# Assessing efficiency profiles of UK commercial banks: a DEA analysis with regression-based feedback

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## Abstract

Data envelopment analysis (DEA) has witnessed increasing popularity in banking studies since 1985. In this paper, we propose a new DEA-based analysis framework with a regression-based feedback mechanism, where regression analysis provides DEA with feedback that informs about the relevance of the inputs and the outputs chosen by the analyst. Unlike previous studies, the DEA models used within the proposed framework could use both inputs and outputs, only inputs, or only outputs. So far, the UK banking sector remains relatively under researched despite its crucial importance to the UK economy. We use the proposed framework to address several research questions related to both the efficiency of the UK commercial banking sector and DEA analyses with and without regression-based feedback. Empirical results suggest that, on average, the commercial banks operating in the UK—whether domestic or foreign—are yet to achieve acceptable levels of overall technical efficiency, pure technical efficiency, and scale efficiency. On the other hand, DEA analyses with and without a linear regression-based feedback mechanism seem to provide consistent findings; however, in general DEA analyses without feedback tend to over- or under-estimate efficiency scores depending on the orientation of the analyses. Furthermore, in general, a linear regression-based feedback mechanism proves effective at improving discrimination in DEA analyses unless the initial choice of inputs and outputs is well informed.

## Keywords

Data envelopment analysis Efficiency UK commercial banks DEA models without explicit inputs DEA models without explicit outputs## 1 Introduction

The banking sector plays a crucial socio-economic role at the regional, national and international levels. Banks are at the heart of financial systems in that they act as financial intermediaries; to be more specific, they borrow money by accepting deposits and issuing debt securities, and lend money both directly to their customers and indirectly through capital markets by investing in debt securities. Banks play an important role in money supply and the efficient allocation of financial resources in an economy. Banks make profits in exchange for their services including risk management. Nowadays, banks have a diversified portfolio of activities that range from personal, corporate and investment banking to trading of currency, commodities, and financial securities on stock markets. Because of the crucial importance of banking systems to the economy and the financial risks they face, banks are required to comply with both national and international regulations, and their performance is constantly monitored by both regulatory bodies and investors. In fact, poor performance often leads to distress which might lead to bankruptcy under some circumstances along with substantial financial, economic and social undesirable consequences.

In this paper, we assess the efficiency profiles of UK commercial banks. The UK banking system has specific distinctive features which distinguish it from other banking systems. In fact, the UK banking system is relatively big compared to the banking systems of other countries. Its size is the result of a combination of factors including its history, as the UK has been a financial centre since the eighteenth century. As a financial hub, the UK banking system offers the benefits of clustering such as higher productivity and wage. The robustness of the UK legal and regulatory structure along with the implicit government subsidy and its openness to trade and capital flow seem to provide attractive incentives and flexibility for foreign banks to do business in the UK and for domestic banks to do business abroad. As a result of some of these features, UK has the largest banking sector on a residency basis compared to US, Japan and the ten largest EU Economies with foreign banks on a residency basis, from 56 different countries, owning approximately 50% of the UK banking sector assets. In addition, nearly 1/5 of the global banking activity is booked in the UK. The contribution of foreign banks to the UK banking system and its economy is substantial as suggested by a growth from around 100% of nominal GDP in 1975 to around 450% of nominal GDP in 2013. This growth of 350% is due to the relatively large assets and liabilities account of foreign banks residing in the UK and representing more than four times the median figure for OECD countries. Last, but not least, the international nature of the UK banking system—foreign banks have a large operation in the UK and UK banks have a large operation abroad—along with the continuous reengineering of UK banking regulations enhances its banking system resilience. For more details on the features of the UK banking system, we refer the reader to the Bank of England publications (e.g., Davies et al. 2010; Bush et al. 2014; Burrows et al. 2015).

In this paper, we propose a revised methodological framework; namely, Data Envelopment Analysis (DEA) with a regression-based feedback mechanism along with new DEA models (i.e., DEA models without explicit inputs or outputs), and use it to assess the efficiency profiles of UK commercial banks. The proposed methodology is useful for variable selection especially when the lack of discrimination is a concern. It is used to address three research questions: (1) how do DEA analyses with and without a linear regression-based feedback mechanism compare? (2) how effective is a linear regression-based feedback mechanism in improving discrimination in DEA? and (3) when a feedback mechanism is used to inform the researcher or analyst about the relevance of the choices of inputs and outputs in a DEA analysis, how do radial models (e.g., CCR, BCC) and non-radial models (e.g., SBM) compare? From a practical perspective, we are questioning whether the efficiency determinants identified in previous studies (i.e., inputs and outputs in DEA analysis under the intermediation approach) are actually (empirically) contributing to efficiency or not and whether methodological choices (e.g., choice of DEA model to use, choice of metrics or proxies of performance criteria) have something to do with it. For the sake of completeness and update of analyses, we also address two conventional research questions: (4) are UK commercial banks managed efficiently? and (5) what are the drivers of UK Commercial Banks’ efficiency? However, unlike previous contributions, which focus on the few largest UK commercial banks, these last two research questions are addressed for the whole UK commercial banking system. In our application, it turned out that the UK banking dataset we used requires and justifies the use of DEA models without explicit inputs or outputs when variable selection is informed by a feedback mechanism. Note that the feedback mechanism does not need to be regression-based.

The remainder of this paper is organised as follows. In Sect. 2, we classify the literature on efficiency assessment in banking according to several criteria and critically discuss some of the choices made in the literature. In Sect. 3, we propose a DEA-based sequential decision making process with regression-based feedback adjustment mechanisms along with new DEA models. In Sect. 4, we summarise our empirical investigation and its findings. Finally, Sect. 5 concludes the paper.

## 2 Landscape of research on efficiency assessment in banking

Research papers on efficiency assessment in banking could be classified into several categories depending on one’s choice of the classification criterion. In this paper, we use three criteria to classify the literature on static DEA analyses; namely, type of analysis, type of approach, and country of focus.

*Single Stage Analysis*—see Fig. 1 for a flow chart of a typical single stage analysis (e.g., Ferrier and Lovell 1990; Elyasiani and Mehdian 1992; Yue 1992; Grabowski et al. 1993; Fukuyama 1993; Zaim 1995; Pastor et al. 1997; Barr et al. 1993; Lozano-Vivas et al. 2002).

*Two*-

*Stage Analysis*to overcome environment bias—see Fig. 2 for a flow chart of a typical two-stage analysis (e.g., Rangan et al. 1988; Elyasiani and Mehdian 1990; Aly et al. 1990; Favero and Papi 1995; Miller and Noulas 1996; Bhattacharyya et al. 1997; Chen 1998; Chu and Lim 1998; Barr et al. 1994; Barr and Siems 1997; Pasiouras 2008; Wanke and Barros 2014; Kwon and Lee 2015; Du et al. 2018). Note however that the efficiency scores obtained with a two-stage analysis would still be environmentally-biased, because the inputs and outputs used in the first stage are not adjusted for environment. In order to properly control for these environmental variables, one could use a three-stage methodology. Finally the third category of studies uses

*Three*-

*Stage Analysis—*see Fig. 3 for a flow chart of a typical three-stage analysis (e.g., Pastor 2002; Drake et al. 2006; Liu and Tone 2008; Avkiran 2009; Liu 2018).

