Stata's Capabilities for Efficiency and Productivity Assessment
Topic
:
Stata’s Capabilities For Frontier Efficiency Assessment
Dr. Rachita Gulati
Associate Professor (Economics), Department of Humanities and Social Sciences
Indian Institute of Technology Roorkee Uttarakhand, India
Abstract :
The development of a comprehensive suite of packages and commands within
Stata has empowered researchers to conduct frontier efficiency and productivity
assessments effectively.
Presented By
P
o
i
n
t
s
t
o
D
i
s
c
u
s
s
Productivity
Estimation
and
De
c
ompo
s
i
t
ion
Malmquist Productivity
Index
(MPI)
Malmquist
Luenberger Productivity
Index (MLPI)
Efficiency
Mea
s
u
r
emen
t
Approaches
N
o
n
p
a
r
a
m
e
t
r
i
c
a
p
p
r
o
a
c
h
e
s
(
D
E
A
,
F
D
H
)
P
a
r
a
m
e
t
r
i
c
a
p
p
r
o
a
c
h
e
s
(
S
F
A
,
T
F
A
,
R
T
F
A
)
T
e
c
h
n
i
c
a
l
e
f
f
i
c
i
e
n
c
y
(
T
E
)
•
Radial
vs. Non-radial
TE
measures
•
Orientations in
DEA(Input
and
Output
Orientations)
•
R
eturns-to-
S
cale
A
ssu
m
ptions
30/11/2023
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Stata
User
Meeting
2023_Rachita
Gulati
2
P
r
o
d
u
c
t
i
o
n
t
e
c
h
n
o
l
o
g
y
•
Inputs:
x
(
x
1
,
x
2
,
...,
x
m
)
•
Output:
y
(
y
1
,
y
2
,
...,
y
s
)
•
Feasible
production
plan, if
y
can be
produced
from
x
T
x
,
y
:
y
ca
n
be
p
r
odu
ce
d
fr
om
x
S
i
n
g
l
e
-
i
n
p
u
t
a
n
d
s
i
n
g
l
e
-
o
u
t
p
u
t
c
a
s
e
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
3
O
r
i
e
n
t
a
t
i
o
n
s
i
n
e
f
f
i
c
i
e
n
c
y
m
e
a
s
u
r
e
m
e
n
t
I
n
p
u
t
-
o
r
i
e
n
t
e
d
O
u
t
p
u
t
-
o
r
i
e
n
t
e
d
T
w
o
p
o
p
u
l
a
r
a
p
p
r
o
a
c
h
e
s
30/11/2023
India
Stata
User
Meeting
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Gulati
4
H
e
r
e
,
w
e
a
r
e
t
r
y
i
n
g
t
o
a
s
s
e
s
s
S
t
a
t
a
’
s
c
a
p
a
b
i
l
i
t
i
e
s
i
n
e
f
f
i
c
i
e
n
c
y
a
n
d
p
r
o
d
u
c
t
i
v
i
t
y
e
s
t
i
m
a
t
i
o
n
u
s
i
n
g
t
h
e
s
e
t
w
o
p
o
p
u
l
a
r
a
p
p
r
o
a
c
h
e
s
.
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5
E
f
f
i
c
i
e
n
c
y
a
n
d
P
r
o
d
u
c
t
i
v
i
t
y
M
o
d
e
l
s
•
T
e
c
h
n
i
c
a
l
E
f
f
i
c
i
e
n
c
y
M
o
d
e
l
s
D
E
A
-
b
a
s
e
d
•
Charnes,
Cooper,
Rhodes
(CCR)
(1978)
Model
•
Banker,
Charnes,
Cooper
(BCC)
(1984)
Model
•
Slacks
and
RTS-based
Estimation
•
Directional Distance Function (DDF)
Model (Radial and Nonradial)
(Chung
et
al.,
1997)
S
F
A
-
b
a
s
e
d
•
Stevenson
(1980)
Model
•
Meeusen
and
Van
den
Broeck
(MB)
(1977)
•
Aigner
et
al
.
(1977)
Model
•
P
r
o
d
u
c
t
i
v
i
t
y
C
h
a
n
g
e
M
o
d
e
l
s
•
Malmquist
Productivity
Index
(MPI)
•
Malmquist
Luenberger
Productivity
Index
(MLPI)
30/11/2023
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User
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2023_Rachita
Gulati
6
E
f
f
i
c
i
e
n
c
y
v
s
.
P
r
o
d
u
c
t
i
v
i
t
y
•
Decision making
units (DMUs) are
used
to
describe
the
production
entity
responsible
for
turning
inputs
into
outputs in instances
when
the firm may not be entirely
appropriate, like
power
plants, schools, banks, states or
countries,
etc.
j
1
,
2
,
...,
n
•
Productivity is
a
ratio of output to input
Productivity= output/input
•
E
f
f
i
c
i
e
n
c
y
i
s
a
r
e
l
a
t
i
v
e
c
o
n
c
e
p
t
Efficiency=
productivity/maximum
productivity
30/11/2023
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7
D
a
t
a
E
n
v
e
l
o
p
m
e
n
t
A
n
a
l
y
s
i
s
(
D
E
A
)
DEA
is
a
data-oriented
and
non-parametric
based
frontier
approach
First
originated
by
Charnes,
Cooper
and
Rhodes
(CCR)
(1978)
Charnes,
A.,
Cooper,
W.W.,
and
Rhodes,
E.
(1978),
“Measuring
efficiency
of
decision
making
units”,
European
Journal
of
Operational
Research,
2,
429-444.
CCR
generalized
Farrell’s
(1957)
radial
measure of
technical
efficiency
to
multiple
input,
multiple-output
cases.
Farrell,
M.
J.,
(1957),
“The
measurement
of
productive
efficiency”,
Journal
of
the
Royal
Statistical
Society
,
Series
A,
Vol.120,
No.
3,
pp.
253-290.
DEA
is
a
n
o
n-par
a
me
t
ric,
f
ron
t
ie
r
-ba
s
e
d
app
r
oa
c
h
f
or
m
e
a
s
ure
m
ent
of
efficiency
and
productivity.
I
.
C
h
a
r
n
e
s
,
C
o
o
p
e
r
a
n
d
R
h
o
d
e
s
(
C
C
R
)
(
1
9
7
8
)
•
CCR
gives information
on
the technical efficiency
(TE)
of
a
unit
using
Shephard’s
distance
function
Assumptions:
Constant
Returns
to
Scale
Strong disposability
of
inputs
and
outputs
Convexity
of
Production
Possibility Set
•
Two
distinct
variants
of
the
CCR
model-
•
Input-oriented
CCR
(CCR-I
•
Output-Oriented
CCR
(CCR-O)
30/11/2023
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8
C
C
R
M
o
d
e
l
Consider
n
production units (
j
=1,…..,
n
)
x
vector
of
inputs
(where
x
ij
is
quantity
of
i
th
input
(
i
=1,….
m
)
y
vector
of
outputs
(where
y
rj
is
the quantity
of
r
th
output
(
r
=1,….,
s
).
Maximum possible
(radial)
contraction
in
inputs
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10
C
C
R
M
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
Technical
information:
•
Stata
11.2
and
above installation
required
(Badunenko
and Mozharovskyi,
2016))
Badunenko,
O.,
and
Mozharovskyi, P.
(2016). Nonparametric
frontier analysis using
Stata.
Stata
Journal
,
16
(3),
550–589.
Syntax
teradial
outputs
=
inputs
(ref
outputs
=
ref
inputs)
if
in,
rts (rtsassumption)
base
(basetype) ref(varname)
tename(newvar)
noprint
Specification
outputs
=
list
of output
variables
inputs
=
list
of
input
variables
rts
(rtsassumption)
=
specifies
the
returns
to
scale
assumption.
Use
rts(crs) for
CRS,
rts(nirs)
for
NIRS
and
rts(vrs)for
VRS
base
(basetype)
=
specifies
the
type
of
optimization.
Use
base(output)for
output-
oriented
measure
and
base(input)
for
input-oriented
measure
tename(newvar)
=
creates
newvar
containing
the
radial
measures
of
technical
efficiency
30/11/2023
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Stata
User
Meeting
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Gulati
11
C
o
n
t
d
…
Syntax
for
CCRI
and
CCRO
teradial
outputs = inputs (ref outputs
= ref inputs), rts
(rtsassumption) base
(basetype) tename(newvar)
.
teradial
Y=X1
X2,
rts(crs)
base(output)
tename(CCRO)
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12
I
I
.
B
a
n
k
e
r
,
C
h
a
r
n
e
s
a
n
d
C
o
o
p
e
r
(
B
C
C
)
(
1
9
8
4
)
M
o
d
e
l
•
Banker
et
al.
(1984)
as
an
extension
of
the
CCR
Model.
•
Based
on
the
variable
returns
to
scale
•
Banker,
R. D., Charnes, A., &
Cooper,
W. W.
(1984). Some Models for Estimating
Technical
and Scale Inefficiencies in Data Envelopment Analysis.
Management Science
,
30
(9), 1078–
1092.
•
Two
variants:
BCC-I
and
BCC-O
B
C
C
M
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
Syntax
teradial
outputs
=
inputs
(
ref
outputs
=
ref
inputs
)
if
in
,
rts
(
rtsassumption
)
base
(
basetype
)
ref(
varname
)
tename(
newvar
)
noprint
Here,
specify ‘vrs’ in
rts()
Specification
outputs
=
list
of output
variables
inputs
=
list
of
input
variables
rts
(rtsassumption)
=
specifies
the
returns
to
scale
assumption.
