A Stochastic Frontier Analysis of
Estimates and Correlates of the Efficiency
of Solid Waste Management in Welsh
SMEs
James Cordeiro , State University of New York (Brockport), USA
Joseph Sarkis, Clark University, USA
Diego Vazquez & Jeroen Dijkshoorn, BRASS Center, Cardiff
University, Wales, UK
GIN CONFERENCE 2008 (Leeuwarden, Netherlands)
Purpose of the Study
 Primary Purpose: To identify the technical efficiency of 299 solid waste
recycling efforts of Welsh SMEs in 2003 by using a sophisticated
econometric technique called Stochastic Frontier Analysis (SFA)
 Secondary purposes:

To identify the extent to which environmental practices such as waste
monitoring, auditing, publication of environmental policies, and use
of local support groups are related to the efficiency scores

To check whether the SFA efficiency scores correlate significantly
with those obtained using Data Envelopment Analysis (DEA)
approaches
Efficiency Analyses
 Efficiency analyses focus on the efficiency of some production process
(e.g. solid waste processing) in transforming inputs into outputs.
 Frontier methods use an efficient frontier to identify the efficiency of
individual organizations relative to a reference set of organizations


DEA is a non-parametric approach that uses mathematical programming to
identify the efficient frontier.
SFA is a parametric approach that hypothesizes a functional form and use
the data to econometrically estimate the parameters of that function using
the entire set of DMUs
 The measure of efficiency is normally one of either:


The distance between observed and maximum possible output for given
inputs (output efficiency)
The distance between observed and minimum possible input for given
outputs (input efficiency)
Overview of Study Sample and Variables
 Study Focus: Efficiency of solid waste management (a subject of major
environmental significance and legislation in Wales)
 Study sample: 299 Welsh Small and Medium-sized manufacturing
Organizations (SMES) in 19 industries who responded to a BRASS survey
in 2003
 Methodology: Stochastic Frontier Analysis (SFA)
 Study Variables:

Output: Ln (Proportion of total solid waste recycled)

Inputs: (a) Ln (Total recycling cost expended per ton)
(b) Ln(Organization size i.e.,#employees))

Variables use to explain inefficiencies: Dummy indicators (0,1) of waste
auditing, waste monitoring, publication of environmental policies, use of
local support groups
Methodology: SFA and DEA compared to each other
and to OLS Regression
 DEA creates virtual units that Output
serve as benchmarks for
measuring DMUs
comparative efficiency
 SFA uses an hypothesized
function to calculate estimates
of the efficiencies of
individual DMUs
 Only SFA can separate
random noise from efficiency;
DEA incorporates noise as
part of the efficiency score.
 SFA and OLS regression
methods reveal overall
sample-based information.
DEA reveals unit-specific data
type of returns to scale,
productivity change
12
Output Efficiency of F: F0/YO
DEA
C
10
O2
E
D
G
O1
8
SFA
B
6
A
OLS Regression
4
X
F
I1 I2
2
H
0
2
4
Y6
8
Input Efficiency of E: XI/XF Input
10
SFA Results (Base Model)
Model Parameters:
Constant
0.031 ***
Ln (Treatment cost per ton of solid waste) -0.002***
Ln (Number of Employees)
-0.010 ***
Model Statistics:
Lambda
Log-Likelihood
Wald Chi2 (Prob. > Chi2)
0.000
-527.27
460280.49(.000)
Model of Predictors of Variance of Inefficiency Scores
(this section of the model is re-estimated as part of the original model):
Constant
Firm Monitors Waste
Firm Audits Waste
Firm Publishes Environmental Policy
Firm Uses Local Business Support Group
2.493 ***
0.159
-0.418 *
0.192
-0.495 ***
Rank Correlations of SFA with DEA Efficiency
Rankings
 SFA technical efficiency score rankings were correlated (using Kendall’s
Tau) with the DEA efficiency score rankings for two different DEA
approaches:


CCR approach (correlation is .87)
Slack-based CCR approach (correlation is .87)
 Thus our confidence that SFA and DEA provide similar efficiency
rankings of the Welsh SMEs is very high  this is encouraging given
the different assumptions underlying these two approaches
Summary of Findings
 SFA was used to obtain technical efficiency scores for 299 Welsh SMEs
and the two inputs specified were both found to be significant
 The SFA scores correlate with those obtained by DEA  increasing our
confidence in our estimates of the SMEs efficiency rankings
 Some initial evidence that waste auditing and use of local business
support groups may impact efficiencies of the SMEs
 Further research: We are focusing on panel data collection of SME
inputs and outputs to answer important questions like:





