Measuring Bank Efficiency

Prepis

1 Charles University in Prague Faculty of Social Sciences Institute of Economic Studies MASTER THESIS Measuring Bank Efficiency Author: Bc. Zuzana Iršová Supervisor: PhDr. Ing. Petr Jakubík, Ph.D. Academic Year: 2008/2009

2 Declaration of Authorship Hereby I declare that I compiled this thesis independently, using only the listed resources and literature. Prague, May 14, 2009 Signature

3 Acknowledgments I am grateful especially to my supervisor Dr. Petr Jakubík for his help and commentaries, furthermore to Tomáš Havránek for the brilliant overall reviewing. Very contributory insight into the topic of stochastic frontier was provided particularly by prof. Pekka Ilmakunnas from the Helsinki School of Economics, Finland; my thanks belongs also to prof. Timothy Coelli from the University of Queensland, Australia, for his consultancy on Frontier 4.1 software technicalities. I cannot overlook my gratitude for the opportunity of accessing the journal databases of the University of Helsinki; besides, I am thankful to prof. Michal Mejstřík from the Institute of Economic Studies, Charles University in Prague, for granting me the access to the BankScope database, to Juraj Kopecsni for his kind and prompt help on DEA analytical tool, and to Dr. Jozef Baruník. Last but not least, I am indebted to my dearest family members for their support and patience during my studies, without which this work would never come to existence. The usual caveat applies. Typeset in L A TEX 2ε using the IES Thesis Template. Bibliographic Record Iršová, Zuzana: Measuring Bank Efficiency. Master thesis. Charles University in Prague, Faculty of Social Sciences, Institute of Economic Studies, 2009, pages 126. Supervisor: PhDr. Ing. Petr Jakubík, Ph.D.

4 Abstract This thesis provides an empirical insight on the frontier efficiency estimation methods in banking and their sensitivity toward the change in definition of particular characteristics in the techniques used. The two methods, stochastic frontier approach (SFA) and deterministic data envelopment analysis (DEA) are compared over several variations, results of which are supported by the meta-regression part including 32 studies on the USA and 14 on the transitional countries. The main findings of this study include: the efficiency score is highly dependent on the methodological design, the largest variation in the estimated scores of SFA and DEA are due to Fourier-flexible functional form application, and the rank order correlation between these methods raises with an increase of the homogeneity degree in the sample. JEL Classification Keywords Author s Supervisor s C13, C61, G21, L25, P27 Bank Efficiency, Stochastic Frontier Approach, Data Envelopment Analysis, Meta-Regression Analysis zuzana.irsova@gmail.com petr.jakubik@cnb.cz Abstrakt Táto práca poskytuje empirický náhľad na metódy výpočtu efektívnej hranice v bankovníctve a citlivosť tejto metódy na zmeny v definícii špecifických parametrov použitých techník. Závery porovnania dvoch prístupov, stochastického odhadu efektívnej hranice (SFA) a deterministického odhadu obálkovou analýzou dát (DEA), sú podporené meta-regresnou časťou zahŕňajúcou 32 štúdií pre oblasť Spojených štátov Amerických a 14 štúdií tranzitívnych ekonomík. Medzi hlavné výsledky patrí zistenie, že odhady efektivity sú vysoko závislé na metodologickej štruktúre v zadaní, najväčšie rozdiely medzi odhadmi SFA a DEA nastávájú pri použití Fourierovej funkčnej špecifikácie a poradová korelácia medzi týmito metódami vzrastá so zvyšujúcim sa stupňom homogenity v dátach. Klasifikácia JEL Kľúčové slová autora vedúceho práce C13, C61, G21, L25, P27 Bank Efficiency, Stochastic Frontier Approach, Data Envelopment Analysis, Meta-Regression Analysis zuzana.irsova@gmail.com petr.jakubik@cnb.cz

5 Contents List of Tables List of Figures Acronyms Thesis Proposal vii ix x xi 1 Introduction Motivation Theoretical Background Stochastic Frontier Approach Introduction to Stochastic Frontier Data and Methodology Empirical Results Commentaries on the US Dataset Commentaries on the Transitional Dataset Concluding Remarks Data Envelopment Analysis Introduction to DEA Data and Methodology DEA Efficiency Estimation Commentaries on the US Dataset Commentaries on the Transitional Dataset Concluding Remarks Meta-Regression Analysis of Bank Efficiency Measurement Introduction to MRA Highlights of the Frontier Approach Data Collection and Methodology Explanatory Meta-Regression Analysis Results interpretation for the US data set Results interpretation for transitional countries Concluding Remarks

6 Contents vi 5 Conclusions 73 Bibliography 85 A Addendum to SFA B Addendum to DEA C Supplementary Tables to Meta-analysis D Content of Enclosed DVD I XI XIV XXIX

7 List of Tables 2.1 Definition of variables used in regressions of SFA Descriptive statistics on variables utilized for the US dataset (displayed in mil. USD) Descriptive statistics on variables used for the transitional data (in ths. USD) Market share of banks in countries and regions as of (A) Cross-sectional estimates of efficiency scores in the USA (A) Rank order correlations across the US models Determinants of bank efficiency in the USA Signs for determinants of efficiency scores in the USA (B) Efficiency scores in transitional countries for (B) Rank order correlations across trans. models in (B) Comparison of efficiency scores in different models Determinants of bank efficiency in transitional countries Signs for determinants of transitional efficiency scores Descriptive statistics on variables utilized for the US dataset Descriptive statistics on variables used for the transitional data Economic efficiency DEA estimation, VRS LP DEA cost and revenue scores in the USA Rank order correlations across the US models Signs for determinants of efficiency scores in the USA Determinants of bank efficiency in the USA DEA cost, profit and revenue scores in trans-countries Rank order correlations across trans-models Signs for determinants of efficiency scores in trans-countries Determinants of bank efficiency in transitional countries Number of observations relative to the US characteristics Number of observations as to characteristics in trans-economies Meta-regression of bank efficiency in the USA Meta-regression of bank efficiency in transitional countries A.1 (A) Profit efficiency scores in trans-countries for III A.2 (A,B) Stochastic panel cost frontier for trans-countries by years.... IV

