銀行產業效率分析: 網絡隨機邊界模型與其應用 - 政大學術集成

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(1)國立政治大學金融學系研究所博士學位論文. 銀行產業效率分析: 網絡隨機邊界模型與其應用. 政Bank治 Studies on Efficiency:. 立. 大. The Network SFA Model and Its Applications. ‧. ‧ 國. 學 er. io. sit. y. Nat. n. 指導教授：黃台心博士 a v. i l C n hengchi U. 研究生：林忠億撰. 中華民國一百零五年四月.

(2) 誌謝人生一直是不間斷的學習，而隨著博士論文的完成，代表著這段博士求學的時光即將結束，迎接著是未來另一個階段的各種挑戰。感謝人生中所遇到的許多人不斷地鼓勵與扶持，讓這段學習的旅程收穫滿滿。今日的學術環境與當初出國念書返國時相較差異甚大，未來教學與研究路程勢必更加艱辛，但仍願能尋覓到我所企盼且合適的舞台，無私地傳承過去個人國內、外所學與業界的相關經驗給下一代的學子們。回顧這段國內博士班的求學時光，最是苦澀與煎熬的階段，莫過於投稿的等待；所幸這段等待的過程，仍能因為曾秉持著一股對教學的熱誠與執著，到國內多所學校任教服務，如:中研院與陽明大學之人文與社會講座、元智大學、元培醫事科大、佛光大學、實踐大學、台北醫學大學等，這些的教學經驗使我更加體悟到因材施教的必要，也認識不少學術界的前輩與環境。即將畢業的我，珍惜第. 立. 政治大. ‧ 國. 學. 一次面試就決定以國立嘉義大學財金系作為未來舞台的機會，雖然是備取第一名，但我相信這也是一份的肯定，未來在良好的環境下，個人的教學、研究與服務的. ‧. 表現應能展現潛力。. n. al. er. io. sit. y. Nat. 本論文得以順利完成，除了感謝指導教授黃台心老師的悉心指導，奠定學生日後從事金融機構與市場效率分析暨應用計量領域的研究基礎與能量外；同時感謝口試委員傅祖壇教授、胡均立教授、陳忠榮教授與林建秀副教授於口試期間所給予的許多寶貴建議，讓本篇論文在內容上更臻充實與完整。. Ch. engchi. i n U. v. 而最想感謝的，絕對莫過於一路含辛茹苦地養育與栽培我、鼓勵我的父母親林清正先生與劉玉春女士，以及不斷默默關心我的弟弟宜賢；同時，也感謝念博士期間結婚生子的妻子冠臻，在我專注努力完成博士論文的期間，辛勞地照顧著三個可愛的小朋友，我深知這一路走得艱辛，點滴在心頭，對你們的付出由衷地感謝；謹將本博士論文及這份榮耀與喜悅獻給摯愛的你們，由衷地感謝你們長久以來的栽培、照顧與支持，讓我更加珍惜這得來不易的博士學位，並期勉未來能有所發揮，不負所望。. 林忠億謹誌於國立政治大學金融研究所中華民國一零五年四月.

(3) 銀行產業效率分析: 網絡隨機邊界模型與其應用. 摘要銀行產業的效率研究中，網絡資料包絡分析法的顯著貢獻，在於可處理存款在銀行生產過程所扮演的雙重角色。透過一定比例的勞動與實質資本投入創造出存款，因此，存款先被視為中間產出；在接續的生產過程中，存款這項中間產出又將被視為要素投入，結合其餘的勞動與實質資本，共同再創造出最終銀行諸多的產出組合。然而，網絡資料包絡分析法無法處理在銀行生產過程中關鍵的勞動與實質資本的比例參數。有鑒於此，第一篇網絡隨機邊界分析法的理論研究中，我們建構生產函數、成本函數與成本份額方程式以刻劃銀行多階段生產技術的經濟模型，運用最大概似法計量結合關聯結構，有效地估計聯立方程式中的各項參數，尤其是關鍵的勞動與實質資本比例參數。同時，模型亦將以 2009 年美國銀行產業為例，估計隨機生產與成本邊界的技術效率值，進而闡釋所創計量模型的. 立. 政治大. ‧ 國. 學. ‧. 實用與可行性。在第二篇實證論文研究中，即假設銀行的生產過程包括兩個階段: 吸納存款的階段與創造放款擴張的階段，我們將探討 2006 年至 2013 年中國商業銀行的效率;結果發現中國大陸的商業銀行於存款創造階段，勞動與實質資本的配置約在 35%及 50%；且平均在存款創造階段與放款擴增階段的技術效率值約 64%與 69%。此外，實證顯示兩階段皆具規模經濟，但第二階段並未顯現範疇經濟。我們的實證結果也支持相關文獻的發現：中型合資商業銀行最具生產技術效率，然而，主要大型商業銀行，包括傳統的四家大型國有銀行，則在技術效率上表現最差。. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 關鍵字: 網絡資料包絡分析法; 技術效率; 比例參數; 關聯結構法; 隨機生產和成本邊界; 多階段生產技術; 存款創造 ; 放款擴張; 中國銀行; 合資銀行; 國有銀行.

(4) Studies on Bank Efficiency: The Network SFA Model and Its Applications. Abstract The main contribution of network DEA deals with the dual role of deposits in the bank production process. Deposits are first viewed as an intermediate output, produced by, e.g., fractions of labor and capital. This intermediate output is next used as an input in the second process, together with the remaining labor and capital, to produce output combinations. A problem occurs in that network DEA suffers from the difficulty of determining the fractions of labor and capital used in the first process. This first research thus develops an economic model to characterize the underlying multi-stage technologies and proposes a copula-based econometric model to identify parameters of the structural equations, including the fractional parameters, by the maximum. 立. 政治大. ‧ 國. 學. ‧. likelihood. Our model also estimates technical efficiencies of the stochastic production and cost frontiers. We collect data from U.S. banks in 2009 to illustrate the feasibility and usefulness of our modeling, and the results are promising. In the second empirical application, we compile data from the Chinese banking industry over the period 20062013 to exemplify our approach with the help of copula methods. Under the assumption of two production stages - i.e., deposit-gathering and loan-expansion stages - we find. er. io. sit. y. Nat. al. n. v i n that banks allocate roughly 35%Cand 50% of labor and h e n g c h i Ucapital, respectively, to collect deposits in the first stage and that the average technical efficiency scores in both production stages are respectively 64% and 69%. Additionally, both production stages. enjoy economies of scale, however, we do not verify the presence of scope economies Our study supports the previous findings that joint-stock banks are the most technically efficient, while larger commercial banks, including the big four state-owned banks, are the least technically efficient.. Key Words: Network DEA; Technical efficiency; Fractional parameters; Copula method; Stochastic production and cost frontiers; Multi-stage technologies; Depositgathering ; Loan-expansion; Chinese banks; Joint-stock banks; State-owned banks.

(5) Contents List of Tables................................................................................................................ V List of Figures ............................................................................................................ VI Chapter 1. Introduction ............................................................................................. 1. Chapter 2 An Extension from Network DEA to Copula-Based Network SFA ... 4 2.1 Introduction ...................................................................................................... 4 2.2 Economic Model .............................................................................................. 9. 政治大. 2.3 The Data ......................................................................................................... 18. 立. 2.4 Empirical Results ........................................................................................... 19. ‧ 國. 學. 2.6 Conclusion ..................................................................................................... 33. ‧. Chapter 3 Evaluating Efficiencies of Chinese Commercial Banks in the Context of Stochastic Multistage Technologies ...................................................................... 36. y. Nat. io. sit. 3.1 Introduction .................................................................................................... 36. n. al. er. 3.2 Literature Review........................................................................................... 40. Ch. i n U. v. 3.3 Model Specifications ..................................................................................... 44. engchi. 3.3.1 The Theoretical Model ........................................................................ 44 3.3.2 The Econometric Model ..................................................................... 48 3.4 Data ................................................................................................................ 52 3.5 Empirical Results ........................................................................................... 55 3.6 Conclusion ..................................................................................................... 69 Chapter 4 Conclusion ............................................................................................. 71 Bibliography ............................................................................................................... 74 Appendix A. ................................................................................................................ 85 Appendix B. ................................................................................................................ 87 IV.

