Data Envelopment Analysis (DEA)

The Seminal work of Charnes, Cooper, & Rhodes (1978) followed the works of Farrell (1957) and brought the DEA model to prominence. DEA is a non-parametric linear approach, which is capable of employing the use of multiple inputs and multiple outputs. It is a ‘data oriented’ methodology for evaluating the performance of Decision Making Units (DMUs). Avkiran (2011) refers to DEA as a non-parametric linear programming technique used to develop empirical production frontiers and to assess the performance of Decision Making Units (DMUs). For the sake of this study, banks are the Decision Making Units (Farrell, 1957; Charnes et al., 1978).

In the last three decades, the application of DEA to evaluate the performance of various entities and activities in different countries has become commonplace. For the reason that DEA requires little assumptions, it has opened up possibilities for use in instances that have previously appeared resistant to other approaches because of the multifarious nature of the relations between the numerous inputs and outputs involved in DMUs. The DEA approach has been utilized to provide insight into activities that have been examined by other methods. For instance, DEA is used in studies for benchmarking inefficient organizations against efficient ones. Since the seminal work of Charnes et al (1978), academics and researchers have recognised that the DEA is an excellent methodology for modelling organizational activities and operations for performance evaluation (Charnes et al., 1978; Cooper, Seiford, & Zhu, 2011; Cooper, Seiford, Tone, & Zhu, 2007).

Charnes et al (1978) referred to DEA as a mathematical programming model that is applied to observational data to provide novel ways of obtaining empirical estimates of relationship. The DEA methodology is directed to frontiers and not central tendencies. Cooper et al (2011) are of the view that rather than trying to fit a regression plane through the center of data as in statistical regressions, the DEA approach floats a piecewise linear surface to rest on observations. Owing to this assertion, the DEA methodology is able to uncover relationships that would remain hidden from other measures. The DEA technique is able to ascertain the efficiency and performance of organizations in a straightforward manner without the recourse to assumptions and variations that are usually required in other models such as linear and nonlinear regression models (Cooper et al, 2011).

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In evaluating the performance and efficiency of DMUs, the DEA approach does not assign any priori measures or information of relative importance to any input or output variable. The DEA approach calculates the distance of each DMU from the efficient frontier by determining by how much each DMU input should be reduced and each output should be increased for each DMU to reach the frontier. Thus, in line with the ‘Pareto Principle’, a DMU is said to be efficient (100%) if and only if none of its inputs or outputs can be improved without negatively affecting its other inputs or outputs. A DMU is regarded as been fully efficient (100%) on the premise of presented evidence if and only if the performances of similar DMUs do not indicate that some of their inputs or outputs can be improved without negatively affecting some of their other inputs or outputs. Likewise, inefficient DMUs have scores that are less than 100%. Therefore, in a standard DEA approach, a DMU is said to be efficient if its performance relative to other DMUs cannot be enhanced (Cook, Seiford, & Zhu, 2013; Paradi & Zhu, 2013).

Furthermore, the efficiency of a DMU is determined by its ability to transform inputs into desired outputs. Put differently, DEA assigns an efficiency score of 100% to an efficient DMU, and less than 100% to inefficient DMUs. A score less than 100% indicate that a linear combination of other DMUs from the sample could produce the same vector of outputs, using a small vector of inputs. The efficiency score reveals the radial distance from the estimated production frontier to the DMUs under evaluation, that is, the minimum proportional decrease in inputs yielding efficiency. Thus, DEA provides an efficiency rating (efficiency score) for efficient and inefficient DMUs. The efficiency score of DMUs is defined as:

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝑊𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑆𝑢𝑚 𝑜𝑓 𝑂𝑢𝑡𝑝𝑢𝑡𝑠 𝑊𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑆𝑢𝑚 𝑜𝑓 𝐼𝑛𝑝𝑢𝑡𝑠

In essence, the DEA approach considers how much total productivity can be improved, and ranks (efficiency scores) the productivity of individual DMUs (Ho & Zhu, 2004). Therefore, in this study, efficient DMBs will have efficiency scores of 100%, whereas inefficient DMBs will have less than 100%. However, efficiency scores are sensitive to changes in data and hinge greatly on the number and type of input and output factors considered (Casu & Molyneux, 2003). To that end, the specific input and output variables and the chosen period of evaluation will determine the efficiency scores of the examined Nigerian DMBs.

The advantages of the DEA model over other approaches for evaluating bank performance serves as the rationale for the choice of DEA for this study. The ability of the DEA approach

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to concurrently capture the relationship between multiple inputs and multiple outputs is a common motivation for employing the DEA to measure performance. Specifically, DEA is conditioned to a broader view of performance measurement and this is regarded as its superior advantage over traditional financial ratio analysis, which typically depends on two variables in a ratio (Avkiran, 2011).