8 List of Tables viii A.3 (A) Stochastic panel profit frontier for trans-countries by years.... V A.4 (B) Stochastic panel profit frontier for trans-countries by years.... VI A.5 (A) Stochastic panel profit frontier for trans-countries, VII A.6 (A) Efficiency scores in transitional countries for VIII A.7 (A) Rank order correlations across trans-models in VIII A.8 (B) Cost efficiency in transitional countries by models ( )... IX A.9 (B) Profit efficiency in trans-countries by models ( ) X B.1 DEA cost and revenue scores in trans-countries XI B.2 DEA cost and revenue efficiency in the USA XII C.1 List of studies used for the US data XV C.2 List of studies used for the US data cont XVI C.3 List of studies used for data of transitional countries XVII C.4 List of studies used for data of transitional countries cont XVIII C.5 List of studies used for data of transitional countries cont XIX C.6 Comparison efficiency in the USA and trans-countries XX C.7 Meta-regression of bank efficiency in the USA till XXI C.8 Meta-regression of bank efficiency in the USA from XXII C.9 Meta-regression of bank cost efficiency in the USA XXIII C.10 Meta-regression of bank profit efficiency in the USA XXIV C.11 Meta-regression of bank efficiency in trans-countries till XXV C.12 Meta-regression of bank efficiency in trans-countries from XXVI C.13 Meta-regression of bank cost efficiency in transition economies.... XXVII C.14 Meta-regression of bank profit efficiency in transition countries.... XXVIII

9 List of Figures 1.1 Input and output oriented measures of TE and AE, resp CRS & VRS frontier and DEA & SFA frontier, resp (A) Kernel USA cost density (epanechnikov, bandwidth ) (A) Kernel USA profit density (epanechnikov, bandw ) (B) Kernel trans. cost density (epanechnikov, bandw ) (B) Kernel trans. profit density (epanechnikov, bandw ) (B) Box plot for trans. cost and profit scores in , resp Box plot for trans. 2y cost and revenue VRS scores, resp Aveff of the U.S. and transitional banking sector, respectively Histograms of aveff USA and aveff trans, respectively A.1 (3) Box plot of the US Mean cost efficiency I A.2 (3) Box plot of the US Mean profit efficiency I A.3 (A,B) Kernel trans. profit density (epanechnikov, bandwidth & ) II A.4 (A,B) Development of trans. profit scores by years II A.5 (A,B) Development of profit scores in trans-countries II A.6 (A,B) Development of cost scores in trans-countries II A.7 (A,B) Development of profit scores in trans-countries II A.8 (A,B) Kernel trans-cost density (epanechnikov, bandw )..... III A.9 (A,B) Kernel trans-prof density (epanechnikov, bandw )..... III B.1 Cost and revenue efficiency in the USA (epanechnikov, bandwidth & 0.022), respectively XI B.2 Cost and revenue efficiency in trans-countries (epanechnikov, bandwidth & 0.068), respectively XI B.3 Box plot of the US cost efficiency, 2y VRS XII B.4 Box plot of the US revenue efficiency, 2y VRS XII B.5 Box plot for trans. 2y cost and revenue CRS scores, resp XIII B.6 Box plot for trans. 3y cost and revenue VRS scores, resp XIII B.7 Development of trans. 3y cost and revenue VRS scores XIII

10 Acronyms BCC CAMEL CCR CEE CRS DEA DFA DMU FDH FDI FE FGLS GLS HHI IRLS LP ML MRA OLS RE SBM SFA TFA VRS WAPM Banker, Charnes & Cooper (1984) model Capital, Assets, Management, Earnings, Liquidity (rating system) Charnes, Cooper & Rhodes (1978) model Central and Eastern Europe Constant Returns to Scale Data Envelopment Analysis Distribution-Free Approach Decision-Making Unit Free-Disposal Hull Foreign Direct Investment Fixed Error Feasible Generalized Least Squares Generalized Least Squares Herfindahl-Hirschman Index Iteratively Re-weighed Least Squares Linear Programming Maximum Likelihood Meta-Regression Analysis Ordinary Least Squares Random Error Slacks-Based Measure Stochastic Frontier Approach Thick Frontier Approach Variable Returns to Scale Weak Axiom of Profit Maximization

11 Master Thesis Proposal Author Supervisor Proposed topic Bc. Zuzana Iršová PhDr. Ing. Petr Jakubík, Ph.D. Measuring Bank Efficiency Topic characteristics Policy decision-making of such a highly regulated sector as banking has significant impact on the overall country s economical performance. The studies measuring technical or cost efficiency are expected to provide a reliable results of how certain restrictions or managerial proceedings impact financial institutions. In recent years, the research has focused on frontier analysis covering several measurement techniques divided into mathematical linear programming and econometric approaches. The advantage of using frontier methods lies in an objective numerical efficiency value and ranking of firms, deprived of market price effects and other exogenous factors that may influence observed performance of the institution. However, application of different methods does not always provide consistent results. Choosing specific method might then affect the policy or bank strategy based on the study result. Hypotheses H1: The foreign ownership leads to more efficient banks and efficiency increases with a bank size in transitional economies. H2: The choice of estimation procedure and its characteristics definition (country, researched years, functional form, data aggregation, bank type, inputs and outputs used) have a significant impact on the estimated result. Methodology I would like to demonstrate the differences in economic efficiency estimates directly by both econometric (stochastic frontier analysis) and mathematical (data envelopment analysis) approach on a sample of commercial banks of chosen transitional countries. Secondly, I will execute a meta-analysis on how the methodological characteristics of efficiency literature influence the efficiency scores and discuss the results in accordance with the former part of the thesis.

