Effects of heterogeneity on bank efficiency scores

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European Journal of Operational Research 195 (2009) 251–261 www.elsevier.com/locate/ejor

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Effects of heterogeneity on bank efficiency scores J.W.B. Bos a, M. Koetter b,c,*, J.W. Kolari d, C.J.M. Kool a a Utrecht School of Economics, Utrecht University, Janskerkhof 12, 3512 BL, Utrecht, The Netherlands University of Groningen, Faculty of Economics & Business and CIBIF, P.O. Box 800, 9700 AV Groningen, The Netherlands c Deutsche Bundesbank, P.O. Box 100602, 60006 Frankfurt, Germany d Mays Business School, Texas A&M University, 310R Wehner Building, 4218 TAMU, College Station, TX 77843-4218, USA b

Received 15 July 2006; accepted 7 January 2008 Available online 26 January 2008

Abstract Bank efficiency estimates often serve as a proxy of managerial skill since they quantify sub-optimal production choices. But such deviations can also be due to omitted systematic differences among banks. In this study, we examine the effects of heterogeneity on bank efficiency scores. We compare different specifications of a stochastic cost and alternative profit frontier model with a baseline specification. After conducting a specification test, we discuss heterogeneity effects on efficiency levels, ranks and the tails of the efficiency distribution. We find that heterogeneity controls influence both banks’ optimal costs and profits and their ability to be efficient. Differences in efficiency scores are important for more than only methodological reasons. First, different ways of accounting for heterogeneity result in estimates of foregone profits and additional costs that are significantly different from what we infer from our general specification. Second, banks are significantly re-ranked when their efficiency is estimated with a specification other than the preferred, general specification. Third, the general specification gives the most reliable estimates of the probability of distress, although differences to the other specifications are low. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Bank production; Heterogeneity; X-Efficiency; Benchmarking; Distress

1. Introduction There is an abundant literature on the measurement of efficiency measurement for both US and European banking industries.1 The measurement of bank efficiency gained importance among both policy makers and practitioners to explain consolidation (Berger et al., 1999; Focarelli et al., 2002; Wheelock and Wilson, 2004), to assess the effects of mergers (Vander Vennet, 1996; Lang and Welzel, 1999), and to discriminate between troubled and healthy * Corresponding author. Address: University of Groningen, Faculty of Economics & Business and CIBIF, P.O. Box 800, 9700 AV Groningen, The Netherlands. Tel: +31 50 363 7822; fax: +31 50 363 7337. E-mail address: [email protected] (M. Koetter). 1 Overview studies by Berger and Humphrey (1997) and Amel et al. (2004).

0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.01.019

banks and predict probabilities of default (Wheelock and Wilson, 2000; Kick and Koetter, 2007). But efficiency measures can vary substantially across different samples and empirical specifications. This may limit the use of efficiency measures by regulators. Ferrier and Lovell (1990) were the first to compare the cost efficiency of US banks resulting from different specifications. They report substantial efficiency differences. In simple nonparametric models, this may be due to the neglect of measurement error (Mountain and Thomas, 1999). But in a comprehensive comparison across five different measurement methodologies, Bauer et al. (1998) still conclude that efficiency scores vary considerably. Mester (1997) hypothesizes that banks are too different to be compared to a common benchmark. She therefore tests, and rejects, the existence of a single cost function for all US banks, and concludes that heterogeneity in large