One advantage of the DEA model is that it does not institute any biased structure on data when determining efficient organizations (in this case, banks). In other words, DEA approach does not assume a specific production technology or correspondence. It distinguishes inefficient banks from efficient ones by comparing them (that is comparing similar firms), rather than associating or correlating a bank’s performance with statistical averages (Paul & Kourouche, 2008).

In general, the non-parametric DEA approach is simple and easy to compute because it is not required to impose any assumptions about functional form, and it does not take into account the effect of random error and environmental noise (Sharma et al., 2013).

However, as with other techniques, the DEA has some disadvantages. One major downside is the need for homogeneity of the DMUs, which denotes that the analyzed institutions should use the same types of resources, generate similar classes of products and the conditions that contextualize the productive process should be identical (Cooper, Seiford & Tone, 2007). Another drawback is that the DEA analysis technique is a deterministic model. It assumes that resulting inefficiencies are based absolutely and entirely on the mismanagement of the DMUs, thereby ignoring any possibility of random influences. Likewise, the DEA techniques demand that extra care should be taken in the selection of variables, owing to the reason that there are no applicable tests for the selection and evaluation of significance (Coelli, Rao, O’Donnell & Battese, 2005; Fuentes, Fuster & Lillo-Banuls, 2016).

Notwithstanding the disadvantages of the DEA analysis technique, the advantages of employing it to evaluate the performance of banks are greater and more significant than any potential drawbacks. For that reason and due to its use in similar studies, as presented in the literature review chapter, this method was selected for this study.

108 4.7.2.1 DEA Efficiency Measures

Efficiency is referred to as a technical term, and it is a sign of efficacy. The DEA technique holds that any producing unit (bank) is said to be technically efficient when it can produce the maximum amount of output while using a given level of input. The non-parametric frontier requires that for the chosen inputs, the highest level of output realizable from those chosen inputs are in line with the available DMU data under alternative assumptions that could be made about production. For instance, under the DEA methodology, it might be assumed that banks operate under the constant returns to scale (CRS) or under variable returns to scale (VRS) (Cooper et al., 2007).

Farrell (1957) asserted that components of efficiency are technical efficiency and allocative efficiency. Farrell opined that a DMU is technically efficient when it is able to obtain maximum output from a given set of inputs. Whereas allocative efficiency indicates a DMU’s ability to utilize inputs in optimal proportions, given their respective prices and production technology (Coelli et al, 2005). Therefore, the technical efficiency of Nigerian DMBs are evaluated under the variable returns to scale (VRS) assumption.

To further drill down, Farell (1957) referred to ‘technical efficiency’ in terms of an organization’s success at producing the maximum amount of output, given a particular set of inputs. The DEA generated technical efficiency scores for a DMU (bank) is a relative measure indicating the particular DMU’s input-output conversion performance in comparison to what is possible according to the frontier. It is worth noting that the measure is specific to the sample and a DMU is only 100 per cent efficient in the event that there is no evidence of inefficiency when compared against all other DMUs. In line with assumptions of Farrell (1957), the production frontier is composed of the most efficient DMUs evaluated, whereas relatively inefficient DMUs fall below the frontier.

In this study, DMBs are benchmarked against the most efficient DMBs in the industry, that is, the relative efficiency of each DMB is measured against the frontier. The fully efficient DMBs form a best practice production frontier and are ‘benchmark’ peers for inefficient DMBs (Rouse, Harrison, & Chen, 2010). Therefore, a bank is technically efficient when it is able to convert multiple resources at its disposal to multiple financial services at a profit (Bhattacharyya & Kumbhakar, 1997). On the contrary, a DMB is said to be technically inefficient if it operates underneath the frontier.

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Furthermore, in evaluating technical efficiency, two extended DEA models can be operated. These are the CCR model and the BCC model. Brief discussions about the two models are presented below.

4.7.2.2 CCR Model

CCR Model denotes Charnes, Cooper, and Rhodes (1978), the originators of the model. Their work extended the works of Farrell (1957) by incorporating the concept of multiple inputs and multiple outputs. The CCR model is anchored on the assumptions of constant return to scale (CRS). The CCR model assumes that a constant relationship exists between inputs and outputs. For example, if one input yields three outputs, then two inputs would yield six outputs. Additionally, the CCR model does not discern between pure technical inefficiencies and inefficiencies due to variable returns. Hence, the assumption of constant returns to scale (CCR) is only justified when all DMUs in a sample are operating at an optimal scale. Thus if the CRS assumption is made when some of the DMUs are not operating at an optimal scale, the computed technical efficiency scores will be tainted with scale efficiency (SE) (Cooper et al., 2007; Lin et al., 2009; Sufian & Habibullah, 2012). A pictorial representation of the CCR model is shown in figure 4.3 below.