With respect to the type of assessment perspective, which drives the choices of inputs and outputs, we classify the literature into six categories; namely, the *intermediation approach* or perspective (e.g., Rangan et al. 1988; Ferrier and Lovell 1990; Charnes et al. 1990; Elyasiani and Mehdian 1990, 1992; Aly et al. 1990; Yue 1992; Grabowski et al. 1993; Fukuyama 1993; Zaim 1995; Favero and Papi 1995; Miller and Noulas 1996; Taylor et al. 1997; Chen 1998; Drake et al. 2006; Liu 2018), the *asset approach* (e.g., Favero and Papi 1995), the *production approach* (e.g., Drake et al. 2006; Liu and Tone 2008), the *value added approach* (e.g., Bhattacharyya et al. 1997; Pastor et al. 1997; Chu and Lim 1998; Pastor 2002; Das and Ghosh 2006), the *profit*-*oriented approach* (e.g., Berger and Mester 2003; Drake et al. 2006; Liu and Tone 2008), and the *user cost approach* (e.g., Hancock 1985a, b; Fixler and Zieschang 1992).

Recall that the intermediation approach considers banks as intermediation agents who collect funds and provide loans and other assets. The asset approach is a variant of the intermediation approach which consider banks as financial intermediaries between liability holders and those who receive bank funds. The production approach considers banks as production units that transform inputs into outputs, or producers of deposit accounts and loan services. Under the value added approach, the share of value added guides the choice of inputs and outputs. Under the profit approach, profit guides the choice of inputs and outputs. Finally, under the user cost approach, the net contribution to bank revenue determines the nature of inputs and outputs.

As to the country of focus, the literature could be divided into two main categories. The first category consists of *single country focused studies* and covers US Banks (Rangan et al. 1988; Ferrier and Lovell 1990; Elyasiani and Mehdian 1990, 1992; Aly et al. 1990; Yue 1992; Miller and Noulas 1996; Kwon and Lee 2015), UK Banks (Drake 2001; Webb 2003; Webb et al. 2010; Tanna et al. 2011), Italian Banks (Favero and Papi 1995), Turkish Banks (Zaim 1995; Kutlar et al. 2017), Japanese Banks (Fukuyama 1993; Liu and Tone 2008), Taiwanese Banks (Chen 1998; Liu 2018), Hong Kong Banks (Drake et al. 2006), Singaporean Banks (Chu and Lim 1998), Indian Banks (Bhattacharyya et al. 1997), Mozambique Banks (Wanke et al. 2016), and Korean Banks (Lee et al. 2017). The second category consists of *multi*-*country focused studies* and covers banks in several countries such as US, Australian, New Zealand, Austrian, Spanish, German, UK, Italian, Belgian, French, Danish, Luxembourg, Dutch, and Portuguese Banks (e.g., Pastor et al. 1997; Pastor 2002; Lozano-Vivas et al. 2002; Casu and Molyneux 2003; Pasiouras 2008; Avkiran 2009).

To conclude this section, it is worthy to mention that *single country focused studies* on banks using static DEA analyses (Drake 2001; Webb 2003; Webb et al. 2010; Tanna et al. 2011) focused exclusively on the few largest commercial banks in the UK, whereas this paper considers the whole UK commercial banking sector. We also would like to point out that other DEA methodologies have been used to assess the efficiency of banks; for example, Network DEA (e.g., Matthews 2013; Grigoroudis et al. 2013; Akther et al. 2013; Fukuyama and Matousek 2017; Gulati and Kumar 2017), Network DEA with undesirable variables (e.g., An et al. 2015; Liu et al. 2015), Dynamic DEA (e.g., Avkiran and Goto 2011; Fukuyama and Weber 2015, 2017), Dynamic Network DEA (e.g., Avkiran 2015; Chao et al. 2015; Fukuyama and Weber 2015, 2017; Zha et al. 2016; Wu et al. 2016; Fukuyama and Weber 2017b), Fuzzy DEA (e.g., Wang et al. 2014; Wanke et al. 2016; Hatami-Marbini et al. 2017), DEA with Bootstrapping (e.g., Ferrier and Hirschberg 1997), Fuzzy DEA with Bootstrapping (e.g., Wanke et al. 2016), and Stochastic DEA (e.g., Kao and Liu 2009). For a recent survey, we refer the reader to Kaffash and Marra (2017).

In the next section, we propose a DEA analysis with a regression-based feedback mechanism along with new DEA models to assess the efficiency profiles of banks, which we apply in the following section to the UK banking sector.

## 3 A DEA analysis with regression-based feedback mechanism

In this section, we shall describe the methodology and models we propose for assessing the efficiency profile of UK commercial banks. The proposed methodology is a sequential decision making process with a feedback adjustment mechanism; namely, a DEA-based analysis with a regression-based feedback mechanism.

DEA was first proposed by Charnes et al. (1978) as a frontier-based non-parametric approach to the relative performance evaluation of a set of \( n \) entities commonly referred to as decision making units (*DMUs*), where *DMUs* are viewed as production systems that make use of the same set of \( m \) inputs to produce the same set of \( s \) outputs. For each *DMU*, lot sizing decisions of both inputs and outputs are made by its management; that is, the quantity \( x_{i,k} \) of input \( i \) (\( i = 1, \ldots ,m) \) used by \( DMU_{k} \) (\( k = 1, \ldots ,n) \) and the quantity \( y_{r,k} \) of output \( r \) (\( r = 1, \ldots ,s \)) produced by \( DMU_{k} \) (\( k = 1, \ldots ,n) \). Unlike parametric methodologies, DEA does not require an explicit specification of the form of the production function, or equivalently the relationship between inputs and outputs. DEA is a mathematical programming-based methodology—for a detailed text on DEA, we refer the reader to Cooper et al. (2007).

CCR and BCC models

Formulation | Description |
---|---|

| |

\( \theta_{k} \) | \( \theta_{k} \) is to be minimised or maximised depending on whether the analysis is input-oriented or output-oriented |

| |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{i,j} \le \theta_{k} \cdot x_{i,k} , \,\,i = 1, \ldots ,m \) OR \( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{i,j} \le x_{i,k} ,\,\, i = 1, \ldots ,m \) | For each input \( i \) (\( i = 1, \ldots ,m \)), the amount used by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should at most be equal to the amount used by \( DMU_{k} \) adjusted for the degree of technical efficiency of \( DMU_{k} \) or not depending on whether the analysis is input-oriented or not |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{r,j} \ge \theta_{k} \cdot y_{r,k} , \,\,r = 1, \ldots ,s \) OR \( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{r,j} \ge y_{r,k} , \,\,r = 1, \ldots ,s \) | For each output \( r \) (\( r = 1, \ldots ,s \)), the amount produced by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should be at least as large as the amount produced by \( DMU_{k} \) adjusted for the degree of technical efficiency of \( DMU_{k} \) or not depending on whether the analysis is output-oriented or not |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} = 1 \) | The technology is required to be convex in BCC models. This constraint is relaxed in CCR models |

