運用新共同邊界函數探討多重產出下聯立估計銀行競爭度與成本效率之影響 - 政大學術集成

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(1)國立政治大學商學院金融學系博士論文 Department of Money and Banking College of Commerce National Chengchi University Doctoral Dissertation. 治. 政運用新共同邊界函數探討多重產出下大. 立. ‧ 國. 學. 聯立估計銀行競爭度與成本效率之影響 A New Approach to Jointly Estimating the. ‧. Lerner Index and Cost Efficiency for Multi-. y. Nat. n. al. er. io. sit. output Banks under a Meta-Frontier Framework. Ch. i n U. v. e n g c h i 博士指導教授：黃台心研究生：江典霖撰. 中華民國一○四年七月 July, 2015.

(2) 謝辭隨著博士論文的完成，求學生涯即將劃下句點，心中除了無比喜悅外，要感謝的人實在太多，無法一一道盡，但你們的鼓勵與支持永遠是我前進的最大動力。本論文得以順利完成，首先要感謝指導教授. 黃台心老師的悉心指導，承蒙. 恩師在研究上的啟蒙與教導，奠定了日後研究論文之基礎，恩師對於學術研究之專注與熱誠，更讓學生感佩至極。同時，也要感謝口試委員傅祖壇老師、胡均立老師、陳忠榮老師及林建秀老師在口試期間提供諸多寶貴意見，使得本論文內容更臻充實與完整。. 政治大豪、靜宜、俊凱、雅凱等好友在求學的過程中相互扶持與鼓勵。立. 感謝在博士班就讀期間，系上老師給予紮實的學科訓練。另外，還要感謝信. 最後，感謝我的父母，江仁宏先生與陳慧玲女士，含辛茹苦地養育與栽培我，. ‧ 國. 學. 每每在我遇到挫折時給予我最大的鼓勵與關懷；感謝兩位弟弟，在我求學的路上. ‧. 默默支持；謝謝岳父、岳母對我如同親兒一般地關愛，不斷地給我支持與肯定。. y. Nat. 也要感謝我一生的摯愛─雅雯，若不是老婆的支持與鼓勵，這條路我無法獨自完. er. io. sit. 成，感謝生命中有妳。. 僅以本論文獻給我最親愛的家人以及關愛我的師長與友人。. n. al. Ch. engchi. i n U. v. 江典霖. 謹致於. 國立政治大學金融學系中華民國一○四年七月.

(3) 摘要過去文獻大多使用 Lerner 指數來衡量銀行業之市場競爭度，但在計算過程中有可能出現其值為負之問題。為解決上述問題，本文運用關聯結構函數建立聯立隨機邊界模型，它由銀行成本邊界與兩條產出價格邊界所組成，可以同時衡量放款市場及投資市場之市場競爭度與成本效率。另外，為比較西歐五個國家的銀行市場競爭度與成本效率，本文進一步採用 Huang et al. (2014)所提出的新隨機共同邊界模型，此模型除使用共同成本邊界計算技術缺口比率外，還透過產出價格共同. 政治大部分，可以比較不同國家間的市場競爭程度。立. 邊界衡量潛在 Lerner 指數，進一步拆解成 Lerner 指數與 MC gap ratio (MCGR)兩. ‧. ‧ 國. 學. 關鍵詞：Lerner 指數、成本效率、產出價格邊界、共同邊界、關聯結構函數. n. er. io. sit. y. Nat. al. Ch. engchi. I. i n U. v.

(4) Abstract This paper proposes the copula-based simultaneous stochastic frontier model (CSSFM), composed of a cost frontier and two output price frontiers for the banking sector, in order to measure cost efficiency and market power in the markets of loans and investments. The new Lerner index can be estimated by relying on the simultaneous equations model, consisting of three frontier equations, which avoids obtaining negative measures of the Lerner index. We then apply the new meta-frontier model to. 政治大 countries over the period 1998-2010. 立 The salient feature of our proposed approach is. simultaneously estimate and compare cost efficiency and market power across five. ‧ 國. 學. that it allows for calculating the technology gap ratio on the basis of the cost frontier, as well as evaluating the potential Lerner index from price frontiers, which can be. ‧. decomposed into the country-specific Lerner index and marginal cost gap ratio.. n. er. io. sit. y. Nat. al. i n U. v. Keywords: Lerner index; cost efficiency; output price frontier; meta-frontier; copula methods. Ch. engchi. II.

(5) Contents. 1. Introduction ……………………………………………………………………...1 2. Literature review ………………………………………………………………...4 2.1 Market competition ….……………………………………………………..4 2.2 A meta-frontier model …………………..………………………………….6 3. Methodology …………………………………………………………………….8 3.1 Simultaneous equations model of the cost and price frontiers……………..8. 政治大 Meta-frontier cost function………………………………………………..14 立. 3.2 The copula-based joint PDF and the likelihood function…………………11 3.3. 4. Data description ………………………………………………………………..18. ‧ 國. 學. 5. Empirical results ……………………………………………………………….21. ‧. 5.1 Parameter estimates ………………………………………………………21. y. Nat. 5.2 Various efficiency scores………………………………………………….32. er. io. sit. 5.3 Lerner index ……………...……………………………………………….35 6. Conclusion …………………………………………………………………......40. al. n. v i n Ch References .......……………………………………………………………………..42 engchi U. III.

(6) List of Tables. Table 1. Descriptive Statistics……………………………………………………..20. Table 2. Parameter Estimates of the Copula Method for Multiple Outputs..……...22. Table 3. Parameter Estimates of a Single Equation for Multiple Outputs...............25. Table 4. Parameter Estimates of the Copula Method for a Single Output….. …....27. Table 5. Parameter Estimates of a Single Equation for a Single Output…. ……....29. Table 6. Parameter Estimates of the Meta-Frontier Cost Function for Multiple. 政治大 Parameter Estimates of the Meta-Frontier Cost Function for a Single 立. Outputs……..………………………………………………………….....30. Table 7. Output…………………………………………………………………….31. ‧ 國. 學. Various Efficiency Estimates for Multiple Outputs…...…..………….......33. Table 9. Various Efficiency Scores for a Single Output…….. ……………….…...36. ‧. Table 8. sit. y. Nat. Table 10 Summary Statistics of the Lerner Index Measure……………………..…37. io. n. al. er. Table 11 Estimates of the Potential Lerner Index...………………………………..39. Ch. engchi. IV. i n U. v.

(7) 1. Introduction Over the past two decades, banks in the member countries of the European Union (EU) have faced dramatic systematic changes aiming at liberalization of financial markets. The implementation of the Single Banking Market noticeably lowered barriers to competition among European banks, as mergers and acquisitions (M&As) decreased the number of banks, such as mutual savings, cooperative banks, and commercial banks. The number of foreign banks in each country also increased due to the removal of restrictions to foreign bank entry. Whether these financial reforms intensified banking. 政治大. competition and improved banking performance in the EU has drawn much attention from researchers.. 立. The conventional Lerner index (henceforth, the old Lerner index) is a popular. ‧ 國. 學. indicator of market power, ranging between 0 (perfect competition) and 1 (pure. ‧. monopoly). There are two potential problems associated with the measurement of it in. sit. y. Nat. existing studies. First, it may take a negative value for some observations, which lacks. io. er. economic implications. This arises mainly from the fact that its calculation requires using output price and marginal cost (MC), but those variables are derived from two. al. n. v i n Cestimate separate sources. One has to first standardUtranslog cost function to derive h e n the gchi MC by taking the partial derivative of the cost function with respect to output quantity.. Output price is directly computed as the ratio of total revenues to the corresponding output measure. Second, the estimation of market power usually fails to consider potential cost inefficiency, which is likely to severely bias the parameter estimates of the cost function and the subsequent calculation of MC. See, for example, Berg and Kim (1998), Delis and Tsionas (2009), and Koetter and Poghosyan (2009). To our knowledge, no existing works address the correlation between market power and cost inefficiency, except for Huang and Liu (2013) who propose the use of the copula-based simultaneous stochastic frontier model (CSSFM), which is composed of 1.

(8) two equations, i.e., a cost frontier and a price frontier. 1 The estimation of the cost frontier allows one to obtain consistent parameter estimates and assess the cost efficiency. Moreover, the estimation of the price equation leads to the evaluation of the Lerner index for each bank. The measures of cost efficiency and the Lerner index are yielded simultaneously. The salient feature of this procedure is that the resultant Lerner index measure will be non-negative for each bank, in addition to taking the possible cost inefficiency into account, which avoids causing inconsistent coefficient estimates of the cost function. Note that the adoption of the joint estimation procedure is able to. 政治大 This paper extends Huang and Liu (2013) by allowing banks to produce multiple 立. improve the efficiency of the parameter estimates.. outputs, therefore escaping from the aggregation problem. Our model requires the joint. ‧ 國. 學. estimation of a cost frontier with two output price frontiers, corresponding to the. ‧. outputs of loans and investments, respectively. As a result, the market power of each. sit. y. Nat. output can be individually assessed for each bank. The new Lerner indices of loans and. io. er. investments are also immune from getting negative values, since these two measures are internally built into the simultaneous equations model and treated as if they are. n. al. inefficiency terms.. Ch. engchi. i n U. v. This paper also generalizes the three equations model for a single country to the stochastic meta-frontier framework of Huang et al. (2014), which can then be used for comparisons of bank efficiency and market power across countries. We estimate the stochastic meta-frontier cost and two MC functions, which allows us to evaluate TGRs and the two MC gaps for each bank. This study also employs the mixed approach of Battese et al. (2004) and O’Donnell et al. (2008) to assess TGRs for the purpose of a comparison.. 1. Huang and Liu (2013) implicitly assume that each bank produces a single output, which may incur the aggregation problem and appears to be not consistent with reality. 2.