Which industries have been improving their efficiencies faster over time?
Which waste management practices have the most impact over time?
Can more sophisticated input and output models be developed for the
rankings?
Can the model be successfully applied to other waste management
approaches?
How do SMEs compare to larger firms? Different countries?
Thanks!
DEA Numeric Example
表 1 Example Data for CRS DEA Example
Firm
Y
x1
x2
x1/y
x1/y
1
1
2
5
2
5
2
2
2
4
1
2
3
3
6
6
2
2
4
1
3
2
3
2
5
2
6
2
3
1
Five LP models are run (for each firm A to E) to
find the efficient frontier for DEA
min TE k  k
s.t.3 x13  x111  x12 2  x133  x14 4  x155   0
3 x23  x211  x222  x233  x24 4  x255   0
 y3   y11  y 2 2  y3 3  y 4 4  y5 5   0
1 , 2 , 3 , 4 , 5  0
.
CRS Input-Oriented DEA Solution
Firm

表 2 CRS Input-Orientated DEA Results
1
2
4
IS1
IS 2
3
5
OS
1
0.5
-
0.5
-
-
-
-
0.5
-
2
1.0
-
1.0
-
-
-
-
-
-
3
0.833
-
1.0
-
-
0.5
-
-
-
4
0.714
-
0.214
-
-
0.286
-
-
-
5
1.0
-
-
-
-
1.0
-
-
-
CRS Input-Oriented Example of DEA Frontier
x2/y
6
5
1
4
3
1’
2
2
1
3
4
4’
5
FRONTIER
0
0
1
2
3
4
5
6
x1/y
1) TE (technical efficiency) of Firm 3 is 0.833, i.e., Firm 3 could
reduce all inputs (x1,x2) by 16.7% to produce the same amount of y.
2) Thus, firm 3 project to 3’ ,on a line joining 2 and 5 and firm 4 to 4’
3) The line joining 2 and 5 is called the “Frontier” and firm 2 and 5
are referred to as “targets” or “peers” for firms 3 and 4.
Start with a Set of Observed DMUS and their Input Output Correspondences
Data Envelopment Analysis
Output
Input
By virtue of interpolation between C and D all input output correspondences on CD are feasible
Data Envelopment Analysis
Output
D
C
Input
Using interpolations between observed units the set of all feasible
input -output correspondences is constructed and its boundary
identified
Data Envelopment Analysis
Output
D
C
B
A
Set of all feasible inputoutput correspondences
below and to the right of
ABCD
Input
Examining DMU Efficiencies in DEA
Using the set of all feasible input output correspondences the comparative efficiency
and other information in respect of a unit (e.g. unit E) is derived as illustrated here:
Data Envelopment Analysis
Output Efficiency of E: FE/FG
Activity level
D
C
G
I
H
E
B
A
F
Input Efficiency of E: HI/HE
Resource level
Output benchmarks for E: Units C
and D
Scope for output augmentation at
E: EG
Returns to scale (increasing,
decreasing, constant): Revealed by the
intercepts of the segments of the
efficient boundary.
Scale elasticity revealed by the
slope of the segments on the efficient
boundary.
Scope for resource conservation at E: IE
PARAMETRIC METHODS FOR COMPARATIVE
EFFICIENCY MEASUREMENT
Consider a production function for I DMUs and K inputs:
k K
y i      k x ik  e i i  1 n
K inputs
k 1
Where y is output, xik are inputs, and ei is the residual for DMU I
It is the residual ei the captures any inefficiency in this model
The residual also captures other noise or random effects (e.g. omitted
variables, measurement error, etc.)
SFA attempts to decompose the error term into inefficiency and noise
components for each DMU i
[1]
SFA Model for I DMUs and K inputs
k K
ln y i      k x ik  [ v i  u i ] i  1 n
k 1
We decompose the error term into two components:
v is an identically distributed conventional two-sided error term with zero
mean. It stands for random noise, omitted variables etc.
u is an identically distributed one-sided error term with a non-zero mean. It
stands for inefficiency.
u is typically assumed to be exponential, half-normal or truncated normal
Stochastic Frontier Example
 Frontier:
y= exp(xβ)
yi= exp(xβ +vi)
 vi is noise due to
random events
 if vi>0 
above frontier