Use
rts(crs)
for
CRS,
rts(nirs)
for
NIRS and rts(vrs)for
VRS
base
(basetype)
=
specifies
the
type
of
optimization.
Use
base(output)for
output-
oriented
measure
and
base(input)
for
input-oriented
measure
tename(newvar)
=
creates
newvar
containing
the
radial
measures
of
technical
efficiency
30/11/2023
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14
B
C
C
M
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
.
teradial
Y=X1
X2,
rts(vrs)
base(output)
tename(BCCO)
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16
C
o
m
p
u
t
a
t
i
o
n
o
f
s
l
a
c
k
s
a
n
d
R
T
S
p
r
o
p
e
r
t
i
e
s
min
(Objective)
i
1
,
2
,
...,
m
;
(
I
npu
t
)
r
1
,
2
,
...,
s
;
(
O
u
t
pu
t
)
o
n
n
*
=
,
,
s
,
s
j
subject
to
j
x
i
j
s
x
io
j
1
j
y
r
j
s
y
ro
j
1
s
,
s
,
j
0
j
1
,
2
,
...,
n
(Non-negativity)
dea:
command
A
l
t
e
r
n
a
t
i
v
e
c
o
m
m
a
n
d
,
t
o
o
b
t
a
i
n
e
f
f
i
c
i
e
n
c
y
a
n
d
s
l
a
c
k
s
Ji, Yong-Bae and Choonjoo Lee (2010), “Data envelopment analysis”,
Stata Journal
10(2): 267-
280.
st0193.pkg
dea ivars = ovars [if] [in] [, rts(crs
|
vrs
|
drs
|
nirs) ort(in
|
out)
stage(1
|
2) trace
saving(
filename
)]
where
the options
are:
rts(crs|vrs|drs|nirs)
specifies
the
returns
to
scale.
The
default
is
rts(crs)
ort(in|out)
specifies
the
orientation.
The
default
is
ort(in)
stage(1|2)
specifies
the
way
to
identify
all
efficiency
slacks.
The
default
is
stage(2)
trace
save
all
sequences
and
results
from
Results
window
to
dea.log
saving(filename)
save
results
to
filename.
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17
dea:
command
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18
I
I
I
.
D
i
r
e
c
t
i
o
n
a
l
D
i
s
t
a
n
c
e
F
u
n
c
t
i
o
n
(
D
D
F
)
•
Chung
et
al.
(1997)
developed
this
model
•
Simultaneous adjustment
of
undesirable inputs
and
outputs
along
with
the
desirable
inputs
and
outputs.
•
Chung,
Y.
H., Fare, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: a
directional distance function approach.
Journal of Environmental Management
,
51
, 229–
240.
•
Weak
disposability
of
undesirable
output.
R
a
d
i
a
l
D
D
F
M
o
d
e
l
30/11/2023
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19
30/11/2023
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20
D
D
F
-
D
E
A
m
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
Technical
information: st0665.pkg, Available for
Stata
16
and
above
installation
required
(Wang
et
al.,
2022)
Wang,
D.,
Du,
K.,
&
Zhang,
N.
(2022).
Measuring
technical
efficiency
and
total
factor
productivity
change
with
undesirable
outputs
in
Stata.
Stata Journal
,
22
(1),
103–124.
D
D
F
-
D
E
A
M
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
30/11/2023
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2023_Rachita
Gulati
21
I
V
.
A
l
t
e
r
n
a
t
i
v
e
v
a
r
i
a
n
t
s
:
N
o
n
-
r
a
d
i
a
l
D
D
F
teddf X1 X2=Y:B,dmu(DMU)
time(Year)
nonradial
saving(ddf_result)
teddf X1 X2=Y:B, dmu(DMU)
nonradial vrs
saving(ddf_result,
replace)
teddf X1 X2=Y:B, dmu(DMU)
time(t)
nonradial sequential
saving(ddf_result,replace)
N
o
n
-
r
a
d
i
a
l
D
D
F
w
i
t
h
u
n
d
e
s
i
r
a
b
l
e
s
(
Y
2
)
(
d
i
r
e
c
t
i
o
n
a
l
v
e
c
t
o
r
i
s
(
-
X
1
-
X
2
-
X
3
-
X
4
Y
1
-
Y
2
)
a
n
d
w
e
i
g
h
t
v
e
c
t
o
r
i
s
(
1
1
1
1
1
1
)
)
30/11/2023
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V
.
M
a
l
m
q
u
i
s
t
P
r
o
d
u
c
t
i
v
i
t
y
I
n
d
e
x
•
First proposed
in
Caves
et al. (1982) and
later modified
by
Fare
et
al.(1992)
,
Fare
et
al.(1994)
and
Ray
&
Desli
(1997)
•
Based
on
Shephard’s
distance
function
•
It
provides
a
decomposition
of
productivity
change
into
its
sources,
i.e.;
efficiency
change
and
technology
change
•
Assumption: Standard
assumptions
of
DEA
and
Constant returns
to
scale
𝑀𝑃𝐼
=
𝑡
𝑡
𝑑
𝑡
(𝑦
,
𝑥
)
𝑑
𝑡
+
1
(
𝑦
𝑡
+
1
,
𝑥
𝑡
+
1
)
𝑑
𝑡
(
𝑦
𝑡
+
1
,
𝑥
𝑡
+
1
)
𝑑
𝑡+1
(𝑦
,
𝑥
𝑡+1
𝑡+1
)
∗
𝑑
𝑡
(
𝑦
𝑡
,
𝑥
𝑡
)
𝑡
𝑡
𝑡
𝑑
𝑡
+
1
(𝑦
,
𝑥
)
Te
c
hn
ic
al
Change
E
ffici
en
cy
Change
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
23
M
a
l
m
q
u
i
s
t
P
r
o
d
u
c
t
i
v
i
t
y
I
n
d
e
x
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
24
M
P
I
a
n
d
i
t
s
D
e
c
o
m
p
o
s
i
t
i
o
n
i
n
S
T
A
T
A
•
T
e
c
hni
c
al
in
f
o
r
ma
t
ion:
A
va
i
l
a
ble
f
or
St
a
t
a
16
a
nd
a
b
o
v
e
(i
n
s
t
alla
t
io
n
required
- ssc
install
malmq2)
Syntax
malmq2
inputvars
=
outputvars
[ if ]
[ in ],
ort(
string
)
dmu(
varname
)
win
dow(
#
)
bi
ennial
seq
uential
global
fgnz
rd
[..other
options]
Specification
inputvars,
outputvars
=
data
for
inputs and
outputs
ort()
=
defines
orientation,
ort(output),
ort(input),
default
is
output
dmu()=
specifies
DMU
names
reference
technology options
=
bi,
seq,
glo
decomposition
=fgnz
((Fare
et
al.,
1994)),
rd
((Ray
& Desli,
1997))
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
25
T
F
P
C
H
u
s
i
n
g
M
P
I
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
Syntax
malmq2
inputvars
=
outputvars
[ if
]
[
in ],
ort(
string
)
dmu(
varname
)
win
dow(
#
)
bi
ennial
seq
uential
global
fgnz rd
[..other
options]
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
26
malmq2
X1
X2
=
Y,
dmu(dmu)
global
malmq2
X1
X2 =
Y,
dmu(dmu)
seq
ort(o)
fgnz
malmq2
X1 X2
=
Y,
dmu(dmu)
ort(o)
rd
sav(tfp_result,replace)
V
I
.
M
a
l
m
q
u
i
s
t
L
u
e
n
b
e
r
g
e
r
P
r
o
d
u
c
t
i
v
i
t
y
I
n
d
e
x
(
M
L
P
I
)
•
First proposed
by Chung et al. (1997),
MLPI
allows
to estimate
productivity
changes while
accounting
for
generation
of
undesirable
products
(e.g.,
pollution)
along
with
desirable
products
•
Chung,
Y.
H.,
Fare,
R.,
&
Grosskopf,
S.
(1997). Productivity
and
undesirable
outputs:
a
directional
distance
function
approach.
Journal of
Environmental
Management
,
51
,
229–240.
