Orientation Issues in Data Envelopment Analysis and Unit Invariance

1
Orientation Issues in Data Envelopment Analysis
and Unit Invariance

• By
• Andy Johnson

2
Outline

• Brief Background
• Directional Distance Function
• Alternative 1 for the estimation of the
  parameters of the distance function
• Alternative 2
• Unit invariant directional model

Introduction
3
Efficient Frontier
output
Efficient Frontier
d
f
i
c
e
b
h
g
a
j
input
Introduction
4
Measuring Efficiency and Orientation
output
Efficient Frontier
d
f
i
c
e
o
b
h
g
a
j
input
Introduction
5
Efficient Frontier and Orientation
output
Efficient Frontier
d
c
e
b
a
input
i
Introduction
6
Orientation in DEA
output
Efficient Frontier
e
input
Introduction
7
Orientation in DEA

• The orientation decisions impacts both the
  measure of efficiency (assuming variable returns
  to scale) and the benchmark identified

Introduction
8
Directional DEA
Luenberger, D. G. (1992). "New Optimality
Principles for Economic-Efficiency and
Equilibrium." Journal of Optimization Theory and
Applications 75(2) 221-264. Chambers, R. G., Y.
H. Chung, et al. (1996). "Benefit and distance
functions." Journal of Economic Theory 70(2)
407-419. Chambers, R. G., Y. Chang, et al.
(1998). "Profit, directional distance function,
and Nerlovian efficiency." Journal of
Optimization Theory and Application 98(2)
351-364.
Directional DF
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Directional DEA
output
Efficient Frontier
e
input
Directional DF
10
Directional DEA

• This allows the direction to be flexible
• Is still a linear program
• However, it does not preserve unit invariance

Directional DF
11
Directional DEA

• How does one decide the values for and ?
• The direction not only impacts the efficiency
  estimate but also impacts the virtual observation
  identified for benchmarking purposes

Directional DF
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Directional DEA Dual
Directional DF
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Estimating and

• Additional information is needed
• One source of information could be multiple
  observations of the same production units over
  time (time series data)
• Given time series data and assuming production
  units are effectively working toward increased
  technical efficiency over time

Directional DF
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Directional DEA
output
e
input
Alter. 1
15
Estimation Procedure

• For a single output multiple input model
• Define a value for .
• Ex.
• Next regress
• Set

Alter. 1
16
Limitations of this Approach

• Only allows for a single input multiple outputs
  or a single output multiple inputs
• If the production unit is growing rapidly over
  time so that not only are output levels
  increasing, but input levels are increasing
  significantly, then could be negative which
  is inconsistent with the modeling assumptions
  (and efficiency measurement in standard models of
  performance)

Alter. 1
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Alternative Approach to Estimating and

• Want to estimate the directional parameters in
  such a way that variables that are difficult to
  adjust receive low parameters and variables that
  are easy to adjust are given larger parameters

Alter. 2
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Alternative Approach to Estimating and

• Use the ratio of the variance to the mean over
  the time periods observed

Alter. 2
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Alternative Approach

• Benefits
• and always have the proper sign
• Estimates are based on the flexibility of the
  inputs and outputs
• Drawbacks
• Even if a variable is very flexible, if the firm
  does not change the usage level, the directional
  parameter assigned to that input or output would
  be small

Alter. 2
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Unit Invariance

• Even though we have parameters for certain cases,
  the problem still exists the directional distance
  function model is not unit invariant

Unit Invariant
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Fare-Lovell Technical Efficiency Measure
Briec, W. (2000). "An extended Fare-Lovell
technical efficiency measure." International
Journal of Production Economics 65 191-199.
Unit Invariant
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Fare-Lovell Technical Efficiency Measure

• Benefit
• Directional while maintaining unit invariance
• Drawbacks
• Combines the Russell input and Russell output
  measures which weight each input and output
  equally
• The solution are on the range of 0 to -1 with 0
  being efficient, which is counter intuitive

Unit Invariant
23
New Directional DEA
Unit Invariant
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New Directional DEA

• Benefits
• Unit invariant and allows for flexibility in the
  direction of improvement
• Generalizes Shephards input and output distance
  functions and the directional distance function
  when is chosen
• Ranges from 0 to 1 with 1 indicating efficient
  operation

Unit Invariant
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Summary

• Directional DEA models are important because they
  capture the flexibility (or lack there of) of the
  inputs and outputs
• Two alternatives have been developed for
  estimating the directional parameters
• Each has benefits and drawbacks
• A directional model that is unit invariant has
  been developed incorporating one of the most
  important properties of efficiency measure into
  directional models

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Any Questions?

• Andy Johnson
• ajohnson_at_tamu.edu