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samples of banks can lead to poor estimates of individual banks’ efficiency.2 However, without a common benchmark it is difficult to compare efficiency levels and rankings (Coelli et al., 2005; Bos and Schmiedel, 2007). Instead of estimating separate frontiers, most recent studies therefore estimate a common benchmark, but seek to control for systematic differences across banks that are not due to inefficiency. For example, Valverde et al. (2007) find that cost inefficiencies vanish almost completely when they include increasingly many control variables. The present paper aims to contribute to this literature in two ways: (i) we find that test results clearly point at a preferred specification among a number of well-established specifications; (ii) we discuss the various costs of using a specification other than our preferred specification. Thereby, we address the following issues. First, while the (in)stability of cost efficiency scores is well-documented, evidence on the reliability of profit efficiency is virtually absent from the literature. Second, we present a large, high quality sample of banks that share the same business model, objectives and production set, and can thus be expected to have a common (optimal) production technology. Third, in analyzing the robustness of efficiency scores, we not only consider mean scores, but also rankings, as well as the upper and lower tails of the efficiency distribution. We employ a data set that consists of a homogenous sample of German cooperative and savings banks. We estimate cost and profit frontiers using five nested specifications. Specification tests reveal that factors accounting for systematic differences across banks affect both technology and the ability to operate efficiently. The use of anything but the preferred, general specification significantly alters both efficiency levels and ranks. It also (somewhat) decreases the reliability of estimates of the probability of bank distress. Our paper is organized as follows. In the next section we introduce a baseline SFA specification and four variants. The latter are used to assess the stability of efficiency scores. Section 3 presents the data and discusses whether accounting for heterogeneity matters. Section 4 reports the results. Section 5 concludes. 2. Methodology We start by outlining a baseline cost minimization model for banks. Then we discuss a number of specifications that have been developed to account for sample heterogeneity. Since the alternative profit model of Humphrey et al. (1997) differs only in a few respects, we summarize it in footnotes.

2 Mester cites studies by Mester (1993), Kolari and Zardkoohi (1995) and Akhavein et al. (1997) that find different cost functions for banks with different product mixes, individual banks, and mutual versus stock savings and loan associations, respectively.

2.1. Basic model We model bank production following the intermediation approach (Sealey and Lindley, 1977). To produce outputs y, banks minimize costs, TOC by choosing optimal input quantities x ðy; w; zÞ, which are conditional on input prices w and the available level of equity z, and subject to the technology constraint.3 The vector of input prices, w, is given, since banks are assumed to be price takers on input markets. In addition, we follow the literature and already account for heterogeneity in capital structures by including equity, z (Hughes and Mester, 1993). The minimum cost level is then obtained by substituting the optimal input demand functions into the total cost function to obtain TOC ¼ w0  xðy; w; zÞ ¼ TOC ðy; w; zÞ.4 Deviations from optimal cost in year t can be due to either random noise or sub-optimal employment of inputs. We can therefore write a baseline stochastic cost frontier for a bank k in logs and add a composed error term e to the deterministic kernel f ðy kt ; wkt ; zkt ; bÞ as5: ln TOCkt ¼ f ðy kt ; wkt ; zkt ; bÞ þ ekt ;

ð1Þ

where b is a vector of parameters to be estimated. The total error in Eq. (1) is ekt ¼ vkt þ ukt , where vkt denotes random noise, and ukt stands for deviations due to inefficiency. For a cost frontier inefficient input use entails higher than optimal cost and therefore ukt is strictly positive.6 In all specifications, the random error term vkt is assumed i:i:d: with vkt  N ð0; r2v Þ and independent of the explanatory variables. The distribution of the inefficiency term ukt is i:i:d: N jð0; r2u Þj in the baseline specification (Stevenson, 1980). It differs across specifications and is independent of vkt . The point estimator of technical efficiency is given by Eðukt jekt Þ, the conditional distribution of u given e (Jondrow et al., 1982). Estimates of bank-specific cost efficiency are obtained by calculating CEkt ¼ ½expðukt Þ. Cost (profit) efficiency equals one for a fully efficient bank that operates on the efficient stochastic frontier.

3 Since the core function of a bank is to collect deposits against interest as to convert these into loans, interest expenses are interpreted as operational costs for a bank. 4 In the alternative profit model, maximizing profits, PBT, also yields optimal output prices p ðy; w; zÞ, since banks can have some pricing power on the output side, subject to an additional pricing opportunity constraint Hðp; y; w; zÞ. Maximum profits p ðy; w; zÞ depend on given input prices, available equity, and output quantities. 5 The most established parametric approach is stochastic frontier analysis (SFA) (Kumbhakar and Lovell, 2000). While SFA distinguishes random noise from inefficiencies, it requires an a priori assumption on the error term (Ali and Seiford, 1993). Non-parametric methods avoid this rigidity but neglect random noise and are very sensitive to outliers. Alternative parametric methods, such as the thick frontier approach (Berger and Humphrey, 1991, 1992) and the distribution free approach (Berger, 1993), also impose less structure on the error than SFA, but do not provide bank-specific point estimates or require an assumption of constant core inefficiency. We therefore focus on methods to account for heterogeneity in SFA. 6 In the profit frontier the total error is ekt ¼ vkt  ukt .

