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Common Methodological Choices in Nonparametric and Parametric Analyses of Firms’ Performance

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The Palgrave Handbook of Economic Performance Analysis

Abstract

This chapter discusses common methodological, theoretical and empirical choices that scholars face when undertaking productive and economic efficiency analyses. After summarizing the main results of duality theory in Section 2, we outline in Section 3 the most popular empirical methods available to undertake efficiency analyses, namely nonparametric data envelopment analysis (DEA) and parametric stochastic frontier analysis (SFA). We discuss in Section 4 several strategies aimed at reducing the dimensionality of the analysis, either by relying on dimension reduction techniques that aggregate the original variables into a smaller set of composites, or by selecting those that better characterize production and economic processes. Section 5 discusses how to control for environmental or contextual z-variables that do not fall within managerial discretion, as well as the implications that each option has for researchers, managers and policy makers. Section 6 presents a series of recent models addressing endogeneity issues in the DEA and SFA approaches. In this section, we also discuss the endogenous nature of the distance function when assessing firms’ efficiency. Finally, Section 7 summarizes the guiding principles of the chapter and draws the main conclusions.

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Notes

  1. 1.

    For an extended and augmented version of this chapter, the reader is referred to Orea and Zofío (2017).

  2. 2.

    The DDF by Chambers et al. (1996) corresponds to the concept of shortage function introduced by Luenberger (1992, p. 242, Definition 4.1), which measures the distance of a production plan to the boundary of the production possibility set in the direction of a vector g. In other words, the shortage function measures the amount by which a specific plan falls short of reaching the frontier of the technology. Chambers et al. (1996) redefine the shortage function as efficiency measure, introducing the concept of DDF.

  3. 3.

    In empirical studies approximating the technology through DEA, the global CRS characterization is assumed for analytical convenience because relevant definitions such as profitability efficiency and the Malmquist productivity index require this scale property, and therefore, their associated distance functions are defined with respect to that benchmark technology.

  4. 4.

    The input and output distance functions define respectively as \(D_{I} (x,y) = \hbox{max} \left\{ {\lambda :(x/\lambda ,y) \in T} \right\}\) and \(D_{O} (x,y) = \hbox{min} \left\{ {\theta :(x,y/\theta ) \in T} \right\}\). If the technology satisfies the customary axioms, the input distance function has the range \(D_{I} \left( {x,y} \right) \ge 1\). It is homogeneous of degree one in inputs, non-decreasing in inputs and nonincreasing in outputs. In contrast, the output distance function has the range \(0\,{ < }\,D_{O} \left( {x,y} \right)\; \le \, 1\). It is homogeneous of degree one in outputs, nondecreasing in outputs and nonincreasing in inputs. Färe and Primont (1995, pp. 15, 22) show that weak disposability of inputs and outputs is necessary and sufficient for the input and output distance functions to completely characterize technology.

  5. 5.

    The hyperbolic distance function inherits its name from the hyperbolic path that it follows towards the production frontier. The range of the hyperbolic distance function is \(0 < D_{H} \left( {x,y} \right) \le 1\). It satisfies the following properties: it is almost homogeneous of degrees k1, k2 and k3: \(D_{H} (\lambda^{{k_{ 1} }} x,\lambda^{{k_{ 2} }} y;\alpha ) = \lambda^{{k_{3} }} D_{H} (x,y;\alpha )\), \({\text{for}}\,{\text{all}}\,\lambda > 0\), \(k = ( - 1, \, 1, \, 1)\) (Aczél 1966, Chs. 5 and 7; Cuesta and Zofío 2005), nondecreasing in outputs and nonincreasing in inputs.

  6. 6.

    Debreu’s (1951) “coefficient of resource utilization” is the corner stone upon which Aparicio et al. (2016) introduce the concept of loss distance function, identifying the minimum conditions necessary to derive a dual relationship with a supporting economic function. They obtain specific normalizing sets of the loss function that correspond to the most usual distance functions.

  7. 7.

    The counterpart to the input distance function corresponds to the cost function, defined as \(C(y,w) = \mathop {\hbox{min} }\limits_{x} \left\{ {wx:x \in L(y)} \right\}\), where \(L(y) = \left\{ {x:(x,y) \in T} \right\}\) is the input requirement set. It represents the minimum cost of producing a given amount of outputs, yielding the input demand functions by applying Shephard’s lemma. Correspondingly, the revenue function \(R(x,p) = \mathop {\hbox{max} }\limits_{y} \left\{ {py:y \in P(x)} \right\}\), where \(P(x) = \left\{ {y:(x,y) \in T} \right\}\) is the output production possibility set, represents the maximum possible revenue of using a given amount of inputs, yielding the output supply functions.

