA
• Greene(2005), TRE
yit    x it'   (i  vit  uit )
SF
i
N (0,  2 ),
vit
N (0,  v2 ),
uit
N  (0,  u2 )
(18.45)
inEff
• 估计方法: MLE
– 相对于传统的线性 RE 模型，只是增加了一个参数而已
Panel SFA: Generalized TRE SFA
• Tsionas and Kumbhakar (2013), G-TRE
yit    xit'   i  vit  (i  uit )
SF
(18.47)
inEff
• 对比: TRE
yit    xit'   (i  vit  uit )
SF
inEff
(18.45)
Panel SFA: Scaling-TFE SFA
• Wang and Ho (2010), Scaling-TFE
yit  i  x it'   vit  uit，
(18.52)
uit  git  ui，
git  f ( zit )，
ui
N  (  ,  u2 )，
• git：scaling function, 是公司特征变量(zit)的函数
– git：可以使非效率具有异质性；
– git：缩放性质使得我们可以用FD或组内去心去除个体效应 i
Panel SFA: dynamic SFA
• Ahn and Sickles (2000), Dynamic-SFA
yit  xit'   vit  uit，
(18.53)
uit  (1  i )uit 1  it
– i ：用于衡量第 i 家公司对非效率项的调整能力(speed)
– i 越大，表明公司克服其非效率行为的能力越强
异质性 SFA: Heterogeneous SFA
• 基本思想
2
Ëæ»ú±ß½ç
1.5
y
ЧÂʵÄÓ°ÏìÒòËØ
1
*²»È·¶¨ÐÔµÄÓ°ÏìÒòËØ
*
.5
0
0
1
2
3
x
4
5
异质性 SFA: Heterogeneous SFA
• 模型设定思想
yi  xi   vi  ui ,
vi
N (0,  v2i ),
ui
N  ( i ,  u2i )
(18.53)
• 异方差的设定(不确定性)
 v2i  exp( zit )
(18.57)
 u2i  exp( wit )
(18.58)
• 均值的设定(无效率水平)
 i  sit 
(18.59)
双边随机边界模型: two-tier SFA
• 基本思想
2
Ëæ»ú±ß½ç
*Over
y
1.5
1
* Under
.5
0
0
1
2
3
x
4
5
双边随机边界模型: two-tier SFA
• 模型设定
yi  xi   (vi  wi  ui )
SF
inEff
vi ~ i.i.d . N (0,  v2 )
wi ~ i.i.d . Exp( w ,  w2 )
(18.60)
ui ~ i.i.d . Exp( u ,  u2 )
• 效率的估计
%over-invest  E (1  e wi |  i )
%under-invest  E (1  e ui |  i )
(18.66)
Thanks
New Course:
http://baoming.pinggu.org/Default.aspx?id=93
References 1
•
•
•
•
•
•
•
•
•
Aigner, D., C. Lovell, P. Schmidt, 1977, Formulation and estimation of stochastic
frontier production function models, Journal of Econometrics, 6 (1): 21-37.
Arellano, M., S. Bond, 1991, Some tests of specification for panel data: Monte carlo
evidence and an application to employment equations, Review of Economic Studies,
58 (2): 277-297.
Arellano, M., O. Bover, 1995, Another look at the instrumental variable estimation of
error-components models, Journal of Econometrics, 68 (1): 29-51.
Battese, G., T. Coelli, 1992, Frontier production functions, technical efficiency and
panel data: With application to paddy farmers in india, Journal of Productivity
Analysis, 3 (1): 153-169.
Battese, G. E., T. J. Coelli, 1988, Prediction of firm-level technical efficiencies with a
generalized frontier production function and panel data, Journal of Econometrics, 38
(3): 387-399.
Battese, G. E., T. J. Coelli, 1995, A model for technical inefficiency effects in a
stochastic frontier production function for panel data, Empirical Economics, 20 (2):
325-332.
Belotti, F., S. Daidone, G. Ilardi, V. Atella, 2013, Stochastic frontier analysis using
stata, Stata Journal: forthcoming.
Chang, S. K., Y. Y. Chen, H. J. Wang, 2012, A bayesian estimator for stochastic
frontier models with errors in variables, Journal of Productivity Analysis, 38 (1): 1-9.
Chen, N.-K., Y.-Y. Chen, H.-J. Wang, 2011, Asset prices and capital investment–a
panel stochastic frontier approach, Working Paper.
References 2
•
•
•
•
•
•
•
•
Coelli, T., D. Prasada Rao, G. E. Battese. An introduction to efficiency and
productivity analysis[M]. Boston: Kluwer Academic Publishers 1998.
Colombi, R., G. Martini, G. Vittadini, 2011, A stochastic frontier model with short-run
and long-run inefficiency, Working Paper, Department of Economics and Technology
Management, Universita di Bergamo, Italy.
Emvalomatis, G., 2012, Adjustment and unobserved heterogeneity in dynamic
stochastic frontier models, Journal of Productivity Analysis, 37 (1): 7-16.
Feng, G., A. Serletis, 2009, Efficiency and productivity of the us banking industry,
1998–2005: Evidence from the fourier cost function satisfying global regularity
conditions, Journal of Applied Econometrics, 24 (1): 105-138.
Fried, H. O., C. Lovell, S. S. Schmidt. 2008, Efficiency and productivity[C], in H. O.
Fried, C. Lovell,S. S. Schmidt eds, The measurement of productive efficiency and
productivity change (Oxford University Press, New York) 3-92.
Greene, W., 2005a, Fixed and random effects in stochastic frontier models, Journal of
Productivity Analysis, 23 (1): 7-32.
