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随机前沿分析模型简明操作.pdf
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by Timothy J.Coelli, et al
´R n
2017 c 3 21 F
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).d£C)D⁄)…ŒØŒ£Translog⁄)…Œ§Aigner and Chu
£1968⁄˜kƒ^ø«{O).d)…Œ" b‰¡Œ„N
ln qi = x
ߧ(‰5)c.–L«Xe
iβ − ui,
(1)
iL«d\ØŒ|⁄k × 1§ui·N
“¥§qiL«1iߧx
u)|"ˇqi–¯£=(
EˆK=§ˇ~bl'|N (0, σ2
‰5⁄exp(x
iβ) §⁄–(1) “L«·(‰5)c"·§øˆ{
¯D(Eˆ'lm5§
1 ¡Œ¯)c.
).d¯)c.L«Xe
ln qi = β0 + β1 ln xi + vi − ui
qi = exp(β0 + β1 ln xi + vi − ui)
× exp(vi)
qi = exp(β0 + β1 ln xi)
¯D(
(‰'
2
(3)
(4)
(5)
× exp(−ui)
ˆ5
Xª1⁄«§ßAB\§c.(‰'N5´ˆ4
~35" ¶L«\§p¶L«" ßA |^\Y†xA)
qA§ßB |^\Y†xB)qB £3ª¥§ø*^Ik×:5
L«⁄" XJvkˆA£=uA = 0, uB = 0⁄§@oßAßB c'
O·
q∗
A = exp(β0 + β1 ln xA + vA) –9 q∗
B = exp(β0 + β1 ln xB + vB)
ª 1 ¯)c
3ª1¥§ßc^Ik⊗:5L«"w,§ßAcu(‰c
§ˇ¯D(K·£=vA > 0⁄¶ßBcu(‰c
e§·ˇ¯D(<·K£=vB < 0⁄" –w§dußA¯D
(KˆAK£=vA − uA < 0⁄§⁄–ßA¢SqAu(‰c
e"
Ø¢S§c·*"n§ck7ø(‰c
e!'“‡§@o¢Sdu¯D(ˆAp^'3(‰
ce·kU" ,§y¢¥¢Sı/“u'3(‰c
e§·ˇıˆAØr" k¯D(Øk¨4Ku
iβ)=εi = vi − ui > 0⁄§¢S*Uu(
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‰c"
i > exp(x
1 ¡Œ¯)c.
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¯c'83uˆA£–¡Eˆ˙§⁄" ~^
E˙u*¢SA¯c’
T Ei =
qi
exp(x
iβ + vi)
=
exp(x
iβ + vi)
exp(x
iβ + vi − ui)
= exp(−ui)
(6)
d(6)“⁄E˙1ißk߃^\⁄U
mا·[0,1]" w,§E˙T Ei§˜kI
O¯)c.(2)“¥ºŒ"
1.3 '¯)c.ºŒO
Aigner, Lovell and Schmidt£1977⁄’u.bXe
vi ∼ iidN (0, σ2
v)
ui ∼ iidN +(0, σ2
u)
(7)
(8)
b“(7)L†¯D(vi·Æ'¯C§0σ2
v"
b“(8)L†§ˆ˙ui ·Æ'¯C§¥”ºŒσ2
u§
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u V˙…Œ
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?1.ºŒO§8cT^#·4.1"Battese and Corra £1977⁄§
Frontier4.1Øø.ØŒq,…ŒUσ2 = σ2
u/σ2 ?1ºŒ
z"ºŒγ·[0,1]§XJγ = 0§K.3EˆA§⁄kØuc
l·d¯D("
uγ = σ2
v + σ2
1
2
3
...
58
59
60
1
1
1
...
1
1
1
L 1 Œ'EG1.dta
2.547725
3.189859
3.037594
...
3.061426
3.300419
2.646529
2.242410
1.535361
1.628260
...
2.233128
2.058473
1.726510
3.559169
4.347655
4.497574
...
4.467332
4.099995
3.789132
e¡–').d)c.~§øªXƒ^Frontier4.1^
?1.ºŒOE˙"duFrontier4.1 U?n5§§ˇdIØ
).d)…Œ>ØŒ§N.§Xe
ln(Qi) = β0 + β1 ln(Ki) + β2 ln(Li) + (vi − ui)
(9)
1 ¡Œ¯)c.
4
“¥§Qi! KiLi'O·! ]N˜§viui'Ol'" Œ
'eg1.dta¥„60ß1 c¡*§Uß?£firm-id⁄! ˇ?