12 Master Thesis Proposal xii Outline 1. Theoretical introduction 2. Stochastic Frontier Approach 3. Data Envelopment Analysis 4. Meta-analysis of the efficiency measurement sensitivity toward the choice of methodology 5. Conclusion Core bibliography Berger, A. N. & Mester, L. J. (1997): Inside the black box: What explains differences in the efficiencies of financial institutions? Journal of Banking & Finance, vol. 21, pp Coelli, T., Estache, A., Perelman, S. & Trujillo, O. (2003): A primer on efficiency measurement for utilities and transport regulators. WBI Development Studies, Washington D.C., ISBN: Berger, A. N. & Humphrey, D. B. (1997): Efficiency of financial institutions: International survey and directions for future research. European Journal of Operational Research, vol. 98, pp Charnes, A., Cooper, W. W., & Rhodes, E. (1978): Measuring the efficiency of decision making units. European Journal of Operational Research, vol. 2, pp Battese, G. E. & Coelli, T. J. (1995): A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics, vol. 20, pp Author Supervisor

13 Chapter 1 Introduction 1.1 Motivation The importance of efficiency measurement in the financial sector is related to the extremely extensive impact that an efficient financial system has on the microeconomic as well as macroeconomic level. Financial sector deeply affects the allocation of financial resources, helping to find their best productive employment in the most effective way, reducing misallocation and unnecessary wastes. In order to properly allocate the economic resources, the financial system, banks included, needs to be efficient. Efficiency in banking then supports the fruitfulness of implemented macroeconomic policies, generating durable development, economic growth and welfare (e.g., by reducing the transaction costs). Recent trends in the market development of the banking industry include the growing demand for banking services and financial activities on the large international scale, cumulative impact of the fast technological development, decrease in regulation of the sector and interventions but also an increasing competition on the market. The regulatory weakening gave a starting point to the emergence of acquisitions and mergers, creating larger institutions utilizing the scaling effect especially on the cost level. On the other hand, banks striking against wider competition face a decrease in average profits. Bank management is therefore struggling for an enhancement of efficiency, while regulators and lawmakers have to ascertain the efficiency before globalization of the market. Banks have to design their strategic moves with respect to many variables to survive, prosper and be rewarding, so that their politics and interests involve interests of the regulators, lawmakers, supervisory and antitrust agencies. Both managers and external decision/law-makers need to have the accurate information about the effects of their acts on performance of these institutions. The concept of efficiency as a performance indicator began to formalize in the early works of Edgeworth (for example, Edgeworth 1881) and Pareto (1927), and recorded its empirical implementation in the book of Shephard (1953). Regarding banks, the standard view of efficiency measurement in ratio analysis can be misleading as the cross-sectional differences in inputs and outputs combinations and their prices are not properly accounted for; moreover, the interpretation requires great caution and extensive knowledge of the local bank conditions. In 1957, Mr. Farrell

14 1. Introduction 2 was the first to propose a measure of firm efficiency in terms of the frontier analysis, which is believed to provide an objective numerical efficiency value and ranking of firms. From this occurance, researchers developed a number of different methodologies applying frontier approach. However, the estimated efficiency scores including the exact definition of certain frontier estimation characteristics differ throughout the studies. It is a goal of this work to enlighten the variations behind the estimates. This introductory part does not contain a literature survey as the common practice dictates author believes that a large part of the relevant literature is cited through the whole thesis; but more importantly, the entire Chapter 4 is dedicated to the summary and analysis of works of our interest regarding bank efficiency in transitional countries and the USA. More general overview of the literature on financial institutions is covered in the article of Berger & Humphrey (1997). The following Section 1.2 provides the reader with the basics in general microeconomic background behind the efficiency estimation. The two main methods of the frontier efficiency estimation, stochastic frontier approach in Chapter 2 and deterministic data envelopment analysis in Chapter 3, are going to be compared over several variations, results of which will be supported by the meta-regression part including 32 studies on the USA and 14 on the transitional countries, inserted in Chapter 4. Chapter 5, the conclusion, summarizes and discusses the outcomes of this analysis. 1.2 Theoretical Background Let us very briefly refresh the basic microeconomic theory behind the distance functions and the functions of cost, revenue and profit orientation with multiple inputs and output, with focus on the fundamental measures of technical, cost and allocative efficiency including their interrelations. This theoretical section partially builds on the chapter 2 and 3 from Coelli et al. (2005), and the works of Cooper et al. (2004), Cook & Zhu (2005) and Ray (2004). Consider a firm using n inputs x to produce a single output y. Then, the production function of this firm (or production frontier) is y = f(x), where x is n 1 vector of inputs. The main properties of this function are non-negativity, weak essentiality, monotonicity and concavity in x. Having two outputs only, the isoquant gives all combinations x 1 and x 2 that produce the same amount of the output y, and the slope of the isoquant is called the marginal rate of technical substitution. Marginal product is a derivation of the production function with respect to a particular input, it measures what happens with one input change, ceteris paribus (if the production function is twice-continuously differentiable). On the other hand, the influence of a proportional change in all inputs to the change in outputs reflects the returns to scale. If this impact is the same (changes in inputs and outputs are the same), we call it the constant returns to scale (CRS) case. The short-run production function may view some inputs as fixed; however, this is not the case of the long-term horizons. The production function concept is generalized to the concept with multiple-outputs, summarized by the transformation