(6) List of Tables Table 1.1：Sample statistics ........................................................................................ 19 Table 1.2：Biases and MSEs of the parameter estimates of the production function . 24 Table 1.3：Biases and MSEs of the parameter estimates of the cost function ............ 24 Table 1.4：Parameter estimates of the production function ........................................ 26 Table 1.5：Parameter estimates of the cost function ................................................... 26. 政治大. Table 1.6：The mean TE scores of various sets of 1 and  2 ................................ 30. 立. Table 2.1：Descriptive statistics ................................................................................. 53. ‧ 國. 學. Table 2.2：Descriptive statistics by different types of banks ...................................... 54. ‧. Table 2.3：Parameter estimates of the production frontier ......................................... 56 Table 2.4：Parameter estimates of the cost frontier .................................................... 57. y. Nat. io. sit. Table 2.5：Parameter estimates of the production frontier with bank dummies ........ 59. n. al. er. Table 2.6：Parameter estimates of the cost frontier with bank dummies ................... 60. Ch. i n U. v. Table 2.7：Estimates of fractional parameters by types ............................................. 61. engchi. Table 2.8：Average TE scores by types ...................................................................... 61. V.

(7) List of Figures Figure 1.1：Kernel densities of estimated TE scores .................................................. 32 Figure 2.1：The network system for banking industry ............................................... 44 Figure 2.2：Average efficiency scores over time for all sample banks ...................... 64 Figure 2.3：Average efficiency scores over time for LNCBs ..................................... 65. 政治大 Figure 2.5：Average efficiency scores over time for CRCBs ..................................... 66 立 Figure 2.4：Average efficiency scores over time for JSCBs ...................................... 66. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. VI. i n U. v.

(8) Chapter 1 Introduction Performance evaluation is quite essential for academic researchers and practitioners in the fields of economics and management, with the trivial financial ratios as performance indicators being the most common tool in accounting and financial management. While other alternative methods such as the balanced scorecard in similar areas are also considered, the frontier efficiency methodology, extended since Farrel. 政治大. (1957) first presented it, has drawn a lot of attention as a practical analysis tool because. 立. it overall considers multiple input and output combinations for the whole production. ‧ 國. 學. process of the decision making unit of interest.. ‧. Traditional efficiency studies mainly employ two distinct methodologies. One is. y. Nat. io. sit. data envelopment analysis (DEA), which originated from mathematical linear. n. al. er. programming, and the other is stochastic frontier analysis (SFA), which performs. i n U. v. statistical econometric models. Over the past six decades, research has been fully. Ch. engchi. interested in uncovering the black box of the production process, because earlier studies only treated the whole input into the model and the output produced from it, without considering multiple production stages. Thus, the origin of inefficiency cannot be clearly resolved.. Ever since the endeavors of Färe and Grosskopf (2000), different modified network DEA models have been developed in practice, however, to our best knowledge, SFA still plays no part in this arena. In fact, in the banking efficiency literature, the factor of deposits can be subjectively treated as an input in the asset (intermediation) 1.

(9) approach or an output in the value-added (production) approach which may possibly result in conflicting efficiency results. Holod and Lewis (2011) use this concept of a network production structure to overcome the dilemma of the dual role of bank deposits. Therefore, in the first essay, we look to extend the network DEA framework to develop a system econometric model that can characterize the production network process through the use of U.S. banking cross-sectional data. Under this theoretical structure, we further perform a Monte Carlo simulation to confirm the importance of estimating the inputs’ pivotal fractional parameters in the efficiency analysis of the network structure.. 立. 政治大. We employ banks in the U.S. as an example. Here, labor and capital are used to. ‧ 國. 學. produce deposits, which serve, along with the rest of labor and capital, as intermediate. ‧. inputs in the production of the final outputs-loans etc. For this case, we specify four. sit. y. Nat. equations ： a production function for deposits, an expenditure equation for loans. io. er. production, and two expenditure share equations for inputs in loans production. This is done in order to identify the parameters as well as the inefficiency of both production. al. n. v i n stages. Our model’s innovationCover traditional network h e n g c h i U DEA is that network SFA. allows for an estimation of the effect of input fractions, which is the effect of inputs that. are used in each stage of production. Methodologically, the network SFA employs a parametric copula to join the system of equations. We use a Gaussian copula herein to build a likelihood for the four equation errors and estimate all the parameters jointly by MLE or by nonlinear iterative SURE.. In the second essay, different from the aforementioned theoretical framework, we modify the previous proposed network multivariate SFA model to perform our empirical studies. Additionally, we evaluate efficiency scores for the China banking 2.

(10) industry using panel data instead of previous cross-sectional data. In the first decade of the 21st t century, China’s overall economic position became more and more important around the world. Thus, it is worth investigating whether or not its banking industry efficiently allocates resources in each operational stage and how to further improve upon it. Using fractional and slope parameter estimates, we are able to depict a more complete picture of the internal operational performances in its banking sector. Moreover, we calculate divisional efficiency scores and characterize banks’ production technology. Finally, we examine whether Chinese commercial banks exhibit economies. 政治大. of scale and scope during the sample period in our empirical study.. 立. The remainder of this dissertation is organized as follows. Chapter 2 proposes the. ‧ 國. 學. theoretical system equations of network SFA and provides an initial promising result. ‧. using U.S. banking data. We then conduct a Monte Carlo simulation to illustrate the. sit. y. Nat. importance of estimating the parameter of shared inputs. Chapter 3 examines the. io. er. feasibility of the proposed novel network SFA model and applies it to the case of the China banking industry. We shall categorize different types of Chinese commercial. al. n. v i n C hof inputs in eachUproduction process. We further banks and examine their allocation engchi. investigate whether Chinese banks exhibit economies of scale and scope. Chapter 4 concludes this dissertation.. 3.

(11) Chapter 2 An Extension from Network DEA to Copula-Based Network SFA. 2.1 Introduction There is long-standing controversy in the literature on how to define a bank’s outputs and inputs. The asset (intermediation) approach characterizes banks as financial. 政治大 considered to be inputs to the 立intermediation process, since banks can turn these funds. intermediaries between depositors and borrowers, whose liabilities such as deposits are. ‧ 國. 學. into various loans. Bank assets such as loans and investments are taken to be outputs that generate various forms of revenues. On the other hand, the value-added (production). ‧. approach identifies the major categories of produced deposits (demand, time, and. sit. y. Nat. savings) and loans (real estate, commercial, and installment) as important outputs,. n. al. er. io. defines purchased funds (federal funds purchased, large CDs, foreign deposits, other. i n U. v. liabilities for borrowed money) as financial inputs, and treats government securities and. Ch. engchi. other non-loan investments as unimportant outputs. See Berger and Humphrey (1992) for details.. Although each approach is capable of characterizing some aspects of the bank production processes and hence has its own advantages, such a classification often causes contradictory and incomparable results in, e.g., measures of technical efficiency (TE), as well as scale and scope economies across studies. Even worse, under the intermediation approach a bank with few deposits and more loans will be identified as technically efficient, but may be identified as inefficient on the basis of the production 4.

(12) approach.. This dilemma of whether to consider deposits as an input or output is partially resolved by the previous literature under the framework of network data envelopment analysis (network DEA), which has been introduced in several forms. 1 See, for example, Färe and Grosskopf (2000). Conventional DEA fails to describe dynamic effects among multi-stage production processes, hindering one from examining the performance of individual sub-units and forcing one to ignore the internal operations in. 政治大 “black box” by concerning only the inputs hired and outputs produced by the firm. 立. the subsectors of a firm such that the entire production process must be viewed as a. Conversely, network DEA allows for the assessment of efficiency for each sub-unit of. ‧ 國. 學. an organization and hence offers guidance to managers on where and how to improve. Nat. sit. y. ‧. each sub-unit’s managerial ability.. io. er. In a two-stage production process the intermediate outputs produced by employing some input mix in the first stage are specifically utilized as inputs in the second stage. al. n. v i n C hTE scores of bothUstages for a bank can then be to manufacture other outputs. The engchi respectively evaluated. Holod and Lewis (2011) propose using network DEA to treat deposits as intermediate outputs in the first production stage by employing some. 1. One of the forms divides the entire production process into a series of interdependent production stages performed by different sectors of a firm, or sub-units. The network model has been widely applied in the literature due to its capability in providing insight regarding the locations of inefficiency and processspecific direction to help managers promote the firm’s TE. Eppen et al. (1998) and Färe et al. (2007) introduce various network models. Pardalos et al. (1997) and Cook et al. (2010) address some advances in this research field. There are many extensions and interesting applications of network DEA models in the literature. See, for example, Färe and Whittaker (1995), Seiford and Zhu (1999), Castelli et al. (2001), Sexton and Lewis (2003), Lewis and Sexton (2004), Prieto and Zofío (2007), Avkiran (2009), Bogetoft et al. (2009), Chen (2009), Chen et al. (2009), Kao (2009, 2014), Tone and Tsutsui (2009, 2014), Kao and Hwang (2008, 2010), Vaz et al. (2010), Holod and Lewis (2011), Yang and Liu (2012), Lewis et al. (2013), and Matthews (2013). Seiford and Zhu (1999) are pioneers in applying two-stage network DEA to analyze the TE scores of U.S. banks, followed by Holod and Lewis (2011) who study TE of U.S. bank holding companies with the non-oriented network DEA method. 5.