Greene, W., 2005b, Reconsidering heterogeneity in panel data estimators of the
stochastic frontier model, Journal of Econometrics, 126 (2): 269-303.
Greene, W., 2008, The econometric approach to efficiency analysis, The
Measurement of Productive Efficiency and Productivity Change, 1 (5): 92-251.
References 3
•
•
•
•
•
•
•
Habib, M., A. Ljungqvist, 2005, Firm value and managerial incentives: A stochastic
frontier approach, Journal of Business, 78 (6): 2053-2094.
Hadri, K., 1999, Estimation of a doubly heteroscedastic stochastic frontier cost
function, Journal of Business & Economic Statistics, 17 (3): 359-363.
Huang, C. J., J.-T. Liu, 1994, Estimation of a non-neutral stochastic frontier
production function, Journal of Productivity Analysis, 5 (2): 171-180.
Jondrow, J., K. Lovell, I. Materov, P. Schmidt, 1982, On the estimation of technical
inefficiency in the stochastic frontier production function model, Journal of
Econometrics, 19 (2-3): 233-238.
Koutsomanoli-Filippaki, A., E. C. Mamatzakis, 2010, Estimating the speed of
adjustment of european banking efficiency under a quadratic loss function, Economic
Modelling, 27 (1): 1-11.
Kumbhakar, S., F. Christopher, 2009, The effects of bargaining on market outcomes:
Evidence from buyer and seller specific estimates, Journal of Productivity Analysis,
31 (1): 1-14.
Kumbhakar, S., G. Lien, J. B. Hardaker, 2012a, Technical efficiency in competing
panel data models: A study of norwegian grain farming, Journal of Productivity
Analysis: 1-17.
References 4
•
•
•
•
•
•
•
Kumbhakar, S., C. Lovell. Stochastic frontier analysis[M]. Cambridge: Cambridge
University Press, 2000.
Kumbhakar, S., R. Ortega-Argilés, L. Potters, M. Vivarelli,P. Voigt, 2012b, Corporate
r&d and firm efficiency: Evidence from europe’s top r&d investors, Journal of
Productivity Analysis, 37 (2): 125-140.
Kumbhakar, S. C., 1990, Production frontiers, panel data, and time-varying technical
inefficiency, Journal of Econometrics, 46 (1): 201-211.
Kumbhakar, S. C., S. Ghosh, J. T. McGuckin, 1991, A generalized production frontier
approach for estimating determinants of inefficiency in us dairy farms, Journal of
Business & Economic Statistics, 9 (3): 279-286.
Kumbhakar, S. C., C. F. Parmeter, E. G. Tsionas, 2013, A zero inefficiency stochastic
frontier model, Journal of Econometrics, 172 (1): 66-76.
Kumbhakar, S. C., E. G. Tsionas, 2011, Some recent developments in efficiency
measurement in stochastic frontier models, Journal of Probability and Statistics, 2011:
forthcoming.
Lai, H.-p., C. J. Huang, 2011, Maximum likelihood estimation of seemingly unrelated
stochastic frontier regressions, Journal of Productivity Analysis: 1-14.
References 5
•
•
•
•
•
•
Lee, Y. H., P. Schmidt. 1993, A production frontier model with flexible temporal
variation in technical efficiency[C], in H. Fried, C. Lovell,S. Schmidt eds, The
measurement of productive efficiency: Techniques and applications (Oxford
University Press, Oxford, UK) 237-255.
Lian, Y., C.-F. Chung, 2008, Are chinese listed firms over-investing?, SSRN working
paper, Available at SSRN: http://ssrn.com/abstract=1296462.
Meeusen, W., J. Van den Broeck, 1977, Efficiency estimation from cobb-douglas
production functions with composed error, International Economic Review, 18 (2):
435-444.
Peyrache, A., A. N. Rambaldi, 2012, A state-space stochastic frontier panel data
model, working Paper.
Pitt, M. M., L.-F. Lee, 1981, The measurement and sources of technical inefficiency in
the indonesian weaving industry, Journal of Development Economics, 9 (1): 43-64.
Tsionas, E. G., S. C. Kumbhakar, 2013, Firm-heterogeneity, persistent and transient
technical inefficiency:A generalized true random effects model, Journal of Applied
Econometrics: forthcoming.
References 6
•
•
•
•
•
•
•
•
Wang, E. C., 2007, R&d efficiency and economic performance: A cross-country
analysis using the stochastic frontier approach, Journal of Policy Modeling, 29 (2):
345-360.
Wang, H., 2003, A stochastic frontier analysis of financing constraints on investment:
The case of financial liberalization in taiwan, Journal of Business and Economic
Statistics, 21 (3): 406-419.
Wang, H. J., C. W. Ho, 2010, Estimating fixed-effect panel stochastic frontier models
by model transformation, Journal of Econometrics, 157 (2): 286-296.
Yélou, C., B. Larue, K. C. Tran, 2010, Threshold effects in panel data stochastic
frontier models of dairy production in canada, Economic Modelling, 27 (3): 641-647.
白俊红, 江可申, 李婧, 2009, 应用随机前沿模型评测中国区域研发创新效率, 管理世界,
(10): 51-61.
林伯强, 杜克锐, 2013, 要素市场扭曲对能源效率的影响, 经济研究, (9): 125-136.
刘海洋, 逯宇铎, 陈德湖, 2013, 中国国有企业的国际议价能力估算, 统计研究, (5): 4753.
卢洪友, 连玉君, 卢盛峰, 2011, 中国医疗服务市场中的信息不对称程度测算, 经济研究,
(4): 94-106.
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