•
Based
on radial
directional
distance function
(DDF)
•
Provides
a
decomposition
of
productivity
change
into
its
sources, i.e.,
efficiency
change
and
technology
change
•
Assumption:
same
as
MPI,
along
with
weak
disposability
of
undesirable
output
𝑡
,
𝑡
+
1
•
𝑀
𝐿
=
1+
𝐷
𝑡+1
𝑥
𝑡+1
,𝑦
𝑡+1
,𝑏
𝑡+1
∗
1+
𝐷
𝑡
𝑥
𝑡
,𝑦
𝑡
,𝑏
𝑡
1+
𝐷
𝑡+1
𝑥
𝑡
,𝑦
𝑡
,𝑏
𝑡
1+
𝐷
𝑡
𝑥
𝑡
,𝑦
𝑡
,𝑏
𝑡
1+
𝐷
𝑡+1
𝑥
𝑡+1
,𝑦
𝑡+1
,𝑏
𝑡+1
1+
𝐷
𝑡
𝑥
𝑡+1
,𝑦
𝑡+1
,𝑏
𝑡+1
Te
c
hn
ic
al
Change
Efficiency
Change
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
27
M
a
l
m
q
u
i
s
t
L
u
e
n
b
e
r
g
e
r
P
r
o
d
u
c
t
i
v
i
t
y
I
n
d
e
x
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
28
M
L
P
I
M
o
d
e
l
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
T
A
T
A
Technical
information:
Available
for Stata 16 and above (installation
required
(Wang
et
al.,
2022))
S
y
n
t
a
x
gtfpch
inputvars
=
desirable_outputvars
:
undesirable_outputvars
[
if
] [
in ],
dmu
(
varname
)
luen
berger
ort
(
string
)
gx
(
varlist
)
gy
(
varlist
)
gb
(
varlist
)
nonr
adial
w
mat
(
name
)
win
dow(
#
)
bi
ennial
seq
uential
global
fgnz
rd
[…other
options]
Specification:
Same
as
for
DDF
model
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
29
T
F
P
C
H
u
s
i
n
g
M
L
P
I
:
A
p
p
l
i
c
a
t
i
o
n
i
n
S
t
a
t
a
gtfpch
X1
X2
=
Y:B, dmu(dmu)
nonradial
saving(ddf_result)
gtfpch
X1
X2
=
Y:B,
dmu(dmu)
seqential
ort(input)
leunberger saving(ddf_result,
replace)
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
30
P
a
r
a
m
e
t
r
i
c
A
p
p
r
o
a
c
h
e
s
t
o
F
r
o
n
t
i
e
r
A
n
a
l
y
s
i
s
Stochastic
Frontier
Models
Cross-Sectional
Models
Aigner et al.
(1977)
Model
Meeusen
and
van
den
Broeck
(1977)
Model
Stevenson
(1980)
Model
Greene
(2003)
Model
Panel
Data
Models
Panel
Data
Time-
invariant
Models
Schmidt
and
Sickles
(1984)
FE
Model
Schmidt
and
Sickles
(1984)
RE Model
Pitt
and
Lee
(1981)
Battese
and
Coelli
(1988)
Panel
Data
Time
Variant
Models
Kumbhakar
(1990)
Battese
and
Coelli
(1992)
Battese
and
Coelli
(1995)
S
t
o
c
h
a
s
t
i
c
F
r
o
n
t
i
e
r
A
n
a
l
y
s
i
s
(
S
F
A
)
Popular
parametric
approach
for
estimating
efficiency
and
productivity.
SFA
allows
the deviation from the frontier
and
decomposes
as; one is
the
random
error
𝑣
𝑖𝑖
and
other
is
technical
inefficiency
𝑢
𝑖𝑖
.
Aigner, Lovell
and
Schmidt (1977)
and
Meeusen
and
van
den Broeck
(1977)
independently proposed the stochastic frontier production function
model
of
the
form.
𝑙𝑛𝑦
𝑖𝑖
=
𝛽
0
+
𝛽
1
𝑙𝑛𝑥
𝑖𝑖
+
𝑒
𝑖𝑖
(Assuming
Cobb-Douglas
functional
form
)
l
𝑛𝑦
𝑖𝑖
=
𝛽
0
+
𝛽
1
𝑙𝑛𝑥
𝑖𝑖
+
Deterministic
Component
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
(
𝑒
𝑖𝑖
=
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
)
noise
inefficiency
Here,
𝑦
𝑖𝑖
is
the output,
𝑥
𝑖𝑖
is
the vector
of
inputs,
β
is
vector
of
technological
parameters to
be
estimated,
𝑒
𝑖𝑖
=
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
is
the composite
error
that
has
two
components.
𝑣
𝑖𝑖
that accounts for
random
disturbance
and
𝑢
𝑖𝑖
that accounts
technical
inefficiency.
TE
=
Observed
Output
𝑀𝑎𝑥𝑖𝑖𝑚𝑢𝑚
𝑝𝑜𝑠𝑠𝑖𝑖𝑏𝑙𝑒
𝑜𝑢𝑡𝑝𝑢𝑡
TE
=
𝑏
𝑏
𝑦
⩘
=
𝑦
𝑖𝑖
+
𝛽
𝛽
𝑖𝑖
𝑖𝑖
𝑙𝑛𝑥
+
𝑣
−
𝑢
𝑖𝑖
𝛽
𝛽
0
1
0
1
+
𝑙𝑛𝑥
+
𝑣
𝑖𝑖
𝑖𝑖
0
1
𝑒
(
𝛽
+
𝛽
𝑙𝑛𝑥
𝑖𝑖
)
∗𝑒
𝑣
𝑖𝑖
(
+
𝑙𝑛𝑥
𝑖𝑖
)
𝑣
𝑖𝑖
−𝑢𝑢
𝑖𝑖
=
𝑒
𝛽
𝛽
∗𝑒
∗𝑒
=
𝑒
−𝑢
0
1
𝑖𝑖
S
F
A
:
C
r
o
s
s
-
S
e
c
t
i
o
n
a
l
M
o
d
e
l
s
I.
Aigner
et
al
. (1977)
Model
(Normal-Half
Normal
Model)
𝑙
𝑛
𝑦
𝑖𝑖
=
𝛽
0
+
𝛽
1
𝑙
𝑛
𝑥
𝑖𝑖
+
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
Assumptions
𝑣
1.
v
i
(random
error)
is
assumed
to be
normally
distributed,
v
i
~
iid
N
(0,
𝜎
2
)
2
.
u
(
t
e
c
h
n
i
c
a
l
i
n
e
f
f
i
c
i
e
n
c
y
)
w
h
i
c
h
i
s
a
s
s
u
m
e
d
t
o
b
e
h
a
l
f
n
o
r
m
a
l
l
y
d
i
s
t
r
i
b
u
t
e
d
,
u
~
i
iid
𝑁
+
𝑢
(0,
𝜎
2
)
i
3.
v
i
is
independent
of
u
i
4.
Assuming
Cobb-Douglas
functional
form,
where
𝑦
𝑖𝑖
is
the
output,
𝑥
𝑖𝑖
is
the
vector
of
inputs,
β
is
vector
of
technological
parameters
to
be
estimated.
To estimate
β
,
Aigner
et
al
.
used
maximum
likelihood
estimation
and
parametrized
the
log-
likelihood
function
for
the
normal-half
normal
model.
l
n
L(
𝑦
|
𝛽
,
𝜎
,
𝜆
)
=
−
𝑁
l
n
𝑖
𝑖
=
1
+
∑
𝑁
𝑙
𝑛
𝑙
(
−
𝑒
𝑖
𝑖
𝜆
𝜆
)
−
𝜋𝜎
2
1
2
2
𝜎
2𝜎
2
∑
𝑁
𝑒
2
𝑖𝑖=1
𝑖𝑖
𝑖𝑖
where,
𝑒
=
v
-
u
i
i
𝑒
2
,
𝜎
is
variance
of
composite error
term
,
𝜆
=
𝜎
𝑢𝑢
𝜎
𝑣
,
𝜎
2
=
𝜎
2
+
𝜎
2
,
and
𝑙
.
is
𝑒
𝑣
𝑢
the
cumulative
distributive
function
(cdf)
of
the
standard
normal
variate.
Aigner,
D.,
Lovell,
C.K.
and
Schmidt,
P.,
1977.
Formulation
and
estimation
of
stochastic
frontier
production
function
models.
Journal
of
Econometrics
,
6
(1),
pp.21-37
.
S
F
A
i
n
S
t
a
t
a
S
t
a
t
a
p
a
c
k
a
g
e
-
f
r
o
n
t
i
e
r
Technical
Information:
Available
for
Stata
15
and
above
(no
installation required)
Syntax
frontier
depvar
[indepvars], [options]
Specification
depvar = dependent variable
indepvars
=
independent
variables
[options]
distribution(hnormal)-
half-normal
distribution
for
the
inefficiency
term
distribution(exponential)
-exponential
distribution
for
the
inefficiency
term
distribution(tnormal)-
truncated-normal
distribution
for the
inefficiency
term
Post
Estimation
Commands
for
predicting
inefficiency
of
each
firm.
predict
(file_name),
u
(For
predicting
inefficiency
of
each
firm)
predict
(file_name),
te
(For
predicting
efficiency
of
each
firm)
R
e
s
u
l
t
s
f
r
o
m
A
i
g
n
e
r
e
t
a
l
.
(
1
9
7
7
)
M
o
d
e
l
u
s
i
n
g
f
r
o
n
t
i
e
r
Source:
Coelli,
T.
(1996).
A
guide
to
Frontier
4.1:
A
computer
program
for
stochastic
frontier
production
function
and cost
function
estimation.
Department
of
Econometrics
University
of
New
England
Armidale
NSW
,
2351
C
o
n
t
d
.
.
S
t
a
t
a
p
a
c
k
a
g
e
–
s
f
c
r
o
s
s
b
y
B
e
l
l
o
t
i
e
t
a
l
.
(
2
0
1
3
)
B
e
l
o
t
t
i
,
F
.
,
D
a
i
d
o
n
e
,
S
.