110 Figure 4.3: The CCR Production Frontier

Adapted from Cooper, Lawrence, & Tone (2006)

In relation to figure 4.3 above, it is assumed that there is only one input and one output. Centered on the constant returns to scale assumption (CCR model), the DMU at point C situated on the efficient (production) frontier is the sole CCR-efficient DMU for the reason that its efficiency score equal 100% or 1. The other DMUs (i.e. A, D, E, G, and N) are inefficient owing to their efficiency scores being less than 100% or 1. Moreover, the CCR model implies that no DMU situated under the frontier (straight line) is more efficient than DMU C. In, like manner, no input/output combination in the inefficient DMUs could produce efficiency scores higher than that of DMU-C.

4.7.2.3 BCC Model

Likewise, the BCC Model, Banker, Charnes, Cooper (1984) broadened the previous works of Farrell (1957) and the 1978 CCR model. They were of the view that if efficiency is

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hypothetically measured under the CRS assumption, a DMU may be inefficient and might not have allocative efficiency, scale efficiency, and technical efficiency. Consequently, Banker et al (1984) introduced the variable returns to scale (VRS) hypothesis that breaks down technical efficiency (TE) into pure technical efficiency (PTE) and scale efficiency (SE), and it is called the BCC model. Theoretically, the VRS assumption offers the measurement of pure technical efficiency (PTE), and it is the measurement of technical efficiency (TE) without the effects of scale efficiency (SE). The BCC model assumes that a variable relationship subsists between inputs and outputs. Therefore, if a proportional increase (or decrease) in inputs transpires into a different proportional increase (or decrease) in outputs, it can be opined that variable returns to scale are present (Cooper et al., 2007). For instance, if one input yields three outputs, but two inputs yields five outputs, it can be suggested that variable returns to scale subsist.

Furthermore, in view of the BCC model (variable returns to scale assumption), the best practice frontier encases the inefficient points more tightly. Hence, the distance between the inefficient DMUs and the efficient DMUs (best practice frontier) is less. As such, the efficiency scores under the BCC model are usually higher than the CCR efficiency scores (Cooper et al., 2007).

112 Figure 4.4: The BCC Production Frontier

Adapted from Cooper et al. (2006)

Assuming the case of one input to one output, the production frontier of the BCC model depicted in figure 4.4 shows three efficient DMUs, which are DMUs A, C, and E. The line section that connects point A and point C indicates the increasing returns to scale (IRS) portion of the efficiency frontier, whereas the line segment that joins point C to E shows the decreasing returns to scale (DRS) portion of the efficiency frontier. And just like figure 4.3, C indicates the constant returns to scale (CRS) portion of the efficiency frontier. Apart from showing DMUs at increasing, decreasing, and constant returns to scale, figure 4.4, indicates that more DMUs appear efficient under the BCC model.

Therefore, based on the above theoretical descriptions, the mathematical depictions of the CCR and BCC models are presented below, as the variant of the DEA analysis (DEA Window Analysis) adopted for this study is discussed. However, even though both the CCR and BCC efficiency scores are calculated, this study relies largely on the BCC model for the analysis of efficiency. The BCC model is specifically adopted because it is viewed as an

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upgrade on the CCR model. The CCR model is a specific type of the BCC model (Toloo & Nalchigar, 2009) and as such the BCC model is utilised in this study due to its capacity to take into consideration scale effects in order to ascertain the most productive scale size for each DMU. Put differently, by adjusting for “scale effects”, the BCC model is in a better position to evaluate “pure” technical efficiency. Likewise, the BCC model is adopted because it ensures the discretization of technical efficiency from the effects of scale efficiency (Sahin, Gokdemir, & Ozturk, 2016). To that end, the BCC model is better than the CCR model in relation to providing policy recommendations, like the introduction of performance measures to embolden operations at the most productive scale size or the fine-tuning of performance outcomes in order to be able to control for scale differences (Alrashidi, 2015). More so, DMBs are determined to be more efficient under the BCC model which is based on the variable returns to scale assumption results in higher efficiency scores than under the CCR model. As such, DMBs may be efficient under the BCC model but not under the CCR model. Equally, an inefficient DMB (DMU) may have different efficiency scores depending on the model adopted.