15 1. Introduction 3 function T (y, x) = 0, where y is m 1 vector of outputs. The transformation function T (y, x) = y f(x) = 0 faces the same properties as the production frontier. One way of dealing with multiple outputs is to simply aggregate them into a single measure and use the index numbers methodology, or make an advantage of the available price information to create the cost, efficiency and profit functions: c(w, y) = min x w x such that T (y, x) = 0, r(p, x) = min y p y such that T (y, x) = 0, π(p, w) = min y,x p y w x such that T (y, x) = 0, where w is the vector of input prices, and cost function c twice-continuously differentiable behaves according to Shephard s Lemma, that is x n (w, y) = c(w, y)/ w n (= dual approach), cost function is non-negative, non-decreasing in w and y, homogeneous and concave in w. The firm/bank can influence output price vector p in the revenue function, which satisfies the property of non-negativity; moreover it is non-decreasing in p and x, convex in p and homogeneous. The profit function π is non-negative, nondecreasing in p and non-increasing in w, homogeneous and convex in (p, w). If profit function is twice-continuously differentiable, then Hotteling s Lemma applies: x n (p, w) = π(p, w)/ w n and y m (p, w) = π(p, w)/ p m. This implies that the cost, profit and revenue frontiers can be by optimization derived from the transformation functions. In fact, they can be transformed back to the production functions, meaning that they contain the same information as production functions. This relation is called a principle of duality. However, the results so far presented assume that the firm knows how to maximize outputs from given inputs or how to choose inputs and outputs to optimize costs and revenues. As in Coelli et al. (2005), relaxing the previous mentioned assumption, consider a multiple-output production process as a multiple-output production technology, a set S. The non-negative real numbers, elements of n 1 input vector x and m 1 output vector y form the technology set: S = {(x, y) : x can produce y}. The output set P (x) and input set L(y) are defined equivalently: P (x) = {y : x can produce y} = {y : (x, y) S}, L(y) = {x : x can produce y} = {x : (x, y) S}, where these sets have certain assumptions: for P (x), it is possible to produce zero output; moreover, from zero inputs nothing is produced. P (x) satisfies strong disposability of outputs and inputs, 1 P (x) is closed, bounded and convex. L(y) satisfies weak disposability of inputs, 2 strong disposability of outputs, is closed and convex. If a production set refers to some particular year, index t is added. 1 Coelli et al. (2005): strong disposability of outputs: if y P (x) and y y, y P (x). For inputs: if y can be produced from x, then y can be produced from any x x. 2 Inputs are weakly disposable if x L(y) then for all λ, λx L(y), Coelli et al. (2005).

16 1. Introduction 4 x 2 /y S P Q A R y 2 /x D C Z B B A Q S D 0 A x 1 /y 0 Z y 1 /x Figure 1.1: Input and output oriented measures of TE and AE, resp. Distance functions describe multi-input and multi-output production technology without the behavioral problem (cost minimization or profit maximization). Output distance function d o on P (x) and input distance function d i on L(y) is defined as: d o (x, y) = min {δ : (y/δ) P (x)}, d i (x, y) = min {ρ : (x/ρ) L(y)}, for which these axioms are valid: d o (x, y) is equal 0 for all non-negative x, it is non-decreasing in y, non-increasing in x, linearly homogeneous in y, quasi-convex in x, convex in y; if y P (x) then d o 1 and d o = 1 if y belongs to the frontier of P (x). Function d i (x, y) is non-decreasing in x, non-increasing in y, homogeneous in x, concave in x and quasi-concave in y, if x L(y) then d i 1 and d i = 1 if x belongs to the frontier of isoquant y. Distance functions have various applications, especially in index number estimates, but they also back the theory of various efficiency and productivity measures. The more complex mathematical insight can be found in Färe et al. (1998). Following Farrell (1957), Coelli et al. (2005) define efficiency of a firm composed of two components: technical efficiency (TE) as an ability of a firm to obtain maximal output given a set of inputs, and allocative efficiency (AE) as an ability of a firm to use the inputs in optimal proportions, given their respective prices and the production technology. A product of these two measures is so-called economic efficiency. Farrell (1957) illustrated input-oriented measure in the space of two inputs x 1 and x 2 to produce one output y, see Figure 1.1 on the right from Coelli et al. (2005). Knowing SS, the unit isoquant of fully efficient firms and given that a firm uses input quantity of point P to produce a unit of output, technical efficiency of this firm is a distance QP, the amount by which all inputs can be reduced proportionally, without reducing ouptut; i.e., T E = 0Q/0P in % or T E = 1/d i (x, y). When market prices of inputs w are available, it is possible to measure cost efficiency (CE). If ˆx of technically efficient point Q and x is an cost-minimizing input vector at Q, and x is the vector at P, the cost efficiency is defined as a ratio 0R/0P, or CE = (w x )/(w x). If the isocost line AA is known, AE = 0R/0Q or AE = (w x )/(w ˆx), so that the cost

