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

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.


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 B Jamal Ouenniche Jamal.Ouenniche@ed.ac.uk 1 Business School, University of Edinburgh, 29 Buccleuch Place, Edinburgh EH8 9JS, UK 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.

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.
With respect to the type of analysis, the literature could be divided into three categories. The first category of studies uses 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).
The second category of studies uses 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).
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.
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.

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, . . . , m) used by DMU k (k 1, . . . , n) and the quantity y r,k of output r (r 1, . . . , s) produced by DMU k (k 1, . . . , 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).
In this paper, we are concerned with measuring overall technical efficiency, pure technical efficiency, and scale efficiency of UK commercial banks. Unlike previous studies, the particular features of UK banking data require additional types of DEA models. Therefore, we shall use both input-and output-oriented CCR models (Charnes et al. 1978); both inputand output-oriented BCC models (Banker et al. 1984); BCC models without explicit inputs, BCC-WEI, or without explicit outputs, BCC-WEO (Lovell and Pastor 1999); input-oriented, output-oriented, and non-oriented SBM models (Tone 2001); and SBM-WEI model (Liu et al. 2011) and SBM-WEO model that we propose. CCR and BCC models are described in Table 1, BCC models without explicit inputs or outputs are described in Table 2, SBM models are described in Table 3, and SBM models without explicit inputs or outputs are described in Table 4, where θ k denotes the technical efficiency of DMU k and measures the efficiency with which DMU k transforms inputs into outputs, which reflects the quality of its management decisions, λ j denotes the weight assigned to DMU j in constructing the "ideal" benchmark of DMU k ; that is, its projection on the efficiency frontier, and s − i,k and s + r,k denote the slacks in input i and output r , respectively, which represent input excess and output shortfall. Recall that most DEA analyses make use of one or several inputs and one or several outputs; however, in some situations one might not have to use any inputs or any outputs-these situations or models are referred to as DEA models or analyses without explicit inputs or without explicit outputs. In a DEA analysis with a regression-based feedback mechanism one might have to discard all inputs or all outputs when regression analysis suggests that they do not drive or explain differences in efficiency profiles. However, in general, in DEA applications the use of DEA models without explicit inputs could be justified when one assumes that inputs are considered similar and equal for all DMUs as they operate, for example, in the same market (e.g., Halkos and Salamouris 2004). On the other hand, the use of DEA models without explicit outputs could be justified when one assumes that outputs are considered similar and Objective function θ k θ k is to be minimised or maximised depending on whether the analysis is input-oriented or output-oriented For each input i (i 1, . . . , 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 For each output r (r 1, . . . , 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 The technology is required to be convex in BCC models. This constraint is relaxed in CCR models λ j ≥ 0, j 1, . . . , n θ k unrestricted Other requirements including non-negativity Table 2 BCC models without explicit inputs or outputs

Formulation Description
Objective function θ k θ k is to be minimised or maximised depending on whether the analysis is without explicit output or without explicit inputs For each input i (i 1, . . . , 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, . . . , 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 n j 1 The technology is convex Other requirements including non-negativity Objective function One of these ρ k formulations is to be minimised depending on whether the analysis is non-oriented, input-oriented, or output-oriented For each input i (i 1, . . . , 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 n j 1 λ j y r, j − s + r,k y r,k , r 1, . . . , s For each output r (r 1, . . . , 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 n j 1 λ j 1 This constraint requires the technology to be convex; however, it could be relaxed Non-negativity requirements equal for all DMUs as they operate, for example, under specific legislation or supply markets with fixed shares on which DMUs could not act upon in the short to medium term. The flowchart of the proposed methodology is outlined in Fig. 4. Within this methodological framework, given a set of relevant environment-independent inputs and outputs specified by the analyst or researcher, DEA analysis with both inputs and outputs is first performed to compute the relevant efficiency scores for the analysis under consideration (e.g., overall technical efficiency, pure technical efficiency, scale efficiency) as well as slacks by solving the appropriate DEA models (e.g., CCR, BCC, SBM models).
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 One of these ρ k formulations is to be minimised depending on whether the analysis is without explicit output or without explicit inputs For each input i (i 1, . . . , 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, . . . , 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 n j 1 λ j 1 This constraint requires the technology to be convex; however, it could be relaxed The weights λ 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 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 regressionbased 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) (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.

