DESCRIPCIÓN DE LOS PROBLEMAS DE CALIDAD PERCIBIDA

The data envelopment analysis occurred under the assumption that all observed inputs and outputs can be controlled. However, in practice this may not necessarily be the case. One

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common problem reported in the literature has been in relation to the handling of “exogenous”, “non-discretionary”, “environmental” or “contextual” variables, which determine observed variables that are exogenously-determined and, therefore, “uncontrollable” (Banker and Morey, 1986). Indeed, there are different ways of handling this problem, which related to the one-stage modelling; the two-stage modelling; and the adjusted-values modelling.

The one-stage model includes environmental variables directly in DEA to obtain efficiency scores with an additional restriction in the standard formulation. The first attempt of such a one stage model was Banker and Morey (1986), which remains the most representative model in terms of one stage for handling environmental variables. Another alternative one-stage model was demonstrated by Ruggiero (1996), which may consider as an extension of the model of Banker and Morey (1986), to treat environmental categorical variables, to the situation where these environmental factors are continuous.

Although the one-stage model has the simplicity advantage, there are many problems that have been noticed. Firstly, one needs to know a priori, which are the “environmental” variables that may positively or negatively influence the production frontier. In addition to that, the efficient units obtained by this approach are not different from those calculated using conventional approaches in which all variables were controllable. Furthermore, the increase in the number of environmental variables and constraints included in the model, although they facilitate the linear programming problem, may decrease the discrimination power of DEA results. Finally and most importantly, the one-stage models have been criticised due to the fact that environmental factors are not true economic inputs into the production process; instead they only influence technical efficiency. Comparatively, the two-stage modelling applies the DEA by including only controllable variables in the first stage. Therefore, the calculation of the technical efficiency may involve influence from “environmental” variables, which is temporarily ignored.

In the second stage of the analysis, environmental variables are introduced in a regression as independent variables, while the efficiency score, which was obtained from the first stage, is the dependent variable. The aim of this second stage is to explain the differences in efficiency scores that could be caused by environmental factors and not to correct efficiency scores. In addition, although an ordinary least squares (OLS) estimation process may be appropriate choice, some authors recommended the Tobit model (Tobin, 1958) in the second stage, which

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allows the dependent variable to be treated as a latent variable (McCarty and Yaisawarng, 1993; Hoff, 2007). Thus, the tobit model may provide more consistent and efficiency coefficient estimates because it can take into account the fact that the efficiency score is bounded between 0 and 1. However, there are other options for the choice of regression that have been implemented (Hoff, 2007; Ramalho et al., 2010).

The two-stage approach has the advantage of testing the influence of different environmental variables, which may be helpful in terms of recognising the possible source of inefficiency. However, there is a strong possibility of multicollinearity characterising the set of DEA scores, which may lead to biased and inefficient estimates, and can ultimately be solved by using bootstrapping (Simar and Wilson, 2007, 2011a). This is another option to avoid such a problem of treating the DEA scores in the second stage as descriptive measures of the relative technical efficiency of the DMUs, as proposed and supported by McDonald (2009), which will be discussed in detail in the following sections.

Multi-stage modelling is another way to deal with environmental factors, as this approach basically evaluates DEA efficiency by using controllable factors only and then correcting the efficiency scores obtained in further stages in order to account for environmental factors. Subsequently, in the final stage the efficiency scores are corrected by running a DEA model with data adjusted for these environmental variables.

Multi-stage modelling aims to decompose the possible effect of “slacks” associated with the technical inefficiency of DMUs and influence of environmental factors, which has not been included in the first stage. In other words, the idea is for the second-stage to distinguish between the effect of “slacks” associated with the first stage and the impact of such environmental variables which have been included in this stage. The DEA can subsequently be run using the ‘corrected’ variables in order to obtain new efficiency scores.

Different multi-stage models have been proposed in the literature depending on the adhered to approach in order to distinguish between the “slacks” and environmental factors that associated to inferences, such as the semi-parametric model recommended by Fried et al. (1999, 2002) or the non-parametric model proposed by Muñiz (2002). The latter uses input- oriented DEA in the second stage. In this stage, the slacks from the first DEA stage are considered as inputs and the environmental factors are considered as outputs. The aim of this stage is to reduce the slacks, while taking the value of the environmental variables to be fixed.

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Non-parametric methods do not require a specific structural form of the objective equation, and therefore, estimation problems, such as mis-specification error and heteroscedasticity and other issues, which could lead to biased estimates, are avoided. However, it is possible to provide biased results due to the deterministic nature of the method as it uses DEA mode in all stages (Cordero et al., 2009). Furthermore, it is unable to identify which environmental factor is the most relevant, and therefore, it is possible that part of the predictive power of the model can be lost, despite the fact that certain environment variables may not be statistically significant.

In addition, in this non-parametric, there is a possibility that efficient DMUs will become inefficient after including environmental effects on the final stage. However, this change cannot be true from the methodological point of view, as discussed in Fried et al. (2002) and Cordero et al. (2009). Finally, with increasing the number of environmental variables, the discrimination power will be reduced and most DMUs tend to be efficient. This disadvantage shares the one stage model, as has been mentioned previously.

Regarding semi-parametric multi-stage methods aimed at estimating a separate regression involving each “slack” variable for inputs or outputs (depending on the orientation of DEA in the first stage), and by incorporating environmental factors as independent variables, the estimation process may follow the Tobit model because “slack” variables are censored at zero. This could allow the identification of the statistical significance of environmental factors on the slacks separately. Therefore, this approach would allow adjustment of the original values of variables.

More importantly, this approach would allow the prediction of new slacks for each variable that takes into consideration the environmental variables on each unit by using the regression coefficients. Thus, the original values of variables could be corrected using these predicted values by taking the original value of the outputs and subtracting the difference that is present between the most elevated value that is predicted and each units’ predicted value, or by adding it in the case of inputs. Following this, the final DEA is run using these adjusted variables.

The previous approach was described as the four-stage model, which was proposed by Fried

et al. (1999) with significant improvements in the calculation of the efficiency scores. However, there is a possibility of a bias result through its two-stage counterpart, since the total slacks is also predicted by using the information of the whole sample. Indeed, this

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problem could be treated by using bootstrap to estimate unbiased regressions to predict total slacks, as applied in Cordero et al. (2009).

Even though the previous multi-stage models (parametric or non-parametric) appear to be attractive methods, as they distinguish slack results from technical efficiency or from environmental factor, Estelle et al. (2010) point out that taking account of these slacks is misguided due to the empirical evidence that there is no additional slack for any benchmark locates in the Farrell projection neighbourhood.

Overall, the two stage approach is the most common form in DEA applications, even though there is no agreement on which is the best method to treat uncontrollable factors in DEA, which explains to managers and policy makers why some DMUs perform better or worse than others, as well as what is the sources of such inefficiencies. In such cases, environmental factors such as ownership types and organisational characteristics, which could also influence DMUs' technical efficiency, need to be taken into account.