(13) portions of labor and capital. In the second stage, deposits and the remaining fractions of labor and capital are defined as inputs to produce the outputs of loans, investments, and non-interest revenues. In this manner, deposits are considered neither an input nor an output and hence the aforementioned dilemma is avoided.. Although the idea of Holod and Lewis (2011) is novel, the fractions of labor and capital consumed in the first stage are usually unknown2, unless disaggregate data for each sub-unit are available. These fractions are not estimable under the network DEA. 政治大 stochastic frontier approach (SFA) are popular methods and have their own strengths 立. framework. In the past literature of efficiency analysis for firms, both DEA and the. and weaknesses.3 However, to date, only DEA can formulate and gauge the production. ‧ 國. 學. efficiency that accounts for inter-sectoral linkages, while the role of SFA played in this. ‧. area is ignored. This paper develops an economic model that embodies the two-stage. sit. y. Nat. network production processes for banks. This dynamic modeling recognizes multiple. io. er. bank production processes and is free from judging whether more or less deposits are consistent with higher bank efficiency. We then formulate an econometric model with. al. n. v i n simultaneous equations, on the C basis of the economicU h e n g c h i model. A salient feature of our. Intuitionally, the fractions of a bank’s labor and capital consumed in the first-deposit stage production are expected to be less than 0.5 and more than 0.5, respectively. The deposit gathering stage requires less labor force, when compared to the labor input used in providing other various final financial activities, including different loans, investment, and non-interesting operating services. In order to absorb deposits, commercial banks need to provide a large amount of physical capital, such as tangible branches or ATMs, to satisfy customers’ basic transaction needs or offer other information facilities related to depositors’ information and needs, as compared to subsequent other production processes in the banking system. 3 Numerous studies have investigated efficiency-related issues ever since Farrel (1957). They are broadly classified into two major categories: SFA and DEA. The former dates back to Aigner et al. (1977) and Meeusen and Van Den Broeck (1977), while the latter is pioneered by Charnes et al. (1978) and Banker et al. (1984). A salient feature of non-parametric DEA is that it does not require assuming specific functional forms for a representative firm, but since DEA is deterministic in essence, it is unable to separate the inefficiency term from the noise component. Contrary to DEA, SFA is capable of distinguishing inefficiency from the noise component, but requires specifying a particular functional form and distributional assumptions on the composed errors. Readers can reference the excellent books written by Kumbhakar and Lovell (2000) and Coelli et al. (2005) for SFA and DEA, respectively. 6 2.

(14) modeling is that the fractions of labor and capital, used in the first stage, can be directly estimated, along with the TE scores in the two production stages for each bank. More importantly, the fractional parameters of the share inputs may not need to rely on disaggregate data, i.e., individual sub-sector data. This feature is particularly crucial, because almost all available databanks compile aggregate data of each firm, with few databanks collecting accounting items on the basis of the sub-sectors of firms. Naturally, our model is also suitable for other forms of firms characterized by a series of production processes.. 政治大 The simultaneous equations model consists of the first-stage production frontier 立. of deposits and a cost frontier in the second production stage, under the assumption that. ‧ 國. 學. banks try to maximize the intermediate output quantity of deposits and pursue cost. ‧. minimization in the production of their final outputs. This appears to be the first attempt. sit. y. Nat. in the literature to extend efficiency measurement accounting for either or both. io. er. vertically- and horizontally-integrated production technologies from network DEA to network SFA. In the context of network SFA, the fractional parameters of the inputs. al. n. v i n explicitly enter both frontiers. ItC is noteworthy that costUshare equations must be jointly hengchi. estimated in order to help specifically identify the fractional parameters that are not estimable under a DEA setting, e.g., Holod and Lewis (2011).. Since the two-stage production processes are interrelated, the production and cost frontiers must be jointly determined. Their disturbance terms should also be correlated so as to embody the joint decision process. In this manner, it is difficult to construct a joint distribution for the two sets of error components without relying on the copula method. This method is based on Sklar’s theorem that enables one to measure the dependence between random variables, and is suitable here for considering the inter7.

(15) sectoral linkage between the two production stages.4 The copula methodology allows us to derive the joint distribution from the margins of a set of random variables on the basis of the copula family. Following Lai and Huang (2013), we choose the Gaussian copula to derive the copula-based likelihood function. Lai and Huang (2013) assert that omitting the dependency component will lead to biased technical efficiency measures and a loss of efficiency for parameter estimates.. The estimation of the fractional parameters is very important, because they are the. 政治大 show the distribution of some inputs among production stages, on the other hand. This 立. core link between sub-units adopting various sub-technologies, on the one hand, and. enables us to deeply investigate the “black box” issue and describe a bank’s entire. ‧ 國. 學. production procedure in a more complete and correct manner. Without them, dynamic. ‧. effects among multi-stage production processes disappear and inter-sectoral production. sit. y. Nat. processes are no longer interdependent. Consequently, the network model collapses to. io. er. a more appropriate conventional DEA or SFA for illustrating a simple, one-stage production process. As mentioned above, the classical models ignore internal structures. al. n. v i n C hmanagers on improving and hence provide little help to bank organizational efficiency. engchi U. The rest of this article is organized as follows. Section 2 develops a theoretical model that splits a bank’s production into two interrelated processes. Two stochastic frontiers, a production frontier and a cost frontier, are established, exemplifying a. 4. Sklar’s (1959, 1973) theorem offers a sound theoretical basis for copula methodology. The theorem has invoked many research fields of application, such as statistics, risk management, finance, economics, etc. Readers can refer to Nelson (2006) and Trivedi and Zimmer (2007) for a detailed and comprehensive description of the copula method. A number of recent studies attempt to shed light on the interdependence issues into area of efficiency analysis, e.g. Bandyopadhyay and Das (2006), Smith (2008), Shi and Zhang (2011), Carta and Steel (2012), El Mehdi and Hafner (2013), Tsay et al. (2013), Lai and Huang (2013), Amsler et al. (2014), and Repkine (2014). It is notable that these articles aim to relax the independence assumption between the inefficiency term and disturbance term in the context of a single equation. 8.

(16) relational network model. A simultaneous equations model is next derived and some estimation issues are discussed. Section 3 briefly describes our data. Section 4 performs the empirical study to demonstrate the usefulness of our model, while the last section concludes the paper.. 2.2 Economic Model The first stage assumes a bank, without loss of generality, employs two inputs, i.e., labor and capital, denoted by a 2-vector x, in order to produce deposits (Z). Under this. 政治大 function to characterize the立 technological relationship between inputs and the output. A. assumption of a single output, it seems to be a natural choice of employing a production. ‧ 國. 學. cost function may alternatively be chosen as the cost function contains basically the same information that the production function contains, by relying on the principle of. ‧. duality. The stochastic production frontier with conventional error components is. sit. y. Nat. expressed as:. n. al. where  x  1 x1 ,  2 x2  ,. (1). er. io. Z = f  x  ev1 u1 ,. C 1, 2 . i n U. v. denotes the fractions of the bank’s total. hengchi. employment of labor and capital that are specifically consumed for the first-stage production of Z, v1 ~ N (0,  v21 ) is the random disturbance term uncontrollable by bank managers, and u1 ~ N  (0,  u21 ) is the output-oriented technical inefficiency and a nonnegative random variable. Variables v1 and u1 are conventionally assumed to be mutually independent. The remaining fractions, 1    1  1 ,1   2  , of the inputs, together with the intermediate output Z , are used to manufacture final outputs, such as loans, investments, and off-balance sheet activities in the second stage. In this 9.

