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ด�วัยแบบจำ!าลัอง DEA

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Lecture 4: ขอบเขตเนื้�อหา

• Metafrontier

• การวั�เค้ราะห% Metafrontier ด�วัยแบบจำ!าลัอง DEA

• การวั�เค้ราะห% Metafrontier ด�วัยแบบจำ!าลัอง SFA

Metafrontier

• When all producers in different groups of a given indu stry have a potential access to the same technology but each producer may choose to operate on a differe

nt part of their technologies depending on circumsta nces such as the natural endowments, relative prices

of inputs and the economic environment, then the as sessment of producer’s efficiency and producti

vity can be measured using a metafrontier concept .

• Hayami and Ruttan (1970) initially proposed a meta production function which is defined as the env

elope of commonly conceived neoclassical prod uction functions. Thus, it is a common underlying p

-roduction function that is used to represent the input output relationship of a given industry.

Metafrontier• Consider there are two different groups of technologi

es, namely A and B.

• Let points A1

, A2

, A3

and A4

-indicate the input outpu t bundles of four producers in group A. These points a

re used to construct a frontier for production technolo gy in group A or TA .

• Similarly, points B1

, B2

, B3

and B4

-show the input ou tput bundles of four producers in group B. These poin

ts are used to construct a frontier for production tech nology in group or TB .

• If each group of producers has potential access to the same technology, the grand frontier which envelo

- ps the two group specific frontiers can be repre sented by line AoA1A

2B2

B3

Bo.

• This line is referred as a metafrontier function or T*.

Metafrontier

y

x

A1

T*

Ao

BoB3B2

B1

B4

A3

A2

A4

A1

TA

TB

Metafrontier

• The metafrontier function can be measuring using both nonparametric and parametric approaches.

• The metafrontier function using DEA construct - s piece wise linear convex production technology b -y enveloping all observed data from each group sp

ecific technology. It does not require specified f - unctional form for each group specific technol

ogy.

• T he metafrontier function using SFA constructs a smooth production technology by tangenting a sp

ecified functional form of production functions from - each group specific technology. The metafrontier

using SFA is a smooth function and not a seg - mented envelope of each group specific techn

ology.

Metafrontier

y

x

A3A1

A2 A4

B2B3

B4

T*

TA

TB

Ao

B1

y

x

A1

T*

TB

TA

Ao

BoB3B2

A3

A2

B1

A4

B4

A1

x

DEA SFA

D ecomposition of TE under metatechnology

B1** B4

A3***

6.8

5.6

3.1

Boo

A4*Ao

Aoo

T *

T A

T B

y

x

A3

A2

B1

B2

B3

Bo A3*

B4**

B4*

A4**

A4

A3**

A1

D ecomposition of TE under metatechnology

• The metatechnology (T* ) which is constructed from t he two production technologies, TA and TB , is represe

nted by line AoA1A2

B2

B3

Boo . The boundary of the metaechnology represents a metafrontier.

• Consider the production technology T A where p oint A1

, A2

and A4

lie on the frontier but point A3

lie s below the frontier.

TEoA of the point A

1 , A2

and A4

= 1 TEo

A of the point A3

= A3

*A3

/ A3

*A3

***.

• When the metafrontier (T* ) is considered,TEo

* of the point A

1 , A2

= 1 TEo

* of the point A3

= A3

*A3

/ A3

*A3

**

TEo* of the point A

4 = A

4

*A4

/ A4

*A4

**.

D ecomposition of TE under metatechnology• Similarly, consider the production technology TB

where point B1

, B2

and B3

lie on the frontier but pointB3

lies below the frontier.

TEoB of the point B

1 , B2

and B3

= 1 TEo

B of the point B4

= B4

*B4

/ B4

*B4

**

• When the metafrontier (T * ) is considered ,TEo

* of the point B2

, B3

and B4

is still the same as TEoB

TEo* of the point B

1 = BoB1 / BoB1**

• When the TEo - is measured relative to the group specific technology and metatechnology, it can occur a gap bet

ween the two technologies used as a reference. This g ap is called a technology gap which is defined as the

ratio of the distance function using an observed data ba sed on the metotechnology T* - to the group specific tec

hnology Tk.