17 1. Introduction 5 y CRS Frontier y DEA Frontier C B VRS Frontier B SFA Frontier G F E D noise A noise inefficiency D E F 0 x 0 x Figure 1.2: CRS & VRS frontier and DEA & SFA frontier, resp. efficiency is a product CE = T E AE. The illustration in Figure 1.1 assumes constant returns to scale (CRS) for simplicity. The right side of the picture is a projection of an output oriented measure. It is again a simple example involving two outputs y 1 and y 2 and a single input x. Because of CRS, ZZ curve is the unit production possibility curve, an upper bound of production possibilities. For an inefficient firm in point A, the measure of outputoriented TE is 0A/0B or T E = d o (x, y). Having an information on output prices, revenue efficiency (RE) can be defined by an output price vector y represented by DD. For y of point A, ŷ the technically efficient vector of B and revenue-efficient vector y associated with point B, the score RE = 0A/0C or RE = (p y)/(p y ), allocative efficiency is 0B/0C ratio or AE = (p ŷ)/(p y ). Overall revenue efficiency is a product of AE and TE. More on this topic including revenue and scale efficiency will be introduced in Chapter 3. The relaxed assumption of CRS, that is an allowance for the variable returns to scale VRS, is compared to CRS in Figure 1.2 on the left. Using an input orientation on the exemplary picture, the scale efficiency SE reflects the amount by which a firm has to improve to reach the point B, the technically optimal productive scale. Because 0D/0E = GE/GD, these distance measures can be used to estimate productivity differences. Technical efficiency of the firm D, the distance from D to VRS technology is T E V RS = GE/GD, while the distance from technically efficient point E to the CRS is equal to SE = GF/GE, the fraction of T E CRS = GF/GD on T E V RS. The multi-input and multi-output case is only a generalization of the idea behind this explanation. In the right panel of Figure 1.2, the SFA and DEA frontiers are projected to compare the stochastic and deterministic frontiers, lining the maximum outputs produced by given set of inputs for the firms A to F. DEA creates a piece-wise linear combination of the best-performing firms assuming VRS (therefore the outer envelope) while SFA creates a fictional deterministic estimate of the frontier driven by the inefficiency component u and the noise v [see Chapter 2, (2.2)]. The deterministic part of SFA has diminishing returns to scale. Firm D uses x D to produce y D if there was no inefficiency u D = 0, the frontier output would be sited at the meeting

18 1. Introduction 6 point of the inefficiency and noise sketch. The noise effect v B in B case is positive, while v D effect is negative. If the difference of inefficiency and noise v i u i < 0, the observed output of a firm stays below the deterministic SFA part. Firms B and C are fully efficient and theirs deviation from the frontier is a stochastic noise. The extent to which noise is recognized from inefficiency is given by the joint distributional assumption of these disturbance components. Although the theory presented before resembles rather the deterministic estimation techniques such as data envelopment analysis (DEA) or free disposal hull (FDH), the estimation through stochastic techniques is in principle the same; the common output-oriented technical efficiency ratio is a fraction of the observed output to the corresponding stochastic frontier output (the one by which a frontier is created). Production function that is estimated by econometric regression with a special disturbance consisting of the inefficiency term and the noise cannot be estimated by simple OLS as this regression would bias the intercept coefficient downwards, because there are standard assumptions for noise (nonzero mean, uncorrelated, homoscedastic) as well as inefficiency (non-negative mean, uncorrelated, homoscedastic) term left. The solution to the bias was developed by COLS estimator in the paper of Olson et al. (1980). Still, maximum likelihood method is used more often, because it produces more efficient estimates and has generally better properties. A dual technology representation like cost or profit function stands in the core of neoclassical production economics. Researchers introduced a number of innovative specifications of the dual functionals (translog or generalized Leontief form) to better simulate the real characteristics of technology, building on works of Hotelling (1932) and Shephard (1953). This thesis draws an attention mainly on the economic efficiency assuming some optimization behavior, such as minimization of costs or maximization of revenues as this is of a crucial importance to the banking industry. Continuously, we will start the following chapter with a more detailed insight into the stochastic frontier approach.

19 Chapter 2 Stochastic Frontier Approach 2.1 Introduction to Stochastic Frontier Benchmarking with parametric techniques of efficiency estimation is based on regression analysis. The most common econometric method, independently developed by Aigner et al. (1977) and Meeusen & van den Broeck (1977), is the stochastic frontier approach (SFA). With use of explicit assumptions about the inefficiency component s distribution, it tries to decompose the residual of the frontier into inefficiency and noise. The direct estimation of production function is a primal approach; but recently, empirical frontier analysis turned to dual approach using cost and profit functions, reasons for which are provided by Battese & Coelli (1995). The SFA assumes the production function of the fully-efficient firm to be known. In general, firms maximize produced output vector Q it (or maximize observed profit, or minimize costs in the dual approach) generated by input variables matrix X (a function of output quantity y it, input prices w it and other variables such as equity capital z it, time variable or country(bank)-specific variables in the dual approach) given the circumstances and technology, with sensitivity indices β. Ideally, Q it = f(x it, β), 1 but in reality it does not hold because of inefficiencies ξ it and other causes, the random shocks exp(v it ). If ξ it (0,1 = 1, a firm is producing optimally, if ξ it < 1, it indicates that a firm with the technology embodied in f(x it, β) can do better. Output Q it > 0 is strictly positive, therefore the degree of technical efficiency is assumed to be ξ it > 0, strictly positive as well. Q it = f(x it, β)ξ it exp(v it ), (2.1) where Q it stands for the output, and X it is a set of independent variables here a function of input quantity. In the beginnings, the Cobb-Douglas function was used for estimation of (2.1) 2, later (see, for example, Berndt & Christensen 1973) its generalized form came to usage, less restrictive transcendental logarithmic (translog) function, extending the original function by more flexible terms including combina- 1 i denotes cross-sectional dimension, t denotes time dimension. These indices are different from the i and t in the equations from Definition 2.1 in the next section. 2 Cobb-Douglas production function: Q it = X βn nit, logarithmized to ln Qit = β n ln X nit. n=0 n=0