(17) manner,  plays a pivotal role due to its ability to connect the two production processes.. It is noteworthy that vector  differs from the measure of input-oriented technical efficiency, say, b, as proposed by Atkinson and Cornwell (1993, 1994). The production function of an output y with input-oriented technical inefficiency can be formulated as: y = f  bx  ,. (1). 政治大 value of b is, the more technically 立 efficient is the firm, and vice versa. Constant b where 0<b  1 is a scalar and signifies the input-oriented TE measure. The higher the. ‧ 國. 學. represents the degree of TE, while  represents the distribution of resources between sub-units. The estimation of  and b relies on the dual cost function, since the. ‧. production function alone is unable to identify them. See below. For the case of multiple. n. al. Ch. engchi. er. io. production transformation function, i.e., F Y , bX   0 .. sit. y. Nat. outputs, the above single-stage production function should be re-expressed as a. i n U. v. Since the second production stage constitutes the primary and conventional stage of a bank, we assume that the bank attempts to minimize its production cost PC *   in this stage, by employing the remaining fractions of labor and capital, as well as the intermediate output from the first stage, i.e., deposits, to produce an m-vector of final outputs Y:. W W    PC *  Y ,  b X  F Y , b X   0   min   b X b  b  . 1 min W   b X  F Y , b X   0 b b X  10.

(18) . 1 PC Y ,W  , b. (2). where  X  1 x1 ,  2 x2 , 3 Z  with 1  1  1 ,  2  1   2 , and  3  1 , W is the corresponding factor prices, F   is the implicit production transformation function, and b scales down the actual input mix arising from input-oriented managerial inability.. Equation (2) is distinguishable from the previous literature in that it incorporates both the TE measure of b and fractional parameter  , in addition to the traditional. 政治大 technology parameters in a cost function. The assumption of cost minimization enables 立 us not only to consider multiple outputs for a bank, but also to estimate all of the. ‧ 國. 學. parameters of interest, including b,  , and the underlying technology parameters in. ‧. the setting of simultaneous equations. The particular identification problem of  will. er. io. sit. y. Nat. be disentangled shortly.. A bank’s demand function for the ith input can be derived by Shephard’s Lemma,. n. al. i.e.:. Ch. PC  W  b i X i  Y ,   Wi b   b *. engchi. i n U.  bi X i Y ,W . v. (3). and. PC * PC *  Wi b  1   b i X i   i X i . Wi b  Wi b  Wi. 11. (4).

(19) Taking a partial derivative of (2) with respect to Wi , we obtain:. PC * 1 PC  . Wi b Wi Equation (4) implies that:. Xi . 1 PC * .  i Wi. The share equation of the ith input can be written as:. Si  Y , W  .  XW  ln PC * PC * Wi   i i * i  Si *  ln Wi Wi PC PC.  W Y ,  b.  ln PC *   .    ln Wi b . (5). 政治大 Equation (5) indicates that the inclusion of technical inefficiency proportionally raises 立 the expenditures of each input such that these input shares remain unchanged.. ‧ 國. 學. The actual expenditure and share equations are defined as:. ‧. PC * Si E  Wi X i  Wi  PC *  Si  i1  PC *G Y ,W   iWi i 1. al. i  1, 2, 3. (6). er. io. 3. ,. y. Nat. Wi X i Wi Si PC * Si  i1   , E E  iWi G. sit. 3. v. i 1. n. where G Y ,W    Si  i1 . Taking the natural logarithm with respect to E and adding. Ch. engchi. i n U. error terms of v2 and  j , j  3, 4,5, to (6) to account for random shocks out of the control of bank managers, we obtain: ln E  ln PC Y ,W   ln b  ln G Y ,W   v2 = ln PC Y ,W   ln G  v2  u2 ,. (7). where u2   ln b is assumed to be a non-negative random variable independent of v2 , reflecting the additional cost incurred by technical inefficiency. Terms v2 and u2 are distributed as N (0,  v22 ) and N  (0,  u22 ) , respectively. In addition, the actual share 12.

(20) regression equation becomes:. Wi X i Si i1  + j , E G. j  3, 4, 5 .. (8). As the three cost shares must sum up to unity, only two of them can be included to avoid the singularity problem of the covariance matrix of the disturbances. Herein, we arbitrarily choose to preclude the last input share.5. Recall that deposits are regarded as an intermediate product that is an output of the first production stage and is treated as an input to the second stage. The presence of this. 政治大 endogenous (choice) variable, 立 and has to be jointly determined within the two stages intersectoral linkage implies that the intermediate output should be considered as an. ‧ 國. 學. by managers. Therefore, the structural equations of (1), (7), and (8) must be simultaneously estimated to embody the treatment of deposits as neither an input nor. ‧. an output by, e.g., the maximum likelihood. At the same time, the fractions of 1 and. y. Nat. er. io. sit.  2 can be estimated only under the framework of simultaneous equations, where (8) plays the key role of identifying them. The additional parameters of 1 and  2. n. al. Ch. engchi. i n U. v. cannot be estimated without (8). We are able to estimate 1 and  2 for each bank, provided panel data are available.6. It is noteworthy that the individual subsectors’ input and output data of a firm are usually not available from accounting statements. In other words, the fractional. 5. In the conventional system estimation of (7) and (8), iterating on seemingly unrelated regressions until convergence generates maximum likelihood estimates (Kmenta and Gilbert, 1968), and Barten (1969) has shown that these estimates are invariant to which share equation is deleted. We also estimate (7) and (8) by dropping the first (second) share equation and the estimation results are indeed robust. 6 Naturally, these estimators are subject to the problem of incidental parameters if the time period of each bank is finite. See, for example, Neyman and Scott (1948) and Hsiao (2003). 13.

(21) parameters, e.g., of 1 and  2 , are not directly observable and have to be estimated. This frequently impedes empirical researchers from using network DEA to investigate the performance of different subunits in a firm, and thus some more of less ad hoc assumptions have to be made. The structural equations of (1), (7), and (8) are quite useful, since they provide a feasible way of jointly estimating all parameters of interest, including those fractional parameters, without the need for collecting sub-units’ data. Instead, the firm’s aggregated data for relevant variables give enough information to identify the structural parameters and allow for the estimation of TE scores for each production stage or subsector.. 立. 政治大. ‧ 國. 學. In the context of system regression equations, the joint distribution of. 1 ( v1  u1 ) ,  2 ( v2  u2 ) ,  3 , and  4 has to be constructed by the copula method.. ‧. Based on Sklar’s theorem, the joint cumulative distribution function (CDF) of a set of. Nat. er. io. sit. y. random variables, F () , can be associated with the copula function, C () , as:. F (1i ,  2i ,  3i ,  a 4i )  C ( F1 (1i ), F2 ( 2i ), F3 ( 3i ), F4 ( 4i );  ) ,. n. v ni. where Fj () ,. l C hengchi U j  1, 2,3, 4 , denotes the one-dimensional marginal CDF, and. (9).  is a. vector of dependence parameter that measures dependence among the marginal CDFs. Therefore, the copula function allows us to model the correlated stochastic frontier regressions and is shown to be unique if all of the marginal CDFs are continuous. Since. 0  Fj ()  1, the copula function can also be viewed as a multivariate distribution of uniform U[0, 1] variables with the dependence parameter  .. The corresponding joint probability density function (PDF) can be obtained by 14.

(22) taking derivatives of equation (9) with respect to 1i ,. ,  4i  , i.e.: 4. f (1i ,  2i ,  3i ,  4i )  c( F1 (1i ), F2 ( 2i ), F3 ( 3i ), F4 ( 4i );  )   f j ( ji ) .. (10). j 1. There are many forms of copula functions, e.g., the multivariate Student’s t copula, Archimedean copula, Gumble n-copula, Clayton n-copula, etc. Each of them imposes a different dependence structure. See Cherubini et al. (2004) for a complete review of the copula functions. Further extensions of our proposed approach to other copula functions should follow the same procedure with a similar calculation. Following Lai. 政治大. and Huang (2013), we select the Gaussian copula to derive the joint CDF of. 立. 1i ,  2i ,  3i ,  4i , expressed as:. ‧ 國. 學. F (1i ,  2i ,  3i ,  4i )  4 (1 ( F1 (1i )), 1 ( F2 ( 2i )), 1 ( F3 ( 3i )), 1 ( F4 ( 4i ));  ) . (11). ‧. io. sit. y. Nat. 1 Here,  () is the inverse CDF of the standard normal, and  4 () is the CDF. n. al. er. of a standard 4-variate normal distribution of the random variables with a mean vector. i n U. of zeros and 4  4 correlation matrix     jk  , i.e.:. 12.  1      12  13   14. 1.  23  24. 13  23 1. 34. Ch. engchi. v. 14   24  . 34   1 . The corresponding Gaussian copula density can be written as: 4. f (1i ,  2i ,  3i ,  4i )  c( F1 (1i ), F2 ( 2i ), F3 ( 3i ), F4 ( 4i );  )   f j ( ji ) j 1. . 1. . 1/ 2.  1  exp   i' (  1  I 4 ) i    f j ( ji ) 2  j 1 4. 15. ,. (12).

