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. 估計技術和規(guī)模的希臘商業(yè)銀行效率： 信用風險、 資產(chǎn)負債表的活動和 國際業(yè)務的影響 原文出處及作者：巴斯大學管理學院2007年碩士畢業(yè)論文，作者Fotios Pasiouras 1.介紹 希臘銀行業(yè)經(jīng)歷了近幾年重大的結構調(diào)整。重要的結構性、政策和環(huán)境的變化經(jīng)常強調(diào)的學者和從業(yè)人員有歐盟單一市場的建立，歐元的介紹，國際化的競爭、利率自由化、放松管制和最近的兼并和收購浪潮。 希臘的銀行業(yè)也經(jīng)歷了相當大的改善，通信和計算技術，因為銀行有擴張和現(xiàn)代化其分銷網(wǎng)絡，其中除了傳統(tǒng)的分支機構和自動取款機，現(xiàn)在包括網(wǎng)上銀行等替代分銷渠道。作為希臘銀行（2004 年）的年度報告的重點，希臘銀行亦在升級其信用風險測量與管理系統(tǒng)，通過引入信用評分和概率默認模型近年來采取的主要步驟。此外，他們擴展他們的產(chǎn)品/服務組合，包括保險、 經(jīng)紀業(yè)務和資產(chǎn)管理等活動，同時也增加了他們的資產(chǎn)負債表操作和非利息收入。 最后，專注于巴爾干地區(qū)（如阿爾巴尼亞、保加利亞、前南斯拉夫馬其頓共和國、羅馬尼亞、塞爾維亞）的更廣泛市場的全球化增加的趨勢已添加到希臘銀行在塞浦路斯和美國以前有限的國際活動。在國外經(jīng)營的子公司的業(yè)績預計將有父的銀行，從而對未來的決定為進一步國際化的嘗試對性能的影響。 本研究的目的是要運用數(shù)據(jù)包絡分析（DEA）和重新效率的希臘銀行部門，同時考慮到幾個以上討論的問題進行調(diào)查。我們因此區(qū)分我們的論文從以前的希臘銀行產(chǎn)業(yè)重點并在幾個方面，下面討論添加的見解。 首先，我們第一次對效率的希臘銀行的信用風險的影響通過檢查其中包括貸款損失準備金作為附加輸入Charnes et al.(1990 年)、 德雷克（2001 年）、 德雷克和大廳 (2003 年)，和德雷克等人（2006 年）。作為美斯特 (1996) 點出"除非質(zhì)量和風險控制的一個人也許會很容易誤判一家銀行的水平的低效 ；例如精打細算的銀行信用評價或生產(chǎn)過高風險的貸款可能會被貼上標簽一樣高效，當相比銀行花資源，以確保它們的貸款有較高的質(zhì)量"(p.1026)。我們估計效率的銀行和無此輸入調(diào)整為不同的信用風險水平和對效率的影響。 第二，以往的研究中，希臘銀行業(yè)，我們考慮資產(chǎn)負債表活動期間估計的效率得分。幾個最近的研究審查效率的 DEA 或隨機前沿技術的銀行，承認銀行在非傳統(tǒng)的活動中更多地參與，包括任何非利息 （即費） 收入 （e.g. Lang和Welzel，1998年；德雷克，2001 年；托爾托薩Ausina，2003年） 或資產(chǎn)負債表項目（例如阿爾通巴什等人，2001 年 ；阿爾通巴什和查克，2001年；架和 Hassan，2003a、 b ；Bos 和 Colari，2005 年 ；饒，2005年） 作為額外的輸出。然而，盡管他們希臘銀行的重要性上升，這種活動沒有被考慮在過去。再次，我們估計，銀行的效率在我們的示例與無負債表外活動，以觀察是否它將會對效率有影響。 第三，我們比較所得的中介方法隨之而來的銀行的效率與利潤導向的做法，最近在 dea 方法，提出了由德雷克等人（2006 年），在他們隨機前沿方法的上下文中杰和美斯特 （2003 年） 的做法是一致的最新研究的結果。這使我們能夠觀察是否不同的輸入/輸出定義影響效率分數(shù)。 第四，我們比較效率得分的希臘銀行，擴大了其海外的業(yè)務（即國際希臘銀行，以下簡稱 IGBs），與那些希臘銀行的業(yè)務在國內(nèi)市場都有限的（即純粹的國內(nèi)銀行，以下簡稱 Pdb）。為了最好的我們的知識，沒有研究開展了這種分析對于希臘。然而，在土耳其銀行業(yè)的研究中，Isik和Hassan（2002 年）發(fā)現(xiàn)的證據(jù)，跨國公司的國內(nèi)銀行均優(yōu)于純粹國內(nèi)銀行的所有提高效率的措施（即成本效率、資源配置效率、技術效率、純技術效率）除了規(guī)模效率。從我們的研究得出的結論可能是有用的希臘銀行或其他正在考慮他們的業(yè)務的國際化的中型銀行部門的經(jīng)理。 第五，我們運行回歸來解釋銀行效率的一直在希臘 （赫里斯托普洛斯等人，2002年；Rezitis，2006年）。但是，在我們的例子中我們檢查最近一段時間，遵循上文所述的許多變化。 本文的其余部分是，如下所示：第2節(jié)文獻側重于希臘銀行部門的效率。第 3 節(jié)規(guī)定 DEA 的簡短的討論。第4節(jié)給的數(shù)據(jù)和變量。第 5 節(jié)討論實證分析的結果，并節(jié) 6 總結研究。 2.文獻綜述 Karafolas和Mantakas (1996) 使用二階超越對數(shù)成本函數(shù)估計（第一次）在希臘銀行部門的費用的一種計量形式和調(diào)查的規(guī)模經(jīng)濟。十一銀行從 1980年至 1989 年期間使用的數(shù)據(jù)，他們發(fā)現(xiàn)雖然經(jīng)營成本規(guī)模經(jīng)濟確實存在，但總成本規(guī)模經(jīng)濟并不存在。由銀行的大?。创?、小銀行）和時間段的子樣本數(shù)據(jù)集的參與 (即1980年—1984 年，1985年-1989 年) 并沒有改變結果。最后，結果表明技術變革中，降低平均成本不發(fā)揮了統(tǒng)計學意義的作用。 Noulas（1997 年）檢查生產(chǎn)率增長的十個私營和十個國有銀行經(jīng)營在希臘在 1991 年和 1992 年，期間使用的Malmquist生產(chǎn)率指數(shù)和 DEA 測量效率。作者遵循調(diào)解方法，并發(fā)現(xiàn)生產(chǎn)率平均增長 8%左右，與國有銀行表現(xiàn)出較高的增長比私人的。結果還表明增長的來源不同跨銀行的兩種類型。國有銀行生產(chǎn)率增長是進步的由于技術，而私人銀行的增長是進步的提高效率的結果。 赫里斯托普洛斯和Tsionas (2001)在1993年—1998年期間估計在希臘的商業(yè)銀行業(yè)效率使用同方差與異方差性的前沿。他們發(fā)現(xiàn)平均技術效率約 80%的異方差模型和總體平均值的分布之一的 83%。他們還發(fā)現(xiàn)技術和資源配置低效率降低隨時間較小，以及較大的銀行。回歸的低效率措施反對趨勢指示在技術和資源配置效率低下的小改進銀行同等 19.7%和39.1%，因此，大型銀行的相應數(shù)字是10.4%和21.1%。 赫里斯托普洛斯（2002 年）檢查同一個多輸入、多輸出的柔性成本函數(shù)代表部門和差異方差前沿方法來測量技術效率的技術相同的樣本。提高效率的措施對銀行的各種特性的回歸表示較大的銀行都是比較小的效率較低和經(jīng)濟績效、銀行貸款和投資都呈正相關，成本效率。 在后者的研究中，Tsionas et al.(2003) 使用赫里斯托普洛斯和Tsionas（2001 年）和赫里斯托普洛斯 et al.（2002 年） 相同的樣本，但雇用DEA測量技術和資源配置效率和 Malmquist 全要素生產(chǎn)率方法來衡量生產(chǎn)力的變化。結果表明，大多數(shù)銀行經(jīng)營接近最佳的市場實踐與整體效率水平達到 95%以上。較大的銀行似乎比較小的效率更高，而資源配置低效率成本似乎要比技術效率低成本更重要。他們還記錄正面但不是堅固的技術效率變化的主要原因是生產(chǎn)效率的提高，為中等規(guī)模的銀行和大型銀行的技術變化改進。 Halkos 和Salamouris（2004 年）也使用 DEA 但按照不同的方法，對比以往的研究，通過使用財務比率作為輸出和沒有輸入的措施。