(9) Since our price frontiers relate output prices to the corresponding MCs, we can measure the gaps between MCs of the group frontiers and those of the meta-frontiers, which is analogous to TGR. The MC gap ratio (MCGR) of an output is thus defined as the ratio of the MC gap to the corresponding output price, which represents a bank’s potential profitability. An efficient bank operating under a superior production technology has a higher value of MCGR and can reduce its production cost and MC more than a less efficient bank that has a lower value of MCGR, so as to earn higher profit in a competitive market.. 政治大 context of multiple outputs, together with the new approach to calculating that index. 立. The purpose of this paper is three-fold. First, we estimate the Lerner index in the. This appears to be new in the literature. Second, a joint estimation procedure of market. ‧ 國. 學. power and cost efficiency is applied to yield the Lerner indices of various outputs for. ‧. individual banks, which solve the problem of a negative value in the Lerner index.. sit. y. Nat. Finally, we employ a new two-step SFA to simultaneously estimate the meta-frontier. io. er. cost function and two MC frontiers, allowing us to compare bank efficiency, market power of different outputs, and profitability across countries, respectively.. al. n. v i n C has follows. SectionU2 briefly reviews some relevant The rest of the paper is organized engchi. studies. Section 3 formulates the simultaneous equations suitable for estimating the cost and price frontiers, followed by deriving their joint probability density function (PDF). using the copula method. We also introduce the meta-frontier cost function and MCGR. Section 4 briefly describes the data and variable definitions. Section 5 conducts the empirical study, while the last section concludes the paper.. 3.

(10) 2. Literature review 2.1 Market competition There are two main approaches to measuring the market power of the banking industry in the literature:. structure and non-structure. The structure approach to. measuring market power is based on the Structure-Conduct-Performance (SCP) paradigm, which posits that market structure affects a bank’s behavior, which in turn determines its performance. Banks in a more concentrated market are inclined to be more collusive and thereby are less efficient. Several market concentration measures. 政治大 concentration ratios for the largest firms (CR ratios), and the Herfindal-Hirschman 立. have been used to proxy for the measure of market competition, including market shares,. index (HHI). See, for example, Berger et al. (2004), Beck et al. (2006), and Alegria and. ‧ 國. 學. Schaeck (2008). The structure approach has some disadvantages. For example, the CR. ‧. ratio ignores the influences of smaller firms, and HHI is subject to the control of a few. sit. y. Nat. large firms. In addition, market concentration measures fail to consider the effect of. io. er. regulation on a banking sector.. According to the new empirical industrial organization (NEIO) theory, the non-. n. al. Ch. structure method relies on the direct observation. engchi. v i n of a firm’s U. behavior and uses. econometric models to estimate the degree of competition. Bresnahan (1982) and Lau (1982) examine banks’ behavior on an aggregate level and estimate the average conjectural variation of banks, which lies between zero (perfect competition) and unity (monopoly). Panzar and Rosse (1987) are the first to develop the H-statistic (henceforth the PR model), based on the idea that market power can be gauged by the extent to which changes in input prices are reflected by the equilibrium revenues received by a specific firm. The H-statistic is equal to the sum of the elasticities of the reduced-form (log) gross revenues with respect to the (log) input prices. A negative value of the Hstatistic corresponds to monopoly or perfect collusion, while a unitary value of it 4.

(11) implies perfect competition. If its value ranges between zero and unity, then the market under study is either oligopoly or monopolistic competition. See, for example, Molyneux et al. (1994, 1996), De Bandt and Davis (2000), Bikker and Haff (2002), Casu and Girardone (2006), and Turk-Ariss (2009). The PR model is also subject to two drawbacks:. (1) one can only estimate a single H-statistic for the whole sample,. rather than for each observation; (2) Bikker et al. (2012) prove that the H-statistic fails to properly measure the degree of market competition. Boone (2007) is the first to develop the Boone indicator, defined as the percentage. 政治大 that in a more competitive market, firms are punished more harshly (in terms of profits) 立. fall in profits due to a percentage increase in marginal costs. The essential insight is. for being inefficient. The larger the absolute value is of the Boone indicator, the. ‧ 國. 學. stronger is the competition. This indicator is associated with the efficiency hypothesis. ‧. (EH), which asserts that more efficient firms have better performance and gain larger. sit. y. Nat. market shares. See, for example, Boone (2007) and Leuvensteijn et al. (2011, 2013).. io. er. Ever since 2000, many researchers adopt the conventional Lerner index to explain the competition behavior among banks. This index reflects a bank’s ability to set its. al. n. v i n output price over MC, which isCpositively related to U h e n g c h i the firm’s market power. Many empirical researchers have applied the Lerner index to investigate the degree of. competition, particularly in banking industries of various countries. The higher the Lerner index is, the larger the difference between output price and MC is, and hence the stronger the bank’s market power is. Under the condition of perfect competition, banks must set output price to be equal to MC, so that the Lerner index is equal to zero. Conversely, a monopolist can charge an output price over MC and in the extreme case the Lerner index is equal to unity. If the value of the Lerner index lies between zero and unity, then the market is either oligopoly or monopolistic competition. Different from the H-statistic and Boone indicator, the merit of the Lerner index is that one can 5.

(12) evaluate one value for each sample. See, for example, Angelini and Cetorelli (2003) for Italian banks, Maudos and de Guevara (2004, 2007) and Carbó et al. (2009) for the cases of European countries, Berger et al. (2009) for the case of 23 different countries, Turk-Ariss (2010) for developing countries, Williams (2012) for Latin American banks, and Koetter et al. (2012) for U.S. banks.. 2.2 A meta-frontier model There are two main approaches to measuring a firm’s technical efficiency in the. 政治大 and van den Broeck (1977) are the first to introduce the former, by measuring the 立. literature:. parametric and non-parametric. Aigner et al. (1977) as well as Meeusen. deviation of a firm’s actual output (cost or profit) level from that of the best-practice. ‧ 國. 學. firm. Numerous researchers have popularly applied this method to evaluate banks’. ‧. efficiency scores, e.g., Huang (2000), Altunbas et al. (2001), Kasman and Yildirim. sit. y. Nat. (2006), Fitzpatric and McQuinn (2008), Berger et al. (2009), Koutsomanoli-Filippaki. io. er. et al. (2009), and Akhigbe and Stevenson (2010), to name a few. Charnes, Cooper and Rhodes (1978) are the first to propose the latter approach, which involves mathematical. al. n. v i n C h constant returnsUto scale (CRS). Banker, Charnes programming techniques and assumes engchi. and Cooper (1984) suggest an extension of the CRS DEA model to take variable returns. to scale (VRS) into account. The previous studies on the comparisons of technical efficiency among different groups (or countries) do not take the differences in regulation, supervision, and technology adopted into account, which affect banks’ efficiency. To compare the efficiency of banks across countries, Altunbas et al. (2001) and Vennet (2002) estimate a common frontier for all banks in different countries and compute efficiency scores against the common frontier. This procedure implicitly assumes that all sample banks from different countries undertake the same production technology. Such a strong 6.

(13) assumption may result in undesirable consequences on parameter estimates and the measures of scale, scope, and technical efficiencies. Battese et al. (2004) and O’Donnell et al. (2008) solve the foregoing problem by proposing a meta-frontier production function model (henceforth, the old meta-frontier), which is constructed on the basis that all firms have potential access to the same production technology, but each may choose a different process, relying on specific circumstances, such as regulation, environments, production resources, and relative input prices. To find the meta-frontier production function, they propose a mixed two-. 政治大 then the mathematical programming techniques are employed to estimate the meta立 step procedure, in which SFA is first used to estimate the group-specific frontier and. frontier that is used to compute the technical gap ratio (TGR). See, for example, Bos. ‧ 國. 學. and Schmiedel (2007), Huang et al. (2010), and Chen and Yang (2011).. ‧. There are two potential disadvantages for estimating the meta-frontier model in the. sit. y. Nat. two-step mixed approach. The meta-frontier estimators have no statistical properties,. io. er. as these results derive from the second-step estimation with mathematical programming techniques. Moreover, potential production environment conditions fail to be taken into. al. n. v i n C h The difficultyU is addressed by Huang et al. account in the meta-frontier estimation. engchi. (2014), who propose a novel two-step SF approach to estimating the meta-frontier production function (henceforth, the new meta-frontier). The main difference between. the new two-step SF approach and that of Battese et al. (2004) and O’Donnell et al. (2008) is that the former’s second-step estimation of the meta-frontier is under the SF framework rather than depending on the mathematical programming technique. The new stochastic meta-frontier (SMF) regression has the following advantages. The estimated parameters of SMF using the conventional maximum likelihood (ML) method have the usual statistical properties. Moreover, the technology gap ratio (TGR) can be specified as a function of potential exogenous variables to capture group-specific 7.

