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Cross redundancy and sensitivity in DEA models

Journal of Productivity Analysis volume 34, pages 151–165 (2010)Cite this article

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Abstract

Data envelopment analysis (DEA) measures the efficiency of each decision making unit (DMU) by maximizing the ratio of virtual output to virtual input with the constraint that the ratio does not exceed one for each DMU. In the case that one output variable has a linear dependence (conic dependence, to be precise) with the other output variables, it can be hypothesized that the addition or deletion of such an output variable would not change the efficiency estimates. This is also the case for input variables. However, in the case that a certain set of input and output variables is linearly dependent, the effect of such a dependency on DEA is not clear. In this paper, we call such a dependency a cross redundancy and examine the effect of a cross redundancy on DEA. We prove that the addition or deletion of a cross-redundant variable does not affect the efficiency estimates yielded by the CCR or BCC models. Furthermore, we present a sensitivity analysis to examine the effect of an imperfect cross redundancy on DEA by using accounting data obtained from United States exchange-listed companies.

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Fig. 1
Fig. 2

Notes

  1. Nunamaker (1985), however, points out that even the addition of a highly correlated variable may alter the resulting DEA efficiency estimates substantially.

  2. Principal component analysis is based on the covariance matrix between variables.

  3. If there are no restrictions on the sign of a i , such a dependence is called a linear dependence. Readers who are not familiar with the term “conic combination” are referred to Barvinok (2002, p.65) and Boyd and Vandenberghe (2004, p.24) for examples.

  4. In the jargon of statistics, if there is a cross redundancy between input and output variables, then the absolute value of the first canonical correlation equals one.

  5. Dulá (1997) examined relationships between DEA and the theory of redundancy in linear systems, but he studied the redundancy of a set of DMUs with respect to the other DMUs. The current study focuses on the redundancy of variables.

  6. Some prior studies, including Smith (1990), Thore et al. (1993), and Halkos and Salamouris (2004), used DEA to examine a firm’s performance by using accounting data and including net income as one of the output variables.

  7. The current study focuses on the CCR and BCC models. For the output-oriented BCC models, to be precise, inputs are not restricted to be nonnegative. But, for the output-oriented CCR models, inputs are restricted to be nonnegative [refer to page 102 in Cooper et al. (2000)]. Hence, to accommodate both output-oriented CCR and BCC models, we consider only nonnegative inputs and outputs when we deal with output-oriented models. Also, practically, negative inputs are not useful.

  8. We analyze the output-oriented models separately because we believe that the proof of Theorem 2 is not easily deduced from the proof of Theorem 1.

  9. For sensitivity analyses and simulation studies of DEA models, refer to Banker et al. (1996a, 2004), Charnes et al. (1985), Cooper et al. (2001), and Seiford and Zhu (1998) for examples.

  10. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.

  11. For a detailed explanation of the selection of input variables, readers are referred to Section 3 in Demerjian et al. (2006).

  12. The item number represents the numeric code used in the Compustat database.

  13. This is a slight abuse of notations in the sense that we are using subindices, not superindices, to enumerate input or output variables. But, using these vector notations simplifies the presentation of our results.

  14. In this study, we use an absolute efficiency change rather than a percentage efficiency change to measure sensitivity. Our rationale for this is as follows. Suppose there are only two DMUs and efficiency scores change from 0.1 to 0.2 for one DMU and from 0.9 to 1.0 for the other DMU after the inclusion of additional output variables. If we use a mean absolute efficiency change as a summary measure, we will report a value of 0.1. If we use a mean percentage efficiency change as a summary measure, we will report a value of 55.6%. Which one between 0.1 and 55.6% is more informative as a single summary measure? Although answers would depend on analysts’ subjective tastes, our opinion is that, if we use a percentage change, then summary measures usually employed, such as the arithmetic or geometric mean, would put disproportionately larger weights on efficiency changes from lower efficiency scores than on those from higher efficiency scores. For this reason, we believe that it would be more difficult to interpret summary measures based on percentage change than those based on absolute change. Also, we suspect that this would be one main reason that several prior studies (e.g., Banker et al. 1993, 1996a, and 2004) use an absolute change (the mean absolute deviation, to be precise) as a measure of sensitivity.