20 2. Stochastic Frontier Approach 8 tions of the product of two variables. Another modification to the original form is the augmentation of translog by trigonometric terms, the so-called Fourier functional form, which is considered by some authors to be the most appropriate in estimations of efficiency in banking sector (as in McAllister & McManus 1993). Rewriting production function (2.1) linear in logarithm with N inputs in logarithmic terms yields: N ln Q it = β n ln X nit + v it u it (2.2) n=0 ( N ) or Q it = exp β n ln X nit exp (v it ) exp ( u }{{} it ), }{{} n=0 }{{} noise inefficiency deterministic component defining ln ξ it = u it, and restriction of u it 0 stemming from u it substracted from ln Q it, which implies that ξ it (0, 1. An analogy for the derivation of production function (2.2) is the cost functional, provided by Kumbhakar & Lozano-Vivas (2000). This function will be the key element of the chapter. They specify the problem as: ln C it = β 0 + j=0 β y j ln y jit + k=0 β w k ln w kit + v it au it, (2.3) where C it is the cost/profit, y jit stands for an output, w kit is the price of an input, a = 1 for production functions (alternative profit function here, the standard profit function would be f(w, p) with output prices p instead of y), a = 1 for cost functions. u it N + (µ, σu) 2 truncated at 0 is the function of firm-specific factors believed iid to determine technical inefficiency. 3 iid v it N(0, σv) 2 stands for disturbances (luck, weather, strikes). Now, the frontier output from (2.1) is X it β + v it and observed output is X it β + ε it, where ε it is a composite error equal to v it au it. According to Battese & Corra (1977) parametrization, the above mentioned σv 2 and σu 2 are replaced by σ 2 = σv 2 + σu, 2 the variance of composed error ε it, and new variable γ = σu/(σ 2 v 2 + σu) 2 is defined, so that γ (0, 1) using ML procedure. For the time-varying model used in this study, the log-likelihood function has the form of: ln L = 1 ( ln 2π + ln σ 2 ) N T i 1 N (T i 1) ln (1 γ) 2 2 i=0 i=0 { ( 1 N Ti ) } ln 1 + ηit 2 1 γ N ln {1 φ( z)} N z2 + i=0 t=1 N ln {1 φ( zi )} i=1 N i=1 z 2 i 1 2 N T i i=1 t=1 ε 2 it (1 γσ 2 ) 3 For cost function, au it 0; 0 implies the most cost-efficient firm and increasing values connote inefficiency. If function is in natural logarithm (ln C it, i.e. values between 0 and 1), the most costefficient firm has a value of 1, the closer to 0 the more cost inefficient. For profit function, au it 0; 0 implies the most profit-efficient firm; in logarithm, 0 denotes firm with the highest profit efficiency.

21 2. Stochastic Frontier Approach 9 ε it is the composite error, η it = exp { η (t T i )}, z = µ/ ( γσ 2) 1/2, and φ(.) is the cumulative distribution function of the standard normal distribution, a is the parameter differentiating between production and cost functions from (2.3), and z i = µ (1 γ) aγ T i t=1 η itε it { ( [γ (1 γ) σ 2 Ti ) }] 1/ t=1 η2 it 1 γ The likelihood function maximization provides us with the estimates of coefficients η, µ, σ v and σ u, the output of Stata 10 software (see more in the publication of Stata Corporation 2005 and 2007) used in this chapter. 4 Note that the inefficiency term can be calculated in one or two-step procedure (more in Coelli 1996). The general form of bank-specific efficiency is determined by vector of variables G it (determinants) and the technological progress (t, t 2 ) explaining technical inefficiency. Either the correlates of inefficiency term are put in the mean (or variance or both) of the truncated error term in dependence on the distribution of inefficiency term assumed; or they may be applied inside the standard functional form as standard explanatory variables besides output quantities and input prices, see Definition 2.1 in the next section. The mean-conditional model (one-step procedure) is defined as: E(µ it ε it ) = t it τ 1 + t 2 itτ 2 + G it δ + ω it. (2.4) where t denotes annual index of time, G is a set of the other variables (conventionally called z-variables) expected to be correlated with the mean-inefficiency term µ it from iid u it N + (µ it, σu), 2 and ω it, the white noise errors. On the other hand, estimating u it by two-step procedure yields: u it = exp ( η (t T i )) u i (2.5) for i-bank and t-time, t = 1,..., T and η parameter signifies inefficiency changes iid over time and u i N + (µ, σu). 2 After estimating the inefficiency term, researchers usually run a second-step individual regression in form of (2.4) but with E(u it ε it ) as a dependent variable. The estimates of technical efficiency term from equation (2.3) are obtained via: [ ] ( 1 φ {aηit σ i ( µ i / σ i )} E {exp( au it ) ε it } = exp aη it µ i + 1 ) 1 φ ( µ i / σ i ) 2 η2 it σ i 2, (2.6) where µ i = µσ2 v a T i t=1 η itε it σ 2 u σ 2 v + T i t=1 η2 it σ2 u and σ 2 i = σvσ 2 u 2 σv 2 +. T i t=1 η2 it σ2 u Replacing η it = 1 and η = 0 changes the time decay model into time-invariant model, so that the estimated efficiencies differ only on the cross-sectional level (for banks), not in the time dimension (through years), i.e. u it = u i. 4 Note that due to software limitations, only truncated normal distribution will be used for the panel estimates.