,
I
l
a
r
d
i
,
G
.
a
n
d
A
t
e
l
l
a
,
V
(
2
0
1
3
)
,
S
t
o
c
h
a
s
t
i
c
f
r
o
n
t
i
e
r
a
n
a
l
y
s
i
s
u
s
i
n
g
S
t
a
t
a
,
S
t
a
t
a
J
o
u
r
n
a
l
,
1
3
,
(
4
)
,
7
1
8
-
7
5
8
Technical
Information:
Available
for
Stata
15
and
above
(installation
required)
Syntax
sfcross
depvar
[indepvars],
[options]
Specification
depvar
=
Dependent
Variable
Indepvars
=
List
of
independent
variables
[options]
distribution(hnormal)-half-normal
distribution
for
the
inefficiency
term
distribution(exponential)-exponential
distribution
for
the
inefficiency
term
distribution(tnormal)-truncated-normal
distribution
for
the
inefficiency
term
Post Estimation
Commands
for
predicting
efficiency
and
inefficiency
of
each
firm.
predict
(file_name),
u
(For
predicting
inefficiency through Batesse
and Coelli, 1988
Method)
predict
(file_name), bc
(For
predicting
efficiency
through
Batesse
and Coelli,
1988
Method )
predict
(file_name), jlms
(For
predicting
efficiency
through
Jondrow
et
al.,
1982
Method)
R
e
s
u
l
t
s
f
r
o
m
A
i
g
n
e
r
e
t
a
l
.
(
1
9
7
7
)
M
o
d
e
l
u
s
i
n
g
s
f
c
r
o
s
s
sfcross lny lnx1 lnx2, distribution(hnormal)
predict
ineff,
u
predict eff_jlms, jlms
predict
eff_bc,
bc
I
I
.
M
e
e
u
s
e
n
a
n
d
V
a
n
d
e
n
B
r
o
e
c
k
(
M
B
)
(
1
9
7
7
)
(
N
o
r
m
a
l
-
E
x
p
o
n
e
n
t
i
a
l
M
o
d
e
l
)
Meeusen,
W.
and
van
Den
Broeck,
J.
(1977).
Efficiency
estimation
from
Cobb-Douglas
production
functions
with composed
error.
International
Economic
Review
,
pp.435-444.
M
e
e
u
s
e
n
a
n
d
V
a
n
d
e
n
B
r
o
e
c
k
(
M
B
)
(
1
9
7
7
)
𝑙𝑛𝑦
𝑖𝑖
=
𝛽
0
+
𝛽
1
𝑙𝑛𝑥
𝑖𝑖
+
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
Assumptions
𝑢
𝑖𝑖
(technical
i
n
e
f
f
i
c
i
e
n
c
y
)
w
h
i
c
h
i
s
a
s
s
u
m
e
d
t
o
b
e
e
x
p
o
n
e
n
t
i
a
l
l
y
i
𝑢
d
i
s
t
r
i
b
u
t
e
d
u
~
𝑒
𝑒
𝑥
𝑥
𝑝
𝑝
𝑜
𝑜
𝑛
𝑛
(
𝜃
𝜃
𝜃
𝜃
)
𝜎
𝜎
2
)
T
o
e
st
ima
t
e
β
,
Meeu
s
en
and
van
den
Broeck
(MB)
u
s
ed
MLE
a
nd
parametrized the
log-likelihood
function
for
the
Normal-
Exponential
model.
ln
L
=
𝑙𝑛𝐿
=
−𝑁𝑙𝑛𝜎
𝑢
+
𝑁
𝑣
𝜎
2
𝑢𝑢
2𝜎
2
𝑖𝑖
+
∑
𝑙𝑛𝑙
−𝐴
+
∑
𝑒
𝑖𝑖
𝑖𝑖
𝜎
𝑢𝑢
i
i
𝑒
𝑒
𝑣
𝑢
where,
𝑒
𝑖𝑖
=
v
-
u
,
𝜎
2
is
variance
of
composite
error
term
,
𝜎
2
=
𝜎
2
+
𝜎
2
,
𝑙
.
is
the
cumulative
distributive
function
(cdf)
of
the
standard
normal
variate,
𝐴
=
−𝜇
�
�
𝜎
𝑣
𝑣
𝑢
,
𝜇
�
=
−𝑒
𝑖𝑖
−
𝜎
2
⁄
𝜎
and
N
is
number
of
producers.
R
e
s
u
l
t
s
f
r
o
m
M
e
e
u
s
e
n
a
n
d
V
a
n
d
e
n
B
r
o
e
c
k
(
M
B
)
(
1
9
7
7
)
predict
eff,
te
predict
eff,
bc
I
I
I
.
S
t
e
v
e
n
s
o
n
(
1
9
8
0
)
M
o
d
e
l
(
N
o
r
m
a
l
-
T
r
u
n
c
a
t
e
d
M
o
d
e
l
)
Model
is
formulated
as
𝑙𝑛𝑦
𝑖𝑖
=
𝛽
0
+
𝛽
1
𝑙𝑛𝑥
𝑖𝑖
+
𝑣
𝑖𝑖
−
𝑢
𝑖𝑖
Assumptions
𝑢
𝑢
𝑖
𝑖
𝑖
𝑖
(
t
e
c
h
n
i
c
a
l
i
n
e
f
f
i
c
i
e
n
c
y
)
w
h
i
c
h
i
s
a
s
s
u
m
e
d
t
o
t
r
u
n
c
a
t
e
d
n
o
r
m
a
l
l
y
d
i
s
t
r
i
b
u
t
e
d
,
u
i
~
i
i
d
𝑁
+
𝑢
(
𝜇
,
𝜎
2
)
To
estimate
β
,
Stevenson
(1980)
Model
used
maximum
likelihood
estimation
and
parametrized
the
log-likelihood function
for
the
Normal-Truncated
Normal
Model.
𝑒
ln
𝐿
=
−𝑁𝑙𝑛𝜎
−
𝑁𝑙𝑛𝑙
𝑖𝑖
−
𝜇
𝜇
−
𝑒
𝑖𝑖
𝜆𝜆
𝜎
𝑢𝑢
𝜎𝜆𝜆
𝜎
2
+
∑
𝑙𝑛𝑙
−
1
∑
𝑖𝑖
𝑒
𝑖𝑖
+𝜇
𝜎
i
where,
μ
(mode)
is
a
d
di
t
io
n
al
para
m
e
t
er
t
o
be
e
sti
ma
t
ed,
𝑒
𝑖𝑖
=
v
-
u
i
𝑒
,
𝜎
2
is
varian
c
e
of
composite
error
term
,
𝜎
2
𝑒
𝑣
𝑢
=
𝜎
2
+
𝜎
2
,
𝑙
.
is
the
cumulative
distributive
function
(cdf)
of
the
standard
normal
variate,
and
N
is
number
of
producers.
S
ou
r
c
e
:
S
te
ven
s
o
n,
R.E
.,
198
0.
Li
k
el
ih
o
o
d
fu
n
c
ti
o
ns
f
o
r
g
e
ner
ali
z
ed
s
t
oc
h
a
stic
fron
t
ier
estimation.
Journal
of
Econometrics
,
13
(1),
pp.57-66.
R
e
s
u
l
t
s
f
r
o
m
S
t
e
v
e
n
s
o
n
(
1
9
8
0
)
M
o
d
e
l
predict
eff,
te
predict
eff,
bc
R
e
s
u
l
t
s
o
f
B
C
,
J
L
M
S
,
M
B
,
a
n
d
S
t
e
v
e
n
s
o
n
’
s
M
o
d
e
l
s
I
V
.
T
i
m
e
-
I
n
v
a
r
i
a
n
t
S
t
o
c
h
a
s
t
i
c
F
r
o
n
t
i
e
r
M
o
d
e
l
s
Schmidt
and
Sickles
(1984)
proposed
both
Time
invariant
Stochastic
Frontier
Fixed
effect
and
Random
effect Models.
F
i
x
e
d
E
f
f
e
c
t
s
M
o
d
e
l
=
𝑦
𝑖𝑖𝑡
𝛽
0
it
+
𝛽
1
𝑥
′
+
𝑣
𝑖𝑖𝑡
−
𝑢
𝑖𝑖
𝑦
𝑖𝑖𝑡
=
𝑖𝑖
it
′
𝛽
+
𝛽
1
𝑥
+
𝑣
𝑖𝑖𝑡
Here,
β
i
=
β
0
-
u
i
is
the individual effects
of
fixed-effects
model.
In FE-Model,
𝑢
𝑖𝑖
is
assumed
to
be
correlated
with the regressor
or
with
𝑣
𝑖𝑖𝑡
R
a
n
d
o
m
E
f
f
e
c
t
s
M
o
d
e
l
𝑙𝑛𝑦
𝑖𝑖𝑡
=
𝛽
0
−
𝐸
𝑢
𝑖𝑖
+
∑
𝑛
𝛽
𝑛
𝑙𝑛𝑥
𝑛𝑖𝑖𝑡
+
𝑣
𝑖𝑖𝑡
−
[𝑢
𝑖𝑖
−
𝐸
𝑢
𝑖𝑖
]
0
𝑖𝑖
𝑙𝑛𝑦
𝑖𝑖𝑡
=
𝛽
∗
+
∑
𝑛
𝛽
𝑛
𝑙𝑛𝑥
𝑛𝑖𝑖𝑡
+
𝑣
𝑖𝑖𝑡
−
𝑢
∗
Source:
Schmidt,
P.
and
Sickles,
R.C.,
1984.
Production
frontiers
and
panel
data.
Journal
of
Business
&
Economic
Statistics
,
2
(4),
pp.367-374.