(23) where  i  (1 ( F1 (1i )), 1 ( F2 ( 2i )), 1 ( F3 ( 3i )), 1 ( F4 ( 4i ))) , and. I 4 is the. 4  4 identity matrix.. After taking the natural logarithm, we write the log-likelihood function of the 4 multiple regression equations for a sample of n observations as: N. N. 4. N. ln L( )   ln f (1i ,  2i ,  3i ,  4i )   ln c( F1 (1i ), F2 ( 2i ), F3 ( 3i ), F4 ( 4i );  )   ln f j ( ji ) i 1. . i 1. j 1 i 1. N 1 ln     i' (  1  I 4 ) i   ln f j  ji  , 2 2 i 1 j 1 i 1 N. 4. N. (13). 政治大. where  is the unknown parameter vector of the system equations. Under the. 立. regularity conditions for the asymptotic maximum likelihood (ML) theory, the ML. ‧ 國. 學. estimators can be shown to be consistent, asymptotically efficient, and asymptotically normal, provided that the copula function is correctly specified. The objective function. ‧. of (13) can be reduced to the case that separately estimates the 4 equations by keeping only the last term, i.e., L     ln f j   ji  , since only the first term of. y. j 1 i 1. ), F2 ( 2i ), F3 ( 3i ), F4 ( 4i );  ) captures the correlation between equations.. er.  ln c( F (. io. N. N. sit. Nat. 4. al. 1. 1i. n. v i n C hthe separate regression Lai and Huang (2013) claim that e n g c h i U may still give consistent i 1. estimates and valid standard errors under correctly specified marginal densities, but the standard errors are inefficient.. We present the PDFs of the composite errors of  1 and  2 as:. f1  1i  . 2. 1.  1i   1i 1   ,  1   1 . . and. 16.

(24) f 2   2i  . 2. 2.   2i    2i 2   ,  2   2 . . where  j   uj2   vj2 , j = 1, 2,  j   uj /  vj , and    is the standard normal PDF. Since those PDFs have no closed form, the computation of the corresponding CDFs is not an easy job. Some numerical integration or simulated ML procedures, e.g., Greene.  . (2003, 2010), may be used to approximate the integration in computing Fj  ji . Thanks to Tsay et al. (2013), we employ the mathematical approximation functions to.  . derive the closed form of Fj  ji , which incurs an approximation error within 105 .. 政治大 We undertake the same procedure to derive the closed forms of 立. ‧. ‧ 國. 學. are presented in the appendix A.. Fj   ji  , j = 1, 2, which. One caveat is worth mentioning, i.e., the log-likelihood function of (13) is highly. sit. y. Nat. non-linear as it contains CDFs of  j and the inverse functions of those CDFs, along. er. io. with the PDFs of  j being skewed normal distributions. The estimation of such a. al. n. v i n C h Some simplifiedUprocedures may be required for likelihood function is not that simple. engchi. the purpose of making the likelihood function converge easier in estimation and to at least have the corresponding parameter estimates be consistent. After getting all of the parameter estimates by maximizing equation (13), we evaluate the measure of technical efficiency (TE), proposed by Battese and Coelli (1988)7, for each sample bank.. 7. The efficiency scores of the first production function of deposit and the second cost function are based. on the formula of.    *i        E  ui |  i   *i   *   *        1    *i   TE  exp u i    *   , i.  .. 17.

(25) 2.3 The Data We compile the data for commercial banks of the U.S. in 2009 from the Report of Condition and Income (Call report). The cross-sectional data contain 6,182 observations after deleting some banks with either missing or extreme values. All nominal variables have been deflated by the consumer price index (CPI) with the base year of 1982. We identify three outputs, two inputs, and an intermediate output on the basis of the intermediation approach. The output variables include total loans ( Y1 ),. 政治大. investments ( Y2 ), and non-interest income ( Y3 ). The input variables are labor ( X 1 ) and. 立. ‧ 國. 學. physical capital ( X 2 ), while the intermediate output ( Z ) is purchased funds. The price of labor ( W1 ) is defined as the ratio of personnel expenses to the total number of full-. ‧. time employees. The price of physical capital ( W2 ) is obtained by dividing the expenses. sit. y. Nat. io. al. er. on premises and fixed assets over the total dollar value of premises and fixed assets.. n. The price of purchased funds ( W3 ) is given by the ratio of total interest expenses to total. Ch. engchi. i n U. v. funds. Total expenditure ( E ) is equal to the sum of the above three items of expenditures on hiring the three inputs. The price of labor is arbitrarily chosen as the numeraire to impose the homogeneity constraint of the first degree in input prices.. Table 1.1 provides the descriptive statistics for the aforementioned variables. In 2009, an average U.S. bank granted US$188 million in loans, engaged in investment up to US$57 million, and earned non-interest income of US$2 million. The average price of labor is about US$60,000, the average price of physical capital is 0.3668, and the average price of funds is 0.0177. Most of the variables have quite large standard 18.

(26) deviations relative to their sample means, meaning that the sample banks differ substantially in their output quantities and input prices.. Table 1.1 Sample Statistics Variable Name. Mean. Standard Dev.. Minimum. Maximum. Total loans* ( Y1 ). 188313. 264669. 6194. 2800000. Investments* ( Y2 ). 57971. 86385. 685. 919083. Noninterest income* ( Y3 ). 2220. 4442. 24. 55958. 67. 83. 5. 751. 5155. 7409. 41. 68274. 248938. 334912. 12505. 3400000. Labor ( X 1 ) Physical capital* ( X 2 ) Funds* ( X 3 ) Price of labor* ( W1 ). 12. 307. Price of physical capital ( W2 ). 61 治18 政 0.3668 0.4866 大. 0.0213. 8.8389. Price of funds ( W3 ). 0.0177. 0.0008. 0.0637. 立. 0.0058. ‧. ‧ 國. 學. *：Measured in thousands of US dollars.. 2.4 Empirical Results. Nat. sit. y. We specify both production and cost functions of a bank as the translog form,. n. al. er. io. because it is a flexible functional form providing a second-order approximation,. i n U. v. commonly used by numerous practitioners.8 The cost function is required to satisfy the. Ch. engchi. regularity conditions suggested by the fundamental microeconomic theory. The homogeneity and symmetry conditions can be directly imposed on equations (7) and (8), while the monotonicity conditions in prices and outputs and concavity in prices can be verified after the parameters in equations (7) and (8) are estimated.. We extend the copula-based maximum likelihood approach, first proposed by Lai and Huang (2013) under the framework of the SFA, to estimate a 4-equation system,. 8. Other functional forms, such as the Cobb-Douglas, constant elasticity of substitution, and generalized Leontief forms, are also potential candidates. Moreover, the production and cost functions may take different forms. 19.