根據(jù)正在審議的今年15和 18 銀行之間的樣本范圍。結果表明在1997 年—1999 年期間平均效率寬變化與大小和效率之間的積極關系。此外，還有非系統(tǒng)的關系之間通過私有化公共銀行的所有權轉讓和最后一期的性能。 Apergis 和 Rezitis (2004) 指定超越對數(shù)成本函數(shù)的全要素生產(chǎn)率分析希臘銀行部門、 技術變化率和增長率的成本結構。1982—1997年期間，他們使用中介和生產(chǎn)方法和樣本的六家銀行。這兩個模型表明，重要的規(guī)模經(jīng)濟和技術變化和全要素生產(chǎn)率增長的負年度利率。 Rezitis（2006年）使用相同的數(shù)據(jù)集，但運用的Malmquist生產(chǎn)率指數(shù)和 DEA 測度與分解生產(chǎn)力的增長和技術效率，分別。他還比較 1982年—1992 年和 1993年—1997 年的分時段，并雇用 Tobit回歸來解釋銀行間效率上的差異。結果表明，總體技術效率的平均水平為91.3%，而生產(chǎn)率增長的整個期間平均上升2.4%。生產(chǎn)力的增長在二子期較高，歸因于技術進步與效率是主要驅(qū)動力，直到1992年中的改進。此外，在第二次分時段純效率較高，和規(guī)模效率較低，表明雖然銀行取得較高的純技術效率，但他們搬離最優(yōu)規(guī)模。回歸結果表明大小和專業(yè)化純兩方面產(chǎn)生積極的影響和規(guī)模效率。 3.研究方法 從方法論的角度來看，有幾種方法可以用于檢查的銀行，如隨機前沿分析(SFA)、厚厚的前沿方法 (TFA)、自由的分配辦法（DFA）和DEA 效率。Et al.伯杰（1993 年），伯杰和漢弗萊 (1997年) 和戈達德等人（2001年）提供關鍵討論和比較這些方法在銀行業(yè)的上下文中。 在本研究中，以下幾個最近的研究我們使用 DEA 估計銀行的效率。Dea方法，這是有關對我們的學習，知名的優(yōu)點之一是它特別好與小樣本工程。作為Maudos et al.(2002 年) 指出的那樣，所有的技術測量效率，需要觀測的最小數(shù)目的那個是的非參數(shù)和確定性的 DEA，作為參數(shù)技術指定大量的參數(shù)，使它有必要可用很大數(shù)量的觀測。(p.511）。其他的 DEA 的優(yōu)點是它不需要任何的假設做出關于分布的低效率，它不需要特定功能窗體上的數(shù)據(jù)在確定最有效決策單元 (動車組)。另一方面，DEA 的缺點是它假定數(shù)據(jù)是免費的測量誤差，這是敏感的異常值。 我們只簡要的勾勒 DEA 在這里，而更詳細和技術的討論可以發(fā)現(xiàn)在Coelli et al.(1999 年)、庫珀等人（2000年）和Thanassoulis（2001年）。通過下面的符號是那些用于Coelli（1996 年）和Coelli et al.(1999年)，由于我們使用他們的電腦程序深 2.1 估計效率得分。 DEA 是使用線性規(guī)劃法的生產(chǎn)前沿發(fā)展和的測量效率相對發(fā)達的前沿 （Charnes 等人，1978 年）。通過分段線性組合的樣品（Thanassoulis，2001年）中的所有決策單元的輸入——輸出對應的實際輸入——輸出對應集構造決策單元 (動車組)，在我們的案例銀行，樣品的最佳實踐生產(chǎn)前沿。每個 DMU 被分配一個范圍0 和 1 之間，與分數(shù)等于 1 指示針對其余部分動車組在樣品中的有效決策單元的效率得分。 DEA 可以由假設（CRS）規(guī)模收益不變或變量返回到規(guī)模（VRS）執(zhí)行。在他們的開創(chuàng)性研究，Charnes et al.(1978) 提出了模型輸入的方向，并假定CRS。因此，此模型的輸出是一個指示每個DMU的下CRS的總體技術效率 (OTE) 的分數(shù)。 在更多的技術術語討論 DEA，讓我們假設是有 K 的輸入數(shù)據(jù)和 M 輸出每個決策單元 N （即銀行） 上。為 ith DMU 它們都分別由向量xi和yi表示。K N 輸入的矩陣、 X 和 Y，M N 輸出矩陣表示的數(shù)據(jù)的所有 N 內(nèi)燃動車組。特定的 DMU，CRS 下的輸入為導向的測量計算如下： Minθ,λθ, s.t.?yi +Yλ≥0,θxi ?Xλ≥0, λ≥0 θ≤1 是高效率的標量得分和λis N 1 向量的常數(shù)。如果θ = 的 1 銀行是高效，它位于邊境上，而 ifθ?1 銀行是低效的需要輸入 1?θ 減少各級以到達邊境。線性規(guī)劃是解決 N倍，一次在示例中，每個DMU的和θ的值獲取為每個DMU代表其效率得分。 銀行家et al.（1984 年）建議使用規(guī)模（VRS）變量返回的公司將OTE分解為兩個組件產(chǎn)品。第一次是下 VRS 技術效率或純技術效率 (PTE)，涉及的管理者利用企業(yè)的給定的資源的能力。第二是規(guī)模效率 （SE），指的是利用規(guī)模經(jīng)濟，在哪里生產(chǎn)前沿展品 CRS 點經(jīng)營。CRS 線性規(guī)劃修改，以考慮VRS通過添加N1λ=1，whereN1isaN1向量的部分。根據(jù)VRS取得的成績都高于或等于那些得到下 CRS 和 SE 的技術效率可以得到（即 SE = OTE/PTE）。 ESTIMATING THE TECHNICAL AND SCALE EFFICIENCY OF GREEK COMMERCIAL BANKS: THE IMPACT OF REDIT RISK, OFF-BALANCE SHEET ACTIVIES, AND INTERNATIONAL OPERATIONS 1. Introduction The Greek banking sector has undergone major restructuring in recent years. Important structural, policy and environmental changes that are frequently highlighted by both academics and practitioners are the establishment of the single EU market, the introduction of the euro, the internationalization of competition, interest rate liberalization, deregulation, and the recent wave of mergers and acquisitions. The Greek banking sector has also experienced considerable improvements in terms of communication and computing technology, as banks have expanded and modernized their distribution networks, which apart from the traditional branches and ATMs, now include alternative distribution channels such as internet banking. As the Annual Report of the Bank of Greece (2004) highlights, Greek banks have also taken major steps in recent years towards upgrading their credit risk measurement and management systems, by introducing credit scoring and