\( \lambda_{j} \ge 0,\,\,j = 1, \ldots ,n \) \( \theta_{k} \) unrestricted | Other requirements including non-negativity |

BCC models without explicit inputs or outputs

Formulation | Description |
---|---|

| |

\( \theta_{k} \) | \( \theta_{k} \) is to be minimised or maximised depending on whether the analysis is without explicit output or without explicit inputs |

| |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{i,j} \le \theta_{k} \cdot x_{i,k} ,\,\, i = 1, \ldots ,m \) OR \( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{r,j} \ge \theta_{k} \cdot y_{r,k} , \,\,r = 1, \ldots ,s \) | For each input \( i \) (\( i = 1, \ldots ,m \)), the amount used by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should at most be equal to the amount used by \( DMU_{k} \) adjusted for the degree of technical efficiency of \( DMU_{k} \), or for each output \( r \) (\( r = 1, \ldots ,s \)), the amount produced by \( DMU_{k} \)’s “ideal” benchmark should be at least as large as the amount produced by \( DMU_{k} \) adjusted for the degree of technical efficiency of \( DMU_{k} \) depending on whether the analysis is without explicit output or without explicit inputs |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} = 1 \) | The technology is convex |

\( \lambda_{j} \ge 0,\,\,j = 1, \ldots ,n \) \( \theta_{k} \) unrestricted | Other requirements including non-negativity |

SBM models

Formulation | Description |
---|---|

| |

\( \rho_{k} = {{\left( {1 - \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} \frac{{s_{i,k}^{ - } }}{{x_{i,k} }}} \right)} \mathord{\left/ {\vphantom {{\left( {1 - \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} \frac{{s_{i,k}^{ - } }}{{x_{i,k} }}} \right)} {\left( {1 + \frac{1}{s}\mathop \sum \limits_{r = 1}^{s} \frac{{s_{r,k}^{ + } }}{{y_{r,k} }}} \right)}}} \right. \kern-0pt} {\left( {1 + \frac{1}{s}\mathop \sum \limits_{r = 1}^{s} \frac{{s_{r,k}^{ + } }}{{y_{r,k} }}} \right)}} \) OR \( \rho_{k} = 1 - \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} \frac{{s_{i,k}^{ - } }}{{x_{i,k} }} \) OR \( \rho _{k} = {1 \mathord{\left/ {\vphantom {1 {\left( {1 + \frac{1}{s}\sum\limits_{{r = 1}}^{s} {\frac{{s_{{r,k}}^{ + } }}{{y_{{r,k}} }}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \frac{1}{s}\sum\limits_{{r = 1}}^{s} {\frac{{s_{{r,k}}^{ + } }}{{y_{{r,k}} }}} } \right)}} \) | One of these \( \rho_{k} \) formulations is to be minimised depending on whether the analysis is non-oriented, input-oriented, or output-oriented |

| |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{i,j} + s_{i,k}^{ - } = x_{i,k} , \,\,i = 1, \ldots ,m \) | For each input \( i \) (\( i = 1, \ldots ,m \)), the amount used by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should at most be equal to the amount used by \( DMU_{k} \) |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{r,j} - s_{r,k}^{ + } = y_{r,k} , \,\,r = 1, \ldots ,s \) | For each output \( r \) (\( r = 1, \ldots ,s \)), the amount produced by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should be at least as large as the amount produced by \( DMU_{k} \) |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} = 1 \) | This constraint requires the technology to be convex; however, it could be relaxed |

\( \lambda_{j} \ge 0,\,\,j = 1, \ldots ,n \) \( s_{i,k}^{ - } \ge 0,\,\, i = 1, \ldots ,m \) \( s_{r,k}^{ + } \ge 0, \,\,r = 1, \ldots ,s \) | Non-negativity requirements |

SBM models without explicit inputs or outputs

Formulation | Description |
---|---|

| |

\( \rho_{k} = 1 - \frac{1}{m}\mathop \sum \limits_{i = 1}^{m} \frac{{s_{i,k}^{ - } }}{{x_{i,k} }} \) OR \( \rho_{k} = 1/\left( {1 + \frac{1}{s}\mathop \sum \limits_{r = 1}^{s} \frac{{s_{r,k}^{ + } }}{{y_{r,k} }}} \right) \) | One of these \( \rho_{k} \) formulations is to be minimised depending on whether the analysis is without explicit output or without explicit inputs |

| |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{i,j} + s_{i,k}^{ - } = x_{i,k} ,\,\, i = 1, \ldots ,m \) OR \( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{r,j} - s_{r,k}^{ + } = y_{r,k} ,\,\, r = 1, \ldots ,s \) | For each input \( i \) (\( i = 1, \ldots ,m \)), the amount used by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should at most be equal to the amount used by \( DMU_{k} \), or for each output \( r \) (\( r = 1, \ldots ,s \)), the amount produced by \( DMU_{k} \)’s “ideal” benchmark; i.e., its projection on the efficiency frontier, should be at least as large as the amount produced by \( DMU_{k} \) depending on whether the analysis is without explicit output or without explicit inputs |

\( \mathop \sum \limits_{j = 1}^{n} \lambda_{j} = 1 \) | This constraint requires the technology to be convex; however, it could be relaxed |

\( \lambda_{j} \ge 0,\,\,j = 1, \ldots ,n \) \( s_{i,k}^{ - } \ge 0,\,\, i = 1, \ldots ,m \) OR \( s_{r,k}^{ + } \ge 0, \,\,r = 1, \ldots ,s \) | The weights \( \lambda_{j} \) s are required to be non-negative as well as the relevant slacks depending on whether the analysis is without explicit output or without explicit inputs |

For our banking application, inputs and outputs are supplied from banks’ financial statements (i.e., balance sheet and income statement). These inputs and outputs are environment-independent because the study is performed on UK banks only, on one hand, and we do not test any specific event-related hypotheses, on the other hand. Then, the DEA scores are regressed on the initial inputs and output supplied by the analyst to find out whether they are statistically significant or not; that is, whether they drive the efficiency scores or not—any inputs or outputs which are not relevant (i.e., not statistically significant) are then discarded and the DEA analysis with both inputs and outputs is performed with a reduced set of inputs and outputs. When regression analysis suggests that none of the inputs or none of the outputs chosen by the analyst are relevant, DEA analysis without explicit inputs or without explicit outputs is performed using the relevant DEA models mentioned above. In sum, regression analysis provides DEA with feedback that informs DEA about the relevance of the inputs and outputs chosen by the analyst.