(27) consisting of equations (1), (7), and (8). The highly non-linear nature of the loglikelihood functions, as mentioned above, challenges empirical researchers. Kumbhakar and Lovell (2000, p.165) suggest a two-step procedure to consistently estimate the system equations, which is a less efficient but computationally simpler method. We adopt their idea here and conduct simulations to confirm the appropriateness of the method.. In the first step, we view the 4 system equations as the seemingly unrelated. 政治大 disturbance terms for the time being, which bias the intercepts of the two equations as 立. regression models. The composed errors in (1) and (4) are assumed to be the standard. the means of the two composed errors are constants differing from zero, by assumption,. ‧ 國. 學. which are absorbed by the intercepts. The application of the non-linear least squares. ‧. potentially leads to consistent parameter estimates for the fractional parameters, as well. sit. y. Nat. as all of the slope parameters in (1) and (7). Monte Carlo simulations will be conducted. io. er. and the results are suggestive of consistency for those estimates.. al. n. v i n C h and slope parameters In the second step, these fractional are treated as given, and engchi U. the share equations of (8) are overlooked since they merely function for identifying the. fractional parameters. The constant terms of (1) and (7), along with the parameters that characterize the distributions of v and u and the dependence in the copula function, i.e.,  u1 ,  v1 ,  u2 ,  v2 , and  , are simultaneously estimated with respect to equations (1) and (7) by the copula-based ML approach. Huang et al. (2014) examine the consistency property of the two-step procedure - similar to Kumbhakar and Lovell (2000) in essence - by Monte Carlo simulations under the framework of a semiparametric regression model. Since the second-step procedure here is analogous to 20.

(28) Huang et al. (2014), we shall not carry out simulations once more.. One problem in the second step worth mentioning arises from the treatment of the estimated fractional and slope parameters as given, so that the error terms contain estimation errors that are functions of explanatory variables shown in the first step and hence result in the variances of the error terms being heteroskedastic and the estimated standard errors of the parameter estimates being inconsistent. To correct for the inconsistency, caused by possible misspecification, we suggest using the procedure. 政治大 covariance matrix of estimators in order to obtain correct standard errors. 立. proposed by White (1982), which requires computing the sandwich-form of the. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. 9. 9. Ch. engchi. i n U. v. Based on (13), the standard ML estimator has the inverse of the Fisher information matrix   ln L    ˆ I     E   as the covariance matrix of the estimator  . However, the covariance matrix     of the quasi-maximum likelihood estimators has the so-called sandwich form: 2. T.   ln L    is the score function. Johnston and DiNardo     (1997), pages 428-430, provide a brief discussion of the quasi-maximum likelihood estimation of misspecified models and the derivation of the covariance matrix. There are two problems to be solved before deriving the partial derivative of  ln L /  . First, I. 1.   S   S   I   , where T. 1. S    E . formula I ( A) and I ( B ) contain the sign functions of sign( A) and sign( B) , which causes the app. app. log-likelihood function to be discontinuous with respect to  . We re-express lnL for different values of the sign function, i.e., sign()  0 and sign()  0 . Second, it is difficult to calculate the partial derivative of  ln L /  , since lnL includes the function of  1 () . We adopt the popular software of Mathematica to do the job. 21.

(29) 2.5 Simulation Results We now briefly describe our design of experiments, followed by evaluating the biasness and mean square error (MSE) of the proposed estimators using Monte Carlo simulations. This helps gain further insight into the performance of our estimators. Here, we specify the production function in the first production stage as the Cobb-Douglas form for simplicity and the cost function in the second stage as the translog form with a single output and three inputs. Following Olson et al. (1980), Fan et al. (1996), and. 政治大. Huang et al. (2014), we consider three sets of variance ratios and variances, i.e. (  ,  2 ). 立. = (1.24, 1.63), (1.66, 1.88), and (0.83, 1.35), where  2   u2   v2 and    u /  v .. ‧ 國. 學 ‧. Table 1.2 presents the simulation results, for the cases of (1 , 12 ) = (1.24, 1.63). sit. y. Nat. and (2 ,  22 ) = (1.66, 1.88), in terms of biases and MSEs for the parameter estimates. er. io. of the production function.10 We see that the estimators of 1 and  2 perform well. al. n. v i n even for the case of the smallestC sample size, i.e., N =U h e n g c h i 300. Their biases and MSEs are. quite small and decrease as the sample size increases. The estimator of ln(1 X1 ) has similar properties to those of 1 and  2 . Although the estimator of ln( 2 X 2 ) has a little larger bias, this bias is relatively small to its true value and diminishes with larger sample sizes, while its MSEs are relatively large, implying that this parameter cannot be accurately estimated. As expected, the estimator of the constant term exhibits quite. 10. We also conduct simulations assuming (1 ,  1 ) = (1.66, 1.88) and (2 ,  2 ) = (0.83, 1.35); (1 ,  1 ) = 2. 2. 2. (0.83, 1.35) and (2 ,  2 ) = (1.24, 1.63). Since the simulation results are quite similar, we choose not to 2. report them to save space. 22.

(30) large biases, irrespective of the sample sizes. Moreover, its MSEs are also large, which can be attributed to the fact it is confounded with the mean value of the inefficiency term.. Table 1.3 shows the simulation results for the cost parameters. All estimators, except for the intercept, have quite small biases and MSEs. However, the estimator of. ln Y1 has a little larger MSE. The above Monte Carlo simulations provide evidence supporting our proposed estimation procedure in the first estimation step, which relies. 政治大 therefore should be revised in the second estimation step. Further recall that we treat 立. on the NISUR. Note that the estimators of the constant terms perform poorly and. the estimates of the fractional and slope parameters as given in the second step.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 23. i n U. v.

(31) Table 1.2 Biases and MSEs of the parameter estimates of the production function N. 300. 1 2 constant ln(1 X1 ) ln( 2 X 2 ). 500 MSE. 1000. True Value. Bias. Bias. MSE. Bias. MSE. 0.26 0.68 0.50 0.30. -0.0032 -0.0053 -0.7019 0.0010. 0.0027 0.0080 2.0471 0.3355. -0.0020 -0.0030 -0.7228 0.0051. 0.0017 0.0047 1.4993 0.2020. 0.0003 0.0007 -0.7430 -0.0019. 0.0008 0.0023 1.0446 0.0932. 0.70. -0.0922. 3.0351. -0.0808. 1.8690. -0.0662. 0.9417. Table 1.3 Biases and MSEs of the parameter estimates of the cost function. 立Bias. True Value. 1000. 0.9541 -0.0053 0.0006. 16.7176 5.7325 0.0155. 0.8966 0.0323 -0.0012. 0.02. 0.0061. 0.1098. 0.0095. 0.10. 9.16E-06. 0.5165. -0.0122. 0.5*ln(W2 / W1 )2. -0.03. -0.0003. 0.0009. -0.0002. 0.5*ln(W3 / W1 )2 ln(W2 / W1 )*ln(W3 / W1 ) ln(W2 / W1 )*ln Y1 ln(W3 / W1 )*ln Y1. -0.05. -0.0001. 0.0016. -0.0009. 0.5*(ln Y1 ). 2. io. 0.20 0.70. a l0.0002 0.0008 -4.39E-05i v -0.0003 n C h 0.0035 U0.0004 -0.0021 e n 0.0105 g c h i -0.0024. n. 0.02. MSE. 9.6189 3.1953 0.0087. 0.8949 0.0231 0.0020. 5.4052 1.6716 0.0046. 0.0638. 0.0133. 0.0303. 0.2879. -0.0068. 0.1508. 0.0005. 0.0007. 0.0003. 0.0009. -3.74E-05. 0.0004. 0.0005 0.0021 0.0060. -0.0007 -0.0017 -0.0026. 0.0003 0.0010 0.0030. er. ln(W3 / W1 ). Bias. ‧. ln Y1 ln(W2 / W1 ). MSE. 學. 0.80 0.20 0.08. Nat. ‧ 國. constant. 政治 500 大 MSE Bias. y. 300. sit. N. Having presented that the parameter estimates have the desirable property of consistency, we proceed to perform an empirical study on the U.S. commercial banks. Tables 1.4 and 1.5 report estimation results from the two-step procedures. Most of the parameter estimates of the production and cost functions achieve the 1% level of significance. Note that the standard errors of 1 , 2 ,  1 ,  2 ,  , and the two constant terms are corrected by the sandwich form, as shown in footnote 7. Using these estimates, we verify the regularity conditions required by the microeconomic theory 24.

