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)

A Stochastic Frontier Analysis of Estimates and Correlates of the

Efficiency of Solid Waste Management in Welsh SMEs

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 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

2 4 6 8 10

2

4

6

8

10

12

0

Input

A

B

E

F

D

C G

H

DEA

OLS Regression

Output

SFA

I1 I2

O2

O1

X

Y

Output Efficiency of F: F0/YO

Input Efficiency of E: XI/XF

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 0.000 Log-Likelihood -527.27Wald Chi2 (Prob. > Chi2) 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 2.493 ***Firm Monitors Waste 0.159 Firm Audits Waste -0.418 *Firm Publishes Environmental Policy 0.192 Firm Uses Local Business Support Group -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

.

0,,,,

0

0

0s.t.

min

54321

55443322113

525424323222121233

515414313212111133

yyyyyy

xxxxxx

xxxxxx

TE kk

CRS Input-Oriented DEA Solution

表 2 CRS Input-Orientated DEA Results

Firm 1 2 3 4 5 1IS 2IS 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

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.

x2/y

6

5 1

4

3 1’

2 2 3 4

1 4’ 5

0 0 1 2 3 4 5 6

x1/y

FRONTIER

Start with a Set of Observed DMUS and their Input -Output Correspondences

By virtue of interpolation between C and D all input - output correspondences on CD are

feasible

Using interpolations between observed units the set of all feasible input -output correspondences is constructed

and its boundary identified

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:

Output Efficiency of E: FE/FG

Output benchmarks for E: Units C and DScope for output augmentation at E: EGReturns 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.

Input Efficiency of E: HI/HE Scope for resource conservation at E: IE

Examining DMU Efficiencies in DEA

n1iexy i

Kk

1kikki

K inputs

Consider a production function for I DMUs and K inputs:

PARAMETRIC METHODS FOR COMPARATIVE EFFICIENCY MEASUREMENT

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

n1i]uv[xyln ii

Kk

1kikki

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 underfrontier;

one-parameter probability density functions

0

0.5

1

1.5

2

2.5

0 1 2 3

random variable, u

f(u

)

f(u) exp f(u) half-normal

The exponential or half-normal distributions often assumed for SFA

inefficiencies (u) acknowledge that larger inefficiency values are less likely

The Stochastic Frontier

)exp()exp(

)exp()exp(

uvxuvx

vxy

TE i

ii

ii

ii

ii

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 = 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

iiuE

We plug into this formula the values of and other values we derive from the to arrive at an estimate of the conditional inefficiency

ui of the ith DMU.

The formulae for differs depending on the distribution assumed for ui and are coded in software such as Limdep and STATA.

ii

uE

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 multiple inputs

Typically can only accommodate single 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 0.000 Log-Likelihood -527.27Wald Chi2 (Prob. > Chi2) 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 2.493 ***

Firm Monitors Waste 0.159 Firm Audits Waste -0.418 *Firm Publishes Environmental Policy 0.192 Firm Uses Local Business Support Group -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?