(14) environmental differences.. 3. Methodology 3.1 Simultaneous equations model of the cost and price frontiers We specify here a bank’s cost frontier as the standard translog form: 3. 3. ln C   0   k ln Yk    m ln Wm k 1. m 1. 3 3  1    kh ln Yk ln Yh    mn ln Wm ln Wn  2  k 1 h 1 m 1 n 1  3. 3. (1). 3 3 3 3 1    km ln Yk ln Wm  k ln YkT   m ln WmT  1T   2T 2   3 2 k 1 m 1 k 1 m 1 ,. 立. 政治大. where C represents the bank’s actual expenditures, Yk (k  1,2,3) stands for the. ‧ 國. 學. output quantities, including loans, investments, and non-interest income, and Wm. ‧. (m  1,2,3) corresponds to the input prices of labor, physical capital, and borrowed.  3  v3  u3 is the. sit. y. Nat. funds, respectively. Notation T represents the time trend, and. n. al. er. io. composite error term, in which random variable v3 ~ N  0,  v23  is assumed to be. i n U. v. statistically independent of the technical inefficiency term of u3 ~ N  0,  u23  .. Ch. engchi. According to the microeconomic theory, we impose some restrictions, such as symmetry and homogeneity of degree one in input prices, on (1) before performing any estimation. Once the unknown parameters of (1) are estimated, we can derive the implied MC function for the two outputs by taking the partial derivatives of (1) with respect to loans ( Y1 ) and investments ( Y2 ), respectively.2 We write the implied MC functions as:.  Y  , the corresponding Lerner index fails to be computed. Therefore, we do not derive the implied MC of Y3 . 2. Due to the lack of price information for non-interest income. 8. 3.

(15) MC1  1  11 ln Y1  12 ln Y2  13 ln Y3  12 ln W2  13 ln W3  1T . C Y1. C MC2   2   22 ln Y2  12 ln Y1   23 ln Y3   22 ln W2   23 ln W3  2T  Y2 .. (2). The old Lerner index is defined as the ratio of the disparity between a firm’s output price and its MC to the output price:. LOld _ Y1  Old. L. P1  MC1 P1. (3). P  MC2 _ Y2  2 P2 .. 政治大. It is noteworthy that the so-derived measures of the Lerner index are likely to be. 立. negative for some observations, indicating that these firms set their output prices below. ‧ 國. 學. MC. This is inconsistent with firms’ profit maximization behavior. In compliance with the principle of profit maximization, a profit maximizing firm. ‧. decides to produce an output level at which the marginal revenue (MR) of selling the. y. Nat. io. sit. last unit of the output is equal to that unit’s MC. Therefore, the following inequality. er. must hold in equilibrium:. n. a l P  MR  MC v (4) i . n Ch e n equality g c h ibyUadding a non-negative random Equation (4) can be transformed into an. variable:. Pi  MCi   i ,  i  vi  ui , i = 1, 2,. (5). Where vi ~ N 0,  vi2  , i  1, 2, denotes the random disturbance, capturing the statistical noises uncontrollable by firms, and ui ~ N 0,  ui2  , i  1, 2, measures the gap between the output price and MC. Both vi and ui are mutually independent of each other. Variable ui , i  1, 2, can be treated as if a firm is technical inefficient. 9.

(16)  . Hence, it can be estimated by taking the conditional expectation of E ui  i , which is similar to the computation of technical inefficiency. The larger the discrepancy is, the greater ability the firm has to exercise its market power to set output price farther over MC, and the higher profit it can earn. We propose to simultaneously estimate (1) and (5) under the assumption that.  i 's ,. i = 1, 2, 3, have a joint distribution. This is justified by the fact that bank managers usually make decisions on the employment of inputs together with output prices and quantities. Such an interrelationship can only be embodied under the framework of a. 治政 simultaneous equations model. Since the composite error 大of 立.  i is a skewed normal,. their joint PDF is difficult to be deduced. Following Lai and Huang (2013), our paper. ‧ 國. 學. suggests using copula methods to overcome the problem. See the next sub-section for. ‧. details. Differing from the traditional Lerner index of (3), our new Lerner index is. y. sit. n. al. er. io. construction.. Nat. calculated by Lnew  E  u   P  0, which must be non-negative since u  0 by. i n U. v. Compared with the conventional one, the new Lerner index has several advantages.. Ch. engchi. First of all, the new Lerner index is certified to be non-negative since we impose the condition on the inequality P  MC of (5). Second, we employ a joint estimation of market power and cost efficiency at the individual bank level by the ML method so that the estimated parameters are more efficient and at the same time the new Lerner index takes cost efficiency into account. Finally, the random shocks have less effect on the new Lerner index, which allows for the presence of vi in separation from MC.. 10.

(17) 3.2 The copula-based joint PDF and the likelihood function Equations (1) and (5) form a three equations seemingly unrelated stochastic frontier model. Let  j   Tj , vj , uj  , j = 1, 2, 3, be a vector of parameters in the T. j th. regression equation, and Fj  j   F  j ; j  is the marginal cumulative distribution function (CDF) of the composite error  j . There are two ways of deriving the approximate CDF of  j . One of them is proposed by Greene (2003, 2010), who uses the simulated ML to approximate the integration in computing Fj  j  . Alternatively,. 政治大. the current paper derives a closed-form formula to approximate CDF of  j , based on. 立. Tsay et al. (2013). According to Sklar’s theorem, the joint CDF of the composite errors. ‧ 國. T. 學.    1 ,  2 ,  3 . is expressed as:. ‧. F  1 ,  2 ,  3   C  F1  1  , F1  2  , F3  3  ;   ,. (6). Nat. io. sit. y. where C () is a unique copula function if F1   to F3   are all continuous, and. er.  is a 3  3 matrix containing all of the dependence parameters between those. n. al. Fj   's , j = 1, 2, 3.. We yield the joint PDF of. Ch . engchi. i n U. v. by taking the partial derivatives of (6) with respect to. 1 ,  2 ,  3  , which is expressed as follows: f  1 ,  2 ,  3   c  F1  1  , F2  2  , F3  3  ;     f j  j  . 3. j 1. . . Here, c F1  1  , F2   2  , F3   3  ;  .  3C  F1  1  , F2   2  , F3   3  ;   F1  1  F2   2  F3   3 . (7). is the copula. density function and f j   j  , j = 1, 2, 3, is the marginal PDF. Following Lai and Huang (2013), this article selects the Gaussian copula to derive the 3-dimensional CDF, which 11.

(18) takes the form:. . C F1 1 , F2  2 , F3  3  ;     3  1 F1 1 ,  1 F2  2 ,  1 F3  3  ;  . 1. .  F1  1    F2  2    F3  3  1. 1. . . . . . . . 1. 2 3 / 2 . e 1/ 2.   . 1  Z T  1 Z 2. dz1dz2 dz3 (8). Here,  1 () is the inverse of CDF of the standard normal distribution function, and  3 is a 3-dimensional multivariate standard normal distribution function with the. 3  3 correlation matrix  , whose diagonal elements are all equal to unity, and the off-diagonal elements are the correlation coefficients between the ith and jth variables.. 政治大. The corresponding Gaussian copula density function of (8) is written as:. 立. ‧ 國. . . . . 1 1 1 where    F1  1  ,  F2   2  ,  FJ   3 . e. . . ,. . T. (9). , and I 3 is a 3  3 identity. ‧.  . . 1/2. 1   T  1  I3  2. 學. c  F1  1  , F2   2  , F3   3  ;   . 1. sit. y. Nat. matrix. The joint PDF of the composite errors of  can be expressed as follows:. f  1 ,  2 ,  3   c  F1  1  , F2   2  , F3   3  ;     f j  j  3. n. 1/2. er. io. al 1  Ce  h. j 1. v   f  n i engchi U . . 1   T  1  I 3  2. 3. j. j 1. (10). j. The foregoing derivation leads us to get the log-likelihood function of the simultaneous SFA model for a sample of N observations as: ln L    ln cF1  1i , F2  2i , F3  3i  ;     ln f j  ji  N. N. i 1. 3. i 1 j 1. . . 3 N 1   ln     iT  1  I 3    ln f j  ji  2 2 i 1 i 1 j 1 N. N. (11). Here,   1T , 2T , 3T ,  T  , and  j are the vectors of unknown parameters of the T. SFA regression equation.. 12. j th.

(19) The ML estimation of (11) requires using f j  ji  and its cumulative distribution function F j  ji  . We already know PDF of f j  ji  as:. f j   ji   where  j   uj  vj and  j. 2   ji   j   j.   v2  u2 j j.    ji  j       j.   , j  1, 2,3 , . (12). . Since f j  ji  has no closed-form, its CDF of. F j  ji  cannot be exactly deduced. We obtain an approximate closed-form formula for. CDF of F j  ji  in line with Tsay et al. (2013). The CDF can be alternatively written. 政治大. as:. 立F Q  . . Qi. . f  i d  i . 2. . I (Qi ) .. (13). 學. Here, I Qi  is defined by:. Qi a i I (Qi )      ( ) d  (b i ) d  i ,     . ‧. ‧ 國. i. (14). sit. y. Nat. in which a   /  and b  1/  . The derivation of the approximate function to. al. n.  bQ erf  i  2 I app  Qi   2b. er. io. I Qi  , I app  Qi  , is more involved and tedious, which can be expressed as:.   1  sign  Q  i   2. Ch. engchi.    a 2 c12 1  1  erf exp   4  b 2  a 2c2    4 b 2  a 2c2  . i n U. v.  ac1  2Qi  b 2  a 2c2  sign  Qi      2 2   2 b  a c 2  . (15). Here, the error function, erf ( z ), is defined as:. erf  z . 2. . z. 0. et 2 dt  2. 0. 2z.   t dt ,. (16). and the sign function takes the values of sign(Qi )  1, 0, or 1 , as Qi >, =, or < 0. Tsay et al. (2013) find the two constants 13. c1  1.09500814703333. and.