(32) and find that most of the observations have the correct signs, indicating that those parameter estimates can theoretically characterize the underlying production technologies used in the two production stages.. The fractional parameters of 1 and  2 are estimated to be equal to 0.26 and 0.68, respectively. This tells us that the sample banks allocate respectively 26% and 68% of their entire workforce and capital stock to luring deposits in the first production. 政治大 involves a fund collection process, where banks are committed to offer safe and 立. process. The results appear to be acceptable, since the first production stage mainly. convenient banking services to satisfy all customers’ needs. Hence, a bank must. ‧ 國. 學. accumulate a large amount of tangible assets like branches, ATMs, data processing and. ‧. storage facilities, security technology equipment, etc. in this stage. The second. sit. y. Nat. production stage corresponds to revenue generation, where various funds collected in. io. er. the first stage are transformed into miscellaneous loans and investments in corporate and government securities, and used to engage in some off-balance sheet activities. In. al. n. v i n C h interest revenues this stage, banks earn their traditional and capital gains from loans engchi U. and investments, as well as non-traditional income, fees, and commissions, from providing financial services to, e.g., credit card holders, wealth management clients, letters of credit issuers in international trade, and new financial derivative products. These services appear to require more labor input. Here, the estimated values of 1 and  2 lead us to infer that a representative bank allocates 74% and 32% of its entire workers and capital, respectively, to fulfill the second production process.. 25.

(33) Table 1.4 Parameter estimates of the production function Variable Parameter Estimates 1 0.2618*** 2 0.6761***. Standard Errors 0.0067 0.0091. constant ln(1 X1 ). 9.4628***. 0.0018. 0.9812***. 0.0081. ln( 2 X 2 ). 0.0593***. 0.0067. 0.5*ln(1 X1 )*ln(1 X1 ). -0.0047*. 0.5*ln( 2 X 2 )*ln( 2 X 2 ). 0.0027. 0.0050***. ln(1 X1 ) ln( 2 X 2 ). 0.0013. -0.0005. 1 1. 0.0016. 治政0.5581*** 大 1.5041***. 立. 2.23E-05 2.28E-05. al. n. 0.5*ln Y2 *ln Y2. 0.1858***. 0.5*ln Y3 *ln Y3 ln Y1 *ln Y2. Ch. 0.1223*** 0.0181*** U e-0.1148*** ngchi. y. 0.0184. sit. io. 0.5*ln Y1 *ln Y1. 0.3161***. 0.0182. er. lnY3. 0.0288. 0.3176***. Nat. ln Y2. Standard Errors 0.0101. ‧. Table 1.5 Parameter estimates of the cost function Variable Parameter Estimates constant 6.4443*** ln Y1 0.0779***. 學. ‧ 國. Note: *** and * denote significance at the 1% and 10% levels, respectively.. v ni. 0.0037 0.0019 0.0018 0.0021. ln Y1 *ln Y3. -0.0354***. 0.0021. ln Y2 *ln Y3. -0.0008. 0.0015. lnW3. 2.1701***. 0.0052. 0.5*ln W3 *ln W3. 0.2302***. 0.0004. -0.0200***. 0.0004. ln W3 *ln Y1. 0.0105***. 0.0003. ln W3 *ln Y2. 0.0028***. 0.0002. ln W2 *ln W3. ln W3 *ln Y3. -0.0001. 0.0002. lnW2. -0.1072***. 0.0034. 0.5*ln W2 *ln W2. 0.0057***. 0.0003. ln W2 *ln Y1. 0.0026***. 0.0003. ln W2 *ln Y2. -0.0002 26. 0.0002.

(34) ln W2 *ln Y3. 2 2 . -0.0012***. 0.0002. 1.8053***. 0.0842. 0.3393***. 0.0082. -0.7840***. 0.0129. Note: *** denotes significance at the 1% level.. We use the foregoing parameter estimates to evaluate the individual TE scores in the two production stages, denoted by TE1 and TE2, respectively. The average value of TE1 from the production frontier is equal to 0.6717 with a standard deviation of 0.1127, reflecting that an average bank could be technically efficient if it can produce around. 政治大 equal to 0.7758 with a standard deviation of 0.0919. This implies that an average bank 立. 33% more intermediate output. The average TE2 score implied by the cost frontier is. ‧ 國. 學. is fully cost efficient, should it cut roughly 28.9% (=1/0.7758 - 1) of its current input mix. Both TE1 and TE2 are accurately estimated due to their small standard deviations. ‧. relative to the individual means. Our model allows for identifying the fractional. sit. y. Nat. parameters and estimating the technical efficiency scores in the two production. io. er. processes, enabling bank managers to adopt valid strategies to improve operational. al. performance in both stages. Moreover, the value of the correlation coefficient between. n. v i n C himplying that the more is as high as 0.8861, e n g c h i U efficient the production of. TE1 and TE2. deposits is, the more efficient is the second production stage for final products. Bank managers are recommended to increase their managerial ability at the first stage, optimizing the allocation of inputs, since this may foster cost efficiency in the second stage.. We can calculate the measure of output elasticity by summing up the partial 2. derivatives of (log)output with respect to each (log)input, i.e.,    ln f /  ln xi . If i 1.  is greater than, equal to, or less than unity, then the technology exhibits increasing, 27.

(35) constant, or decreasing returns to scale (IRS, CRS, or DRS). Our empirical result regarding the mean value of scale economies is equal to 1.93, reflecting that an average U.S. bank exhibits IRS in the first stage. We may conclude that the sample banks should keep expanding their production scale in order to enjoy the advantage of economies of scale, since doubling all their inputs would raise outputs by more than double.. The coefficient estimates of the cost frontier in the second stage permit us to evaluate the measures of scale economies (SE) and scope economies (SC).11,12 The. 政治大 banks are, on average, operating under increasing returns to scale technology, which is 立. average SE measure is equal to 1.05 with a standard deviation of 0.0408. The U.S.. consistent with the findings of Wheelock and Wilson (2012), Hughes and Mester (2013),. ‧ 國. 學. and Restrepo et al. (2013), to mention a few. The sample banks benefit from economies. ‧. of size and therefore are suggested to expand their production scale in order to reduce. sit. y. Nat. their long-run average cost. The average SC measure is equal to 0.07, indicating the. io. er. presence of scope economies. It is preferable for those U.S. banks producing multiple outputs to concentrate on a single output or a few outputs. This finding tends to support. al. n. v i n Ch the formation of financial conglomerates that are capable of providing an array of engchi U. financial products within an organization in such a way as to share various resources, like computer equipments and clients’ information. The application of the network DEA model requires knowing the distribution of. 11. The formula of scale economies is written as SE  C (Y , W ) /. 3.  Y C (Y ,W ) , where j. j. C j (Y , W ). j 1. 12. is the partial derivative of C (Y , W ) with respect to the jth output. The measure of returns to scale is increasing, constant, or decreasing, when the SE is greater than, equal to, or less than unity. Following Kim (1986), we formulate scope economies as:. SC  C (Y1  21 , 2 , 2 )  C (1 , Y2  22 , 3 )  C (1 , 2 , Y3  23 )  C (Y1 , Y2 , Y3 )  / C (Y1 , Y2 , Y3 ) , where.  j (j =1, 2, 3) denotes 10% of the mean value of the jth output. If SC is greater than (less than). zero, then the economies (diseconomies) of the product mix prevail. 28.

(36) some inputs among alternative production stages - that is, the fractional parameters must be known, a priori. This requirement does not generally hold and cannot be estimated in the context of DEA. Holod and Lewis (2011) thus propose a modified model that avoids this requirement and can assess the efficiency of each bank at the expense of failing to evaluate the efficiency of each stage. Kao and Hwang (2010) impose restrictions on the values of fractional parameters in the range of [0.6, 0.9]. These restrictions may not be consistent with the true condition and hence give rise to undesirable estimation results. To validate this assertion, we re-estimate our model. 政治大 combinations of them. Table 立6 shows the results of average TE scores in the two stages,. under the assumption that the values of 1 and  2 are arbitrarily given for several. ‧ 國. 學. which are denoted by TE1* and TE2* , respectively, for different sets of 1 and  2 . The column “Diff” represents the difference in the average TE scores between ( TE1 ,. ‧. TE2 ) and ( TE1* , TE2* ). For the case of 1  0.5 and  2  0.5 , i.e., half the labor and. sit. y. Nat. io. n. al. er. physical capital are consumed in the first stage, the average values of TE1* and TE2*. i n U. v. are roughly 74% and 80%, respectively, which are significantly different from the. Ch. engchi. corresponding average values of TE1 and TE2. The results reveal that the fractional parameters play important roles in determining efficiency scores in different stages. We also consider other cases, such as ( 1 = 0.26,  2 = 0.26), ( 1 = 0.68,  2 = 0.68), and ( 1 = 0.68,  2 = 0.26), and the results are presented in the second to fourth rows of Table A.6. All of the paired differences between ( TE1 , TE2 ) and ( TE1* , TE2* ) attain statistical significance at the 1% level. We conduct another experiment, where the intersectoral production processes are assumed to be mutually independent, i.e., setting  = 0. The conclusion of the 29.

