Methods for Estimating Efficiency-A Taxonomy of Frontier Models

Frontier Models

The following section cites some of the principal techniques in the literature for estimating efficiency level. Therefore, the aim of this section is to propose a general taxonomy of efficient frontier models that gives an overview of the different approaches presented in the literature for estimating the efficient frontier of a PPS and to explore the methodological advances applied in the English HE system. Analysis of the existent literature is a necessary step for the advancement of a discipline. This is particularly evident in the field of efficiency and productivity research that, in the last decades, has experienced an exponential increase in the number of methodological and applied works (Tavares, 2003; Daraio and Simar, 2007).

The first distinction in a methodological grounding for efficiency estimation approaches is between the statistical (or econometric) approach and the non-statistical (or programming) approach. The distinction between the two approaches derives from the underlying assumptions. The non-statistical approach makes no assumptions regarding the distribution of inefficiencies. In addition, it is often (but not always) non-parametric, which means that the input and output data are used to compute a convex hull to represent the efficiency frontier (Sengupta, 1999) using linear programming methods. The non-parametric frontier approach, based on envelopment techniques (DEA, FDH45) has been used extensively for estimating the efficiency of DMUs as it relies on very few assumptions for the production possibility set (PPS). The non-parametric approach relies on linear programming or some other form of mathematical programming to characterise the set of efficient producers and then derive estimates of efficiency for inefficient observations based on how far they deviate from the most efficient ones, rather than estimating values for selected parameters. Another competitive superiority of the non-statistical, non-parametric approach is the lack of misspecification problems,46 since neither distributions are specified, nor is there a particular functional form47 _{for the frontier function (Johnes, 2004).}

Furthermore, programming methods can easily be used in a production situation in which multiple inputs and multiple outputs are handled and ensure robustness in model choice. For a comprehensive DEA bibliography covering 1978–1992, see Seiford (1994, 1996), and for an extension until 2001, see Gattoufi et al. (2004). More than 1,500 DEA references are reported by Cooper et al. (2000), despite the highlighted disadvantages of the non-statistical, non-parametric approaches since they are barren of

45 _{The Free Disposal Technique introduced by Deprins et al. (1984) relies only on the free disposability assumption of the PPS and}

does not restrict itself to convex technologies. The FDH estimator, proposed by Deprins, et al. (1984), is a more general version of the DEA estimator.

46 _{Both in the production function and the distribution of efficiencies.} 47 _{See appendix 4 chapter 3 for a useful insight into different functional forms.}

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estimates or significance tests of parameters (Geva-May, 2001). Another limitation is that the convex hull is defined using information on only a small number of observations in the sample. Further shortcomings shared by many non-parametric methods concern the curse of dimensionality. This is to avoid large variances and wide confidence interval estimates; therefore, the analysis becomes ‘hungry for data’, which means that a large amount of data is needed (Daraio and Simar, 2007).

The statistical approach is often (but not always) parametric, which means that a specific functional form48 _{for the production frontier function 𝑔(𝑥, 𝛽) is assumed}

(Sengupta, 1999). Therefore, statistical, parametric methods use a simple mathematical form depending on some 𝑘 unknown parameters, since 𝛽 ∈ ℛ𝑘 _{represents the} production technology set 𝑆. Hence, it provides estimates on the parameters of the frontier, the significance of which can be tested using standard errors (Schmidt, 1985– 6). The main methodological advances of this approach are the economic interpretation of parameters and the statistical properties of estimators; more critical are the choice of the function 𝑔(𝑥, 𝛽) and the handling of multiple input and multiple output cases (Daraio and Simar, 2007).

A further classification between alternative methods is based on the criterion of noise presence. Hence, the distinction between deterministic and stochastic models is attributed to whether deviations from the production function are a consequence not only of inefficiency. In terms of the deterministic approach, deviation in observed output from the production frontier is solely a consequence of inefficiency (Lovell, 1993; Ondrich and Ruggiero, 2001) since it assumes that all observations (𝑥_𝑖, 𝑞_𝑖) belong to the production set, so:

𝑃𝑟𝑜𝑏{(𝑥𝑖, 𝑞𝑖) ∈ S} = 1 ∀ 𝑖 = 1 … … … 𝑛

The main limitation of this approach is that any errors in measurement or stochastic errors are incorporated into the measurement of efficiency and, therefore, the event of sensitivity to ‘super-efficient’ outliers49 _{is standard. With regard to stochastic models,}

there might be noise in the data, i.e. some observations might lie outside 𝑆. The main weakness on this ground is the identification of noise from inefficiency.According to the stochastic approach, deviations from the production function have a bilateral explanation since they are not attributed solely to inefficiency. The objective of stochastic models is, therefore, to decompose the residual into two components: one stemming from inefficiency and one random. In practice, this implies an assumption of a specific distribution for each error component, which constitutes an important limitation. Stochastic methods are preferable on events of random shocks or measurement errors, giving them the comparative advantage of curtailing any distortions on the efficiency estimates; however, they may be affected by misspecification errors.

48 _{Various specifications can be used: e.g. the CBD, the flexible fixed quadratic function, the hybrid translog function, the CES}

function, etc.

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In particular, SFA allows the presence of noise, but it demands parametric restrictions on the shape of the frontier and on the data-generating process (DGP) in order to permit the identification of noise from inefficiency and the estimation of the frontier. The statistical approach assumes that inefficiencies (the difference between the firm’s observed output and the output that could be achieved if it was producing on the production frontier) follow a specific distribution (Førsund et al., 1980). However, any misspecification errors (either of the production function or of the inefficiency distribution) are incorporated in the efficiency measure (Lovell, 1993). Furthermore, the statistical, parametric approach is not easily applied in a situation in which there are multiple inputs andmultiple outputs (Johnes, 2004).

Both techniques have strengths and limitations, so an extensive review and updated presentation of both approaches is considered appropriate (Fried et al., 2006). In the next section, an overview of the DEA and SFA methodologies is discussed. A statistical approach that unifies the parametric and non-parametric approaches can be also found in Simar and Wilson (2006b).

From a theoretical perspective, the available methodologies for measuring efficiency vary from statistical to non-statistical, parametric to non-parametric, and deterministic to stochastic.50 Among the various versions in the literature, two are the most frequently used approaches: statistical parametric methods (deterministic or stochastic) and deterministic non-statistical non-parametric methods (Johnes, 2004). In the next section, the most common methods, including more recent developments in the context of efficiency measurement in HE, are discussed as an introduction to the whole scope of this thesis.