Before we proceed with the application of the proposed DEA analysis with regression-based feedback, we hereafter position our contribution with respect to the literature on variable selection in DEA. So far, such literature could be divided into (1) Judgemental Screening or Expert Opinions such as Fuzzy Delphi Method (Arsad et al. 2017); (2) Statistical Tests and Bootstrapping (e.g., Banker 1996; Olson et al. 1980; Simar and Wilson 2001; Nataraja and Johnson 2011); (3) Dimensionality Reduction Techniques such as Principal Component Analysis (Ueda and Hoshiai 1997; Adler and Golany 2001, 2002; Cinca and Molinero 2004; Adler and Yazhemsky 2010; Nataraja and Johnson 2011); and (4) Variable Reduction Techniques such as Correlation Analysis and Variants (Nunamaker 1985; Jenkins and Anderson 2003; Eskelinen 2017; Adler and Yazhemsky 2010), Copula (Alpay and Akturk Hayat 2017), Efficiency Contribution Measure (Pastor 2002; Nataraja and Johnson 2011; Eskelinen 2017), Stepwise Procedures (Norman and Stoker 1991; Sigala et al. 2004; Wagner and Shimshak 2007; Subramanyam 2016; Sharma and Yu 2015), Akaike’s Information Criterion rule (Li et al. 2017), Directional Technology Distance Function (Guarda et al. 2013), Regression Analysis (Lewin et al. 1982; Fanchon 2003; Ruggiero 2005; Luo et al. 2012; Golany and Roll 1989); Decision Tree Analysis (Lim 2008; Jain et al. 2016), and Genetic Algorithms (Madhanagopal and Chandrasekaran 2014). Our contribution falls into the subcategory of Regression Analysis; however, unlike previous contributions, ours use regression analysis within a feedback mechanism and allows for no-inputs or no-outputs situations.

In the next section, we shall apply the proposed methodology to assess the efficiency profile of UK commercial banks.

## 4 Empirical study

In our empirical investigation, we used all UK commercial banks for which data is available from Bankscope, provided by Bureau van Dijk, over a period of 29 years; namely, 1987–2015. Our dataset includes 109 commercial banks and consists of a total number of 1171 bank-year observations or decision making units.

Choices of measures of inputs and outputs for DEA analyses

Setup | Inputs | Output | |||||||
---|---|---|---|---|---|---|---|---|---|

Personnel expenses | Fixed assets | Equity | Total interest expense | Total expenses not including personnel expense | Gross loans | Total customer deposits | Gross interest and dividend income | Total income | |

1 | x | x | x | x | x | x | |||

2 | x | x | x | x | x | x | |||

3 | x | x | x | x | x | x | |||

4 | x | x | x | x | x | x |

Statistics on all UK commercial banks in our dataset

Statistics | Inputs | Output | |||||||
---|---|---|---|---|---|---|---|---|---|

Personnel expenses | Fixed assets | Equity | Total interest expense | Total expenses not including personnel expense | Gross loans | Total customer deposits | Gross interest and dividend income | Total income | |

Minimum | 145 | 3 | 496 | 17 | 176 | 155 | 2 | 156 | 592 |

1st quartile | 3084 | 952 | 37,825 | 4430 | 7824 | 79,031 | 88,376 | 11,060 | 29,423 |

2nd quartile | 8900 | 5184 | 184,532 | 27,658 | 45,300 | 559,289 | 566,231 | 57,864 | 151,137 |

3rd quartile | 102,738 | 44,899 | 788,590 | 182,050 | 339,226 | 3,971,509 | 3,188,962 | 352,378 | 1,272,411 |

Maximum | 20,018,117 | 28,031,677 | 104,117,263 | 57,559,609 | 63,550,393 | 1,113,372,106 | 887,561,640 | 73,422,162 | 164,071,334 |

Mean | 587,769 | 523,085 | 4,136,704 | 1,279,699 | 1,907,944 | 42,454,571 | 36,800,174 | 2,295,624 | 6,341,943 |

SD | 2,207,552 | 2,217,200 | 14,238,836 | 4,567,619 | 6,356,927 | 143,155,173 | 123,979,458 | 7,477,161 | 20,947,952 |

Statistics on the largest UK commercial banks in our dataset

Statistics | Inputs | Output | |||||||
---|---|---|---|---|---|---|---|---|---|

Personnel expenses | Fixed assets | Equity | Total interest expense | Total expenses not including personnel expense | Gross loans | Total customer deposits | Gross interest and dividend income | Total income | |

Minimum | 399,483 | 406,371 | 4,013,775 | 302,794 | 769,471 | 51,423,159 | 28,122,407 | 1,204,932 | 3,390,042 |

1st quartile | 2,603,403 | 2,244,140 | 21,507,171 | 5,308,666 | 10,213,000 | 275,469,780 | 259,803,728 | 13,710,215 | 40,757,916 |

2nd quartile | 5,818,473 | 4,407,448 | 33,460,279 | 10,110,862 | 16,524,428 | 404,934,564 | 372,874,000 | 20,017,169 | 57,496,490 |

3rd quartile | 8,911,865 | 7,688,636 | 63,639,245 | 15,534,106 | 25,620,037 | 680,819,266 | 547,769,283 | 30,769,484 | 93,498,785 |

Maximum | 20,018,117 | 28,031,677 | 104,117,263 | 57,559,609 | 63,550,393 | 1,113,372,106 | 887,561,640 | 73,422,162 | 164,071,334 |

Mean | 6,378,980 | 5,842,691 | 42,820,658 | 12,205,320 | 18,872,460 | 441,836,080 | 393,482,462 | 22,820,758 | 64,658,760 |

SD | 4,833,962 | 5,421,576 | 28,886,641 | 10,099,372 | 12,450,209 | 262,830,397 | 213,471,361 | 13,940,763 | 39,175,418 |

Additional analyses of the UK commercial banking sector

# Banks | Percentage (%) | Personnel expenses | Percentage (%) | Fixed assets | Percentage (%) | Gross loans | Percentage (%) | Total customer deposits | Percentage (%) | Total income | |
---|---|---|---|---|---|---|---|---|---|---|---|

All commercial banks | 109 | 100 | 688,278,073 | 100 | 612,533,104 | 100 | 49,714,302,196 | 100 | 43,093,003,273 | 100 | 7,426,415,455 |

5 Largest UK banking groups | 8 | 7 | 606,003,076 | 88 | 555,055,662 | 91 | 41,974,427,620 | 84 | 37,380,833,843 | 87 | 6,142,582,203 |

Local banks | 68 | 62 | 644,949,192 | 94 | 577,851,194 | 94 | 45,167,923,292 | 91 | 39,630,979,746 | 92 | 6,793,674,300 |

Foreign banks | 41 | 38 | 43,328,882 | 6 | 34,681,910 | 6 | 4,546,378,903 | 9 | 3,462,023,527 | 8 | 632,741,155 |

Local banks–largest banks | 60 | 55 | 38,946,115 | 6 | 22,795,532 | 4 | 3,193,495,672 | 6 | 2,250,145,903 | 5 | 651,092,097 |

DEA analyses of the UK commercial banking sector, as represented by the 109 commercial banks in our dataset, are summarised as follows.