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,ˇLFrontier4.1JłBlank.ins'M§S$1⁄I-'£ø
p•¶eg1.ins⁄§-'SN’)”L2" 11 1ƒ^rJ
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˙flK¶.2·E˙K.§)ß¡Œ^eE˙=
ˇKflK¶w,§øp9ØE˙Kˇ'§ˇdJ.1"
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§I.'/5߉·˜DXJb‰ui l'§AT
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1111˛fl^r·˜ØETAD" Øu.1§ETANE˙·˜‹
mu)Cz"w,§Øu¡Œ§I˜mCzflK§ˇdøpAJn"
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1
1=Error Components Model, 2=TE Effects Model
J.
L 2 -'EG1.ins
EG1.dta Data File Name
EG1.out Output File Name
1=Production Function, 2=Cost Function
Logged Dependent Variable (y/n)
Number of Cross-sections
Number of Time Period
Number of Observations in Total
‰Œ'¶¡
‰'¶¡
)…Œ·⁄…Œ
ˇC·˜LØŒ
¡Œ
ˇŒ
*Œ
1
y
60
1
60
2
n
n
n
Number of Regressor Variables (Xs)
MU (y/n)[Or Delta0 (y/n) if Using TE Effects Model] ·˜u‰δ0D
ETA (y/n)[Or Number of TE Effects Regressors (Zs)] ·˜ηD‰WKˇŒ
)”CŒ
Starting Values (y/n)
·˜ˆ˜‰—'§Jn
,$1Frontier4.1^§.¡DOSI"øp‹˛fl^r·3“£=^
I¥⁄\-£\t⁄§·ˇL-'5$1§S£\f⁄" XJ\t§
@oI¥‹ƒ^r¯£f-'¥y⁄kflK¶XJ\f§KI
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w«3ø'¥"
d uFrontier4.1 ' L § · n r 1 . ˚ ˇ ƒ O
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—'?1|¢§,?\SzL§¶1n§SL§“§=
.ºŒq,O£MLE⁄"ˇd§'·Uøn‰w«(J§¿
‹3w«E˙O"L3!L4L5 'Ow«OLSO(J!†L
|¢O(J“q,O(J"
L 3 'EG1.out1'OLSO
the ols estimates are:
beta0
beta1
beta2
sigma-squared
coefficient
standard-error
t-ratio
0.24489834E+00
0.21360307E+00
0.11465114E+01
0.28049246E+00
0.53330637E+00
0.11398496E+00
0.48066617E-01
0.51498586E-01
0.58354940E+01
0.10355748E+02
log likelihood function = -0.18446849E+02
L3¥§beta0ØA.§beta1·]\£ln Ki⁄ºŒO§beta2·
N˜\£ln Li⁄ºŒO§sigma-squaredL«.¥d¯D(ˆA˛
^" l(J5w§]\N˜\5'O0.280.53§
¿31% w˝5Y†ˇLtu" ·§duO·k§ˇ
1 ¡Œ¯)c.
6
dOLSO(J¿vk¢S¿´§Ue|¢—'"
L 4 'EG1.out1'|¢O(J
the estimates after the grid search were:
beta0
beta1
beta2
sigma-squared
gamma
coefficient
0.58014216E+00
0.28049246E+00
0.53330637E+00
0.22067413E+00
0.80000000E+00
mu is restricted to be zero
eta is restricted to be zero
Øu|¢ºŒO'§•w“q,O
(J£MLE⁄§L5" –w§OLS O(J’§MLE O¿§
Ok5" Frontier4.1γ = σ2
u/σ2 ØŒq,…ŒºŒz§
Oγ = 0.7972§ø’p31%w˝5Y†ˇLtu§L†E
C·dˆ˙£u⁄"ºŒβ1 = 0.2811 β2 = 0.5365L«]N
˜'OU)”28.11% 53.65%§˜˛†L˘¿´§¿UˇL
Ou"
L 5 'EG1.out1n'q,O
the final mle estimates are:
beta0
beta1
beta2
sigma-squared
gamma
coefficient
standard-error
t-ratio
0.56161963E+00
0.20261668E+00
0.27718331E+01
0.28110205E+00
0.53647981E+00
0.11398496E+00
0.79720730E+00
0.47643365E-01
0.45251553E-01
0.63909106E-01
0.63909106E-01
0.59001301E+01
0.11855501E+02
0.33954545E+01
0.58436004E+01
mu is restricted to be zero
eta is restricted to be zero
log likelihood function = -0.17027229E+02
LR test of the one-sided error = 0.28392402E+01
with number of restrictions = 1
[note that this statistic has a mixed chi-square distribution]
Frontier4.1'¥E˙O"œ“(6)ØßE˙
‰´T Ei = exp(−ui)§d^z*ßE˙O§–
gEˆ˙§ui"L6¥c·Frontier4.1$1'¥60
ßE˙O§•øpOAßEˆ˙§§w«31
n¥",§˙–w⁄·¥⁄kß˙§–w⁄·3‰
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L 6 'EG1.out1o'E˙OEˆ˙§
firm
eff.-est.