probability default models. Furthermore, they have expanded their product/service portfolio to include activities such as insurance, brokerage and asset management, and at the same time increased their off-balance sheet operations and non-interest income. Finally, the increased trend towards globalization that focused on the wider market of the Balkans (e.g. Albania, Bulgaria, FYROM, Romania, Serbia) has added to the previously limited international activities of Greek banks in Cyprus and USA. The performance of the subsidiaries operating abroad is expected to have an impact on the performance of parent banks and consequently on future decisions for further internationalization attempts. The purpose of the present study is to employ data envelopment analysis (DEA) and reinvestigate the efficiency of the Greek banking sector, while considering several of the issues discussed above. We therefore differentiate our paper from previous ones that focus on the Greek banking industry and add insights in several respects, discussed below. First of all, we examine for the first time the impact of credit risk on the efficiency of Greek banks by including loan loss provisions as an additional input as in Charnes et al. (1990), Drake (2001), Drake and Hall (2003), and Drake et al. (2006)among others. As Mester (1996)points out “Unless quality and risk are controlled for, one might easily miscalculate a bank’s level of inefficiency; e.g. banks scrimping on credit evaluations or producing excessively risky loans might be labelled as efficient when compared to banks spending resources to ensure their loans are of higher quality” (p. 1026). We estimate the efficiency of banks with and without this input to adjust for different credit risk levels and examine its impact on efficiency. Second, unlike previous studies in the Greek banking sector, we consider off-balance sheet activities during the estimation of efficiency scores. Several recent studies that examine the efficiency of banks, with DEA or stochastic frontier techniques, acknowledge the increased involvement of banks in non-traditional activities and include either non-interest (i.e. fee) income (e.g.Lang and Welzel, 1998; Drake, 2001; Tortosa-Ausina, 2003) or off-balance sheet items (e.g. Altunbas et al., 2001; Altunbas and Chakravarty, 2001; Isik and Hassan, 2003a,b; Bos and Colari, 2005; Rao, 2005) as an additional output. However, despite their increased importance for Greek banks, such activities have not been considered in the past. Again, we estimate the efficiency of the banks in our sample with and without off-balance sheet activities to observe whether it will have an impact on efficiency. Third, we compare the results obtained from the intermediation approach that has been followed in most recent studies of banks’ efficiency with a profit-oriented approach that was recently proposed by Drake et al. (2006)in the context of DEA, and is in line with the approach of Berger and Mester (2003)in the context of their stochastic frontier approach. This allows us to observe if different input/output definitions affect efficiency scores. Fourth, we compare the efficiency scores of Greek banks that have expanded their operations abroad (i.e. international Greek banks, hereafter IGBs), with those of Greek banks whose operations are limited in the domestic market (i.e. purely domestic banks, hereafter PDBs). To the best of our knowledge, no study has undertaken such an analysis