Summary statistics on input-oriented scores of overall technical, pure technical and scale efficiencies without regression feedback

Statistics on CCR-IO scores | Statistics on BCC-IO scores | Statistics on SE-IO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0318 | 0.033 | 0.0357 | 0.0632 | 0.044 | 0.0381 | 0.0358 | 0.0637 | 0.0641 | 0.2525 | 0.041 | 0.263 |

1st quartile | 0.1937 | 0.3091 | 0.2908 | 0.4773 | 0.2845 | 0.3815 | 0.3904 | 0.5557 | 0.4721 | 0.6363 | 0.6609 | 0.8195 |

2nd quartile | 0.2612 | 0.3902 | 0.418 | 0.6057 | 0.4423 | 0.5129 | 0.5441 | 0.6894 | 0.689 | 0.8529 | 0.854 | 0.9431 |

3rd quartile | 0.3729 | 0.4999 | 0.5875 | 0.7514 | 0.7094 | 0.7578 | 0.7944 | 0.867 | 0.895 | 0.9665 | 0.9674 | 0.9887 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.3144 | 0.4297 | 0.4512 | 0.6119 | 0.5132 | 0.5733 | 0.591 | 0.6976 | 0.667 | 0.7846 | 0.7847 | 0.8796 |

SD | 0.1978 | 0.1829 | 0.2203 | 0.1967 | 0.2716 | 0.237 | 0.2502 | 0.1965 | 0.2466 | 0.1997 | 0.2189 | 0.1473 |

Summary statistics on output-oriented scores of overall technical, pure technical and scale efficiencies without regression feedback

Statistics on CCR-OO scores | Statistics on BCC-OO scores | Statistics on SE-OO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

1st quartile | 2.6815 | 2.0002 | 1.7022 | 1.3308 | 1.3562 | 1.3233 | 1.2774 | 1.1791 | 1.0808 | 1.0266 | 1.0326 | 1.0078 |

2nd quartile | 3.8284 | 2.5626 | 2.3925 | 1.6511 | 2.4235 | 2.026 | 1.9246 | 1.5049 | 1.3652 | 1.1287 | 1.1204 | 1.0455 |

3rd quartile | 5.162 | 3.2357 | 3.4386 | 2.0952 | 3.6867 | 2.7762 | 2.8629 | 1.919 | 2.0277 | 1.4916 | 1.3582 | 1.1365 |

Maximum | 31.4064 | 30.3305 | 28.02 | 15.822 | 30.698 | 20.6932 | 18.6209 | 11.8291 | 12.0431 | 4.2375 | 14.2522 | 2.2664 |

Mean | 4.3804 | 2.7956 | 3.1031 | 1.8955 | 2.9497 | 2.2414 | 2.43 | 1.7138 | 1.7408 | 1.3365 | 1.3369 | 1.1152 |

SD | 2.9099 | 1.6866 | 2.5788 | 1.0475 | 2.3493 | 1.4107 | 1.8062 | 0.9311 | 1.0381 | 0.4509 | 0.8369 | 0.1736 |

Summary statistics on SBM efficiency scores without regression feedback

Statistics on SBM-IO scores | Statistics on SBM-OO scores | Statistics on SBM scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0011 | 0.0022 | 0.0012 | 0.0023 | 0.0001 | 0.0001 | 0.0002 | 0.0002 | 0.0001 | 0.0001 | 0.0002 | 0.0002 |

1st quartile | 0.157 | 0.2528 | 0.2765 | 0.4 | 0.0933 | 0.1177 | 0.1406 | 0.1546 | 0.0703 | 0.1069 | 0.1246 | 0.1404 |

2nd quartile | 0.2511 | 0.3572 | 0.4168 | 0.5547 | 0.2271 | 0.2559 | 0.299 | 0.3406 | 0.1664 | 0.2247 | 0.2716 | 0.3219 |

3rd quartile | 0.4587 | 0.5607 | 0.6651 | 0.7169 | 0.5096 | 0.533 | 0.5514 | 0.6255 | 0.3947 | 0.4615 | 0.5262 | 0.5947 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.3577 | 0.4373 | 0.4807 | 0.5646 | 0.3366 | 0.3572 | 0.3829 | 0.4104 | 0.2898 | 0.3269 | 0.364 | 0.3931 |

SD | 0.2868 | 0.2586 | 0.2771 | 0.2417 | 0.3091 | 0.3053 | 0.3023 | 0.3068 | 0.3054 | 0.2981 | 0.3035 | 0.303 |

Summary statistics on input-oriented scores of overall technical, pure technical and scale efficiencies with input focused regression feedback

Statistics on CCR-IO scores | Statistics on BCC-IO scores | Statistics on SE-IO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0211 | 0.0085 | 0.0357 | 0.0085 | 0.044 | 0.0091 | 0.0358 | 0.0091 | 0.0372 | 0.0939 | 0.0410 | 0.0939 |

1st quartile | 0.1429 | 0.0855 | 0.2908 | 0.0855 | 0.2845 | 0.1129 | 0.3904 | 0.1129 | 0.3322 | 0.3385 | 0.6612 | 0.3385 |

2nd quartile | 0.2042 | 0.1095 | 0.418 | 0.1095 | 0.4423 | 0.1811 | 0.5441 | 0.1811 | 0.4852 | 0.7046 | 0.8541 | 0.7046 |

3rd quartile | 0.2931 | 0.1451 | 0.5875 | 0.1451 | 0.7094 | 0.3455 | 0.7944 | 0.3455 | 0.7027 | 0.9524 | 0.9674 | 0.9524 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.2424 | 0.1264 | 0.4512 | 0.1264 | 0.5132 | 0.276 | 0.591 | 0.276 | 0.5247 | 0.6477 | 0.7847 | 0.6477 |

SD | 0.1633 | 0.0085 | 0.2203 | 0.0772 | 0.2716 | 0.2371 | 0.2502 | 0.2371 | 0.2408 | 0.3040 | 0.2189 | 0.3040 |

Summary statistics on output-oriented scores of overall technical, pure technical and scale efficiencies with input focused regression feedback

Statistics on CCR-OO scores | Statistics on BCC-OO scores | Statistics on SE-OO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

1st quartile | 4.1603 | 6.8929 | 1.7022 | 6.8929 | 1.3562 | 2.3027 | 1.4465 | 2.3027 | 1.6245 | 1.0380 | 1.0328 | 1.0380 |

2nd quartile | 5.8167 | 9.1365 | 2.3925 | 9.1365 | 2.4235 | 4.8168 | 2.4248 | 4.8168 | 2.3409 | 1.7774 | 1.1205 | 1.7774 |

3rd quartile | 8.2587 | 11.6993 | 3.4386 | 11.6993 | 3.6867 | 8.3935 | 4.1161 | 8.3935 | 3.6181 | 3.7435 | 1.3580 | 3.7435 |

Maximum | 94.342 | 117.8434 | 28.02 | 117.8434 | 30.698 | 49.6054 | 41.4108 | 49.6054 | 21.4901 | 10.6462 | 14.2522 | 10.6462 |

Mean | 7.4828 | 9.9778 | 3.1031 | 9.9778 | 2.9497 | 5.9386 | 3.4743 | 5.9386 | 0.3354 | 2.5006 | 1.3369 | 2.5006 |

SD | 6.2755 | 7.2698 | 2.5788 | 7.2698 | 2.3493 | 4.8377 | 3.6333 | 4.8377 | 0.3089 | 1.7216 | 0.8369 | 1.7216 |

Summary statistics on SBM efficiency scores with input focused regression feedback