4.7.2.4 DEA Window Analysis

As a model, window analysis tries to offer a more comprehensive treatment to the evaluation of efficiency changes over a time period (Charnes, Clark, Cooper, & Golany, 1985; Cook & Seiford, 2009). The DEA window analysis technique is based on the principle of moving averages (Cooper et al., 2007; Gu & Yue, 2011; Yue, 1992). A DMU in each different period is evaluated as if it were a different DMU in a different window. Explicitly, the performance of a DMU in a particular period (window) is compared against its own performance in other periods (windows) in addition to the performance of the other DMUs (Cooper et al., 2011). Webb, Bryce, & Watson (2010) in line with Webb (2003) are of the view that the DEA window analysis technique is of advantage when examining the performance of the entire banking sector as well as units within individual organizations. They enumerated the advantages that also serve as motivation for the adoption of the DEA window analysis technique as follows:

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 The technique makes it easy to analyse the performance of DMUs (banks) over a specified period of time taking into consideration cost/expenses and income/profits.  The technique can also be utilized to examine stability and other properties of

efficiency and performance across as well as within windows.

 DEA is at most useful when analysing closely homogenous organizations that operate in closely homogenous markets. For instance, comparing a bank in 2000 with another one in 2013 could render relative results meaningless. On this account, analysing banks in a three-year period “windows” reduces the problem and may be considered an improvement on the traditional DEA panel model.

 Each bank in the sample is treated as a different bank in a different period. This treatment increases the number of data points. Put differently, each bank in a different period is evaluated as if it was a different bank (independent) but remains comparable in the same period (Repkova, 2014). Therefore, the problem of small sample sizes is solved with this approach.

 The performance of a bank being analysed by the DEA windows technique in a period can be compared to itself and other banks over the course of time (Asmild, Paradi, Aggarwall, & Schaffnit, 2004; Repkova, 2014).

The window analysis has been only used in a handful of studies to ascertain the performance of banking institutions over a time period (examples of studies include Asmild et al., (2004) – Canadian banking industry, Gu & Yue (2011) – Chinese Listed banks, Sufian (2007) – Singapore commercial banking group, Kisielewska, Guzowska, Nellis, & Zarzecki, (2007) – Polish banking industry, Repkova (2014) – Czech banking sector).

Therefore, the performances of Nigerian DMBs have been evaluated using the DEA window analysis based on the advantages enumerated above and following the listed adaptations of the approach in various banking industries.

Consequently, this study has considered the following formulas in line with Asmild et al., (2004), Gu & Yue (2011)and Repkova (2014), where N DMUs (n = 1, 2, …, N) observed in T (t = 1, 2, …, T) periods using r inputs to produce s outputs. Let𝐷𝑀𝑈_𝑛𝑡 _{represents a 𝐷𝑀𝑈}

𝑛 in period t with a r dimensional input vector 𝑥_𝑛𝑡 = (𝑥_𝑛1𝑡, 𝑥_𝑛2𝑡, … 𝑥_𝑛𝑟𝑡) and s dimensional input vector 𝑦_𝑛𝑡 = (𝑦_𝑛1𝑡, 𝑦_𝑛2𝑡, … 𝑦_𝑛𝑠𝑡). If windows start time k (1 ≤ k ≤ T) with window width w (1 ≤ w ≤ t - k), then the metric of inputs is given as follows:

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𝑋𝑘𝑤 = (𝑥1𝑘, 𝑥2𝑘… , 𝑥𝑁𝑘+1, 𝑥2𝑘+1, … , 𝑥𝑁𝑘+1, … , 𝑥1𝑘+𝑤, 𝑥2𝑘+𝑤, … , 𝑥𝑁𝑘+𝑤) And the metric of outputs as:

𝑌_𝑘𝑤 = (𝑦₁𝑘_{, 𝑦}

2𝑘… , 𝑦𝑁𝑘+1, 𝑦2𝑘+1, … , 𝑦𝑁𝑘+1, … , 𝑦1𝑘+𝑤, 𝑦2𝑘+𝑤, … , 𝑦𝑁𝑘+𝑤)

The CCR model (constant returns to scales, CRS) of DEA window problem for 𝐷𝑀𝑈𝑛𝑡 is given by solving the following linear program:

min 𝜃

Subject to 𝜃𝑋_𝑡− 𝜆𝑋_𝑘𝑤≥ 0

𝜆𝑌𝑘𝑤− 𝑌𝑡≥ 0

𝜆𝑛 ≥ 0 (𝑛 = 1, 2, … , 𝑁 × 𝑤).

The BCC model -variable returns to scales (VRS) formulation can be obtained by adding the restriction ∑𝑁𝑛=1𝜆𝑛 = 1 (Banker et al., 1984). The objective value of the CCR model is given