for Greece. However, in a study of the Turkish banking sector, Isik and Hassan (2002)found evidence that multinational domestic banks are superior to purely domestic banks in terms of all efficiency measures (i.e. cost efficiency, allocative efficiency, technical efficiency, pure technical efficiency) except for scale efficiency. The conclusions drawn from our study could be useful to the managers of Greek banks or other medium-sized banking sectors that are considering the internationalization of their operations. Fifth, we run regressions to explain the efficiency of banks, an approach that has been followed in only two of the past studies in Greece (Christopoulos et al., 2002; Rezitis, 2006). However, in our case we examine a most recent period that follows the numerous changes outlined above. The rest of the paper is as follows: Section 2 reviews the literature that focuses on the efficiency of the Greek banking sector. Section 3 provides a brief discussion of DEA. Section 4 presents the data and variables. Section 5 discusses the empirical results, and Section 6 concludes the study. 2. Literature reviews Karafolas and Mantakas (1996)use a second-order translog cost function to estimate (for the first time) an econometric form of the costs in the Greek banking sector and investigate economies of scale. Using data for eleven banks from the period 1980 to 1989, they find that although operating-cost scale economies do exist, total cost scale economies are not present. Participation of the dataset in sub-samples by banks’ size (i.e. large and small banks) and time periods (i.e. 1980–1984, 1985–1989) has not altered the results. Finally, the results indicate that technical change has not played a statistically significant role in the reduction of average cost. Noulas (1997) examines the productivity growth of ten private and ten state banks operating in Greece during 1991 and 1992, using the Malmquist productivity index and DEA to measure efficiency. The author follows the intermediation approach and finds that productivity growth averaged about 8%, with state banks showing higher growth than private ones. The results also indicate that the sources of the growth differ across the two types of banks. State banks’ productivity growth is a result of technological progress, while private banks’ growth is a result of increased efficiency. Christopoulos and Tsionas (2001) estimate the efficiency in the Greek commercial banking sector over the period 1993–1998 using homoscedastic and heteroscedastic frontiers. They find an average technical efficiency about 80% for the heteroscedastic model and 83% for the homoscedastic one. They also find that both technical and allocative inefficiencies decrease over time for smaller as well as larger banks. The regression of inefficiency measures against a trend indicates that the improvement in technical and allocative inefficiencies for small banks equal 19.7% and 39.1%, accordingly. The corresponding figures for large banks are 10.4% and 21.1%. Christopoulos et al. (2002)examine the same sample with a multi-input, multi-output flexible cost function to represent the technology of the sector and a heteroscedastic frontier approach to measure technical efficiency. Regression of the efficiency measures over various bank characteristics indicates that larger banks are less efficient than smaller ones, and that economic performance, bank loans and investments are positively related