Statistics on SBM-IO scores | Statistics on SBM-OO scores | Statistics on SBM scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0011 | 0.0022 | 0.0012 | 0.0001 | 0.0002 | 0.0001 | 0.0002 | 0.0002 | 0.0002 | 0.0001 | 0.0002 | 0.0002 |

1st quartile | 0.1602 | 0.2528 | 0.2765 | 0.0386 | 0.0884 | 0.1177 | 0.1406 | 0.1546 | 0.0704 | 0.1069 | 0.1246 | 0.1221 |

2nd quartile | 0.2449 | 0.3572 | 0.4168 | 0.0782 | 0.188 | 0.2559 | 0.299 | 0.3406 | 0.1478 | 0.2247 | 0.2716 | 0.3085 |

3rd quartile | 0.4734 | 0.5607 | 0.6651 | 0.2046 | 0.4104 | 0.533 | 0.5514 | 0.6255 | 0.3566 | 0.4615 | 0.5262 | 0.5708 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.343 | 0.4373 | 0.4807 | 0.1817 | 0.2913 | 0.3572 | 0.3829 | 0.4104 | 0.2618 | 0.3269 | 0.364 | 0.3737 |

SD | 0.2677 | 0.2586 | 0.2771 | 0.2404 | 0.2799 | 0.3053 | 0.3023 | 0.3068 | 0.2773 | 0.2981 | 0.3035 | 0.2955 |

Summary statistics on input-oriented scores of overall technical, pure technical and scale efficiencies with output focused regression feedback

Statistics on CCR-IO scores | Statistics on BCC-IO scores | Statistics on SE-IO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0011 | 0.023 | 0.0011 | 0.016 | 0.0358 | 0.0283 | 0.0358 | 0.0304 | 0.0099 | 0.1863 | 0.0099 | 0.1628 |

1st quartile | 0.1617 | 0.2632 | 0.1617 | 0.4441 | 0.3581 | 0.3388 | 0.3581 | 0.4404 | 0.3574 | 0.4997 | 0.3574 | 0.8085 |

2nd quartile | 0.2664 | 0.3271 | 0.2664 | 0.5736 | 0.5251 | 0.4492 | 0.5251 | 0.5759 | 0.5805 | 0.6676 | 0.5805 | 0.9329 |

3rd quartile | 0.3837 | 0.4101 | 0.3837 | 0.7234 | 0.762 | 0.7008 | 0.762 | 0.7487 | 0.7440 | 0.8438 | 0.7440 | 0.9875 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.2937 | 0.358 | 0.2937 | 0.5788 | 0.5627 | 0.5278 | 0.5627 | 0.5964 | 0.5570 | 0.6633 | 0.5570 | 0.8693 |

SD | 0.1781 | 0.1533 | 0.1781 | 0.1991 | 0.258 | 0.2403 | 0.258 | 0.2042 | 0.2468 | 0.2027 | 0.2468 | 0.1559 |

Summary statistics on output-oriented scores of overall technical, pure technical and scale efficiencies with output focused regression feedback

Statistics on CCR-OO scores | Statistics on BCC-OO scores | Statistics on SE-OO scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

1st quartile | 2.6815 | 2.4386 | 2.6062 | 1.6638 | 1.4829 | 1.8382 | 1.323 | 1.8382 | 1.1480 | 1.1283 | 1.2511 | 1.1283 |

2nd quartile | 3.8284 | 3.0572 | 3.7533 | 2.1944 | 2.6466 | 3.0555 | 2.0381 | 3.0555 | 1.4734 | 1.4679 | 1.6070 | 1.4679 |

3rd quartile | 5.162 | 3.7992 | 6.191 | 2.8296 | 4.027 | 3.9226 | 3.1517 | 3.9226 | 2.2064 | 1.9900 | 2.4887 | 1.9900 |

Maximum | 31.4064 | 43.5696 | 947.6238 | 257.9347 | 47.3749 | 350.5737 | 47.3749 | 350.5737 | 12.0431 | 5.3689 | 132.4496 | 5.3689 |

Mean | 4.3804 | 3.3063 | 7.665 | 2.7958 | 3.3081 | 3.5811 | 2.8136 | 3.5811 | 1.8552 | 1.6342 | 2.6234 | 1.6342 |

SD | 2.9099 | 2.0551 | 41.5639 | 7.9682 | 3.0596 | 10.8695 | 2.7962 | 10.8695 | 1.1070 | 0.6150 | 6.0733 | 0.6150 |

Summary statistics on SBM efficiency scores with output focused regression feedback

Statistics on SBM-IO scores | Statistics on SBM-OO scores | Statistics on SBM scores | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | Setup 1 | Setup 2 | Setup 3 | Setup 4 | |

Minimum | 0.0003 | 0.0022 | 0.0002 | 0.0000 | 0.0001 | 0.0001 | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0002 | 0.0000 |

1st quartile | 0.0965 | 0.2528 | 0.1421 | 0.0062 | 0.1084 | 0.0815 | 0.1606 | 0.0875 | 0.0474 | 0.1069 | 0.1385 | 0.0640 |

2nd quartile | 0.1873 | 0.3572 | 0.2727 | 0.0397 | 0.2448 | 0.1891 | 0.3672 | 0.2726 | 0.1164 | 0.2247 | 0.3173 | 0.2040 |

3rd quartile | 0.3776 | 0.5607 | 0.487 | 0.1580 | 0.4845 | 0.4292 | 0.6017 | 0.4653 | 0.3141 | 0.4615 | 0.5625 | 0.3701 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.2956 | 0.4373 | 0.3435 | 0.1203 | 0.3375 | 0.2975 | 0.4112 | 0.3265 | 0.2446 | 0.3269 | 0.3837 | 0.2737 |

SD | 1.3633 | 0.2586 | 0.2702 | 0.1795 | 0.2953 | 0.2919 | 0.3024 | 0.2824 | 0.2951 | 0.2981 | 0.3019 | 0.2641 |

Fourth, in addition to enhancing discrimination amongst DMUs and adjusting their DEA scores, which in itself is a major issue in DEA applications, feedback reveals a completely new story on the actual drivers of a range of efficiency measures and exposes the importance of the choice of DEA models in estimating these measures. In the following paragraphs, we shall provide evidence of these claims.

In our empirical analysis, we used two types of regression feedback—see Tables 12, 13, 14, 15, 16 and 17 in “Appendix”. The first regression feedback—referred to as input focus regression analysis—involves regressing DEA scores on inputs. The second regression feedback—referred to as output focus regression analysis—involves regressing DEA scores on outputs. Depending on the statistical significance of inputs (respectively, outputs), some inputs (respectively, outputs) may have to be discarded and the DEA scores recomputed with a reduced set of inputs (respectively, outputs), if necessary. Note however that, in some cases, none of the inputs (respectively, outputs) proves to explain the behaviour of DEA scores in which case DEA models without explicit inputs (respectively, explicit outputs) would have to be solved—as illustrated by Setup 4 in output focus regression. So far, this case has not been encountered by or reported in previous studies, which has motivated the new methodological design in this research.