(20) c2  0.75651138383854 by simulation, which is able to ensure that the error function. erf ( z ). can. be. precisely. approximated. by. the. function. of. g ( z )  1  exp  c1z  c2 z 2  , for z  0 . When performing the empirical study, I  Qi  and F  Qi  are replaced by their respective approximation functions, i.e., I app  Qi  and Fapp  Qi  , which is defined by:. Fapp Qi  . 2. . I app Qi  .. (17). Tsay et al. (2013) conduct Monte Carlo simulations and demonstrate that F  Qi  can. 立. be accurately approximated by. 政治大 F Q  . app. i. ‧ 國. 學. 3.3 Meta-frontier cost function. ‧. The joint estimation of equations (1) and (5) for each country allows one to calculate. sit. y. Nat. the country-specific cost efficiency and the new Lerner index, Lnew . We now turn our. n. al. er. io. attention to the meta-frontier and briefly introduce the mixed approach of Battese et al.. i n U. v. (2004) and O’Donnell et al. (2008). They first use the ML estimation to estimate each. Ch. engchi. country-specific frontier in (1) and then the meta-frontier function is obtained by solving the liner programming (LP) problem or the quadratic programming (QP) problem.3 Using mathematical programming techniques to compute the meta-frontier leads to two potential problems, as mentioned above. To disentangle these problems in the second step we extend the new meta-frontier model of Huang et al. (2014), which is a single equation framework, to the simultaneous equations SFA model containing three equations, rather than two equations like by Lai and Huang (2013). We now briefly. 3. Readers can refer to Battese et al. (2004) and O’Donnell et al. (2008) for a more detailed description. 14.

(21) describe below the construction of the meta-frontier cost and the corresponding MC regression equations. In the first step we estimate each country-specific frontier. Equation (1) can be rewritten as:. ln C  ln f j   3 j，j  1,. ,5 ,. (18). where j denotes the j th group. In the second step, the relation between country j’s frontier and the meta-frontier cost function is:. ln f j  ln f M  U M ,. (19). 政治大 where the non-negative component U  0 represents the gap between the country立 M. ‧ 國. 學. specific frontier and the meta-frontier and is assumed to be a half-normal random variable. In this manner, the meta-frontier envelops all country-specific frontiers from. ‧. al. ,. n. which yields:. Ch. sit. io.  ln fˆj  ˆ3 j. engchi. ln fˆ j  ln f j   3 j  ˆ3 j  V M ,. (20). er. Nat. ln C  ln f j   3 j. y. below. Let ln fˆ j be the fitted values of ln f j . Equation (18) can be rewritten as:. i n U. v. (21). where V M denotes the estimated error. Plugging (21) into (19), we obtain:. ln fˆj  ln f M  V M  U M .. (22). Equation (22) is similar to the conventional stochastic frontier model and hence is called the stochastic meta-frontier regression equation. Moreover, the symmetric error. . . V M ~ N 0 ,  vM 2 and U M ~ N 0 , uM 2  are assumed to be independent of each other. We can deduce the corresponding meta-frontier MC function of output k, MCkM , by 15.

(22) taking the partial derivative of the meta-frontier cost function, f M , with respect to the kth (k =1, 2) output quantity. Note that MCkM should envelop all individual countryspecific MC frontiers of the same product. Country j’s MC frontier of output k, MCkj , is associated with MCkM as follows:. MCkj  MCkM  ukM ,. (23). where ukM  0 stands for the gap between country j’s MC frontier and MCkM . We. 政治大 P  MC  v  u .. re-write (5) by deleting the subscript i and adding the subscript k and superscript j as:. 立. j. k. j k. j k. j k. (24). Pk j  MCkM  ukM  vkj  ukj  MCkM  vkj  U kM. ,. ‧. ‧ 國. 學. Substituting (23) into (24), we get:. (25). io. sit. y. Nat. where U kM  ukM  ukj stands for the overall gap between the output price and the meta-. er. frontier MC. It is important to note that U kM consists of two components, where ukj. al. n. v i n C hprice and MC, which is the difference between the output can be used to calculate the engchi U. new Lerner index of output k for the jth country, and ukM is the MC gap that will be. used to compute the MC gap ratio (MCGR). In the second step, we simultaneously estimate the three stochastic frontiers of (22) and (25). The estimation of the meta-frontier cost function enables one to compute the technology gap ratio (TGR). TGR evaluates the size of the technology gap for country. j , whose currently available technology lags behind the potential technology available for all countries, as illustrated by the meta-frontier cost function. The larger the value of TGR is, the more advanced the technology is that the country has adopted. We define 16.

(23) the overall technical efficiency, CEˆ M , against the meta-frontier cost function by the product of TGR and country-specific technical efficiency, CEˆ : ˆ  CEˆ CEˆ M  TGR. .  E e U.  . Uˆ M  Vˆ M  E e u3 ˆ3. M. . .. (26). Since the estimated TGR and CE are always less than or equal to unity, CEˆ M must be less than or equal to unity. The estimation of the meta-frontier price equation enables one to the measure the potential Lerner index of output k for each bank, which is defined by the ratio of the. 政治大 specifically, the potential Lerner 立 index is defined as: p k, j. Lerner. . . E U kM vˆkj  Uˆ kM Pk j.  0.. 學. (27). ‧. ‧ 國. gap between that output price and MCkM , i.e., U kM , to the same output price. More. We calculate MCGR by the ratio of the MC gap to the output price, which signifies a. Pk. n. al. . Ch. j. engchi. .. er. io. MCGR . E ukM | vˆkM  Uˆ kM. sit. y. Nat. bank’s potential profitability:. i n U. v. (28). An efficient bank adopting a superior production technology can reduce its production cost and MC more than a less efficient bank, so as to earn more profit in a competitive market. The more advanced the technology a country adopts, the lower the meta-frontier MC is and the larger the MC gap, and so the more potential profit a bank can make. Therefore, Equation (27) can be viewed as the sum of the new Lerner index and the MC gap ratio and written as follows: p k, j. Lerner. . . E ukj uˆkj  vˆkj Pk. j.   E u. M k.  Lnew k , j  MCGR 17. vˆkj  Uˆ kM Pk. j.  .. (29).

(24) Note that. Lnew k, j. is empirically calculated in the first estimation step after. simultaneously estimating (1) and (5) for country j.. 4. Data description We compile data mainly from the balance sheets and income statements of the Bankscope database for five west European countries:. France, Germany, Italy,. Luxembourg, and Switzerland. We exclude the UK and Spain due to the lack of complete output price information. All the accounting data are automatically. 政治大 consumer price indices of the 立individual countries with base year 2005. The data consist. transformed into nominal US dollars by the databank and are further deflated by the. ‧. ‧ 國. 1998 to 2010.. 學. of 725 commercial banks with a total of 4455 bank-year observations spanning from. Following the intermediation approach, we identify three inputs and three outputs.. sit. y. Nat. The three inputs are labor, physical capital, and borrowed funds. The price of labor is. n. al. er. io. calculated as the ratio of personnel expenses to total assets (TA).4 The price of physical. i n U. v. capital is identified as the ratio of non-interest expenses minus personnel expenses to. Ch. engchi. physical capital. The price of borrowed funds is defined as the ratio of interest expenses to all deposits and borrowed money. The total costs are the sum of the foregoing three items of expenditures. Total loans and investments are regarded as the conventional outputs. The ratios of their respective revenues to the corresponding outputs are defined as their prices. The interest income is the sum of loan income and investment income. We define noninterest income as the third output, as it represents a universal bank’s degree of product. 4. Since the data on the number of employees are either missing or unavailable for many sample banks, the item of total assets is used as a proxy for the number of employees. Altunbas et al. (2001) and others utilize the same definition. 18.

(25) diversification and becomes a critical source of revenue for modern commercial banks. Obviously, its price is unavailable. We also want to investigate the Lerner index of TA, whereby the price of TA is defined as the ratio of total revenues (interest and noninterest income) to TA. Table 1 reports the descriptive statistics for all variables in the five sample countries, showing that there are considerable differences in the means and standard deviations and implying that the sample banks in the five countries are likely to employ heterogeneous inputs to produce differential outputs by different production technologies. It is thus invalid to directly compare the performance of those banks in. 政治大. different countries as they have distinct benchmarks. The meta-frontier model may be preferable.. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 19. i n U. v.

(26) FRA. GER. ITA. LUX. SWI. 11052.8449. 14320.4684. 10294.6162. 1748.1182. 4292.1085. (43293.5013). (60359.8403). (29568.0407). (3504.0127). (29698.8148). 18175.7162. 17858.7505. 8600.7110. 5527.8754. 5640.2340. (81581.7124). (90047.6587). (35852.4283). (9888.0329). (50364.0868). 268.0048. 294.3079. 258.0840. 57.2079. 162.6026. revenue*. (997.9901). (1442.5626). (746.6936). (94.6002). (1307.9367). Labor (TA). 36029.1722. 38585.2952. 20263.7810. 7595.4882. 11428.0054. (160851.2775). (198532.2096). (64249.2606). (13034.3419). (92553.3368). 79.5134. 71.9536. 132.5649. 21.2113. 57.6915. (311.2670). (262.8593). (394.9989). (52.4468). (381.9769). 23386.7879. 26441.5981. 11915.3287. 6314.1950. 9337.8418. (98542.5932). (122127.8204). (33825.2851). (10624.2919). (75852.4448). 0.0164. 0.0148. 0.0144. 0.0069. 0.0208. (0.0124). (0.0149). (0.0091). (0.0082). (0.0223). 5.9739. 5.4659. 4.4915. 5.2759. 3.0691. (9.7181). (8.7270). (9.0339). (8.1748). (7.1345). 0.0456. 0.0394. 0.0342. 0.0529. (0.0524). (0.0346). (0.0437). ‧. 0.0232. (0.0595). (0.0203). 0.0669. 0.0412. 0.0541. y. Table 1 Descriptive Statistics. 0.2749. 0.0733. (0.0471). (0.0245). (0.0250). io. Variables. 0.0190. 0.0451. Loans*. Investments* Non-interest. Physical capital*. capital Price of borrowed funds. Price of. Nat. Price of loans. 0.0857. n. al. investments. (0.1851). Price of total. 0.0763. assets Total costs*. Number of. (0.0394). 學. Price of physical. 立. sit. Price of labor. 政治大. (0.2190). er. funds*. ‧ 國. Borrowed. i n C h0.0716 i U e n g c h0.0611 (0.0152). (0.0501). 0.0155. v (0.0169). (0.0969) 0.0136 (0.0105). 0.0653. 0.0709. (0.0562). (0.0262). (0.0342). (0.0500). 1461.9805. 1395.9295. 803.5323. 442.8657. 503.1359. (5327.7241). (5795.9485). (2441.4952). (912.9426). (4674.8493). 163. 132. 166. 101. 163. 966. 1013. 648. 552. 1276. banks Number of observations Note: 1. * means measured by millions of real US Dollars with base year 2005. 2. Numbers in parentheses are standard deviations.. 20.