S
t
a
t
a
p
a
c
k
a
g
e
s
:
x
t
f
r
o
n
t
i
e
r
a
n
d
s
f
p
a
n
e
l
Technical
Information:
Available
for
STATA 15
and
above
(no
installation
required)
P
a
n
e
l
D
a
t
a
S
F
A
M
o
d
e
l
s
i
n
S
T
A
T
A
x
t
f
r
o
n
t
i
e
r
a
n
d
s
f
p
a
n
e
l
p
a
c
k
a
g
e
s
f
o
r
t
i
m
e
-
i
n
v
a
r
i
a
n
t
m
o
d
e
l
s
:
S
c
h
m
i
d
t
a
n
d
S
i
c
k
l
e
s
(
1
9
8
4
)
xtset
i
t
xtfrontier
y
x1
x2,
ti
predict
eff,
te
predict
ineff,
jlms
O
t
h
e
r
S
t
o
c
h
a
s
t
i
c
F
r
o
n
t
i
e
r
T
i
m
e
-
I
n
v
a
r
i
a
n
t
M
o
d
e
l
s
P
i
t
t
a
n
d
L
e
e
(
1
9
8
1
)
M
o
d
e
l
it
y
it
=
β
0
+
β
1
𝑥
′
+
v
it
-
u
i
𝑁
+
A
s
s
u
m
p
t
i
o
n
s
:
u
i
(
t
e
c
h
n
i
c
a
l
i
n
e
f
f
i
c
i
e
n
c
y
)
w
h
i
c
h
i
s
a
s
s
u
m
e
d
t
o
b
e
h
a
l
f
n
o
r
m
a
l
l
y
2
(
0
,
𝜎
𝑢
)
an
d c
on
s
tant
∗
where
𝜇
=
−
𝑣
𝑢𝑢
𝜎
2
+𝑇𝜎
2
𝑇𝜎
𝑢𝑢
𝑒
𝑖𝑖
́
1
𝑇
𝑖𝑖
𝑖𝑖
𝑖𝑖
𝑡
,
𝑒
́
=
∑
𝑒
and
𝜎
2
=
∗
𝑣
𝑢
𝑢
𝜎
2
𝜎
2
𝜎
2
+𝑇𝜎
2
𝑣
𝑢𝑢
Pitt,
M.M.
and
Lee,
L.F.
(1981
).
The
measurement
and
sources of technical inefficiency in the Indonesian
weaving
industry.
Journal
of
Development
Economics
,
9
(1),
pp.43-64.)
𝑙𝑛𝐿
=
−
d
i
s
t
r
i
b
u
t
e
d
,
u
i
~
i
i
d
over
time
𝑁(𝑇
−
1)
2
𝑁
2
𝑣
𝑣
𝑢
𝜎
2
−
ln
𝜎
2
+
𝑇𝜎
2
+
𝑖𝑖
𝜇
∗𝑖𝑖
�
ln
1 −
𝑙
−
𝜎
∗
∑
𝑖𝑖
𝑒
𝑖𝑖
𝑒
́
𝑖𝑖
−
𝑣
2𝜎
2
2
+
�
𝑖𝑖
1
𝜇
∗
𝑖𝑖
𝜎
∗
2
2
B
a
t
t
e
s
e
a
n
d
C
o
e
l
l
i
(
1
9
8
8
)
M
o
d
e
l
it
y
it
=
β
0
+
β
1
𝑥
′
+
v
it
-
u
i
A
s
s
u
m
p
t
i
o
n
s
:
u
i
(
t
e
c
h
n
i
c
a
l
i
n
e
f
f
i
c
i
e
n
c
y
)
w
h
i
c
h
i
s
a
s
s
u
m
e
d
t
o
b
e
t
r
u
n
c
a
t
e
d
n
o
r
m
a
l
d
i
s
t
r
i
b
u
t
e
d
,
i
iid
𝑁
+
𝑢
u
~
(0,
𝜎
2
)
and constant over
time.
𝑙𝑛𝐿
=
−
𝑁
(𝑇
− 1)
𝑁
2
2
𝑣
𝑣
𝑢
𝜎
2
−
ln
𝜎
2
+
𝑇𝜎
2
+
𝑁𝑙𝑛
1
−
𝑙
−
𝜇
𝜎
𝑢
𝑖𝑖
+
�
l
n
1 −
𝑙
𝜇̅
𝑖𝑖
−
𝜎
∗
𝑣
∑
𝑖𝑖
𝑒
𝑖𝑖
𝑒́
𝑖𝑖
𝑁
𝜇
−
−
2𝜎
2
2
𝜎
𝑢
2
1
+
�
2
𝑖𝑖
𝜇̅
𝑖𝑖
𝜎
∗
2
where
𝜇̅
𝑖𝑖
=
2
𝜇
𝜎
𝑣
−
𝑇
𝑒
𝑖
𝑖
𝜎
𝑢𝑢
́
2
𝑣
𝑢𝑢
𝜎
2
+𝑇𝜎
2
Battese,
G.E.
and
Coelli, T.J.,
1988.
Prediction
of firm-
level technical efficiencies with
a
generalized frontier
production function and panel data.
Journal of
econometrics
,
38
(3),
pp.387-399)
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
48
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
49
V
.
T
i
m
e
-
V
a
r
i
a
n
t
S
t
o
c
h
a
s
t
i
c
F
r
o
n
t
i
e
r
M
o
d
e
l
s
30/11/2023
India
Stata
User
Meeting
2023_Rachita
Gulati
50
T
h
a
n
k
s
Stata offers a range of tools for conducting frontier efficiency and productivity assessments, including nonparametric and parametric approaches, technical efficiency modeling, different orientations in DEA, productivity estimation techniques, and popular models like MPI and MLPI. The software empowers researchers to analyze efficiency and productivity in various sectors effectively.
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Topic: Statas Capabilities For Frontier Efficiency Assessment Abstract : The development of a comprehensive suite of packages and commands within Stata has empowered researchers to conduct frontier efficiency and productivity assessments effectively. Presented By Dr. Rachita Gulati Associate Professor (Economics), Department of Humanities and Social Sciences Indian Institute of Technology Roorkee Uttarakhand, India
Points to Discuss Nonparametric approaches (DEA, FDH) Parametric approaches (SFA, TFA, RTFA) Technical efficiency (TE) Radial vs. Non-radial TE measures Orientations in DEA(Input and Output Orientations) Returns-to-ScaleAssumptions Efficiency Measurement Approaches Productivity Estimation and Decomposition Malmquist Productivity Index (MPI) Malmquist Luenberger Productivity Index (MLPI) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 2
Production technology Inputs: Single-input and single-output case x = (x1,x2,...,xm) Output: Orientationsin efficiency measurement Input-oriented Output-oriented y = (y1, y2,..., ys) Feasible production plan, if y can be produced from x T = (x, y): y can be produced from x 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 3
Two popular approaches Here, we are trying to assess Stata s capabilities in efficiency and productivity estimation using these two popular approaches. 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 4
Efficiency and Productivity Models Technical Efficiency Models DEA-based Charnes, Cooper, Rhodes (CCR) (1978) Model Banker, Charnes, Cooper (BCC) (1984) Model Slacks and RTS-based Estimation Directional Distance Function (DDF) Model (Radial and Nonradial) (Chung et al., 1997) SFA-based Stevenson (1980) Model Meeusen and Van den Broeck (MB) (1977) Aigner et al. (1977) Model Productivity Change Models Malmquist Productivity Index (MPI) Malmquist Luenberger Productivity Index (MLPI) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 5
Efficiency vs. Productivity Decision making units (DMUs) are used to describe the production entity responsible for turning inputs into outputs in instances when the firm may not be entirely appropriate, like power plants, schools, banks, states or countries, etc. j =1,2,...,n Productivity is a ratio of output to input Productivity= output/input Efficiency is a relative concept Efficiency= productivity/maximum productivity 30/11/2023 IndiaStataUser Meeting2023_RachitaGulati 6
Data Envelopment Analysis (DEA) DEA is a data-oriented and non-parametric based frontier approach First originated by Charnes, Cooper and Rhodes (CCR) (1978) Charnes, A., Cooper, W.W., and Rhodes, E. (1978), Measuring efficiency of decision making units , European Journal of Operational Research, 2, 429-444. CCR generalized Farrell s (1957) radial measure of technical efficiency to multiple input, multiple-output cases. Farrell, M. J., (1957), The measurement of productive efficiency , Journal of the Royal Statistical Society, Series A, Vol.120, No. 3, pp. 253-290. DEA is a non-parametric, frontier-based approach for measurement of efficiency and productivity. 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 7
I. Charnes, Cooper and Rhodes (CCR) (1978) CCR gives information on the technical efficiency (TE) of a unit using Shephard s distance function Assumptions: Constant Returns to Scale Strong disposability of inputs and outputs Convexity of Production Possibility Set Two distinct variants of the CCR model- Input-oriented CCR (CCR-I Output-Oriented CCR (CCR-O) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 8
CCR Model Consider n production units (j=1, ..,n) x vector of inputs (where xijis quantity of ithinput (i=1, .m) y vector of outputs (where yrjis the quantity of rthoutput (r=1, .,s). Input-Oriented CCR Model ?? ???= min????? ? ? ???????? ?????? ? ??=1 ? ??????? ??? ??=1 ??? 0 ?? = 1, .? Output-Oriented CCR Model ?? ???= max????? ? ? ? ??,?? ? ? , ? ? ? ?? = 1, .? ???????? ???? ??=1 ?? = 1, .? ??? ? ??????? ?????? ? = 1, .? ? = 1, .? ? ?? ??=1 ??? 0 ?? = 1, .? ?? ???measures the input-oriented TE of ? production unit o ?? ???measures the output-oriented TE of ? production unit o Maximum possible (radial) contraction in inputs IndiaStata User Meeting2023_RachitaGulati 30/11/2023 9