  8. 8.

    The technology may be characterized by variable returns to scale as in (2), allowing for scale (in)efficiency \(D_{G}^{CRS} \left( {x,y;\alpha } \right) = D_{G} \left( {x,y;\alpha } \right)SE_{G}\), with \(SE_{G} = D_{G}^{CRS} \left( {x,y;\alpha } \right)/D_{G} \left( {x,y;\alpha } \right)\), but the final supporting technological benchmark is characterized by CRS.

  9. 9.

    Here, we take into account that \(T = \left\{ {(x,y):D_{G} (x,y;\alpha ) \le 1} \right\}\) and \(T = \left\{ {(x,y):D_{T} \left( {x,y, - g_{x} ,g_{y} } \right) \ge 0} \right\}\). For the case of the profit and DDFs, the additive overall efficiency measure is normalized by \(pg_{y} + wg_{x} = 1\), ensuring that it is independent of the measurement units as its multiplicative counterparts—see Nerlove (1965). These dual relations are economic particularizations of Minkowski’s (1911) theorem: every closed convex set can be characterized as the intersection of its supporting halfspaces. In fact, the cost, revenue, profit and profitability functions are known as the support functions characterizing the technology for alternative shadow prices—e.g. for the particular case of the cost function, see Chambers (1988, p. 83).

  10. 10.

    The overall cost and revenue efficiencies correspond to \(C(y,w)/wx = \left( {1/D_{I} (x,y)} \right) \cdot AE_{I}\) and \(py/R(x,p) = D_{O} (x,y) \cdot AE_{O}\), respectively.

  11. 11.

    The (strongly) efficient set consists of all firms that are not dominated, requiring monotonic preferences to characterize efficiency (ten Raa 2008, p. 194, Lemma).

  12. 12.

    This in turn implies that the radial framework or choosing as a directional vector the observed amounts of inputs and outputs in the case of the DDF is no longer valid.

  13. 13.

    Regarding denominations, we note that a firm is overall profit efficient when its technical and allocative terms are zero rather than one. This implies that the larger the numerical value of the DDF the more inefficient is the firm, thus the technical and allocative (in)efficiency notation: TI and AI, with \(TI = D_{T} \left( {x,y; - g_{x} ,g_{y} } \right)\). Other authors, e.g. Balk (1998), favour a consistent characterization of efficiency throughout, so the larger the value the greater the firm’s efficiency. This is achieved by defining \(TE = - D_{T} \left( {x,y; - g_{x} ,g_{y} } \right)\).

  14. 14.

    A comprehensive exposition is presented in earlier chapters devoted to the deterministic and stochastic benchmarking methodologies by Subash Ray, and William H. Greene and Phill Wheat, respectively.

  15. 15.

    See Cooper et al. (2007) and Färe et al. (1994) for an introduction to the Activity Analysis DEA within a production theory context.

  16. 16.

    The dual for the GDF envelopment formulation (11) can be determined because it corresponds to a CRS characterization of the production technology, rendering it equivalent, for instance, to the radially oriented output distance function for \(\alpha = 1\)—since the value of \(D_{G}^{CRS} \left( {x,y;\alpha } \right)\) is independent of α.

  17. 17.

    Nevertheless, the computational effort of solving the envelopment problems grows in proportion to powers of the number of DMUs, I. As the number of DMUs is considerably larger than the number of inputs and outputs (N + M), it takes longer and requires more memory to solve the envelopment problems. We contend that except for simulation analyses and the use of recursive statistical methods such as bootstrapping, nowadays processing power allows calculation of either method without computational burdens.

  18. 18.

    Although most early SFA applications used production functions, the distance function became as popular as the production functions since Coelli and Perelman (1996), who helped practitioners to estimate and interpret properly the distance functions. In addition, the distance functions can constitute the building blocks for the measurement of productivity change and its decomposition into its basic sources (see, e.g., Orea 2002). This decomposition can be helpful to guide policy if estimated with precision.

  19. 19.

    To obtain this equation, we have taken into account that the vi and −vi have the same normal distribution.

  20. 20.

    While the flexibility of the functional forms allows a more precise representation of the production technology and economic behaviour, it is prone to some drawbacks. For instance, Lau (1986) proved that flexibility is incompatible with global regularity if both concavity and monotonicity are imposed using standard econometric techniques. That is, imposing regularity conditions globally often comes at the cost of limiting the flexibility of the functional form. It should be pointed out, however, that it is possible to maintain local flexibility using Bayesian techniques. See Griffiths et al. (2000) and O’Donnell and Coelli (2005).