to cost efficiency. In a latter study, Tsionas et al. (2003) use the same sample as in Christopoulos and Tsionas (2001) and Christopoulos et al. (2002) but employ DEA to measure technical and allocative efficiency, and the Malmquist total factor productivity approach to measure productivity change. The results indicate that most of the banks operate close to the best market practices with overall efficiency levels over 95%. Larger banks appear to be more efficient than smaller ones, while allocative inefficiency costs seem to be more important than technical inefficiency costs. They also document a positive but not substantial technical efficiency change which is mainly attributed to efficiency improvement for medium-sized banks and to technical change improvement for large banks. Halkos and Salamouris (2004) also use DEA but follow a different approach, in contrast to previous studies, by using financial ratios as output measures and no input measures. The sample ranges between 15 and 18 banks depending on the year under consideration. The results indicate a wide variation in average efficiency over the period 1997–1999, and a positive relationship between size and efficiency. Furthermore, there is non-systematic relationship between transfer of ownership through privatization of public banks and last period’s performance. Apergis and Rezitis (2004)specify a translog cost function to analyze the cost structure of the Greek banking sector, the rate of technical change and the rate of growth in total factor productivity. They use both the intermediation and the production approach and a sample of six banks over the period 1982–1997. Both models indicate significant economies of scale and negative annual rates of growth in technical change and in total factor productivity. Rezitis (2006) uses the same dataset but employs the Malmquist productivity index and DEA to measure and decompose productivity growth and technical efficiency, respectively. He also compares the 1982–1992 and 1993–1997 sub-periods, and employs Tobit regression to explain the differences in efficiency among banks. The results indicate that the average level of overall technical efficiency is 91.3%, while productivity growth increased on average by 2.4% over the entire period. The growth in productivity is higher in the second sub-period and is attributed to technical progress, in contrast to improvements in efficiency that was the main driver until 1992. Furthermore, during the second sub-period pure efficiency is higher, and scale efficiency is lower, indicating that although banks achieved higher pure technical efficiency, they moved away from optimal scale. The regression results indicate that size and specialization have a positive impact on both pure and scale efficiency. 3. Methodology From a methodological perspective, there are several approaches that can be used to examine the efficiency of banks, such as stochastic frontier analysis (SFA), thick frontier approach (TFA), distribution free approach (DFA), and DEA. Berger et al. (1993), Berger and Humphrey (1997) and Goddard et al. (2001) provide key discussions and comparisons of these methods in the context of banking. In the present study, following several recent studies we use DEA to estimate the efficiency of banks. One of the well-known advantages of DEA, which is relevant to our study, is that it works particularly well with small samples. As Maudos et al. (2002) point out, “Of all