Metafrontier

• Using the output orientation, the technology gap ra tio can be defined as

or it can be written as

• The metafrontier (T*) can be decomposed into the product of the TE measured with respect to the k-th group technology (T k) and the technology gap ratio.

),(

),(

),(

),(),(

**

YXTE

YXTE

YXD

YXDYXTGR

ko

oko

oko

),(),(),(* YXTGRYXTEYXTE ko

koo

Metafrontier

• For example, consider point A3 in the figure, TE with respect to T A can be measured by the ratio of the distances between A3

*A3 to A3*A3

***.

• The TEoA = 3.1/5.6 = 0.554 implying that all outputs

could be possibly produced by 45% more from the given inputs by using TA as a reference.

• The TE with respect to T* can be measured by the ratio of the distances between A3

*A3 to A3*A3

**.

The TEo* = 3.1/6.8 = 0.456 implying that all outputs

could be possibly produced by 54% more from the given inputs by using T* as a reference.

• Therefore, TGRok = 0.456/0.554 = 0.823 implying

that the possible output for the TA is 82.3 percent of that represented by the metafrontier (T*).

DEA Approach to Metafrontier

• First, calculate the group-specific technology• I f the group k consists of data on Ik producers, the lin

ear programming (LP) problem that is solved for the i- - - th producer in the k th group at the t th time period i s given as follows.

st

where θk ≥1. The inverse of θk is used to define an output-

oriented TE scores of the i-th producer in the k-th group at the t-th time period or TEk

o,it(x, y). 0 ≤ TEk

o,it(x, y) ≤ 1.

kkk

,max

0λλ

λθ

k

itkk

t

kktit

k

xX

Yy

,

DEA Approach to Metafrontier

• Second, calculate the metafrontier technology• The metafrontier is constructed based on the pooled dat

a of all producers in all groups. The LP problem that is sol - ved for the i th producer at time period t is given as follo

ws.

st

where θ* ≥1. The inverse of θ* is used to define an output-

oriented TE scores of the i-th producer at the t-th time period using the metafrontier as a reference or TE*

o,it(x, y).

0 ≤ TE*o,it(x, y) ≤ 1 and TE*

o,it(x, y) ≤ TEko,it(x, y).

*

, **max

0λλ

λθ

**

**

,itt

tit

xX

Yy

SFA Approach to Metafrontier

• First, the stochastic production frontier for each group is estimated and compared with that for all producers. Then, a statistical test is performed to examine whether all producers in different groups have potential access to the same technology.

• If the group k consists of data on producers, the stochasti c production frontier model for the - i th producer at time p

- eriod t based on the group specific data and the pooled d ata is given as follows.

where c = k refers to the stochastic group-specific production frontier model when the data for the i-th producer in the k-th group at the t-th time period are used, and c = p refers to the stochastic pooled production frontier model when the data for all producers in all groups for all time periods are used.

cit

cit

ccit

cit uvtXfY );,(lnln

SFA Approach to Metafrontier

• Following Battese and Coelli (1992), the stochastic group-specific and pooled production frontier models, taking the log-quadratic translog functional form under a non-neutral TC assumption can be written as follows.

• - - An estimate of output orientated TE for the i th prod - ucer at the t th time period is given by

,2

1ln

lnln2

1lnln

2

1

1 110

cit

cit

ct

ct

cnit

N

n

cnt

cmit

cnit

N

n

N

m

cnm

cnit

N

n

cn

ccit

uvtttX

XXXY

}exp{ cit

coit uTE

Tests of Hypotheses

• If the stochastic frontiers across groups do not differ, then th e stochastic pooled frontier function can be used as a grand t

echnology for each group. A test based on the hypothesis th at all producers in different groups have potential access to t he same technology can be conducted using the likelihood ra tio (LR) test.