22 2. Stochastic Frontier Approach 10 To ensure that the estimated frontier is well behaved, the duality theorem requires two standard properties of the production function to be met; symmetry of the second-order parameters and linear homogeneity in input prices are imposed via parameter restrictions homogeneity by normalizing C it (e.g., the price of labor and of fixed capital by the price of funds) and symmetry by conditions of β y ij = βy ji and βij w = βw ji, i, j. Standard restrictions of production function as to linear homogeneity would be: N N βk w = 1, and for translog w-product terms βkl w = 0, k=1 here not applied due to price normalization. 5 SFA is not driven by outliers in such an extent as some of the non-parametric methods (for instance, DEA may pronounce a bank to be efficient not because of its low cost management but because it is an outlier). The cost (profit) function is defined by the behavior of a representative cost-minimizing (profit-maximizing) subject, controlling the amount of every input used to produce a given output implication of a need for properties of linear homogeneity and concavity in input prices, and monotonicity in input prices and output. There is a different progression when estimating annual mean efficiencies in certain industries/countries and firms. Authors estimate efficiency measures (between 0 and 1) for individual financial institutions (firms, banks) for each year of the studied period. These are usually aggregated as a weighted average (where the weight is, for instance, equity of specific banks) for each year to get the average annual efficiency estimate for the observed country. In the case of number of countries (with similar background or environment, such as transition countries, OECD countries, etc., for which efficiencies are reasonable to compare), average annual efficiency estimates can be compared between these; however, cross-country comparisons should be used only in the case of common frontier utilization (pooled panel dataset). We will operate on the US and transitional countries banking data. The US data set is of a secondary importance for our purposes. While the transitional countries inefficiencies are going to be estimated as a panel, we will execute cross-sectional analysis for the US data through all the years, where the residuals attain the right kind of skewness (besides other assumptions of the estimation). The models on the United States will then be normal-half normal: ln L = N i=1 { 1 2 ln 2 ln σ + ln φ π ( aεi λ σ k=1 ) } ε2 i 2σ 2 5 The choice of normalizing the prices and C it has some practical reasons as well; it is problematic to assure price homogeneity for trigonometric terms of Fourier-flexible forms estimations intended to be performed in the next section. This is not the only kind normalization to be made, the cost/profit and output quantities are likely to be normalized by equity capital to control for potential heteroscedasticity.

23 2. Stochastic Frontier Approach 11 and normal-truncated normal model: ln L = N { 1 ( )} µ 2 ln 2π ln σ ln φ σ γ i=1 { [ ] N (1 γ)µ aγε i + ln φ {σ 2 γ(1 γ)} 1/2 1 ( ) } εi + aµ 2, 2 σ i=1 where all the variables are defined in the same way as for the panel model and λ = σ u /σ v. Then, the cost/profit efficiency (a = 1) is estimated by E {exp( au i ) ε i } = { } ( 1 φ (aσ µ i /σ i ) exp aµ i + 1 ) 1 φ ( µ i /σ ) 2 σ2, (2.7) where µ i = aε i σ 2 u/σ 2 for half normal, µ i = ( aε i σ 2 u + µσ 2 v)/σ 2 for truncated normal model, and σ = σ u σ v /σ. Assuming heteroscedasticity in half-normal models, σ u and σ v is replaced by σ 2 i = exp(g iδ), in the conditional mean model of truncated normal distribution, µ = G i δ as discussed by (2.4) and log-likelihood function can be rewritten as: ln L = N { 1 ( )} 2 ln 2π ln σ ln φ Gi δ σ γ i=1 { [ ] N (1 γ)g i δ aγε i + ln φ {σ 2 γ(1 γ)} 1/2 1 ( ) } εi + ag i δ 2. 2 σ i=1 Measurement of bank cost or profit efficiency per se is hardly informative for owners, regulators or customers of a bank. Therefore, studies include the regression analysis (with OLS, GLS, FGLS estimates from one or two-step procedure), where dependent variable is the (computed) level of efficiency and independent variables are such as country macroeconomic variables (population density, financial deepening ratio = assets/gdp), structure of banking industry (intermediation ratio = loans/deposits, density of demand, HH index of market concentration, EBRD index of banking sector development, market share of state owned banks, proportion of foreign-owned banks, population per bank, banking deposits per cap), individual bank characteristics (ownership status, return on average equity, return on average assets, net interest margin). Statistical significance (t-statistics) and direction (positive or negative sign) of the variables impact is commented. Such an analysis consecution is naturally not followed when the inefficiency term or the standard errors of inefficiency and white noise are explained right during the stochastic frontier estimation. This chapter is divided into four main sections following this introduction, we will define and make a summary statistics on the US and transitional datasets in Section 2.2, as well as specify the main methodological progression; the next Section 2.2 provides the reader with results on the datasets with relevant commentaries, and the last section concludes this chapter.

24 2. Stochastic Frontier Approach Data and Methodology This study uses banks balance sheet and income statement data for a sample of European and US banks between the years 1995 and 2006, obtained from the BankScope database (Bureau van Dijk Electronic Publishing). The restrictions on the choice of banks were the following (regarding choice in the database): for transitional countries, we decided for the Central Europe and Baltics region the Czech Republic, Hungary, Poland, Slovak Republic and Slovenia; the second region resides the USA. Regarding bank types, the selection for transitional countries includes commercial, savings and cooperative banks, furthermore, real estate and mortgage banks, medium & long term credit banks, and investment banks & securities houses; the US dataset is restricted to commercial banking only. 6 The general procedure for estimating efficiency using (2.3) is to estimate equation coefficients (ML more efficient than OLS) and composed error ε it = v it + au it and to calculate efficiency for each observation in the sample using the regression errors and (2.6). The inefficiency is then measured as the score relative to the estimated frontier. To make this measure comparable across the techniques of SFA and DEA (Chapter 3), stochastic frontier efficiency estimate will be normalized, so that the most efficient bank in the sample will have a score of 1 (Berger & Mester 1997). Table 2.1: Definition of variables used in regressions of SFA Regressands Description OC Operating costs Operating expenses OP Operating profits Operating income Regressors Output variables y 1 y 2 y 3 Loans Deposits Other earning assets Input price variables w 1 Price of labor Personal expenses over total assets w 2 Price of capital Depreciation over fixed assets w 3 Price of funds Interest expenses /(deposits + other funds) Netputs z Equity capital Let us have a closer look on the variables used in the definitions of functional forms above. In the banking literature, there is a controversy regarding the choice of inputs and outputs. We decided for the most common choice of intermediation approach (Sealey & Lindley 1977) in defining the variables. Having three outputs and three inputs at the disposal, there is a possible freedom in combinations of numbers of variables used in the model to see the sensitivity of the estimate. Short definitions of variables can be found in Table 2.1. The costs OC and profits OP are reported as operating expenses and income of the bank, the output variables cover loans y 1, deposits y 2 and other earning assets y 3 6 Institutions with consolidation code preference of C1, C2, U1 and A1; for US dataset, the classic US coverage was used as well, see BankScope database help for details.