the techniques for measuring efficiency, the one that requires the smallest number of observations is the non-parametric and deterministic DEA, as parametric techniques specify a large number of parameters, making it necessary to have available a large number of observations.” (p. 511). Other advantages of DEA are that it does not require any assumption to be made about the distribution of inefficiency and that it does not require a particular functional form on the data in determining the most efficient decision making units (DMUs). On the other hand, the shortcomings of DEA are that it assumes data to be free of measurement error and it is sensitive to outliers. We only briefly outline DEA here, while more detailed and technical discussions can be found in Coelli et al. (1999), Cooper et al. (2000) and Thanassoulis (2001). The notations adopted below are those used in Coelli (1996) and Coelli et al. (1999), since we use their computer program DEAP 2.1 to estimate the efficiency scores. DEA uses linear programming for the development of production frontiers and the measurement of efficiency relative to the developed frontiers (Charnes et al., 1978). The best-practice production frontier for a sample of decision making units (DMUs), in our case banks, is constructed through a piecewise linear combination of actual input–output correspondence set that envelops the input–output correspondence of all DMUs in the sample (Thanassoulis, 2001). Each DMU is assigned an efficiency score that ranges between 0 and 1, with a score equal to 1 indicating an efficient DMU with respect to the rest DMUs in the sample. DEA can be implemented by assuming either constant returns to scale (CRS) or variable returns to scale (VRS). In their seminal study, Charnes et al. (1978)proposed a model that had an input orientation and assumed CRS. Hence, the output of this model is a score indicating the overall technical efficiency (OTE) of each DMU under CRS. To discuss DEA in more technical terms, let us assume that there is data on K inputs and M outputs on each of N DMUs (i.e. banks). For the ith DMU these are represented by the vectors xi and yi, respectively. The K N input matrix , X , and the M N output matrix , Y, represent the data for all N DMUs. The input oriented measure of a particular DMU, under CRS, is calculated as: Minθ,λθ, s.t.?yi +Yλ≥0,θxi ?Xλ≥0, λ≥0 whereθ≤1 is the scalar efficient score andλis N1 vector of constants. Ifθ= 1 the bank is efficient as it lies on the frontier, whereas ifθ?1 the bank is inefficient and needs a 1?θ reduction in the inputs levels to reach the frontier. The linear programming is solved N times, once for each DMU in sample, and a value of θ is obtained for each DMU representing its efficiency score. Banker et al. (1984) suggested the use of variable returns to scale (VRS) that decomposes OTE into a product of two components. The first is technical efficiency under VRS or pure technical efficiency (PTE) and relates to the ability of managers to utilize firms’ given resources. The second is scale efficiency (SE) and refers to exploiting scale economies by operating at a point where the production frontier exhibits CRS. The CRS linear programming is modified to consider VRS by adding the convexityN1’λ= 1, whereN1isaN1 vector of ones. The technical efficiency scores obtained under VRS are higher than or equal to those obtained under CRS and SE can be obtained by dividing OTE with PTE (i.e. SE = OTE/P