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.
The choice of the inputs and outputs with which DEA models are fed is driven by the intermediation approach, where banks are considered as intermediation agents who collect funds and provide loans and other assets. For a discussion on the choice of inputs and outputs in banking applications, we refer the reader to Fethi and Pasiouras (2010). Our survey and classification of the inputs and outputs used in the literature (see Ouenniche et al. 2017) along with an analysis of the balance sheet and the income statement of UK commercial banks revealed that inputs are typically chosen based on resources, costs, or financial burden, whereas outputs are typically chosen based on bank's ability to provide financial services (i.e., Loans and Deposits), generate revenue (i.e., Income and Investments) and acquire more assets (i.e., Investments). However, our critical analysis of such choices suggests that some authors' choices-especially of inputs based on financial burden rather penalise the very means by which banks are able to perform their lending operations. Therefore, we selected inputs based only on resources (i.e., Labour as measured by Personnel Expenses-because the number of employees was not available for all UK banks; Capital as measured by Fixed Assets/Physical Capital or Equity/Financial Capital) and costs (i.e., Total Interest Expense; Total Expenses not including Personnel Expense). As to outputs, we selected them based on the ability of a bank to provide financial services (i.e., Gross Loans; Total Customer Deposits) and generate revenue (i.e., Total Income; Gross Interest and Dividend Income). We did not consider the ability of banks to acquire more assets or to make investments because small UK banks, which are part of our sample, are not quite involved in off-balance sheet activities. These chosen criteria could however be measured in different ways. In our empirical experiments, we used four setups or scenarios each corresponding to a different combination of measures-see Table 5 for details.
A snapshot of the 109 UK commercial banks in our dataset is summarised in Table 6 (see "Appendix"), where the figures are measured in millions of USD. Table 7 provides a snapshot of the leading UK commercial banks (see "Appendix"). Analysis of raw data on UK commercial banks in our dataset revealed that Pareto's Law holds; that is, eight leading banks (i.e., 8/109 ∼ 7% of UK commercial banks); namely, National Westminster Bank Plc-NatWest, The Royal bank of Scotland, Ulster bank, Lloyds bank, Bank of Scotland, Barclays, HSBC Bank Plc, and Standard Chartered Bank, together account for almost 87% of the stock of UK customer lending and deposits. In addition, as highlighted by some statistics on fixed assets, as a proxy for size (i.e., the first quartile of total assets in Table 7 is 400% bigger than the third quartile of total assets in Table 6); the UK commercial banks in our dataset, excluding the largest eight, are altogether smaller than the smallest bank of the largest ones. We also performed several analyses by size (e.g., total assets); market share (e.g., total customer deposits, gross loans), gross profitability (e.g., total income), operational expenses (e.g., personnel expenses), and origin (e.g., domestic, foreign)-see Table 8 in "Appendix". These analyses also support Pareto's Law. In addition, they highlight the importance of foreign banks in the UK; in fact, although foreign banks represent 38% of the total UK commercial banks as compared to 55% of domestic banks but the largest eight, their market share is bigger. Last, but not least, assuming that Personnel Expenses are a good proxy for the number of employees, the largest bank; namely, Barclays Bank Plc, employs about 50% of the labour used by all small domestic banks. We also investigated the UK commercial banks' ownership structure and found out that ownership structure is not a discriminatory feature, since 1 foreign bank in residency and 2 local banks are Limited Liability Corporations, 1 foreign bank in residency and 2 local banks are Mutual/Co-ops, and the remaining banks; that is, 39 foreign banks in residency and 64 local banks are Stock Corporations.
DEA analyses of the UK commercial banking sector, as represented by the 109 commercial banks in our dataset, are summarised as follows.
First, in input-oriented analyses (see Tables 9, 11 in "Appendix"), numerical results suggest that in the UK commercial banking system the combination of choices of measures of inputs matters; in other words, how resources and expenses are proxied as well as the combinations of these proxies matter for banks' levels of efficiencies. To be more specific, equity or financial   4 and 3), as a proxy for resources, enhances on average overall technical efficiency (OTE) or CCR scores, overall technical efficiency adjusted for mix efficiency (adj-OTE) or SBM scores; pure technical efficiency (PTE) or BCC scores, and scale efficiency (SE) better than fixed assets or physical capital (setups 1 and 2); therefore, UK commercial banks are better at managing their equity or liquidity than their fixed assets, which is in line with the intermediation role of the banks. On the other hand, total expenses not including personnel expense (setups 4 and 2), as a proxy for expenses, seems to enhance on average OTE, adj-OTE, PTE, and SE better than total interest expense (setups 1 and 3). Judged on their use of inputs, on average, UK commercial banks fall short on overall technical efficiency, pure technical efficiency, and scale efficiency-see Tables 9 and 11. In fact, depending on the choice of measures of inputs across setups, average CCR scores vary between 0.3144 and 0.6119, average SBM scores (i.e., overall technical efficiency adjusted for mix efficiency) vary between 0.3577 and 0.5646, average BCC scores vary between 0.5132 and 0.6976, and average SE scores vary between 0.667 and 0.8796. In sum, the management of the UK commercial banking sector seems to