Ho : All producers in different groups have potential ac cess to the same technology

Ha : All producers in different groups do not have poten tial access to the same technology

• The test statistic is calculated as

where L(H0) and L(Ha) are the value of the likelihood function under the null and alternative hypothesis.

)]}(ln[)]({ln[2 ao HLHLLR

SFA Approach to Metafrontier

• The second step will involve estimating the met afrontier function. The parameter estimates of the

metafrontier function are estimated by solving the foll owing LP problem.

such that

where ßk s are the estimated coefficients obtained from th - e stochastic group specific frontiers

ß* s are parameters of the metafrontier function to be estimated.

** )ˆ(min βββ1 1

xxxI

i

T

t

kitit

kitit xx ˆ*

SFA Approach to Metafrontier

• Once the ß* parameters of the metafrontier function are estimated, the decomposition of TE under the m

etafrontier can be calculated.

• - - The technology gap for the i th producer in the k th - group at the t th time period can be obtained by

• - Then, a measure of the output oriented TE relative t o the metafrontier, TE*

o(x, y). can be obtained usingequation

*),(

it

kit

x

xkoit

e

eYXTGR

),(),(),(* YXTGRYXTEYXTE ko

koo

Exercise

• Panel data of conventional and organic farms– 28 Farms: 14 farms are conventional farms and

14 farms are organic farms– 15 periods from 1991 to 2005– 1 output: The gross output value of farming aggregates

physical output from seven grain crops and twelve economic crops.

– 6 inputs: capital, labor, chemical fertilizer, pesticide, plastic film and irrigation

Estimated Parameters Parametersa

Stochastic FrontierMetafrontierb

Organic Conventional All

ß0

2.6686 (0.0465)2.579

7(0.053

7) 2.5495(0.043

3)2.629

3(0.015

0)

ß1

0.0420 (0.0317)0.018

4(0.028

9) 0.0439(0.016

4)0.041

3(0.008

5)

ß2

0.3646 (0.0614)0.330

4(0.120

2) 0.2947(0.035

6)0.244

6(0.006

0)

ß3

0.2906 (0.0727)0.529

3(0.114

9) 0.3859(0.055

2)0.434

1(0.016

7)

ß4

0.0051 (0.0519)

-0.014

0(0.065

8) 0.0358(0.031

2)0.053

0(0.011

3)

ß5

0.0678 (0.0392)0.025

5(0.030

9) 0.0203(0.017

7)0.069

0(0.006

4)

ß6

0.5520 (0.1193)0.803

9(0.236

4) 0.4799(0.074

8)0.528

5(0.031

0)

ßt

0.0421 (0.0059)0.020

7(0.007

8) 0.0365(0.003

3)0.027

1(0.001

0)

ß11

0.0211 (0.0355)

-0.029

5(0.026

7)-

0.0067(0.020

4)

-0.002

7(0.011

0)

ß12

-0.2059 (0.0510)

0.0128

(0.0575)

-0.0776

(0.0274)

-0.160

3(0.012

6)

ß13

0.1199 (0.0520)

-0.012

5(0.066

0) 0.0672(0.039

8)0.094

6(0.025

0)

ß14

0.0374 (0.0442)

-0.042

0(0.033

9)-

0.0314(0.022

8)0.023

0(0.012

3)

ß15

0.0408 (0.0359)0.000

9(0.021

8) 0.0176(0.015

9)0.082

5(0.011

6)

ß16

-0.1707 (0.0775)

0.1570

(0.1130)

-0.0315

(0.0663)

-0.216

0(0.023

1)

ß22

0.3070 (0.1112)

-0.294

4(0.274

8) 0.1332(0.068

5)0.083

9(0.028

9)

ß23

-0.0850 (0.1230)