Summary of drivers of efficiency scores after regression feedback

Inputs | Input-focus regression analysis | ||||||
---|---|---|---|---|---|---|---|

CCR-IO | BCC-IO | CCR-OO | BCC-OO | SBM-IO | SBM-OO | SBM | |

| |||||||

Personnel expenses | X | X | X | X | X | X | |

Fixed assets | X | X | X | X | X | X | |

Total interest expense | X | X | X | X | X | X | X |

| |||||||

Personnel expenses | X | X | X | X | X | X | X |

Fixed assets | X | X | X | ||||

Total expenses not including personnel expense | X | X | X | ||||

| |||||||

Personnel expenses | X | X | X | X | X | X | X |

Equity | X | X | X | X | X | X | X |

Total interest expense | X | X | X | X | X | X | |

| |||||||

Personnel expenses | X | X | X | X | X | X | X |

Equity | X | X | |||||

Total expenses not including personnel expense | X |

Outputs | Output-focus regression analysis | ||||||
---|---|---|---|---|---|---|---|

CCR-IO | BCC-IO | CCR-OO | BCC-OO | SBM-IO | SBM-OO | SBM | |

| |||||||

Gross loans | X | X | X | X | X | X | X |

Total customer deposits | X | X | X | X | X | X | X |

Gross interest and dividend income | X | X | |||||

| |||||||

Gross loans | X | X | X | X | X | X | X |

Total customer deposits | X | X | X | X | X | X | |

Total income | X | X | |||||

| |||||||

Gross loans | X | X | X | X | X | X | X |

Total customer deposits | X | X | X | X | |||

Gross interest and dividend income | |||||||

| |||||||

Gross loans | X | X | X | X | X | X | |

Total customer deposits | X | ||||||

Total income |

Furthermore, with respect to the importance of the choice of DEA models in estimating efficiency measures, DEA analyses with input focus regression feedback provides a good example. In fact, empirical results suggest that, in some setups, DEA scores estimated by CCR and BCC models are not driven by the initial choice of inputs. For example, under Setup 2, CCR and BCC scores are only driven by Personnel Expenses. Interestingly, under the same setup, SBM scores are driven by Personnel Expenses, Fixed Assets (physical capital), and Total Expenses not including Personnel Expense. Further investigation of this fact revealed that the slacks associated with Fixed Assets, and Total Expenses not including Personnel Expense turn out to be important in magnitude, but ignored by radial measures of efficiency. SBM scores however take these slacks into account and thus avoid the elimination of Fixed Assets, and Total Expenses not including Personnel Expense through regression feedback. In sum, ignoring slacks might result in the regression-based feedback suggesting that some efficiency determinants should be discarded when they should not. These findings suggest that, in practice, one should use slacks-based measures of efficiency instead of the conventional ones whenever possible, on one hand, and remind us of the importance for the DEA community to design new SBM based metrics to measure pure technical efficiency and scale efficiency, which are yet to be proposed, on the other hand.

Summary of CCR-IO efficiency scores for domestic and foreign banks

CCR-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Set up 4 | |||||

Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | |

Minimum | 0.0211 | 0.0318 | 0.0085 | 0.0189 | 0.0357 | 0.0381 | 0.0085 | 0.0189 |

1st quartile | 0.1439 | 0.1334 | 0.0855 | 0.0855 | 0.3193 | 0.2357 | 0.0855 | 0.0855 |

2nd quartile | 0.2094 | 0.1974 | 0.1084 | 0.1114 | 0.4568 | 0.3536 | 0.1084 | 0.1114 |

3rd quartile | 0.2978 | 0.2779 | 0.1420 | 0.1588 | 0.6216 | 0.5317 | 0.1420 | 0.1588 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.2526 | 0.2233 | 0.1213 | 0.1361 | 0.4860 | 0.3862 | 0.1213 | 0.1361 |

SD | 0.1760 | 0.1345 | 0.0606 | 0.1007 | 0.2163 | 0.2132 | 0.0606 | 0.1007 |

Summary of BCC-IO efficiency scores for domestic and foreign banks

BCC-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | |||||

Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | |

Minimum | 0.0440 | 0.0869 | 0.0091 | 0.0214 | 0.0358 | 0.0778 | 0.0091 | 0.0214 |

1st quartile | 0.2962 | 0.2704 | 0.1118 | 0.1179 | 0.4216 | 0.3455 | 0.1118 | 0.1179 |

2nd quartile | 0.4563 | 0.4322 | 0.1787 | 0.1843 | 0.5990 | 0.4898 | 0.1787 | 0.1843 |

3rd quartile | 0.7545 | 0.6343 | 0.3529 | 0.3303 | 0.8609 | 0.6890 | 0.3529 | 0.3303 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.5295 | 0.4828 | 0.2811 | 0.2667 | 0.6252 | 0.5270 | 0.2811 | 0.2667 |

SD | 0.2784 | 0.2561 | 0.2466 | 0.2182 | 0.2519 | 0.2343 | 0.2466 | 0.2182 |

Summary of SE-IO efficiency scores for domestic and foreign banks

SE-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | |||||

Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | |

Minimum | 0.1027 | 0.0372 | 0.0939 | 0.1627 | 0.1742 | 0.0410 | 0.0939 | 0.1627 |

1st quartile | 0.3190 | 0.3618 | 0.3293 | 0.3637 | 0.6802 | 0.6347 | 0.3293 | 0.3637 |

2nd quartile | 0.4759 | 0.4960 | 0.7037 | 0.7140 | 0.8671 | 0.8282 | 0.7037 | 0.7140 |

3rd quartile | 0.7191 | 0.6735 | 0.9458 | 0.9616 | 0.9640 | 0.9726 | 0.9458 | 0.9616 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.5270 | 0.5204 | 0.6390 | 0.6640 | 0.7998 | 0.7565 | 0.6390 | 0.6640 |

SD | 0.2414 | 0.2398 | 0.3084 | 0.2955 | 0.1975 | 0.2521 | 0.3084 | 0.2955 |

Summary of SBM-io efficiency scores for domestic and foreign banks

SBM-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Setup 4 | |||||

Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | Domestic | Foreign | |

Minimum | 0.0011 | 0.0024 | 0.0072 | 0.0022 | 0.0035 | 0.0012 | 0.0001 | 0.0009 |

1st quartile | 0.1711 | 0.1272 | 0.2747 | 0.2239 | 0.3180 | 0.1955 | 0.0443 | 0.0272 |

2nd quartile | 0.2800 | 0.2148 | 0.3773 | 0.3130 | 0.4583 | 0.3628 | 0.0820 | 0.0678 |

3rd quartile | 0.5165 | 0.3848 | 0.6080 | 0.4890 | 0.7197 | 0.5651 | 0.2151 | 0.1722 |

Maximum | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Mean | 0.3858 | 0.3017 | 0.4651 | 0.3853 | 0.5237 | 0.4005 | 0.1941 | 0.1586 |