Software/Package/P rogramme DEAP DEAFrontier Matlab R Frontier Methods DEA DEA DEA DEA and SFA Reference Coelli (1996) Zhu (2014) DEA Toolbox additiveDEA (Soteriades, 2017); Benchmarking (Bogetoft and Otto, 2010, 2015); FEAR (Wilson, 2014); Frontier- Frontiles (Daouia and Laurent, 2015); Nonparaeff (Oh and Suh, 2013); npsf (Badunenko et al., 2017); Productivity (Dakpo et al., 2016); semsfa (Ferrara and Vidoli, 2015); SFA (Straub, 2015); spfrontier (Pavlyuk, 2016); SSFA (Fusco and Vidoli, 2015); TFDEA (Shott and Lim, 2015); rDEA (Simm and Besstremyannaya, 2016); DJL (Lim, 2016). frontier, xtfrontier Kumbhakar and Wang (2015) Tauchmann(2012) Stata Packages: - DEA (Ji and Lee, 2010); SFA (Kumbhakar et al., 2015); sfcross (Belotti et al., 2013); sfpanel (Belotti et al., 2013); tenonradial, teradial, teradialbc, nptestind, and nptestrts; (Badunenko and Mozharovskyi, 2016); Stata DEA and SFA Additional Software/Packages: Python, GAMS, SAS, DEAExcel, DEA-Solver-Pro, MaxDEA, DPIN, TFPIP, Frontier, LIMDEP, Inverse DEA, PIM Soft Source:Adapted from Daraio et al. (2019) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 10
CCR Model: Application in Stata Technical information: Stata 11.2 and above installation required (Badunenko and Mozharovskyi, 2016)) Badunenko, O., and Mozharovskyi, P. (2016). Nonparametric frontier analysis using Stata. Stata Journal, 16(3), 550 589. Syntax teradial outputs = inputs (ref outputs = ref inputs) if in, rts (rtsassumption) base (basetype) ref(varname) tename(newvar) noprint Specification outputs = list of output variables inputs = list of input variables rts (rtsassumption) = specifies the returns to scale assumption. Use rts(crs) for CRS, rts(nirs) for NIRS and rts(vrs)for VRS base (basetype) = specifies the type of optimization. Use base(output)for output- oriented measure and base(input) for input-oriented measure tename(newvar) = creates newvar containing the radial measures of technical efficiency 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 11
Contd Syntax for CCRI and CCRO teradial outputs = inputs (ref outputs = ref inputs), rts (rtsassumption) base (basetype) tename(newvar) . teradial Y=X1 X2, rts(crs) base(output) tename(CCRO) Hypothetical Sample Data X1 X2 Y 2 5 1 2 4 2 6 6 3 3 2 1 6 2 2 30/11/2023 IndiaStataUser Meeting2023_RachitaGulati 12
II. Banker, Charnes and Cooper (BCC) (1984) Model Banker et al. (1984) as an extension of the CCR Model. Based on the variable returns to scale Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science, 30(9), 1078 1092. Two variants: BCC-I and BCC-O Input-Oriented BCC Model ?? ???= min????? ? Output-Oriented BCC Model ?? ???= max????? ? ? ? ??,?? ? ? , ? ? n n j?ij ??BCC? i = 1, .m j?ij ?io i = 1, .m o io Con vexity j=1 j=1 n n j?rj ??BCC? r = 1, .s j?rj ?ro j=1 ? ???= 1 ??=1 r = 1, .s o ro j=1 ? ???= 1 ??=1 ?? = 1, .? ?? = 1, .? j 0 j 0 30/11/2023 IndiaStata User Meetin g 2023_RachitaGulati 13
BCC Model: Application in Stata Syntax teradial outputs = inputs (ref outputs = ref inputs) if in , rts (rtsassumption) base (basetype) ref(varname) tename(newvar) noprint Here, specify vrs in rts() Specification outputs = list of output variables inputs = list of input variables rts (rtsassumption) = specifies the returns to scale assumption. Use rts(crs) for CRS, rts(nirs) for NIRS and rts(vrs)for VRS base (basetype) = specifies the type of optimization. Use base(output)for output- oriented measure and base(input) for input-oriented measure tename(newvar) = creates newvar containing the radial measures of technical efficiency 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 14
BCC Model: Application in Stata . teradial Y=X1 X2, rts(vrs) base(output) tename(BCCO) Hypothetical Sample Data X1 4 7 12 10 9 X2 9 3 8 6 8 Y 10 8 16 9 7 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 15
Computation of slacks and RTS properties *= min j (Objective) o , ,s ,s+ subject to jxij+s = xio j=1 jyrj s+= yro j=1 s ,s+, j 0 n i =1,2,...,m; (Input) n r =1,2,...,s; (Output) j =1,2,...,n (Non-negativity) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 16
dea: command Alternative command, to obtain efficiency and slacks Ji, Yong-Bae and Choonjoo Lee (2010), Data envelopment analysis , Stata Journal 10(2): 267- 280. st0193.pkg dea ivars = ovars [if] [in] [, rts(crs | vrs | drs | nirs) ort(in | out) stage(1 | 2) trace saving(filename)] where the options are: rts(crs|vrs|drs|nirs) specifies the returns to scale. The default is rts(crs) ort(in|out) specifies the orientation. The default is ort(in) stage(1|2) specifies the way to identify all efficiency slacks. The default is stage(2) trace save all sequences and results from Results window to dea.log saving(filename) save results to filename. 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 17
dea: command X1 4 7 12 10 9 X2 9 3 8 6 8 Y 10 8 16 9 7 dmu 1 2 3 4 5 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 18
III. Directional Distance Function (DDF) Chung et al. (1997) developed this model Simultaneous adjustment of undesirable inputs and outputs along with the desirable inputs and outputs. Chung, Y .H., Fare, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: a directional distance function approach. Journal of Environmental Management, 51, 229 240. Weak disposability of undesirable output. Radial DDF Model ? ?,?,?, ??, ??, ?? = ???? ?,?? ? ???????? ???? ???? ?? = 1, ? where b is undesirable output, o is the unit under consideration and ( ??,??, ??) is the directional vector. ??=1 ? ??????? ???+ ???? ? = 1, ? ??=1 ? ???????= ??? ???? ? = 1, ? ??=1 ??? 0 ?? = 1, ..,? 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 19
DDF-DEA model: Application in Stata Technical information: st0665.pkg, Available for Stata 16 and above installation required (Wang et al., 2022) Wang, D., Du, K., & Zhang, N. (2022). Measuring technical efficiency and total factor productivity change with undesirable outputs in Stata. Stata Journal, 22(1), 103 124. Syntax teddf: Directional distance function (DDF) with undesirable outputs for efficiency measurement teddf inputvars = desirable_outputvars: undesirable_outputvars [ if ] [ in ], dmu(varname) time (varname) gx (varlist) gy (varlist) gb (varlist) nonradial wmat (name) vrs rf (varname) window(#) biennial sequential global [ other options] Specification Inputvars = inputs data (gx,gy,gb) = directional vector for input, good and bad output vrs = specify if variable returns required desirable_outputvars = good output data undesirable_outputvars = Undesirable output data dmu = specifies name of DMUs reference technology options = bi, seq, glo non-radial options = nonr, wmat 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 20
DDF-DEA Model: Application in Stata teddf X1 X2=Y:B,dmu(DMU) saving(ddf_result, replace) DMU D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 X1 327 521 227 1333 1113 530 1104 751 164 669 749 1638 208 342 354 482 X2 67 98 105 425 162 111 194 94 66 210 119 336 83 117 81 6 Y B 1097 1995 1517 2817 3124 2844 3248 2404 1068 2463 2976 2103 915 1245 1270 1710 1678 989 303 21733 1565 1009 2221 2655 212 4470 1163 18807 652 1679 411 148 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 21
IV. Alternative variants: Non-radial DDF Non-radial DDF with undesirables (Y2)(directional vector is (-X1 -X2 -X3 -X4 Y1 -Y2) and weight vector is (1 1 1 1 1 1)) teddf X1 X2=Y:B,dmu(DMU) time(Year) nonradial saving(ddf_result) teddf X1 X2=Y:B, dmu(DMU) nonradial vrs saving(ddf_result,replace) teddf X1 X2=Y:B, dmu(DMU) time(t) nonradial sequential saving(ddf_result,replace) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 22