  21. 21.

    Both variances can also be estimated using the second and third moments of the composed error term taking advantage of the fact that, while the second moment provides information about both variances, the third moment only provides information about the asymmetric random conduct term.

  22. 22.

    Note that, for notational ease, we use \(\sigma_{u}\) to indicate hereafter the standard deviation of the pretruncated normal distribution, and not the standard deviation of the post-truncated variable \(u_{i}\).

  23. 23.

    Other authors have used the copula method in other types of SFA applications. For instance, Amsler et al. (2014) employ them to model time dependence, while Carta and Steel (2012) suggest their use in modeling multi-output models.

  24. 24.

    Empirical application of the state contingent approach has proved difficult for several reasons because most of the data needed to estimate these models are lost in unrealized states of nature (i.e. outputs are typically observed only under one of the many possible states of nature).

  25. 25.

    As aforementioned, firms’ efficiency scores can also be computed without making specific distributional assumptions on the error components using the so-called distribution-free approach. As Kumbhakar et al. (2015, p. 49) remark, the drawback is that the statistical properties of the estimator of \(u_{i}\) may not be ready available.

  26. 26.

    However, Aparicio and Zofío (2017) show that the use of radial measures is inadequate to decompose cost efficiency in the case of nonhomothetic production functions because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—i.e. allocative inefficiency. They demonstrate that a correct definition of technical efficiency corresponds to the DDF.

  27. 27.

    Alternative hypotheses testing methods corresponding to nonparametric and bootstrap-based inference have been proposed in the literature, see the chapter devoted to the statistical analysis of nonparametric benchmarking contributed by Leopold Simar, Camilla Mastromarco and Paul Wilson.

  28. 28.

    A comprehensive discussion about the theoretical implications of different types of separability (e.g. strong vs. weak) can be found in Chambers (1988).

  29. 29.

    For instance, whereas Xia et al. (2002) and Bura (2003) propose semiparametric techniques to estimate the inverse mean function, E(X|Y), Cook and Ni (2005) develop a family of dimension reduction methods by minimizing Quadratic discrepancy functions and derive the optimal member of this family, the inverse regression estimator.

  30. 30.

    The chapter by John Ruggiero discusses environmental variables and how to render observations comparable in performance studies.

  31. 31.

    Although the two-stage method is the most popular one in DEA for identifying inefficiency determinants, three-stage models have also been developed (see, e.g., Fried et al. 2002).

  32. 32.

    Interesting enough, this specification of the way efficiency scores depend on z-variables corresponds to the popular KGMHLBC model in the SFA approach (see next subsection).

  33. 33.

    Daraio and Simar (2005) propose an alternative approach by defining a conditional efficiency measure. This approach does not require a separability condition as demanded by the two-stage approach.

  34. 34.

    The general models introduced by Wang (2002) and Lai and Huang (2010) are similar, but they parameterize the pretruncation mean of the distribution as a linear function of the z-variables.

  35. 35.

    Parmeter and Kumbhakar (2014) show that, if \(z_{i}\) and \(x_{i}\) do not include common elements, the conditional mean \(E\left[ {u_{i} |z_{i} } \right]\) can be estimated in a nonparametric fashion without requiring distributional assumptions for \(u_{i}\).

  36. 36.

    Kumbhakar (2011) also relies on profitability maximization, but he solves the endogeneity of both outputs and inputs first by deriving a particular form of the estimating equation in which the regressors are ratios of inputs and outputs. Thus, his transformed specification can be estimated consistently by ML methods using standard stochastic frontier software.

  37. 37.

    In his model, the distribution of \(u_{i}\) is not allowed to have efficiency determinants.

  38. 38.

    A copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform.

  39. 39.

    Applications of simulations to evaluate a likelihood can be found in Greene (2005, p. 24), Amsler et al. (2016), and Parmeter and Kumbhakar (2014; Sects. 6 and 7).

  40. 40.

    As shown by Parmeter and Kumbhakar (2014, p. 52) using a Translog cost function, if the production technology is homogeneous in outputs, the model can be estimated using simple ML techniques.

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  1. Oviedo Efficiency Group, Department of Economics, School of Economics and Business, University of Oviedo, Oviedo, Spain

    Luis Orea

  2. Department of Economics, Universidad Autónoma de Madrid, Madrid, Spain

    José Luis Zofío

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Orea, L., Zofío, J.L. (2019). Common Methodological Choices in Nonparametric and Parametric Analyses of Firms’ Performance. In: ten Raa, T., Greene, W. (eds) The Palgrave Handbook of Economic Performance Analysis. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-23727-1_12

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