(37) statistical tests is still the same as above, as shown in the fifth row of Table 1.6. We finally consider the traditional treatment, i.e., banks employ a bundle of inputs to produce an array of outputs in a single stage. Efficiency scores can be evaluated either by a production function that contains two inputs, labor and capital, and an output, deposits, or by a cost function that includes three inputs, labor, capital, and funds, and three outputs, loans, investments, and non-interest income. Average efficiency measures are reported in the bottom two rows of Table 1.6. These average values are significantly different from those of TE1 and TE2 .. 政治大. Table 1.6 The mean TE scores of various sets of 1 and  2. -0.0315*** -0.1014*** -0.0757*** -0.1636*** -0.2059*** －. n. al. Ch. engchi U. Note: *** denotes significance at the 1% level.. -0.0267***. 0.90 0.93 0.95 0.85 －. -0.1257*** -0.1511*** -0.1741*** -0.0784*** －. y. 0.70 0.77 0.75 0.84 0.88 －. Diff.. 0.80. sit. ‧ 國. -0.0649***. ‧. io. 0.74. TE2*. 學. Nat. 1 = 0.50,  2 = 0.50 1 = 0.26,  2 = 0.26 1 = 0.68,  2 = 0.68 1 = 0.68,  2 = 0.26 =0 Production Function Cost Function. Diff.. TE1*. er. 立. v ni. 0.93. -0.1546***. Figure 1.1 plots the kernel densities of the estimated TE scores for all cases considered, where Panel A corresponds to our empirical results with estimated ˆ1 = 0.26 and ˆ 2 = 0.68. The remaining panels of B to G draw the kernel density for each of the cases considered in Table 1.6, in which the fractional parameters are not estimated, but given. Obviously, shapes of the kernel densities of the remaining panels deviate substantially away from those in Panel A, implying that the distributions of those TE scores are indeed dissimilar. 30.

(38) Panel A TE1. 2. Density. 0. 0. 1. 1. Density. 2. 3. 4. 3. TE2. 0. .2. .4. .6. .8. 1. 0. .2. .4. .6. .8. 1. .8. 1. .6. .8. 1. .6. .8. 1. α1=0.26, α2=0.68. α1=0.26, α2=0.68. Panel B TE2. 6. 6. 8. TE1. .2. .4. 2 0. ‧ 國. 0. .6. .8. 1. 0. .2. α1=0.5, α2=0.5. .4. 學. 0. 2. 立. 4. Density. Density. 4. 政治大. ‧. Panel C. .2. .4. .6. sit. 15. .8. er. 5. Ch. i n U. engchi 0. 0. 1. n. al. Density. io. Density. 2. Nat. y. 20. TE2. 10. 3. TE1. 0. .6. α1=0.5, α2=0.5. 1. 0. .2. α1=0.26, α2=0.26. v. .4. α1=0.26, α2=0.26. Panel D TE2. 10. 20. Density. 4. 0. 2 0. Density. 6. 30. 8. 40. TE1. 0. .2. .4. .6. .8. 1. 0. α1=0.68, α2=0.68. Panel E 31. .2. .4. α1=0.68, α2=0.68.

(39) TE2. 60 40. Density. 0. 0. 20. 2. Density. 4. 80. 6. 100. TE1. 0. .2. .4. .6. .8. 1. 0. .2. α1=0.68, α2=0.26. .4. .6. .8. 1. .6. .8. 1. .8. 1. α1=0.68, α2=0.26. Panel F. 6. 10. 8. 10. TE2. 15. TE1. 5. 4. Density. Density. 政治大. .4. ‧ 國. .2. .6. 2. .8. 1. 0. .2. .4. ρ=0. Panel G. ρ=0. 學. 0. 0. 0. 立. .2. .4. 25. sit er. 10. Density. y. 20. .6. production function. Ch. .8. 0. 0. n. 2. io. al. 5. 4. Density. 6. Nat. 0. 15. 10 8. TE2. ‧. TE1. engchi 1. 0. .2. i n U. v. .4. .6. cost function. Figure 1.1 Kernel densities of estimated TE scores. 32.

(40) In sum, Table 1.6 and Figure 1.1 present that the mean TE is sensitive to different values of 1 and  2 , depends on the presence of  , and varies with the assumption of a single stage or multiple production stages. If 1 and  2 are given incorrect values, then the resulting TE score tends to be misleading. This suggests that the fractional parameters should be either given directly on the grounds of disaggregated data or estimated by an appropriate econometric model like the one proposed by this article. In addition, employing the copula method is necessary since it is able to account for the dependence between the production frontier and the cost frontier, characterizing. 政治大. the two production processes of banks.. 立. ‧ 國. 學. 2.6 Conclusion. ‧. sit. y. Nat. This essay extends the network DEA model to a copula-based network SFA model. io. er. that embodies multi-stage production processes for banks under the framework of simultaneous equations. These equations are derived from economic models under the. al. n. v i n C h in the first stageUand cost minimization in the assumption of output maximization engchi. second stage. Our model allows for correlated composite errors, which arise from two subunits of a bank that are connected in series by intermediate outputs and reflect the joint decision making for the two subunits made by bank managers. In this manner, the entire production process is no longer treated as a “black box”, as previous studies did, and the efficiency measures of different subsectors can be respectively estimated. More importantly, the fractional parameters are not required to be known, a priori, but rather can be estimated using aggregated, instead of disaggregated, data of firms - a salient feature of our model. The theoretical model can be transformed into an econometric model, consisting of 4 simultaneous equations, for the purpose of identifying the 33.

(41) additional fractional parameters.. In the empirical study, deposits are treated as an intermediate output produced in the first production stage and described by the translog production frontier that is a function of some portions of labor and capital, rather than the entire employment of both inputs. This intermediate output is next viewed as an input in the second production stage, together with the remaining portions of labor and capital, to produce loans, investments, and non-interest incomes, in the context of the translog cost frontier.. 政治大 inputs involved in the first stage can be estimated by relying on additional information 立. Under the copula-based network SFA framework, the fractional parameters of the two. from the cost share equations. The resulting simultaneous equations, formed by a. ‧ 國. 學. stochastic production frontier, a stochastic cost frontier, and two cost share equations,. ‧. can be consistently estimated by the ML whose likelihood function is derived by the. sit. y. Nat. copula methods of Lai and Huang (2013). This allows for estimating the mutual. io. n. al. er. dependence among equations.. Ch. Adopting the idea of a two-step procedure,. engchi. v i n which U is. in essence similar to. Kumbhakar and Lovell (2000), we successfully solve the convergence problem specifically for a highly non-linear simultaneous equations model. To justify this procedure, we conduct Monte Carlo simulations to show the consistency of the parameter estimates. The efficiency score of each subsector and the fractional parameters can then be assessed and identified. The knowledge of the technical efficiencies in and the resources’ allocation among different stages provides detailed information to regulatory authorities, bank managers, and industry consultants, concerning the effects of mergers and acquisitions, capital regulations, deregulation of deposit rates, removal of geographic restrictions on branching and holding company 34.

(42) acquisitions, etc. on bank performance and probability of failure.. Using data of U.S. commercial banks in 2009, we demonstrate that the fractions of a bank’s labor and capital consumed in the first-stage production are estimable and equal to 0.26 and 0.68, respectively. This implies that a representative bank is apt to allocate relatively less of its workforce and more of its physical capital in the first production stage to satisfy customers’ financial needs. This intermediate output, along with the remaining inputs of labor and capital, is next used to produce three final outputs. 政治大 average TE score in the first stage is equal to around 0.6717, meaning that an average 立 in the second production stage, which is primarily a revenue generation process. The. bank is manufacturing around 67.17% of the best-practice bank’s output for a given. ‧ 國. 學. input mix. The average TE measure in the second stage is equal to 0.7758, implying. ‧. that an average bank should cut 28.90% of its current expenditures to reach the cost. sit. y. Nat. frontier. Evidence shows that the major source of inefficiency comes from the first stage,. io. er. rather than the second stage. Banks are suggested to enhance their managerial ability, particularly in the first production process, in order to effectively raise output quantities. n. al. for a given input mix.. Ch. engchi. 35. i n U. v.