be in need of further improvements. Commercial banks in the fourth quartile however seem to be scale efficient to a large extent; therefore, for these banks any further efficiency improvement efforts should be put on pure technical efficiency. Second, most DEA analyses in banking have focused on input-oriented analyses, which is typically justified by the fact that bank managers have more control over the management of inputs than outputs. This is an arguable point of view as some outputs could be acted upon through better and more focused commercial strategies and marketing campaigns. In addition, in practice, the analysis of output-oriented DEA scores could provide important insight. Motivated by these concerns, we also performed output-oriented analyses of the UK commercial banks-see Tables 10 and 11 in "Appendix". In output-oriented analyses (see Tables 10, 11), numerical results suggest that, in the UK commercial banking system, the choices of measures of outputs as well as the combinations of choices of measures of inputs matter; in other words, how income is proxied as well as the combinations of proxies of inputs matter for banks' levels of efficiencies. To be more specific, regardless of the choice of inputs proxies, on average, OTE, PTE and SE are enhanced when total income (setups 2 and 4) is used as a proxy for income compared to gross interest and dividend income (setups 1 and 3). Consequently, on average, the management of UK commercial banks seem to be good at managing total income, but less so in generating gross interest and rewarding their shareholders through dividends. However, average adj-OTE figures are affected by both the choice of income proxies and the combinations of proxies of inputs; in fact, setup 4 enhances adj-OTE more than setup 3 followed by setup 2 then setup 1. Finally, in terms of scale efficiency, output-oriented results are in line with the input-oriented ones.
Third, regression feedback informs the analyst about the relevance of his or her choices of efficiency drivers (i.e., inputs and outputs). Our empirical analysis shows that taking account of regression feedback to revise DEA models always enhances discrimination and adjusts DEA scores downwards or upwards, depending on whether the DEA analysis is input-oriented or output-oriented-see Tables 9, 10, 11, 12, 13, 14, 15, 16 and 17 in "Appendix". Note that, in the case of the UK commercial banks in our sample, the conclusions with respect to the efficiency profiles of banks remain the same. In sum, regardless of whether DEA analyses are performed with or without regression feedback, the UK commercial banking sector is in need of further efficiency improvements.
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.
A summary of the statistically significant input and output drivers of efficiency is provided in Table 18, where Labour, as measured by Personnel expenses, seems to be the most consistent input driver of efficiency scores across all setups and DEA analyses, whereas the provision of financial services, as measured by Gross Loans, seems to be the most consistent output driver of efficiency scores across all setups and DEA analyses. The relevance of remaining drivers however depends on both the setups or combinations of drivers and the DEA analyses. Notice, however, that those setups (i.e., choices of combinations of drivers) that make the UK commercial banking sector look more efficient (e.g., Setup 4 without feedback) are the ones that are most affected by the regression feedback, on one hand, and those setups that lead to more conservative estimates of efficiency scores (e.g., Setup 1 without feedback) are less or not at all affected by the regression feedback, on the other hand. Therefore, the regression feedback serves as a correction mechanism in that it adjusts overand under-estimated scores. These findings have important implications on the relevance of the choices of inputs and outputs and the combinations of their measures; in fact, they often tell the opposite story revealed by DEA analyses without regression feedback. For example, input-oriented DEA analyses without regression feedback suggested that UK commercial banks are better at managing their financial capital than their physical capital, which is in line with the intermediation role of the banks, but when feedback is incorporated the management of UK commercial banks does not seem to be doing such good job anymore in managing equity. In sum, the lessons to be learned could be summarised as follows. From the perspective of banks' managers, DEA analyses without feedback make them look better, and most importantly it backs up their strategies of being intermediation agents in the economy. However, regulators and investors might be better off performing DEA analyses with feedback, alongside DEA analyses without feedback, to unveil different pictures. 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.
Finally, our analysis of DEA scores of domestic and foreign banks suggests that their efficiency profiles are very similar regardless of which DEA models or regression analysis focus is used to estimate the scores-see, for example, Tables 19, 20, 21 and 22 in "Appendix" for illustration. Also, our analysis of DEA scores of large and smaller banks suggests that their efficiency profiles are very different. In fact, large banks are more overall technically efficient and pure technically efficient than the small ones, but the large ones seem to be less scale efficient than the small banks regardless of which DEA models or regression analysis focus is used to estimate the scores-see, for example, Tables 23, 24, 25 and 26 in "Appendix" for illustration.
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 underestimate efficiency scores depending on whether the analyses are input-oriented or outputoriented. 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.

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 regressionbased 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.