V. Malmquist Productivity Index First proposed in Caves et al. (1982) and later modified by Fare et al.(1992) , Fare et al.(1994) and Ray & Desli (1997) Based on Shephard s distance function It provides a decomposition of productivity change into its sources, i.e.; efficiency change and technology change Assumption: Standard assumptions of DEAand Constant returns to scale ??+1(??+1,??+1) ??(??+1,??+1) ??+1(? ?+1 ??(??,??) ??+1(? ,? ) ??? = ?+1) ??(? ,? ) ,? ? ? ? Technical Change Efficiency Change 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 23
Malmquist Productivity Index By (p pe f alternatively, putting the values of , q), we can obtain the required own- riod and cross-period unctions as below: distance [??+???+?,??+?] 1= ?????,???? ?????????? ?,? = (0,0) for solving ????,?? ? for solving ?????+? ????+? ??? ??=1 ? ?????+? ??+? ???? ??=1 ??? 0 ?,? = (1,1) ??+1(??+1,??+1) ? = 1, ? ?? ?,? = (0,1) for solving ??+1(??,??) ?? = 1, .? ??? for solving ?,? = (1,0) ????+1,??+1 ?? = 1, ..,? 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 24
MPI and its Decomposition in STATA T echnical information: Available for Stata 16 and above (installation required - ssc install malmq2) Syntax malmq2 inputvars = outputvars [ if ] [ in ], ort(string) dmu(varname) window(#) biennial sequential global fgnz rd [..other options] Specification inputvars, outputvars = data for inputs and outputs ort() = defines orientation, ort(output), ort(input), default is output dmu()= specifies DMU names reference technology options = bi, seq, glo decomposition =fgnz ((Fare et al., 1994)), rd ((Ray & Desli, 1997)) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 25
TFPCH using MPI: Application in Stata Syntax malmq2 inputvars = outputvars [ if ] [ in ], ort(string) dmu(varname) window(#) biennial sequential global fgnz rd [..other options] DMU Year D1 D2 D3 D4 D5 D6 D7 D8 D1 D2 D3 D4 D5 D6 D7 D8 X1 327 521 227 X2 67 98 105 425 162 111 194 94 72 105 115 376 187 119 193 96 Y 2000 2000 2000 2000 1333 2000 1113 2000 2000 1104 2000 2001 2001 2001 2001 1339 2001 1130 2001 2001 2001 1097 1995 1517 2817 3124 2844 3248 2404 1403 2120 1698 3261 3476 3156 3520 2738 530 751 397 499 227 605 948 746 malmq2 X1 X2 = Y, dmu(dmu) global malmq2 X1 X2 = Y, dmu(dmu) seq ort(o) fgnz malmq2 X1 X2 = Y, dmu(dmu) ort(o) rd sav(tfp_result,replace) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 26
VI. Malmquist Luenberger Productivity Index (MLPI) First proposed by Chung et al. (1997), MLPI allows to estimate productivity changes while accounting for generation of undesirable products (e.g., pollution) along with desirable products Chung, Y. H., Fare, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: a directional distance function approach. Journal of Environmental Management, 51, 229 240. Based on radial directional distance function (DDF) Provides a decomposition of productivity change into its sources, i.e., efficiency change and technology change Assumption: same as MPI, along with weak disposability of undesirable output 1+ ????,??,?? 1+ ??+1??+1,??+1,??+1 1+ ????+1,??+1,??+1 1+ ??+1??,??,?? 1+ ????,??,?? ?? =1+ ??+1??+1,??+1,??+1 ?,?+ 1 Technical Change Efficiency Change 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 27
Malmquist Luenberger Productivity Index The directional vector ( ??, ??, ??) is taken as ( ????, ???, ???) alternatively, putting the values of (p, q), we can obtain the required own- period and cross-period below: ??+???+?, ??+?, ??+?, ??,??, ?? = ???? ?,?? . By s.?. ? ?????+? DDFs as ??+? ????+? ??? ?? = 1, ? ???? ??? ??=1 ?,? = (0,0) for solving ????,??,?? ? ?????+? ??+?+ ????+? ?? ? = 1, ? for solving ?? ?,? = (1,1) ??+1 ??+1,??+1,??+1 ??? ??=1 ? for solving ?????+?= ??+? ????+? ?? ?,? = (0,1) ??+1 ??,??,?? ? = 1, ? ??? ?? ??=1 ?,? = (1,0) for solving ????+1,??+1,??+1 ??? 0 ?? = 1, ..,? 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 28
MLPI Model: Application in STATA Technical information: Available for Stata 16 and above (installation required (Wang et al., 2022)) Syntax gtfpch inputvars = desirable_outputvars: undesirable_outputvars [ if ] [ in ], dmu (varname) luenberger ort (string) gx (varlist) gy (varlist) gb (varlist) nonradial wmat (name) window(#) biennial sequential global fgnz rd [ other options] Specification: Same as for DDF model 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 29
TFPCH using MLPI: Application in Stata DMU Year D1 D2 D3 D4 D5 D6 D7 D8 D1 D2 D3 D4 D5 D6 D7 D8 X1 327 521 227 X2 67 98 105 Y B 2000 2000 2000 2000 1333 425 2000 1113 162 2000 530 2000 1104 194 2000 751 2001 397 2001 499 2001 227 2001 1339 376 2001 1130 187 2001 605 2001 948 2001 746 1097 1678 1995 1517 2817 21733 3124 1565 2844 1009 3248 2221 2404 2655 1403 1813 2120 1698 3261 27987 3476 1784 3156 1054 3520 2486 2738 3073 989 303 111 94 72 105 115 917 337 119 193 96 gtfpch X1 X2 = Y:B, dmu(dmu) nonradialsaving(ddf_result) gtfpch X1 X2 = Y:B, dmu(dmu) seqential ort(input) leunberger saving(ddf_result, replace) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 30
Parametric Approaches to Frontier Analysis Stochastic Frontier Models Cross-Sectional Models Panel Data Models Aigner et al. (1977)Model Panel DataTime- invariant Models Panel DataTime Variant Models Meeusen and van den Broeck (1977) Model Schmidt and Sickles (1984) FE Model Kumbhakar (1990) Stevenson(1980) Model Schmidt and Sickles (1984) RE Model Battese and Coelli (1992) Battese and Coelli (1995) Greene (2003) Model Pitt and Lee (1981) Battese and Coelli (1988)
Stochastic Frontier Analysis (SFA) Popular parametric approach for estimating efficiency and productivity. SFA allows the deviation from the frontier and decomposes as; one is the random error ???and other is technical inefficiency ???. Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) independently proposed the stochastic frontier production function model of the form. (Assuming Cobb-Douglas functional form ) ?????= ?0+ ?1?????+ ??? l????= ?0+ ?1?????+ (?? ?= ?? ? ?? ?) ?? ? ?? ? DeterministicComponent noise inefficiency Here, ???is the output, ???is the vector of inputs, is vector of technological parameters to be estimated, ???= ??? ???is the composite error that has two components. ???that accounts for random disturbance and ???that accounts technical inefficiency.
Observed Output ???????? ????????? ?????? TE = ( + ?????) ??? ???? =? ? ? ? ? = ? ? ??? + ??? + ? ?? ? ??? +? ? ? TE = ? = ? ? + ? ? ? ? ?? 0 1 0 1 ?(?+??????) ???? ? ? ? ? 0 1 0 1
SFA: Cross-Sectional Models I.Aigner et al. (1977) Model (Normal-Half Normal Model) ???? ?= ?0+ ?1???? ?+ ?? ? ?? ? Assumptions 1. vi(random error) is assumed to be normally distributed, vi~iidN(0, ?2) 2. u (technical inefficiency) which is assumed to be half normally distributed, u~i iid?+ i 3. viis independent of ui 4. Assuming Cobb-Douglas functional form, where ???is the output, ???is the vector of inputs, is ? (0, ?2) ? vector of technological parameters to be estimated. To estimate , Aigner et al. used maximum likelihood estimation and parametrized the log- likelihood function for the normal-half normal model. ??2 1 2?2 ? ?2 lnL(?|?,?,?) = ?ln ???( ?? ?? ?) + ? ? ? =1 ??=1 ?? 2 2 ? , ?2= ?2+ ?2, and ?. is ? ? ?? ? ?? where, ? = v - u ,? is variance of composite error term , ? = ?2 i i ? ? ? the cumulative distributive function (cdf) of the standard normal variate. Aigner, D., Lovell, C.K. and Schmidt, P., 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), pp.21-37.