  15. Note, however, that R 2 is not exactly equal to the degree of cross redundancy between input and output variables. In Equation (2), we restricted the sign of coefficients a r and b i to be nonnegative, but in the regression of net income on sales and other inputs, the coefficients are not restricted in sign.

  16. As a perturbation scheme, we can also consider an additive perturbation scheme: y ε3  = y 3 + ε. In this case, to make ρ = corr(y 3, y ε3 ), we need to set \(\sigma^{2}_{\epsilon} = (1 - {\rho ^{2}}) {\sigma^{2}_{y_{3}}}/ \rho^2\). For industry #33, \({\sigma^{2}_{y_{3}}}\) is greater than twenty million. Hence, even if ρ2 = 0.99, we must have σ 2ε  ≥ 200000. However, for a number of DMUs in this industry, the value of y 3 is close to zero. Therefore, the additive perturbation scheme, y ε3  = y 3 + ε, results in a change in the sign of y 3 for a large number of DMUs, which alters the DEA results significantly. Thus, in the current study, we believe that use of the multiplicative perturbation scheme is appropriate.

  17. We repeated our simulation study using data from two other industries, industry #31 (Utilities) and industry #43 (Restaurants, Hotels, and Motels). The results from these industries are much better than the results in Table 2. Here, the term “better” means that the effect of a perturbation on DEA is smaller.

  18. For a sample from any distribution, a larger the sample size results in a larger maximum difference. Hence, a maximum difference increase is expected as the sample size increases.

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Authors and Affiliations

  1. College of Management, Georgia Institute of Technology, Atlanta, GA, 30308-0520, USA

    Kyuseok Lee

  2. College of Hotel and Tourism Management, Kyung Hee University, Seoul, 130-701, South Korea

    Kyuwan Choi

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  1. Kyuseok Lee
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Correspondence to Kyuwan Choi.

Appendix

Appendix

In this appendix, we provide the rationale behind Equation (5). For convenience of notations, we set Y = y 3. Suppose that Y and ε are independent and that ε follows a normal distribution N(0, σ 2ε ). Let

$$ Y^\epsilon = Y +\epsilon Y = (1+\epsilon)Y. $$

Then we have

$$ \begin{aligned} &Cov(\epsilon Y,Y)\\ &\quad =E [(\epsilon Y-E[\epsilon Y] ) (Y-E[Y] ) ]=E [\epsilon Y (Y-E[Y] ) ]\\ &\quad =E [\epsilon ] \cdot E [ Y (Y-E[Y]) ] = 0, \end{aligned} $$

where we used the independence between Y and ε. Using Cov(εY, Y) = 0, we also have

$$ \begin{aligned} &Var(Y^\epsilon ) \\ &\quad =Var((1+\epsilon) Y )=Cov(Y+\epsilon Y,Y+ \epsilon Y)\\ &\quad =Var(Y)+Var(\epsilon Y)=Var(Y)+E[(\epsilon Y-E[\epsilon Y] )^2 ]\\ &\quad =Var(Y)+E[\epsilon^2 Y^2 ]\\ &\quad =Var(Y)+\sigma_\epsilon^2 (Var(Y)+E[Y]^2). \end{aligned} $$

Hence, to make ρ = corr(Y, Y ε), we must set

$$ \sigma_\epsilon^2=\frac{1-\rho^2}{\rho^2} \times \frac{\sigma_Y^2}{\sigma_Y^2+E[Y]^2}. $$

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Lee, K., Choi, K. Cross redundancy and sensitivity in DEA models. J Prod Anal 34, 151–165 (2010). https://doi.org/10.1007/s11123-009-0166-2

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  • DOI: https://doi.org/10.1007/s11123-009-0166-2

Keywords

  • Data envelopment analysis (DEA)
  • Efficiency
  • Cross redundancy
  • Sensitivity analysis
  • Simulation
  • Accounting data

JEL classification

  • C67
  • D20
  • M11