25 2. Stochastic Frontier Approach 13 providing us with 3 possible regressors of Cobb-Douglas specification. Furthermore, there are three inputs to be used: labor x 1, capital x 2 and funds x 3. The prices of labor (personnel on total assets) w 1 and capital w 2 (covering depreciation on fixed assets) are normalized by the price of the funds w 3 (other funds over the sum of interest expenses and deposits). The translog production function applies equity capital as one netput variable. Furthermore, we define the correlates with inefficiency term. The summary statistics appearing in the cost and profit functions on output/input variables as well as correlates are presented in Table 2.2 and Table 2.3. The goal of clarifying the efficiency estimation methodological sensitivity in this study led us to the trial of executing all three mentioned specifications of production function. The intention is to use a multi-product (three inputs & three outputs, and their combinations) function shapes. First, we will define so-called benchmark model which will stand for our preferred specification of the functional form. 7 Definition 2.1 (Benchmark model A). Let us define the Benchmark A model with Fourier-flexible cost/profit functional form, 2 outputs and 2 input prices, not normalized by equity capital z as: 2(3) ln C = α 0 + α i ln y i + w i=1 2 2 k=1 l=1 2 k=1 β k ln w k w δkl w ln w k ln w 2(3) l + w 3 w 3 2(3) + τ 1 t τ 2t 2 + τ y i t ln y i + 2(3) i=1 i=1 i=1 k=1 2 k= τ 2 z ln z 2 + τ zy i ln z ln y i + 4(5) 2(3) 2(3) γ y ij ln y i ln y j (2.8) i=1 j=1 2 ρ ik ln y i ln w k w 3 τ w k t ln w k w 3 + τ z 1 ln z 2 k=1 τ zw k ln z ln w k w 3 + n ξ m G m m=1 + [θ i cos q i + ω i sin q i ] + [θ ij cos(q i + q j ) + ω ij sin(q i + q j )] i=1 4(5) 4(5) 4(5) 4(5) 4(5) i=1 j=i + [θ ijn cos(q i + q j + q n ) + ω ijn sin(q i + q j + q n )] + v au, i=1 j=i n=j where C is the cost (or profit, normalized by w 3 ), w k is the price of an input (prices are normalized by w 3 ) for k = 1, 2, y i stands for the output for i = 1,..., 3; t denotes time variable (not time dimension of panel data but time as a regressor) accounting for technological change over time, G m are bank-specific variables and the q variables are transformation of ln y s and ln w w 3 s according to z-terms from the study of Berger & Humphrey (1997) 8. 7 Note, that the indices denoting cross-sectional and time dimension are not listed, however, we take them as present in all the defined equations. 8 To specify this transformation due to the eligibility of trigonometric terms usage: ln y 1 q 1,..., ln w 2 w 3 q 5, where q i = 0.2π µa + µ ln y i (ln w i w 3 ), µ = (0.9 2π 0.1 2π) / (b a), and a, b is the range of ln y i or ln w i w 3 for i = 1,..., 5.

26 2. Stochastic Frontier Approach 14 Definition 2.2 (Benchmark model B). Define the Benchmark B model with Fourierflexible cost/profit functional form, 2 outputs and 2 input prices, normalized by equity capital z as: ln C 2(3) w 3 z = α 0 + α i ln y i z i=1 2 2 k=1 l=1 2 k=1 β k ln w k w δkl w ln w k ln w 2(3) l + w 3 w 3 2(3) + τ 1 t τ 2t 2 + τ y i t ln y i z + 2(3) τ 2 z ln z 2 + 4(5) i=1 i=1 τ zy i ln z ln y i z + i=1 k=1 2 k=1 2(3) 2(3) 2 i=1 j=1 γ y ij ln y i z ln y j z ρ ik ln y i z ln w k w 3 τ w k t ln w k w 3 + τ z 1 ln z 2 k=1 τ zw k ln z ln w k w 3 + n ξ m G m m=1 + [θ i cos q i + ω i sin q i ] + [θ ij cos(q i + q j ) + ω ij sin(q i + q j )] i=1 4(5) 4(5) 4(5) 4(5) 4(5) i=1 j=i + [θ ijn cos(q i + q j + q n ) + ω ijn sin(q i + q j + q n )] + v au, i=1 j=i n=j (2.9) where variables are defined as in 2.1, the dependent variable C with all output quantities y are normalized by equity capital z to account for heterogeneity. Expression in gray marks the possibility of using these variables either in regression (time and inefficiency correlates) or conditional mean model in other deviations from the benchmark, so that it is not a part of these benchmark models as such. Also, number of outputs and input prices in the sums provided by Definition 2.1 and 2.2 are 2 and 2 respectively, and the ones in brackets denote the possibility of using them in other deviations from these benchmarks; however, these are not parts of benchmark models, too. As postulated in Section 2.1, there are several ways of executing the estimation to get to efficiency scores; after defining the chosen classical shape of the production function, variables that may be explanatory to inefficiency score are added to the production function, or to the regression stating expected value (mean) of inefficiency score or the variance of inefficiency as the regressor, and finally, the impact of possible inefficiency correlated can be examined after the computation of inefficiency scores. The transformed Cobb-Douglas function (first two sums of logarithmized outputs and input prices in definitions) is enlarged by combinations of input prices and output product, forming the translog specification (first two lines in definitions). The prices and cost (profit) are normalized in both definitions to ensure the homogeneity of the functional in prices which is the only way as other specification are considered as deviations from the Fourier-flexible, terms of which are not multiplicative. An insufficient approximation provided by the transcendental-logarithmic functional can be cured by adding trigonometric terms; however, specification prob-