Abstract
In this paper we propose a target efficiency DEA model that allows for the inclusion of environmental variables in a one stage model while maintaining a high degree of discrimination power. The model estimates the impact of managerial and environmental factors on efficiency simultaneously. A decomposition of the overall technical efficiency into two components, target efficiency and environmental efficiency, is derived. Estimation of target efficiency scores requires the solution of a single large non-linear optimization problem and provides both a joint estimation of target efficiency scores from all DMUs and an estimation of a common scalar expressing the environmental impact on efficiency for each environmental factor. We argue that if the indices on environmental conditions are constructed as the percentage of output with certain attributes present, then it is reasonable to let all reference DMUs characterized by a composed fraction lower than the fraction of output possessing the attribute of the evaluated DMU enter as potential dominators. It is shown that this requirement transforms the cone-ratio constraints on intensity variables in the BM-model (Banker and Morey 1986) into endogenous handicap functions on outputs. Furthermore, a priori information or general agreements on allowable handicap values can be incorporated into the model along the same lines as specifications of assurance regions in standard DEA.
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Notes
Simar and Wilson (2007) have recently proposed a coherent data generating process (DGP) that can serve as a rationale for this approach.
The simplifying assumption of one output is made to accommodate the need for a presentation of the basic idea behind the VWBM-model that is as simple as possible. The general case is covered in the next section. In this paper we focus on the output oriented model, since this orientation is used in the application in Sect. 5. It is straight forward to change the various models to an input orientation.
In this paper we focus on produced output as a measure of size. It is left for future research to consider measures of size based on input consumption.
Clearly, the weighting of environmental variables is only meaningful in the case of cardinal data. The treatment of ordinal data is not considered in this paper.
It is of interest to observe that the VWBM- and the BM-model will approach each other for increasing sample size, since facets become smaller which in turn implies a smaller deviation between volume of produced outputs among the DMUs spanning each facet.
The collection of additional data allowing for a transformation of indices to quantities is a more straightforward approach in applications where environmental conditions are not defined in terms of output with certain attributes.
For illustration, consider an example from Ruggiero (2004) with three DMUs A, B & C producing one output Y using one discretionary input X and one non-discretionary input Z measuring harshness of environment. The [Y,X,Z]—vectors from A, B & C are [10,10,20], [10,8,30] and [10,1,40]. Let Y be number of (100) students, let X be number of (10) teachers, and let Z be the percentage of parents with a college education. Using the model proposed by Ruggeiero all DMUs are efficient. B is efficient, since A cannot dominate B on its own. B is inefficient in the BM-model and is dominated by the convex combinations \(\frac{1} {2}A+\frac{1} {2}C.\) In the context of an efficiency evaluation of schools, a precise interpretation of the convex combination \(\frac{1}{2}A+\frac{1}{2}C\) is available and the expected Z-index at this virtual school equals the harshness index of B. The expected number of students at the virtual school \(\frac{1}{2}A+\frac{1}{2}C\) with a parent with a college education is 100 + 200 and the corresponding Z-index is \(\frac{300}{1000}\times 100=30.\) Hence \(\frac{1}{2}A+ \frac{1}{2}C\) should be allowed to dominate B and this is indeed the outcome of the volume weighted BM-model (VWBM-model) to be suggested in next section.
For ease of exposition we have omitted non-Archimedians as lower bound on the virtual multipliers. A stringent approach would require u > 0. Hence u −1 always exists.
The challenging task of incorporating non-linear handicap functions is left for future research.
Analysis of assurance regions such as \(\widehat{h}_{k}=\left[1-u^{-1}g^{T}\left( Z_{j}-Z_{0}\right) \right] \geq 0\) and more generally in the next section g T(Z j − Z 0) ≤ 1 = u T Y 0 is left for future research.
The set of environmental indices may well relate to a subset of the outputs, and two different indices may not relate to the same subset. One must in cases like that select which combinations of the p indices and the s outputs it is meaningful to include in model (5). To simplify we have included all ps constraints in the presentation of the model (5).
Another way to argue for this model is to consider facet inducing dual constraints \(F\left( x,y,\Updelta z\right) =u^{T}y-g^{t}\Updelta zu^{T}y-v^{T}x=0 \) and to argue that the marginal rate of substitution between observed outputs must equal the marginal rate of substitution between handicapped outputs along the facet, i.e. \(\frac{\hbox{d}y_{2}} {\hbox{d}y_{1}}=- \frac{u_{1}-g^{t}\Updelta zu_{1}} {u_{2}-g^{t}\Updelta zu_{2}}=-\frac{\left(1-g^{t}\Updelta z\right) } {\left( 1-g^{t}\Updelta z\right)}\frac{u_{1}} {u_{2}}\;{\text {must\;be\;equal\; to}}-\frac{u_{1}} {u_{2}}=\frac{d\left( 1-g^{t}\Updelta z\right) y_{2}}{d\left( 1-g^{t}\Updelta z\right) y_{1}}.\) This links nicely to the data generating process proposed by Simar and Wilson (2007).