(27) 5. Empirical results This paper employs CSSFM, consisting of a cost frontier and two output price frontiers of loans and investments, in the first and second steps to estimate the individual group frontiers and the meta-frontiers for the five EU commercial banks. The meta-frontier cost function model divides CE M into CE and TGR, which permits us to gain further insights on the sources of a bank’s cost inefficiencies. This provides more regulatory and managerial implications to government authorities, business consultants, and bank managers. The meta-frontier price function can be similarly split into two parts:. 政治大. the Lerner index measure and MCGR, which reflects a bank’s potential. profitability.. 立. ‧ 國. 學. 5.1 Parameter estimates. ‧. We jointly estimate equations (1) and (5) for each individual country to obtain the. sit. y. Nat. parameter estimates of the group frontiers with the results listed in Table 2. We also. io. er. estimate the single cost frontier without considering the price equation, presenting the results in Table 3. The number of parameter estimates of the copula method in Table 2. al. n. v i n C h level exceeds that attaining at least the 10% significance of the single cost frontier in engchi U. Table 3. This arises from the fact that the coefficient estimates are more efficiently estimated by the copula method than the single equation, since more information, such as the potential dependence among equations, is used in estimation. It is worth emphasizing that the dependence parameters  are significant in these countries, indicating that the dependence between the production cost and output prices indeed exists, confirming that the simultaneous equations model of (1) and (5) is preferable to the single equation model. As noted before, the exclusion of . may cause. inconsistent estimated parameters, and the subsequent measures of technical efficiency and the Lerner index measures are also misleading. The parameter estimates are found 21.

(28) to satisfy the regularity conditions, as required by the microeconomic theory on the cost function in Tables 2 and 3.. Table 2 Parameter Estimates of the Copula Method for Multiple Outputs Variables. FRA. GER. ITA. LUX. SWI. constant. 3.4579*** (0.0300) 0.3908*** (0.0200) 0.6256***. 7.4594*** (0.0362) 0.0638*** (0.0086) 0.0546***. 5.8336*** (0.0240) 0.1110*** (0.0291) 0.0728***. 2.3742*** (0.0226) 0.4850*** (0.0658) 0.2389***. 6.2849*** (0.0234) 0.3181*** (0.0132) 0.0091. (0.0299) -0.1786*** (0.0520). (0.0094) -0.055 (0.0719). (0.0226) 0.4599*** (0.0715). (0.0671) 0.2148*** (0.0888). (0.0059) 0.0701* (0.0419). (0.1269) 1.2428*** (0.1591) 0.0079*** (0.0009) 0.0124***. 0.0883 (0.0993) 1.1959*** (0.1302) 0.0120*** (0.0011) 0.0460***. 0.1340 (0.0873) 0.5961*** (0.1197) 0.0417*** (0.0063) 0.1419***. -0.0939 (0.0692) 0.3874*** (0.0650) 0.0596*** (0.0017) 0.0078***. ln y1  ln y3 ln y2  ln y3. ln w2  ln w2 ln w3  ln w3 ln w2  ln w3. ln y1  ln w2. al. n. ln y1  ln y2. (0.0019) (0.0013) (0.0025) (0.0076) (0.0008) 0.0417*** 0.0984*** 0.0165** -0.0125 0.0972*** (0.0072) (0.0074) (0.0079) (0.0096) (0.0041) -0.0785*** -0.0100*** -0.0327*** -0.0945*** -0.0048*** (0.0025) (0.0008) (0.0023) (0.0057) (0.0007) -0.0070** -0.0005 0.0222*** 0.0506*** -0.0520*** (0.0033) (0.0009) (0.0028) (0.0069) (0.0023) 0.0194*** -0.0041*** -0.0090*** -0.0431*** -0.0035*** (0.0042) (0.0014) (0.0016) (0.0062) (0.0005) 0.0534*** -0.0246 0.0311** -0.0090 0.0318*** (0.0097) (0.0242) (0.0163) (0.0125) (0.0095). io. ln y3  ln y3. Nat. ln y2  ln y2. ‧. ln y1  ln y1. 學. ln w3. 立0.3649***. -0.1015 (0.0639) 0.2153*** (0.0680) 0.0723*** (0.0024) 0.0410***. y. ln w2. 政治大. sit. ln y3. er. ln y2. ‧ 國. ln y1. 0.0671*** (0.0104) -0.0155* (0.0090) -0.0014 (0.0027). Ch. engchi. 0.0277 (0.0379) -0.0080 (0.0256) -0.0009** (0.0004). i n U. 0.0074 (0.0306) -0.0166 (0.0173) -0.0052** (0.0023) 22. v. 0.0319** (0.0154) -0.0026 (0.0114) 0.0214*** (0.0054). 0.1608*** (0.0121) 0.0072 (0.0080) -0.0259*** (0.0022).

(29) ln y3  ln y3. t  ln y1 t  ln y2 t  ln y3. (0.0024) 0.0043*** (0.0006) 0.0089*** (0.0007) 0.0344*** (0.0059) 0.0145** (0.0064) 0.0020**. (0.0009) -0.0055*** (0.0008). (0.0002) 0.0005* (0.0003). (0.0009) 0.0032*** (0.0004). (0.0010) -0.0011*** (0.0001). 0.0040 (0.0050) -0.0138* (0.0072) 0.0788** (0.0381). -0.0023 (0.0017) 0.0098** 0.0071** -0.0006 (0.0043) (0.0032) (0.0027) 0.0104 -0.0165*** -0.0085*** (0.0068) (0.0043) (0.0029) -0.0876*** -0.0304 0.0542*** (0.0348) (0.0363) (0.0224). 政治大 0.0011 -0.0060** -0.0085*** 0.0030 立 (0.0018) (0.0030) (0.0025) (0.0033). -0.0056*** (0.0023) 0.0073*** (0.0028) 0.0574*** (0.0221). -0.0040** 0.0016 (0.0017) (0.0035) -0.0398 0.0722 (0.0352) (0.0449) -0.0596* 0.0784*** (0.0323) (0.0325) -0.2300*** -0.2865*** (0.0324) (0.0278) 0.8066*** 1.9263*** (0.1638) (0.0937). 0.0069** (0.0031) 0.1598*** (0.0357) -0.0197 (0.0341) -0.0609 (0.0483) 0.9798 (0.0685). 0.0002 0.0005 (0.0023) (0.0018) 0.3507*** -0.1920*** (0.0521) (0.0279) 0.1185 0.1157*** (0.0842) (0.0272) -0.5046*** -0.1628*** (0.0401) (0.0277) 2.7567*** 1.3648*** (0.2843) (0.1463). 0.4462*** (0.0658) 0.0233*** (0.0008) 0.3482*** (0.0168). 3.2940*** (0.2201) 0.3457*** (0.0736) 0.4230*** (0.0136). 0.7383** (0.1139) 0.0004*** (0.0011) 0.5471*** (0.0325). io. t t. (0.0021) (0.0072) -0.0019** -0.0194*** (0.0008) (0.0052) 0.0289*** -0.0354*** (0.0031) (0.0073) -0.0125** -0.0064 (0.0062) (0.0075) -0.1016*** -0.0085 (0.0119) (0.0094) 0.0046*** -0.0048**. Nat. t. (0.0010) -0.0015 (0.0010) 0.0123*** (0.0018) -0.0212** (0.0103) -0.0130 (0.0142) 0.0002. al. n. 12 13.  23. 1. 2 3. 1. ‧. t  ln w3. (0.0024) -0.0013 (0.0021) 0.0576*** (0.0032) -0.0051 (0.0051) -0.0696*** (0.0067) 0.0047***. (0.0022) 0.0043** (0.0020). 0.0028. 學. t  ln w2. 0.0632***. y. ln y3  ln w2. 0.0118***. sit. ln y2  ln w3. 0.0041***. Ch. engchi. 3.8369*** (0.3032) 1.5078*** (0.0046) 1.1449*** (0.0229). 23. er. ln y2  ln w2. 0.0238***. ‧ 國. ln y1  ln w3. i n U. v. 0.1561*** (0.0184) 1.7816*** (0.0930) 0.49768*** (0.0178).