0.1517

(0.3298)

-0.1045

(0.0962)

-0.121

7(0.058

9)

ß24 -0.1129 (0.0713)

0.0083

(0.1219)

-0.0074

(0.0420)

0.0834

(0.0308)

ß25 -0.0272 (0.0570)

0.0230

(0.0633)

-0.0007

(0.0282)

0.0424

(0.0130)

ß26

0.7263 (0.1500)

-0.527

2(0.430

2) 0.6944(0.113

5)0.526

1(0.041

1)

ß33

-0.1962 (0.2384)

-0.054

0(0.581

5) 0.2132(0.184

0)0.567

0(0.132

6)

Estimated Parameters (continued) Parametersa

Stochastic FrontierMetafrontierb

Organic Conventional All

ß55

-0.1638 (0.0637)

-0.008

4(0.026

4) 0.0120(0.019

4)0.002

9(0.006

4)

ß56

0.0586 (0.0959)

-0.345

9(0.172

8) -0.1349(0.081

9)

-0.045

8(0.060

7)

ß66

0.4344 (0.5484)

-2.627

6(0.991

2) 1.1428(0.416

7)0.315

0(0.162

0)

ß1t

-0.0213 (0.0048)

0.0039

(0.0055) -0.0050

(0.0027)

-0.021

2(0.001

4)

ß2t

0.0324 (0.0085)

-0.006

1(0.014

9) 0.0164(0.004

7)0.000

7(0.002

8)

ß3t

-0.0369 (0.0103)

0.0174

(0.0164) -0.0185

(0.0071)

-0.002

4(0.002

3)

ß4t

0.0115 (0.0058)

-0.003

3(0.010

0) 0.0074(0.003

8)0.010

9(0.003

1)

ß5t

0.0093 (0.0047)0.000

5(0.006

5) -0.0003(0.003

3)0.008

7(0.002

2)

ß6t

0.0501 (0.0122)

-0.031

8(0.036

9) 0.0546(0.010

6)0.044

8(0.005

7)

ßtt

0.0004 (0.0011)0.000

6(0.001

9) 0.0016(0.000

8)0.000

4(0.000

5)

σ2

0.0146 (0.0019)0.012

2(0.001

6)

0.3107(0.454

3)

γ0.7200 (0.0633)

0.6612

(0.0568) 0.9830

(0.0249)

η -0.0075 (0.0120)

0.0136

(0.0089) -0.0082

(0.0056)

Log-L 256.1712 235.9472 438.8973

Average TE and TGR Farm TEk TGR TE* Farm TEk TGR TE*

Organic Conventional

Beijing 0.820 0.948 0.778 Shanxi 0.615 0.903 0.554

Tianjin 0.740 0.938 0.694 Inner-Mongolia 0.976 0.863 0.842

Hebei 0.688 0.960 0.661 Anhui 0.596 0.938 0.558

Liaoning 0.948 0.948 0.898 Jiangxi 0.694 0.844 0.584

Jilin 0.784 0.969 0.760 Henan 0.726 0.858 0.623

Helongjiang 0.839 0.980 0.822 Hunan 0.699 0.789 0.551

Shanghai 0.840 0.847 0.712 Guangxi 0.720 0.934 0.672

Jiangsu 0.793 0.960 0.761 Sichuan 0.980 0.842 0.825

Zhejiang 0.742 0.958 0.710 Guizhou 0.731 0.888 0.650

Fujian 0.771 0.943 0.728 Yunnan 0.711 0.941 0.669

Shandong 0.797 0.951 0.758 Shaanxi 0.649 0.966 0.627

Hubei 0.742 0.950 0.705 Gansu 0.649 0.975 0.633

Guangdong 0.978 0.962 0.940 Qinghai 0.917 0.851 0.781

Xijiang 0.806 0.958 0.772 Ningxia 0.581 0.764 0.443

Average 0.806 0.948 0.764 Average 0.732 0.883 0.644