if vi<0 
under
frontier;
The exponential or half-normal distributions often assumed for SFA
inefficiencies (u) acknowledge that larger inefficiency values are less likely
f(u)
one-parameter probability density
functions
2.5
2
1.5
1
0.5
0
0
1
2
random variable, u
f(u) exp
f(u) half-normal
3
The Stochastic Frontier
exp(x  v u i)
y
TE  exp(x  v )  exp(x  v )  exp(u )
i
i
i
i
i
i
i
i
i
The SFA model is usually fitted using Maximum
Likelihood estimation
We need to estimate the inefficiency of the ith producer (ui) by using its
composed residual ei = vi - ui .
Depending on the assumption we make about the distribution (e.g.
half-normal, exponential) of the inefficiency ui we arrive at a different
formula for the conditional value
E u e
i
i

We plug into this formula the values of ei and other values we derive
from the ei to arrive at an estimate of the conditional inefficiency
ui of the ith DMU.
The formulae for E ui e i  differs depending on the distribution
assumed for ui and are coded in software such as Limdep and STATA.
Summary Comparison of DEA v/s SFA Approaches
DEA
SFA
Non-parametric method  Cannot test
hypotheses
Parametric method  Can test
hypotheses
Uses mathematical programming
Uses maximum likelihood econometric
estimation
Does not accommodate noise (noise is
effectively part of the efficiency score)
Specifies noise (separates noise from
efficiency scores)
Can accommodate multiple outputs and Typically can only accommodate single
multiple inputs
output with multiple outputs
Functional form is not specified
Functional form needs to be specified
Overview of Study Sample and Variables
 Study Focus: Efficiency of solid waste management (a subject of major
environmental significance and legislation in Wales)
 Study sample: 299 Welsh Small and Medium-sized manufacturing
Organizations (SMES) in 19 industries who responded to a BRASS survey
in 2003
 Study Variables:

Output: Ln (Proportion of total solid waste recycled)

Inputs: (a) Ln (Total recycling cost expended per ton)
(b) Ln(Organization size i.e.,#employees))

Variables use to explain inefficiencies: Dummy indicators (0,1) of waste
auditing, waste monitoring, publication of environmental policies, use of
local support groups
SFA Results (Base Model)
Model Parameters:
Constant
0.031 ***
Ln (Treatment cost per ton of solid waste) -0.002***
Ln (Number of Employees)
-0.010 ***
Model Statistics:
Lambda
Log-Likelihood
Wald Chi2 (Prob. > Chi2)
0.000
-527.27
460280.49(.000)
SFA Results -- 2
Model of Predictors of Variance of Inefficiency Scores
(this section of the model is re-estimated as part of the original
model):
Constant
Firm Monitors Waste
Firm Audits Waste
Firm Publishes Environmental Policy
Firm Uses Local Business Support Group
2.493 ***
0.159
-0.418 *
0.192
-0.495 ***
Rank Correlations of SFA with DEA Efficiency
Rankings
 SFA technical efficiency score rankings were correlated (using Kendall’s
Tau) with the DEA efficiency score rankings for two different DEA
approaches:


CCR approach (correlation is .87)
Slack-based CCR approach (correlation is .87)
 Thus our confidence that SFA and DEA provide similar efficiency
rankings of the Welsh SMEs is very high  this is encouraging given
the different assumptions underlying these two approaches
Summary of Findings
 SFA was used to obtain technical efficiency scores for 299 Welsh SMEs
and the two inputs specified were both found to be significant
 The SFA scores correlate with those obtained by DEA  increasing our
confidence in our estimates of the SMEs efficiency rankings
 Some initial evidence that waste monitoring and use of local business
support groups may impact efficiencies of the SMEs
 Further research: We are focusing on panel data collection of SME
inputs and outputs to answer important questions like:





Which industries have been improving their efficiencies faster over time?
Which waste management practices have the most impact over time?
Can more sophisticated input and output models be developed for the
rankings?
Can the model be succesfully applied to other waste management
approaches?
How do SMEs compare to larger firms? Different countries?