(30) 2 3 log L. 0.0542***. 0.0367***. 0.0504***. 0.1513***. 0.0723***. (0.0012) 0.1363*** (0.0002) 1946.87. (0.0007) 0.0180*** (0.0002) 4117.38. (0.0014) 0.0392*** (0.0011) 2124.61. (0.0039) 0.0246*** (0.0007) 1585.15. (0.0015) 0.0114*** (0.0002) 5383.46. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2. Numbers in parentheses are standard errors. 3.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 4.  j.  u / v j j. and  j   v2j  u2j , where. j = 1, 2, 3. 政治大. We next simultaneously estimate equations (1) and (5) again, assuming a single. 立. output (total assets) for each country, and estimate the single cost frontier for a single. ‧ 國. 學. output without considering the price equation. Tables 4 and 5 show the estimated results, respectively. Most of the estimates of the copula method attain the 10% level of. ‧. significance in Table 4, while more estimates fail to be significant in the single cost. Nat. sit. y. frontier in Table 5. The efficiency gain from using the copula method, stemming from. n. al. er. io. the simultaneous estimation procedure, is confirmed. These parameter estimates are. i n U. v. consistent with the requirement of the microeconomic theory, as the partial derivatives. Ch. engchi. of the cost function with respect to the input prices and the output quantities are on average all positive. It is crucial to note that all the dependence parameters  in these countries are significant at least at the 10% level, whereby France, Italy, and Switzerland have negative values and Germany and Luxembourg have positive values, signifying that the simultaneous equation model is preferable to the single equation model. Again, the omission of  may lead to inconsistent parameter estimates, and thereby the subsequent technical efficiency and the Lerner index measures may be misleading.. 24.

(31) Table 3 Parameter Estimates of a Single Equation for Multiple Outputs Variables. FRA. GER. ITA. LUX. SWI. constant. 1.4928*** (0.3698) 0.4475*** (0.0641) 0.6083*** (0.0583) -0.0525 (0.0707) 0.1581***. 0.7626*** (0.3295) 0.8392*** (0.0457) 0.4559*** (0.0755) -0.2368*** (0.0758) 0.1706***. 0.6842 (0.4499) 0.5318*** (0.0693) 0.4133*** (0.0665) 0.1795*** (0.0818) 0.1615***. 0.0606 (1.0111) 1.0305*** (0.1336) 0.1597 (0.1787) 0.0878 (0.1203) 0.1441***. 1.0082*** (0.2632) 0.6723*** (0.0439) 0.5192*** (0.0588) -0.1344*** (0.0616) 0.1522***. (0.0085) 0.1234*** (0.0061). (0.0067) 0.2100*** (0.0101). (0.0065) 0.1868*** (0.0094). (0.0182) 0.2045*** (0.0314). (0.0051) 0.1228*** (0.0093). ln w3. ln y1  ln y1. 0.0022 (0.0545) 0.2165*** (0.0629) 0.0310*** (0.0062) 0.1041*** (0.0097) -0.0079 (0.0070). io. ln y1  ln y3. Nat. ln y1  ln y2. ‧. ln y3  ln y3. 0.0084 0.0035 (0.0103) (0.0111) (0.0133) (0.0168) (0.0081) -0.1435*** -0.2178*** -0.1635*** -0.1807*** -0.1431*** (0.0057) (0.0067) (0.0071) (0.0170) (0.0046) -0.0084 0.0216*** 0.0014 -0.0218 -0.0216*** (0.0077) (0.0061) (0.0081) (0.0126) (0.0052) 0.0081 0.0147 -0.0286*** 0.0210 0.0250*** (0.0083) (0.0099) (0.0093) (0.0189) (0.0087). 學. ln y2  ln y2. 立. al. n. ln y2  ln y3. ln w2  ln w2 ln w3  ln w3 ln w2  ln w3. ln y1  ln w2 ln y1  ln w3. ln y2  ln w2. 政治大 -0.0162 0.0229* -0.0230. y. ln w2. 0.0434 (0.0623) -0.1875 (0.1007) 0.0300*** (0.0088) -0.0260 (0.0280) -0.0213* (0.0118). Ch. sit. ln y3. 0.1422*** 0.1288 (0.0557) (0.0961) -0.2066*** -0.0538 (0.0917) (0.2019) 0.0025 0.0004 (0.0079) (0.0113) -0.0210 -0.0546*** (0.0168) (0.0233) 0.0023 0.0151 (0.0093) (0.0144). er. ln y2. ‧ 國. ln y1. engchi. i n U. -0.0228*** 0.0001 -0.0172*** (0.0053) (0.0050) (0.0052) 0.0169*** 0.0542*** 0.0804*** (0.0066) (0.0083) (0.0072) 0.0009 -0.0135*** 0.0122*** (0.0040) (0.0062) (0.0055) 25. v. 0.0030 (0.0094) 0.0704*** (0.0164) -0.0243* (0.0129). 0.0272 (0.0438) -0.0199 (0.0638) 0.0205*** (0.0040) 0.1155*** (0.0082) 0.0074 (0.0048) -0.0118*** (0.0037) 0.0154*** (0.0060) -0.0095*** (0.0048).

(32) t  ln y1 t  ln y2 t  ln y3. t  ln w2. -0.0043. 0.0194***. (0.0064) (0.0143) (0.0097) 0.0206*** 0.0073 0.0056 (0.0058) (0.0068) (0.0073) -0.0466*** -0.1048*** -0.0851*** (0.0089) (0.0137) (0.0123) 0.0061*** 0.0048*** -0.0008 (0.0020) (0.0020) (0.0023) -0.0002*** 0.0008 -0.0006 (0.0018) (0.0022) (0.0023) -0.0057*** -0.0063*** 0.0005. (0.0243) 0.0164 (0.0112) -0.0032 (0.0130) 0.0071 (0.0043) -0.0043 (0.0059) 0.0002. (0.0074) 0.0154*** (0.0047) -0.0100 (0.0075) -0.0008 (0.0014) -0.0013 (0.0017) 0.0025. (0.0025) (0.0026) (0.0030) -0.0040*** -0.0083*** -0.0068*** (0.0017) (0.0019) (0.0021). (0.0051) 0.0035 (0.0036). (0.0017) 0.0013 (0.0012). (0.0069) -0.0427 (0.0394) 0.0014 (0.0024) 0.9937*** (0.0033). 0.0009 (0.0020) -0.0155 (0.0132) 0.0021*** (0.0009) 0.9214*** (0.0122). 0.3502*** (0.0239) -140.8698. 0.0645*** (0.0033) 576.5571. 立(0.0040). t t. 0.0829*** (0.0054) 214.7036. 0.2008*** (0.0109) -92.1569. 0.0677*** (0.0078) 179.4791. Nat. . io. 2. al. n. log L. Ch. engchi. ‧. 0.0216 (0.0187) 0.0006 (0.0014) 0.9396*** (0.0108). (0.0034) 0.0133 (0.0227) 0.0018 (0.0014) 0.7648*** (0.0668). 學. t. 政治大 0.0013 0.0072*** -0.0173***. 0.0009 (0.0023) 0.0260 (0.0187) -0.0051*** (0.0012) 0.8144*** (0.0258). ‧ 國. t  ln w3. 0.0199***. y. ln y3  ln y3. 0.0817***. sit. ln y3  ln w2. 0.0329***. er. ln y2  ln w3. i n U. v. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2. Numbers in parentheses are standard errors. 3.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 4..    u2  u2   v2 and  2   u2   v2. 26.

(33) Table 4 Parameter Estimates of the Copula Method for a Single Output. ln w2 ln w3. ln w2  ln w2 ln w3  ln w3. (0.0332) 0.0071** (0.0035). 0.1421*** (0.0076) (0.0113) (0.0159) -0.0146** -0.0508*** -0.0272*** (0.0065) (0.0065) (0.0082) -0.01618*** 0.0072*** -0.0111*** (0.0034) (0.0030) (0.0031) 0.0074** -0.0025 0.0191*** (0.0033) (0.0042) (0.0059). 0.0096 (0.0114) -0.0051 (0.0074) -0.0076** (0.0040) 0.0411*** (0.0069). 0.1595*** (0.0044) -0.0104*** (0.0028) -0.0024 (0.0026) -0.0067*** (0.0024). 0.0535*** (0.0185) -0.0028*** (0.0012) -0.0006 (0.0011) -0.0057*** (0.0016) 0.0033 (0.0021). 0.0301*** -0.0126 -0.1050*** (0.0124) (0.0204) (0.0267) 0.0016* -0.0005 0.0015 (0.0009) (0.0015) (0.0014) -0.0012 0.0018 0.0074*** (0.0008) (0.0012) (0.0019) -0.0055*** -0.0054*** 0.0020 (0.0012) (0.0021) (0.0018) 0.0060*** 0.0049 -0.0150*** (0.0019) (0.0034) (0.0030). 0.0059 (0.0113) 0.0016*** (0.0006) -0.0012 (0.0008) -0.0010 (0.0010) 0.0048*** (0.0010). -0.0974*** (0.0196) 1.4763*** (0.0052) 0.9745*** (0.0587). 0.5701**** (0.0460) 2.3640*** (0.0397) 0.2840*** (0.0780). -0.1237*** (0.0167) 1.4962*** (0.1213) 1.4495*** (0.0824). 立. al. n. t t t  ln ta. t  ln w2 t  ln w3. . 1 2. 0.2832*** (0.0066) 1.0307*** (0.0247) 0.0005 (0.0018) 0.0527 (0.0348) 0.4231***. (0.0959) 0.0178*** (0.0076). io. t. (0.0605) 0.0083 (0.0062). SWI. (0.0852) 0.0241*** (0.0077). Nat. ln ta ln w3. (0.0581) 0.0459*** (0.0067). 政治大 0.1919*** 0.0708***. ‧. ln ta  ln w2. -0.4029*** -0.6551*** -1.4950*** (0.0088) (0.0103) (0.0143) 1.0979*** 1.1388*** 1.3548*** (0.0278) (0.0460) (0.0791) -0.0083*** -0.0080*** -0.0287*** (0.0021) (0.0031) (0.0058) -0.0134 0.1975*** 0.0402 (0.0427) (0.0549) (0.0613) 0.5821*** 0.1089 0.1687*. 學. ln w2  ln w3. LUX. y. ln ta  ln ta. -0.1394*** (0.0006) 1.0793*** (0.0393) -0.0007 (0.0026) 0.0922* (0.0553) 0.2800***. ITA. sit. ln ta. GER. er. cons. FRA. ‧ 國. Variables. Ch. engchi. 27. i n U. -0.0348** (0.0215) 1.0713*** (0.0930) 1.1447*** (0.0013). v. 0.0950*** (0.0227) 1.776*** (0.1616) 1.0208*** (0.0888).