SFA in Stata Stata package- frontier Technical Information: Available for Stata 15 and above (no installation required) Syntax frontier depvar[indepvars], [options] Specification depvar = dependent variable indepvars= independent variables [options] distribution(hnormal)- half-normal distribution for the inefficiency term distribution(exponential) -exponential distribution for the inefficiency term distribution(tnormal)- truncated-normal distribution for the inefficiency term Post Estimation Commands for predicting inefficiency of each firm. predict (file_name), u (For predicting inefficiency of each firm) predict (file_name), te (For predicting efficiency of each firm)
Results from Aigner et al. (1977) Model using frontier te i 1 2 3 4 5 6 7 8 9 t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y x1 x2 ineff 0.161 0.855 0.087 0.918 0.504 0.608 0.179 0.840 0.110 0.898 0.166 0.851 0.278 0.762 0.177 0.841 0.073 0.931 0.123 0.887 0.368 0.696 0.056 0.947 0.227 0.801 0.135 0.877 0.205 0.819 15.131 9.416 35.134 26.309 4.643 77.297 6.886 5.095 89.799 11.168 4.935 35.698 16.605 8.717 27.878 10.897 1.066 92.174 8.239 0.258 97.907 19.203 6.334 82.084 16.032 2.35 38.876 12.434 1.076 81.761 2.676 3.432 29.232 4.033 55.096 16.58 7.975 12.903 7.604 10.618 0.344 10 11 12 13 14 15 9.476 73.13 24.35 65.38 Source: Coelli, T. (1996).Aguide to Frontier 4.1:Acomputer program for stochastic frontier production function and cost function estimation. Department of Econometrics University of New England Armidale NSW, 2351
Contd.. Stata package sfcross by Belloti et al. (2013) Belotti, F., Daidone, S., Ilardi, G. and Atella, V (2013), Stochastic frontier analysis using Stata, Stata Journal, 13, (4), 718-758 Technical Information: Available for Stata 15 and above (installation required) Syntax sfcross depvar [indepvars], [options] Specification depvar = Dependent Variable Indepvars = List of independent variables [options] distribution(hnormal)-half-normal distribution for the inefficiency term distribution(exponential)-exponential distribution for the inefficiency term distribution(tnormal)-truncated-normal distribution for the inefficiency term Post Estimation Commands for predicting efficiency and inefficiency of each firm. predict (file_name), u (For predicting inefficiency through Batesse and Coelli, 1988 Method) predict (file_name), bc (For predicting efficiency through Batesseand Coelli,1988 Method ) predict (file_name), jlms (For predicting efficiency through Jondrow et al., 1982 Method)
Results from Aigner et al. (1977) Model using sfcross sfcross lny lnx1 lnx2, distribution(hnormal) predict ineff, u predict eff_jlms, jlms predict eff_bc, bc
II. Meeusen and Van den Broeck (MB) (1977) (Normal- Exponential Model) Meeusen and Van den Broeck (MB) (1977) ?????= ?0+ ?1?????+ ?? ? ?? ? Assumptions (technical inefficiency) which is assumed to be exponentially ??? distributed u~?????(? ? ) ?2) i ? T o estimate , Meeusen and van den Broeck (MB) used MLE and parametrized the log-likelihood function for the Normal- Exponential model. ln L= ??? = ?????+ ? ?2 2?2 ??? ? + ??? ? + ? ??? ? ? ? ? ? where, ???= v - u is the cumulative distributive function (cdf) of the standard normal variate, ? = ? ?? ? ? , ? = ??? ?2 ? and N is number of producers. ,?2is variance of composite error term , ?2= ?2+ ?2, ?. i i ? ? ? ? Meeusen, W. and van Den Broeck, J. (1977). Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, pp.435-444.
Results from Meeusen and Van den Broeck (MB) (1977) predicteff,te predicteff,bc
III. Stevenson (1980) Model (Normal-Truncated Model) Model is formulated as ?????= ?0+ ?1?????+ ?? ? ?? ? Assumptions ???(technical inefficiency) which is assumed to truncated normally distributed, ui~iid (?, ?2) ? ?+ To estimate , Stevenson (1980) Model used maximum likelihood estimation and parametrized the log-likelihood function for the Normal-Truncated Normal Model. ? ? ????? ??? ???+? ? 1 ? ? + ??? ln? = ???? ???? ? ? ? ??? ? 2 where, (mode) is additional parameter to be estimated, ?? ?= v - u of composite error term , ,?2is variance i i ? = ?2+ ?2 , ?. is the cumulative distributive function (cdf) of the standard normal variate, and N is number of producers. Source: Stevenson, R.E., 1980. Likelihood estimation. Journal of Econometrics, 13(1), pp.57-66. ?2 ? ? ? functions for generalized stochastic frontier
Results from Stevenson (1980) Model predicteff,te predicteff,bc
Results of BC, JLMS, MB, and Stevensons Models i t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lny ln x1 0.973866 0.666799 0.707144 0.693287 0.940367 0.027757 0.099681 0.801678 0.371068 0.031812 0.535547 0.605628 0.901731 0.881042 0.369958 ln x2 1.545728 1.888163 1.953272 1.552644 1.445262 1.964608 1.990814 1.914259 1.589682 1.912546 0.976625 1.74112 1.864096 1.386499 1.815445 jlms 0.8510 0.9163 0.6041 0.8360 0.8954 0.8469 0.7573 0.8375 0.9294 0.8838 0.6924 0.9457 0.7969 0.8738 0.8150 bc eff_mb 0.9028 0.9404 0.6389 0.8949 0.9297 0.9027 0.8326 0.8919 0.9493 0.9246 0.7728 0.9576 0.8611 0.9182 0.8803 stevenson 0.8880 0.9343 0.6288 0.8774 0.9203 0.8869 0.8075 0.8769 0.9439 0.9136 0.7392 0.9545 0.8419 0.9055 0.8606 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1.179868 1.420104 0.837967 1.047975 1.220239 1.037307 0.915875 1.283369 1.204988 1.094611 0.427486 1.465859 1.219585 1.110691 1.026043 0.8546 0.9183 0.6077 0.8398 0.8979 0.8505 0.7616 0.8413 0.9309 0.8867 0.6965 0.9467 0.8011 0.8769 0.8190
IV. Time-Invariant Stochastic Frontier Models Schmidt and Sickles (1984) proposed both Time invariant Stochastic Frontier Fixed effect and Random effect Models. Fixed Effects Model + ?1? + ???? ??? ?+ ?1? + ???? ???? ? = ????= it 0 it ? ? Here, i= 0 - ui is the individual effects of fixed-effects model. In FE-Model, ???is assumed to be correlated with the regressor or with ???? Random Effects Model + ??????????+ ???? [??? ? ???] ??????= ? + ??????????+ ???? ? ??????= ?0 ? ??? 0 ? ? Source: Schmidt, P. and Sickles, R.C., 1984. Production frontiers and panel data. Journal of Business & Economic Statistics, 2(4), pp.367-374.
Panel Data SFA Models in STATA Stata packages: xtfrontier and sfpanel Technical Information: Available for STATA 15 and above (no installation required) Syntax xtfrontier depvar [indepvars], ti [ti_options] Syntax sfpanel depvar [indepvars], model () [model_options] Specification depvar = Dependent Variable indepvars= List of independent variables Specification depvar = Dependent Variable indepvars = List of independent variables [options] ti - use time invariant model cost - fit cost frontier model no constant - suppress constant term. constraints- applyspecific linear constraints [model_options] 1. fe- fixed effects model of SS (1984) 2. regls random effects model of SS (1984) 3. pl81 - Pitt and Lee (1981) model 4. bc88 - Battese and Coelli (1988) model Post Estimation Commands for predicting inefficiency of each firm. Post Estimation Commands for predicting efficiency and inefficiency ofeach firm. predict (file_name),u (inefficiency through BS 1988) predict (file_name),bc (efficiency through BS 1988) predict (file_name),jlms(efficiency through Jondrow et al. 1982 ) predict (file_name), u predict (file_name), te (efficiency of each firm) (inefficiency of each firm)
xtfrontier and sfpanel packages for time-invariant models: Schmidt and Sickles (1984) xtset i t xtfrontier y x1 x2, ti predictineff,jlms predict eff,te
Other Stochastic Frontier Time-Invariant Models Pitt and Lee (1981) Model Battese and Coelli (1988) Model yit = 0 + 1 ? + vit- ui yit = 0+ 1? + vit- ui it it Assumptions: ui(technical inefficiency) which is assumed to be truncated normal distributed, u ~ (0, ?2) and constant over time. ??? = ?(? 1) 2 2 ? ?? ? ? 2?2 Assumptions: ui(technical inefficiency) which is assumed to be half normally distributed, ui~iid over time ?(? 1) 2 ln 1 ? (0, ??) and constant 2 i iid?+ ? ?+ ? ? ?? 2 ?2 ln ?2+ ??2 + ??? 1 ? ? ? ? ? 2 ?2 ln ?2+ ??2+ ? ??? ?? ? ? ? 2?2 ??? = 2 ?????? ?? ? ? 1 ? ? ? ? ?? ? + ln 1 ? + ? ?? 2 2 ?? ? ? ?? ? ? ? 2 2 where ? ?? ??? ??? ??? ? ?2+??2 2 1 ? ? ? ? 2 = + ? ? ? 2 ? ? ??????? 1 where ? = , ? = and ? ? ? ? ? ? ? ? ?2= ?2+??2 ? ? ? ? ?2?2 ?2+??2 ? Pitt, M.M. and Lee, L.F. (1981). The measurement and sources of technical inefficiency in the Indonesian weaving industry. Journal of Development Economics, 9(1), pp.43-64.) ? ? ? ? ? Battese, G.E. and Coelli, T.J., 1988. Prediction of firm- level technical efficiencies with a generalized frontier production function and panel data. Journal of econometrics, 38(3), pp.387-399)
Other Stochastic Frontier Time-Invariant Models Model Name Pitt and Lee (1981) half normally distributed Battese and Coelli (1988) truncated normal distributed Syntax Main Command sfpanel depvar [indepvars], model () [model_options] sfpanel depvar [indepvars], model (pl81) sfpanel depvar [indepvars], model () [model_options] sfpanel depvar [indepvars], model (bc88) 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 48
V. Time-Variant Stochastic Frontier Models Stochastic Frontier Time-Variant Models Kumbhakar (1990) [indepvars], model () [model_options] sfpanel depvar sfpanel depvar [indepvars], model (kumb90) Battese and Coelli (1992) sfpanel depvar [indepvars], model () [model_options] sfpanel depvar [indepvars], model (bc92) Battese and Coelli (1995) sfpanel depvar [indepvars], model () [model_options] sfpanel depvar [indepvars], model (bc95) 30/11/2023 IndiaStataUser Meeting2023_RachitaGulati 49
Thanks 30/11/2023 IndiaStata User Meeting2023_RachitaGulati 50