An extensive list of references to applications of such two stage approaches can be found in Simar and Wilson (2007).
An OLS approach is of course problematic since the efficiency scores obtained have a bounded support, either (0,1] or [1,∞). Simar and Wilson (2007) are also concerned with the problems that DEA scores are serially correlated, a phenomenon that creates problems for the use of traditional inference in the regression stage and that the obtained scores are biased.
Without this additional requirement this large optimization model is simply n separable optimization problems, but adding the requirement of one common g implies a non-separable nonparametric estimation of this common handicap function and all scores.
The same idea can be applied to the classical BM-model (Banker and Morey 1986). Solve an estimation of all BM-scores using a simultaneous estimation of n dual programs to the BM-model in (1) with a set of linking constraints. These constraints must force each environmental virtual multiplier relative to all other virtual multipliers to be equal across the set of DMUs.
An important question is of course whether or not the DGP described in Simar and Wilson (2007) is reasonable. In this formulation, the environmental variables influence the mean and variance of the inefficiency process, but not the boundary of its support. A number of different alternatives are discussed in Coelli et al. (1998, pp. 166–171). Among these alternatives are the BM- and the R-model which are coherent with a DGP where the boundary of the production possibility set is dependent on the environment.
The PISA scores from 2003 were given to 15-year-old students. We use the student–teacher ratio from the previous 3 years as a measure of the possible teacher impact on the performance measured by these PISA scores. We aim at an efficiency analysis of the lower and upper secondary educations. Hence, we multiply all the PISA scores (means) and the teacher–student ratio by the average (2000–2002) enrollment in lower and upper secondary educations. This procedure requires an assumption, that the performance in 2003 of the 15-year-old students is representative for the performance of all 5 grades in the secondary education, i.e. for similar performance in the previous 5 years.
This ad hoc procedure provides the environmental index as the conditional mean of the BCC score added to the shift factor. The 2SR_DEA does not provide any estimation of the level of the environmental impact in a decomposition of the BCC-score into an environmental and a managerial part. This is really an econometric identification problem since no information is provided on how to disentangle the estimated intercept term into a part related to the environmental and a part related to the managerial impact. Notice that the estimation using the TE-DEA-model [(10), (11)] with a common handicap function provides all the information needed for the decomposition of the overall index into these two aspects. Ray suggests a shift of the residuals by adding the smallest residual.
It is not obvious that the impact from the environment in general works independently of the choice of input and output mix. Input or output mix may in some cases have significant impact on how environment contracts or expands output. If this is the case, then using the TE-DEA model will provide biased results. This problem corresponds to maintaining an assumption of a Hick-neutral technical progress in a situation where the movement of the frontier over time is different for different output mixes.
This MILP is formulated using a reformulation from William (1989, p. 178).
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Appendices
Appendix 1
Appendix 2: Implementation of the model (a mixed integer linear programming model)
With more than one output, as illustrated in the application, we have to use the more general model (7) which clearly is nonlinear and not even a convex optimization model. Hence, we cannot be sure that solving the model by a non-linear solver will provide a global maximum. To overcome this problem we reformulate (7) as a MILP.Footnote 21 To illustrate the general idea we will focus on the situation with only one environmental non-discretionary variable \((p=1,g\in {\mathbb{R}}_{+}){:}\)
where M kj is a large (but not too large) number, \(\forall k,j.\) The product of the decision variables \(gu^{t}Y_{j}\) in (7) is here substituted for \(w^{t}Y_{j}.\) The binary structure implies that \(w_{kj}=\sum_{j=1}^{50}\widetilde{w}_{kj} \ {\text{and}} \ \widetilde{w}_{kj}=\left\{ \begin{array}{ll} 2^{(j-40)}u_{k} & \hbox{if} \quad b_{j}=1 \\ 0 & \hbox {if} \quad b_{j}=0 \end{array} \right. \)
Hence, we get an optimal value of g from the MILP as \(g^{\ast }=\sum_{j=1}^{50}b_{j}^{\ast }2^{(j-40)}.\) The implications of the two possible values of binaries b j follows from:
Solving (12) we know for sure that we have an optimal (or near optimal solution). However, computational experience shows that it in certain cases is difficult to get the correct optimal values (and dual values) to (7) by solving (12). Hence, we recommend that (12) is solved to get a near optimal solution, which is given as initial solution to a nonlinear solver, which then quickly determines a precise solution satisfying the Kuhn Tucker conditions to (7).
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Olesen, O.B., Petersen, N.C. Target and technical efficiency in DEA: controlling for environmental characteristics. J Prod Anal 32, 27–40 (2009). https://doi.org/10.1007/s11123-009-0133-y
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DOI: https://doi.org/10.1007/s11123-009-0133-y