(34) 1. 2 log L. 0.2669***. 0.4702***. 0.2506***. 0.2427***. 0.1823***. (0.0016) 0.0239*** (0.0000) 2455.31. (0.0120) 0.0453*** (0.0009) 1936.83. (0.0053) 0.0156*** (0.0005) 1907.84. (0.0125) 0.0175*** (0.0005) 1697.28. (0.0055) 0.0230*** (0.0003) 3726.07. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2. Numbers in parentheses are standard errors. 3.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 4.  j.  u / v j j. and  j   v2j  u2j , where. j = 1, 2. 政治大. We now turn to jointly estimate equations (22) and (25) to derive the parameter. 立. estimates of the meta-frontiers with the results presented in Table 6. We also apply the. ‧ 國. 學. mixed approach of the LP model of the single cost function, proposed by Battese et al. (2004) and O’Donnell et al. (2008), to get the meta-frontier. Table 6 shows the results.. ‧. The standard errors of the mathematical programming estimators are obtained through. Nat. sit. y. bootstrapping methods with 1000 replications. The estimated standard error of a meta-. n. al. er. io. frontier parameter is the standard deviation of the 1000 replications. There are. i n U. v. substantial differences between the coefficients of our proposed model and those of the. Ch. engchi. LP model. In addition, most of the standard errors of the proposed model are relatively small versus those of the LP model. We finally jointly estimate a cost frontier with a single output and an output price frontier. Both the new and the old meta-frontiers are estimated, and these parameter estimates are listed in Table 7. Similarly, all the standard errors of the new meta-frontier are smaller than those of the old meta-frontier.. 28.

(35) Table 5 Parameter Estimates of a Single Equation for a Single Output. ln w2 ln w3. ln w2  ln w2 ln w3  ln w3. 0.1159 (0.4628) 1.0360*** (0.0558) -0.0026 (0.0038) 0.1118*** (0.0556) 0.1158. -1.1728 (1.3080) 1.3403*** (0.1921) -0.0327*** (0.0141) -0.0673 (0.1090) -0.0623. 0.7099*** (0.1885) 0.9981*** (0.0228) -0.0005 (0.0017) -0.0019 (0.0335) 0.2650***. (0.0592) 0.0415*** (0.0061). (0.0989) 0.0139 (0.0100). (0.0932) 0.0020 (0.0080). (0.2234) 0.0090 (0.0125). (0.0339) 0.0018 (0.0034). 政治大 0.1340*** 0.1032*** -0.0341*** -0.0739*** 立(0.0199) (0.0184) (0.0220) (0.0077) -0.0106* -0.0280*** (0.0063) (0.0110) -0.0143*** -0.0041 (0.0036) (0.0051) 0.0101*** 0.0215*** (0.0035) (0.0067). -0.0075 (0.0088) 0.0005 (0.0033) 0.0136*** (0.0067). al. n. t t t  ln ta. t  ln w2 t  ln w3. . 2 log L. -0.0003 (0.0141) 0.0030 (0.0082) 0.0681*** (0.0158). 0.0462*** 0.0181 0.0323 -0.0716 (0.0185) (0.0193) (0.0198) (0.0457) -0.0023*** 0.0019 0.0006 0.0003 (0.0011) (0.0014) (0.0015) (0.0028) -0.0003 0.0003 -0.0019 0.0056 (0.0012) (0.0013) (0.0013) (0.0036) -0.0062*** -0.0080*** -0.0056*** 0.0027 (0.0016) (0.0021) (0.0020) (0.0034) 0.0031 0.0054 0.0098*** -0.0219*** (0.0021) (0.0033) (0.0038) (0.0062). io. t. -0.5702 (0.3683) 1.1422*** (0.0457) -0.0091*** (0.0037) 0.0980 (0.0705) 0.1729*. Nat. ln ta ln w3. -0.1714 (0.3484) 1.0924*** (0.0411) -0.0021 (0.0028) 0.0754 (0.0542) 0.2238***. ‧. ln ta  ln w2. SWI. 學. ln w2  ln w3. LUX. y. ln ta  ln ta. ITA. sit. ln ta. GER. er. cons. FRA. ‧ 國. Variables. 0.7258*** (0.0300) 0.0757*** (0.0048) 194.2783. Ch. engchi. 0.9026*** (0.0121) 0.2051*** (0.0115) -155.33415. i n U. 0.9132*** (0.0256) 0.1054*** (0.0088) 126.3285. v. 0.9653*** (0.0078) 0.3570*** (0.0250) -188.5928. 0.1433*** (0.0040) -0.0061*** (0.0027) 0.0027 (0.0026) 0.0081*** (0.0026) 0.0072 (0.0106) 0.0003 (0.0007) -0.0002 (0.0008) -0.0008 (0.0010) 0.0003 (0.0010) 0.7846*** (0.0265) 0.0361*** (0.0021) 774.4010. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 29.

(36) 2. Numbers in parentheses are standard errors. 3.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 4..    u2  u2   v2 and  2   u2   v2. Table 6 Parameter Estimates of the Meta-Frontier Cost Function for Multiple Outputs Copula Stochastic Metafrontier Single LP Metafrontier Method Method. estimates. errors. 6.0478*** 0.1334*** 0.1669*** 0.1752*** 0.0382 0.8345*** 0.0432*** 0.0294*** 0.0517*** -0.0299***. 0.1235 0.0105 0.0120 0.0181 0.0237 0.0266 0.0007 0.0007 0.0025 0.0011. 0.2594 0.7483 0.4941 -0.1347 0.1314 0.1244 -0.0553 -0.1609 0.0233 0.0328. 0.4489 0.0923 0.0726 0.0931 0.0107 0.0120 0.0228 0.0089 0.0133 0.0139. 0.0013 0.0012 0.0038 0.0041 0.0032 0.0010 0.0007 0.0007 0.0013 0.0018 0.0025. 0.2266 -0.3177 -0.0459 -0.1110 0.0584 -0.0254 0.0666 0.0201 0.0416 0.0029 -0.0789. 0.0003 0.0002 0.0006 0.0008 0.0010 0.0078. 0.0068 -0.0004 -0.0055 0.0051 -0.0141 -0.0419. -0.0073*** -0.0043*** 0.0003 0.0920*** -0.0257*** 0.0013 0.0045*** 0.0083*** 0.0338*** -0.0061*** -0.0450***. al. 0.0024*** -0.0027*** -0.0042*** 0.0002 -0.0048*** 0.0653***. Ch. engchi. 30. y. sit. 立. 政治大. ‧. t  ln w2 t  ln w3 t. errors. 學. t  ln y1 t  ln y2 t  ln y3. estimates. n. ln y2  ln w2 ln y2  ln w3 ln y3  ln w2 ln y3  ln w3. Bootstrapped Standard. io. ln w2  ln w2 ln w3  ln w3 ln w2  ln w3 ln y1  ln w2 ln y1  ln w3. Parameter. Nat. ln w2 ln w3 ln y1  ln y1 ln y2  ln y2 ln y3  ln y3 ln y1  ln y2 ln y1  ln y3 ln y2  ln y3. Standard. er. constant ln y1 ln y2 ln y 3. Parameter. ‧ 國. Variables. i n U. v. 0.0834 0.1238 0.0120 0.0217 0.0126 0.0087 0.0112 0.0096 0.0137 0.0114 0.0156 0.0028 0.0023 0.0030 0.0024 0.0038 0.0213.

(37) t t . 1 2 1 2 log L. 0.0008. 0.0006. 0.4541*** 0.9616*** 0.8451*** 0.1332*** 0.1094*** 7384.34. 0.0077 0.0012 0.0003 0.0019 0.0001. -0.0018. 0.0013. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 3.  j.  u / v j j. and  j   v2j  u2j , where. j = 1, 2. 治政 Table 7 Parameter Estimates of the Meta-Frontier Cost Function 大 for a Single Output 立 Copula Stochastic Metafrontier. Single LP Metafrontier Method. 1 2 1 2. 0.0001 0.0081 0.0005. -0.0799 1.0713 -0.0062. 0.0537*** 0.3759*** 0.0109*** 0.1494*** -0.0147*** -0.0015*** 0.0009 0.0186*** -0.0001 -0.0009*** -0.0024***. 0.0102 0.0122 0.0013 0.0017 0.0011 0.0006 0.0008 0.0034 0.0002 0.0002 0.0003. 0.1495 0.0014 -0.0105 -0.1034 0.0184 -0.0027 0.0258 0.0174 -0.0001 -0.0006 -0.0021. 0.0004 0.0001 0.0001 0.0001 0.0001 0.0001. 0.0007. al. 0.0011*** -0.0123*** 0.6566*** 1.1787*** 0.1075*** 0.0252***. Ch. engchi. 31. y. -0.2673*** 1.1036**** -0.0061***. errors. er. estimates. sit. ‧ 國. errors. Bootstrapped standard. ‧. t  ln w2 t  ln w3 . estimates. n. t  ln ta. Parameter. io. ln w2  ln w2 ln w3  ln w3 ln w2  ln w3 ln ta  ln w2 ln ta ln w3 t t t. Standard. Nat. constant ln ta ln ta  ln ta ln w2 ln w3. Parameter. 學. Variables. Method. i n U. v. 0.1423 0.0190 0.0019 0.0296 0.0709 0.0049 0.0109 0.0071 0.0020 0.0068 0.0110 0.0006 0.0009 0.0009 0.0015.

