Malmquist Productivity Analysis
using DEA Frontier in Stata
14-15 July, 2011
Stata Conference Chicago 2011
Choonjoo Lee1, Kyoung-Rok Lee1, Byung-Ihn Lim2
sarang90@kndu.ac.kr, bloom.rampike@gmail.com
Korea National Defense University1, Nextree Soft2
Contents
Malmquist Productivity Index
Malmquist Index using DEA Frontier
The User Written Command “malmq”
Notes and Examples
References
Malmquist Productivity Index
• Productivity = Output / Input
• Productivity (Growth) Index measures the
Productivity changes over Time
• Malmquist (Productivity Growth) Index
measures the productivity changes along with
time variations and can be decomposed into
changes in efficiency and technology.
Malmquist Productivity Index
Output
A2(4,4)
A1(2,1)
0
Input
• Productivity Index = (4/4)/(1/2) = 2
☞ Productivity is improved by 100%
Malmquist Productivity Index
• Malmquist Productivity Index
M
t 1
t
I
t 1
D (x , y )
(x , y , x , y ) 
t
t
t
I
DI ( x , y )
t
t
t
t 1
t 1
Where Input based distance function at time t is defined by
D ( x , y )  max  | ( x /  , y )  P ( x , y )}
t
I
t
t
t
t
t
is measured by production possibility set
Pt
for Production Possibility Set
Input vector
t
t
P( x t , y t )
x  {x1 , x2 , x3 ,..., xm }
Output vector y  { y1 , y2 , y3 ,..., yn }
☞
M It
at time t.
Malmquist Productivity Index
• Malmquist Productivity Index
And accordingly,
DIt 1 ( xt , y t )  max  | ( xt /  , y t )  Pt 1 ( xt 1, y t 1 )}
D ( x , y )  max  | ( x
t 1
t
I
t 1
t 1
/  , y )  P ( x , y )}
t 1
t
for cross period distance function.
Further,
M
can be defined as
t 1
t 1
I
t 1
t
I
t 1
D (x , y )
(x , y , x , y ) 
t
D (x , y )
t 1
I
M
t 1
I
t
t
t 1
t 1
t
t
Malmquist Productivity Index
Output(y)
y6
P t 1 ( x t 1 , y t 1 )
Pt ( xt , y t )
y5
y4
y3
y2
At+1(4,4)
y1
At(2,1)
0
x1 x2 x3 x4 x5 x6
Input(x)
oy4
• Productivity Change =
ox6
oy1
ox3

oy4 ox3
oy1 ox6
=(4/1)*(2/4)=2
Malmquist Productivity Index
• Malmquist Productivity Index
DIt ( x t , y t )  ox2 / ox3
t 1
t 1
D ( x , y )  ox5 / ox6
t
I
t
t 1
t 1
D
(
x
,
y
) ox5 / ox6
t
t
t
t 1
t 1
I
M I ( x , y , x , y )  DIt ( xt , y t )  ox / ox
2
3
ox3 ox5 ox3 oy4


 Productivity Change
ox6 ox2 ox6 oy1
Malmquist Index using DEA Frontier
• Concepts of Malmquist Index using CRS Frontier
Malmquist Index using DEA Frontier
The input oriented CRS Malmquist Index using the observations at
time t and t+1.
The Geometric mean of two input oriented CRS Malmquist Indices
Malmquist Index using DEA Frontier
Decomposition of the input oriented geometric mean of Malmquist
index using the concept of input oriented efficiency change and
input oriented technical change
♨ Malmquist Index can be obtained from the DEA measure
The User written command “malmq”
• Program Syntax
malmq ivars = ovars [if] [in] [, ort(in | out)
period(varname) trace saving(filename)]
– ort(in | out) specifies the orientation. The default is ort(in),
meaning input-oriented DEA.
– period(varname) identifies the time variable.
– trace specifies to save all the sequences displayed in the Results
window in the malmq.log file. The default is to save the final
results in the malmq.log file.
– saving(filename) specifies that the results be saved in
filename.dta.
• See “malmq.ado” file for the details
Notes and Examples
• Notes
– Updated “dea.ado”, “malm.ado” files
– In terms of accuracy and computational efficiency?
Current version is more focused on ‘accuracy’
– Tested for 365DMU data set for dea.ado command
and compared with other DEA programs.
Notes and Examples
• Example
– Data : see “365dmu.dta” for dea command and
“panel_data_for_malmquist_dea.dta” for malmq
command.
– Try the following commands
• dea i_total = o_licnese o_sic o_nsic o_dpatent
o_fpatent, rts(crs) ort(i)
• malmq i_AC = O_SPI O_CPI, ort(i) period( period)
Notes and Examples
– Result
• For dea: Results including the messages “No
Solution(LOOP grather than
maxiter):[DMUi=119][LOOP=16001]CRS-IN-SIPII”.
 See “dea.log” file for details
 Compare with results by other programs
• For malmq
 see “malmquist.log” file for details
 Compare with results by other programs
References
• Lee, C., & Lee, K.(2010). “An Efficient Data Envelopment Analysis with a
large data set in Stata”, Boston10 Stata Conference.
• Ji, Y., & Lee, C. (2010). “Data Envelopment Analysis”, The Stata Journal,
10(no.2), pp.267-280.
• Lee, C., & Ji, Y. (2009). “Data Envelopment Analysis in Stata”, DC09 Stata
Conference.
• Fare, R., Grosskopf, S., Norris, M. & Zhang, Z. (1994). “Productivity Growth,
technical progress and efficiency change in industrialized countries”, American
Economic Review, 84(no.1), pp.66-83.
• Lee, J., & Oh, D.,(2010). “Efficiency Analysis Methodology: Data
Envelopment Analysis”, IB Book(in Korean).