SD | 0.2919 | 0.2674 | 0.2591 | 0.2499 | 0.2754 | 0.2623 | 0.2499 | 0.2197 |

Summary of CCR-IO efficiency scores for large and small banks

CCR-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Set up 4 | |||||

Large | Small | Large | Small | Large | Small | Large | Small | |

Minimum | 0.1300 | 0.0211 | 0.0830 | 0.0085 | 0.3554 | 0.0357 | 0.0830 | 0.0085 |

1st quartile | 0.2020 | 0.1369 | 0.1205 | 0.0833 | 0.4724 | 0.2809 | 0.1205 | 0.0833 |

2nd quartile | 0.2681 | 0.1986 | 0.1423 | 0.1066 | 0.5468 | 0.3990 | 0.1423 | 0.1066 |

3rd quartile | 0.3100 | 0.2853 | 0.1728 | 0.1407 | 0.6769 | 0.5766 | 0.1728 | 0.1407 |

Maximum | 0.6683 | 1.0001 | 0.3385 | 1.0000 | 1.0000 | 1.0000 | 0.3385 | 1.0000 |

Mean | 0.2731 | 0.2397 | 0.1507 | 0.1243 | 0.5681 | 0.4409 | 0.1507 | 0.1243 |

SD | 0.0979 | 0.1676 | 0.0467 | 0.0790 | 0.1292 | 0.2237 | 0.0467 | 0.0790 |

Summary of BCC-IO efficiency scores for large and small banks

BCC-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Set up 4 | |||||

Large | Small | Large | Small | Large | Small | Large | Small | |

Minimum | 0.5697 | 0.0440 | 0.4789 | 0.0091 | 0.6457 | 0.0358 | 0.4789 | 0.0091 |

1st quartile | 0.8125 | 0.2750 | 0.6691 | 0.1085 | 0.9045 | 0.3753 | 0.6691 | 0.1085 |

2nd quartile | 0.9531 | 0.4145 | 0.7886 | 0.1676 | 0.9905 | 0.5117 | 0.7886 | 0.1676 |

3rd quartile | 1.0000 | 0.6304 | 0.9426 | 0.2806 | 1.0000 | 0.7357 | 0.9426 | 0.2806 |

Maximum | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Mean | 0.8988 | 0.4791 | 0.7891 | 0.2306 | 0.9313 | 0.5609 | 0.7891 | 0.2306 |

SD | 0.1211 | 0.2543 | 0.1627 | 0.1830 | 0.0989 | 0.2369 | 0.1627 | 0.1830 |

Summary of SE-IO efficiency scores for large and small banks

SE-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Set up 4 | |||||

Large | Small | Large | Small | Large | Small | Large | Small | |

Minimum | 0.1311 | 0.0372 | 0.0939 | 0.1228 | 0.3636 | 0.0410 | 0.0939 | 0.1228 |

1st quartile | 0.2375 | 0.3670 | 0.1737 | 0.4111 | 0.5094 | 0.7003 | 0.1737 | 0.4111 |

2nd quartile | 0.2818 | 0.5177 | 0.1854 | 0.7496 | 0.6040 | 0.8827 | 0.1854 | 0.7496 |

3rd quartile | 0.3369 | 0.7360 | 0.1936 | 0.9615 | 0.7120 | 0.9721 | 0.1936 | 0.9615 |

Maximum | 0.6683 | 1.0001 | 0.3385 | 1.0000 | 1.0000 | 1.0000 | 0.3385 | 1.0000 |

Mean | 0.3041 | 0.5443 | 0.1907 | 0.6882 | 0.6124 | 0.8000 | 0.1907 | 0.6882 |

SD | 0.1002 | 0.2399 | 0.0368 | 0.2835 | 0.1291 | 0.2188 | 0.0368 | 0.2835 |

Summary of SBM-IO efficiency scores for large and small banks

SBM-IO input-focus regression feedback | ||||||||
---|---|---|---|---|---|---|---|---|

Setup 1 | Setup 2 | Setup 3 | Set up 4 | |||||

Large | Small | Large | Small | Large | Small | Large | Small | |

Minimum | 0.4181 | 0.0011 | 0.5055 | 0.0022 | 0.5229 | 0.0012 | 0.3192 | 0.0001 |

1st quartile | 0.5547 | 0.1486 | 0.6706 | 0.2425 | 0.7071 | 0.2690 | 0.6217 | 0.0354 |

2nd quartile | 0.8138 | 0.2337 | 0.8843 | 0.3404 | 0.9272 | 0.3983 | 0.7432 | 0.0709 |

3rd quartile | 1.0000 | 0.3873 | 1.0000 | 0.5017 | 1.0000 | 0.6097 | 0.8952 | 0.1405 |

Maximum | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Mean | 0.7857 | 0.3185 | 0.8266 | 0.4028 | 0.8538 | 0.4477 | 0.7386 | 0.1324 |

SD | 0.2068 | 0.2602 | 0.1750 | 0.2356 | 0.1598 | 0.2606 | 0.1814 | 0.1735 |

In sum, our empirical analyses provided the following answers to our research questions. First, UK commercial banks need further efficiency improvements. Second, UK commercial banks’ measures of efficiency seem to be driven by the inputs and outputs identified by researchers so far except when the combinations of measures and their interaction along with their slacks and the type of DEA models used for estimating efficiency scores come into play. Third, DEA analyses with and without a linear regression-based feedback mechanism seem to provide consistent findings in terms of inefficiency; however, compared to DEA analyses with feedback, in general DEA analyses without feedback tend to over- or under-estimate efficiency scores depending on whether the analyses are input-oriented or output-oriented. Fourth, in general, a linear regression-based feedback mechanism proves effective at improving discrimination in DEA analyses unless the initial choice of inputs and outputs is well informed. Last, but not least, ignoring slacks might result in the regression-based feedback suggesting that some efficiency determinants should be discarded when they should not, which suggest that, in practice, one should use slacks-based measures of efficiency instead of the conventional ones whenever possible, on one hand, and remind us of the importance for the DEA community to design new SBM based metrics to measure pure technical efficiency and scale efficiency, which are yet to be proposed, on the other hand.

## 5 Conclusions

In this paper, we investigated the efficiency profiles of the UK commercial banking sector using a new DEA-based analysis framework with a regression-based feedback mechanism, where DEA models could use both inputs and outputs, only inputs, or only outputs. Note that the use of DEA models without explicit inputs or outputs is required when the regression-based feedback mechanism informs DEA analysis that all inputs or all outputs should be discarded, because they do not drive efficiency, which turned out to be the case in our empirical analysis of UK banking data. The proposed DEA analysis design was used to address several research questions related to both the UK commercial banking sector and DEA analyses with and without regression-based feedback—see Sect. 4 for details on our findings. Amongst these findings, it tuned out that performing DEA analyses with radial models such as CCR and BCC, which ignore slacks in computing technical efficiency scores, might result in the regression-based feedback suggesting that some efficiency drivers should be discarded when they should not. Therefore, we recommend that, in practice, one should use slacks-based measures of efficiency instead of the conventional ones whenever possible. These findings remind us of the importance for the DEA community to design new SBM based metrics to measure pure technical efficiency and scale efficiency, which are yet to be proposed.

## Notes

### Acknowledgements

The authors would like to express their gratitude to two anonymous referees for their helpful comments and recommendations.

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