(38) log L. 14205.9. Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2.. W1. is arbitrarily selected as the numeraire to satisfy the homogeneity restriction in input. prices. 3.  j.  u / v j j. and  j   v2j  u2j , where. j = 1, 2. 5.2 Various efficiency scores We use the foregoing coefficient estimates to calculate various efficiency scores, including CE, TGR, CE M , the Lerner index, and MCGR. Table 8 shows the results from CSSFM and LP under the assumption that each bank produces three outputs. The. 政治大. overall mean value of CE from CSSFM is equal to 0.6891, which is lower than that. 立. presented by Huang et al. (2010) (0.7136). The mean CE values of the five countries. ‧ 國. 學. range from 0.4402 for Germany to 0.8376 for France, indicating that on average the potential cost savings are around 56% and 17% of their actual costs, respectively.. ‧. German banks have the highest cost inefficiency, which may result from the highest. y. Nat. sit. Lerner index in the markets for loans and investments, as noted below. Since German. n. al. er. io. banks are found to have higher market power with less technical efficiency, the “quiet. i n U. v. life hypothesis” may be applied.5 Banks in France, Italy, Luxembourg, and Switzerland. Ch. engchi. have higher country-specific mean CE scores than the overall average. Note that the mean value of CE from LP is equal to 0.7978, which is not far apart from the results found by Bos and Schmiedel (2007), in which Luxembourg banks attain the lowest mean CE score.6 Bos and Schmiedel (2007) claim that even in a coordinated single European banking market, banks’ CE substantially fluctuate across markets due to country-specific conditions, i.e. regulation, competition, etc. This appears to support the use of the meta-frontier model to compare bank performance across nations.. 5. The quiet life hypothesis asserts that a firm with market power has the luxury of being inefficient, ceteris paribus. 6 Since the results from QP are similar to those of LP, we choose not to show them to save space. 32.

(39) Table 8 Various Efficiency Estimates for Multiple Outputs. Max.. SD. FRA CE TGR. TGR. CE. M. 0.8376. 0.4624. 0.8730. 0.0431. CE. 0.9649. 0.9046. 0.9927. 0.0108. TGR. 0.8082. 0.4347. 0.8665. 0.0430. CE. 0.0878. 0.6882. 0.1394. CE. 0.9600. 0.8681. 0.9938. 0.0172. TGR. 0.4207. 0.0872. 0.6617. 0.1275. CE. M. 0.1461. 0.9786. 0.0874. 0.7368. 0.1084. 1.0000. 0.1564. 0.6112. 0.0507. 0.8657. 0.1454. 0.7491. 0.1476. 0.9775. 0.1571. 0.7846. 0.2398. 0.7711. 0.1371. 0.5896. 0.1065. 0.8718. 0.1660. 0.8389. 0.3448. 0.9704. 0.0825. M. 0.8318. 0.1362. 1.0000. 0.1387. 0.6995. 0.0931. 0.9670. 0.1362. 0.1917. 0.8447. 0.0825. CE. 0.9773. 0.9282. 0.9961. 0.0092. TGR. 0.7402. 0.7060. TGR M. 0.9760 0.6890. SWI. TGR. M. 0.8694. 0.1234. CE. 0.9057. 0.9934. 0.0111. TGR. 0.3073. 0.8637. 0.1204. CE. 0.1168. 0.9868. 0.2169. 0.6023. 0.0812. 1.0000. 0.1554. 0.4219. 0.0625. 0.8646. 0.1811. 0.8397. 0.1333. 0.9832. 0.1018. 0.7250. 1.0000. 0.1589. 0.0496. 0.9815. 0.1591. 0.1168. 0.9868. 0.1411. 0.7414. 0.0812. 1.0000. 0.1633. 0.5960. 0.0496. 0.9815. 0.1741. 0.6915. M. SWI. 0.1744. 0.8275. 0.0952. CE. 0.9741. 0.9167. 0.9926. 0.0088. TGR. 0.7132. 0.1727. 0.8194. al. 0.0935. CE. n. TGR. CE. 0.7321. Overall CE. 0.0793. 0.3131. io. M. 立. 0.8414. LUX. Nat. CE. 0.1905. M. 0.6891. 0.0878. 0.9696. 0.8681. 0.6682. 學. CE. 政治大. 0.7576. LUX. CE. M. 0.4402. ‧ 國. TGR. CE. 0.8265. ITA. CE. CE. SD. GER. ITA. CE. Max.. FRA. GER CE. Min.. 0.0872. M. Overall. C 0.1744 h e n gCEc h i 0.9961 0.0137 TGR 0.8730. 0.8665. 0.1699. CE. M. 0.1489. er. CE. M. Mean. y. Min.. ‧. Mean. LP. sit. CSSFM. 0.6122. i n 0.7978 U. v. The overall mean TGR measure is as high as 0.9696 with a very small standard deviation (0.0137), implying that banks in our sample countries tend to adopt quite similar and advance production technology to provide financial services to their customers in the highly integrated European banking market. The mean TGR measures of the five countries lie in a narrow range, i.e., between 0.96 and 0.9773. Among them, Italian banks adopt the most advanced production technology, because their country 33.

(40) frontier is the closest to the meta-frontier, followed by Luxembourg, Switzerland, France, and Germany.7 A representative Italian bank can save merely 2.27% of its production costs, provided it undertakes the potential advanced technology. Having the lowest mean TGR value, German banks can shave 4% of their costs by exploiting the most advanced production technology available to all countries. The mean value of TGR is close to one, verifying that differences between country-specific frontiers and a meta-frontier are rather small for the single European banking market. Our results are similar to Bos and Schmiedel. 政治大 (2008). However, the average TGR of the LP approach for the five countries range 立. (2007), who employ the mixed approach of Battese et al. (2004) and O’Donnell et al.. widely from 0.6023 to 0.8318 with an overall mean value of 0.7414 and a standard. ‧ 國. 學. deviation of 0.1633. This low mean value of TGR differs from that of Bos and. sit. y. Nat. is as expected.. ‧. Schmiedel (2007), and this larger value of standard deviation than that of the CSSFM. io. er. French banks reach the highest mean meta-cost efficiency score ( CE M ), followed by Italy, Switzerland, Luxembourg, and Germany. The component CE is on average less. al. n. v i n C hcountry, meaningUthat the main source of inputthan the component TGR in each engchi oriented production inefficiency comes from managerial inabilities, rather than. adopting inferior technology. The sample banks are suggested to reduce their input mix for a given output mix in such a way as to improve their cost efficiency scores. Following the convention of a single output bank, we re-estimate the CSSFM and LP models, presenting results in Table 9. The table shows that the mean CE value of CSSFM is equal to 0.8270 and ranges between 0.7070 for Germany and 0.8883 for Switzerland. The mean CE value of the LP model is equal to 0.8038 and varies from. 7. The null hypothesis that Italy and Luxembourg have equal average TGR value is rejected by t-test statistics. 34.

(41) 0.6820 for Luxembourg to 0.8791 for Switzerland. Comparing to Table 8, we see that the mean CE score derived from the case of multiple outputs tends to be lower than that from the single output case. This implies that the aggregation of multiple outputs is apt to distort the estimates of cost efficiency. The overall average TGRs of CSSFM and LP equal 0.9527 and 0.6837, respectively, which are both lower than those in Table 8. Aggregating outputs here is inclined to under-estimate the average TGR. CSSFM still gives quite similar average TGRs among the five countries, while LP suggests that banks in the sample countries adopt quite. 政治大 each country, as expected. The component of TGR of CSSFM is on average larger than 立. different technologies. The standard deviation of CSSFM is smaller than that of LP for. that of CE, but the reverse is true for LP.. ‧ 國. 學 ‧. 5.3 Lerner index. sit. y. Nat. Table 10 reports the summary statistics of the Lerner indices for the outputs of loans. io. er. and investments and for the single output case across countries estimated by CSSFM and conventional models. As far as the loan market is concerned, the Swiss banking. al. n. v i n market is the most competitive C among five sample countries, as its average h e the ngchi U. LNew. is the lowest, followed by France, Luxembourg, Italy, and Germany. 8 German and Italian banks appear to enjoy stronger market power than the remaining three nations. The average LOld gives the same ordering, as shown in the last 5 column of the table. Moreover, the new Lerner index measures tend to be higher than the conventional ones, except for Germany, due partially to the fact that some of the LOld estimates are negative. Consequently, the conventional model inclines to underestimate the index such that the degree of market competition is exaggerated. 8. The pairwise differences in the average new Lerner index among these countries are all significant at the 1% level, except for the difference between Germany and Switzerland in the investment market and the difference between Germany and Luxembourg in the single output case. 35.