随机边界模型
Stochastic Frontier Models
连玉君
中山大学 岭南学院
arlionn@163.com
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
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