1
2
3
...
58
59
60
0.65068880E+00
0.82889151E+00
0.72642592E+00
...
0.66471456E+00
0.85670448E+00
0.70842786E+00
mean efficiency =
0.74056772E+00
u
0.42972378E+00
0.18766600E+00
0.31961877E+00
...
0.40839756E+00
0.15466225E+00
0.30033820E+00
1.4 '¯)c.ºŒO
'.·Øˆ˙bk⁄Cz§dui ∼ iidN +(µ, σ2
u)"
'’§'„eˆ˙20§·~Œ" øp§•
–ØŒ)c.~§ƒ^Frontier4.1^O'c."
.N/“Xe
ln(Qi) = β0+β1 ln(Ki)+β2 ln(Li)+β3 ln(Ki)2+β4 ln(Li)2+β5 ln(Ki) ln(Li)+(vi−ui) (10)
“¥§Qi!Ki!Livi„´.e‰§·ˆ˙uil
'" ~faq§øp•˜kŒ'eg2.dta£L7⁄" du
ØŒ)…Œ’).d)…ŒO\]N˜\†
p§ˇdŒ'¥IO\n"§•flkØ⁄kCØŒ"
L 7 Œ'EG2.dta
1
2
3
...
58
59
60
1
1
1
...
1
1
1
2.547725
3.189859
3.037594
...
3.061426
3.300419
2.646529
2.242410
1.535361
1.628260
...
2.233128
2.058473
1.726510
3.559169
4.347655
4.497574
...
4.467332
4.099995
3.789132
5.028404
2.357333
2.651230
...
4.986860
4.237312
2.980835
12.667690
7.981118
18.902110
6.675219
20.228170
...
7.323218
...
19.957060
9.976124
16.809960
8.439730
14.357520
6.541973
’ ~ f§ - ' I › # ‰ Œ ' ' ¶ ¡§ ' O
EG2.dtaEG2.out¶)”CŒ?U5§ˇO\]N˜\
1 ¡Œ¯)c.
8
†p¶’uMU DJy§ˇXc')”§XJ؈˙b
'y§'n" ƒ-–C§I˜m
CzK"
1
1=Error Components Model, 2=TE Effects Model
J.
L 8 -'EG2.ins
EG2.dta Data File Name
EG2.out Output File Name
1=Production Function, 2=Cost Function
Logged Dependent Variable (y/n)
Number of Cross-sections
Number of Time Period
Number of Observations in Total
‰Œ'¶¡
‰'¶¡
)…Œ·⁄…Œ
ˇC·˜LØŒ
¡Œ
ˇŒ
*Œ
1
y
60
1
60
5
y
n
n
Number of Regressor Variables (Xs)
MU (y/n)[Or Delta0 (y/n) if Using TE Effects Model] ·˜u‰δ0D
ETA (y/n)[Or Number of TE Effects Regressors (Zs)] ·˜ηD‰WKˇŒ
)”CŒ
Starting Values (y/n)
·˜ˆ˜‰—'§Jn
OLS:
beta0
beta1
beta2
beta3
beta4
beta5
sigma-squared
L 9 'EG2.outOLSMLEO(J
coefficient
standard-error
t-ratio
0.55625612E+00
0.36337949E+00
0.15307857E+01
0.37854314E+00
0.19384373E+00
0.19528263E+01
0.34779928E+00
0.21214558E+00
0.16394369E+01
-0.92959011E-01
0.30049984E-01
0.84811417E-02
0.11036711E+00
0.45170712E-01
0.34924524E-01
0.45231451E-01
-0.20579488E+01
0.86042644E+01
0.18750541E+00
log likelihood function = -0.15857202E+02
µ = σ2
u = 0 £\⁄LLF
MLE:
beta0
beta1
beta2
beta3
beta4
beta5
coefficient
standard-error
t-ratio
0.75634273E+00
0.31780239E+00
0.23799152E+01
0.38643648E+00
0.16320617E+00
0.23677810E+01
0.35175493E+00
0.17255941E+00
0.20384569E+01
-0.93929604E-01
0.30592987E-01
0.85664268E-02
0.41900823E-01
0.28655714E-01
0.37942019E-01
-0.22417126E+01
0.10676051E+01
0.22577678E+00
sigma-squared
0.45413086E+00
0.29132405E+00
0.15588513E+01
gamma
mu
0.89276949E+00
0.96460242E-01
0.92553105E+01
-0.12734743E+01
0.12297575E+01
-0.10355491E+01
eta is restricted to be zero
log likelihood function = -0.13910334E+02
ˆLLF
LR test of the one-sided error = 0.38937374E+01
with number of restrictions = 2
[note that this statistic has a mixed chi-square distribution]
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