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Following the literature, we choose the multi-output translog functional form, and write the reduced form of the baseline specification in logs as: ln TOCkt ðw; y; zÞ ¼ a0 þ

I X i¼1

ai ln mikt þ

I X J 1X aij 2 i¼1 j¼1

 ln mikt ln mjkt þ ekt :

ð2Þ

Here m consists of outputs y, input prices w, and control variable z (equity). A time trend t captures technological change (Baltagi and Griffin, 1988).7 All specifications are estimated following the standard procedure outlined in Kumbhakar and Lovell (2000). 2.2. Accounting for sample heterogeneity Our baseline specification relies on two important assumptions. First, in this specification all banks use the same transformation function to convert inputs into outputs, while minimizing costs (maximizing profits). Put differently, all banks share a frontier that has the same slope and intercept. Second, all banks are assumed to share the same efficiency distribution. In this section, we introduce three specifications that either relax one of these assumptions, or both. In doing so, we aim to account for sample heterogeneity. We measure sample heterogeneity in a very simple manner, with a vector of dummy variables hk that captures differences across a variety of banking groups, regions, and size classes. The question of whether a vector of exogenous factors hk should be modeled to influence the position of the frontier or the ability of management to attain that frontier was first recognized by Deprins and Simar (1989). Kumbhakar and Lovell (2000) observe that it can be difficult to determine if an exogenous variable is a characteristic of production technology or a determinant of productive efficiency. Put differently, these systematic differences hk can have two effects on the stochastic frontier: (i) parallel shifts of the frontier, and (ii) systematic different deviations from the frontier. In the baseline specification, exogenous factors hk are completely ignored. The baseline cost frontier is ln TOCkt ¼ f ðy kt ; wkt ; zkt ; bÞ þ ekt as in Eq. (1). In this speci7

For the alternative profit model we face the problem that the log of negative numbers is not defined. As a result, most studies either exclude loss-incurring banks from profit efficiency studies or scale profits up by the sample minimum plus one (Berger and Mester, 1997; Vander Vennet, 2002; Maudos et al., 2002). These approaches either exclude important observations, i.e. those that performed worse in terms of profitability, or may yield biased results due to scaling the data (Greene, 2003). Here, we use an alternative approach, developed and tested in Bos and Koetter (2007), and censor the dependent variable at one for banks incurring losses. At the same time, we specify an additional negative profit indicator variable NPI. The latter equals one for banks with positive profits and the absolute value of losses for banks with negative profits. Finally, results are qualitatively similar if we use the most common approach in the literature and scale up before taking logs, although efficiency ranks are significantly affected (as in Bos and Koetter, 2007).

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fication, efficiency follows a half-normal distribution with mean zero. Hence, the implicit assumption is that the probability mass of the inefficiency distribution is concentrated close to the frontier. Although this implies that most banks are likely to only suffer from a relatively small amount of inefficiency, there is no theoretical reason to make the ex ante assumption that the mean of the truncated half-normal is zero. Therefore, as a minor extension of the benchmark specification, we can estimate the mean l of the truncated half-normal distribution (Stevenson, 1980). By allowing l to be non-zero, we relax the implicit assumption that most banks are highly efficient. Instead, l can now influence the location of the distribution of measured inefficiencies. We refer to this approach as the truncated specification. An important limitation is the fact that we still neglect group-specific sources of heterogeneity. Estimation of a common truncation point for all banks may not suffice to capture the variety of reasons that cause efficiency measures to differ so much. A first way to account for heterogeneity, is by allowing our vector hk to shift the distribution of inefficiency, while the frontier f ðy kt ; wkt ; zkt ; bÞ is the same for all banks similar to the benchmark and truncated models. The difference is that each firm’s ukt now depends on hk (Kumbhakar et al., 1991).8 Inefficiency ukt is still i:i:d: but now drawn from the truncated distribution ukt  N j½ðl þ d0 hk Þ; r2u j, as the ability of banks to reach the efficient frontier now depends on hk . An important implication is that we can account for heterogeneity across banks and still benchmark all banks against an identical frontier.9 A second way to account for heterogeneity, is by including the vector of exogenous variables, hk ; in the deterministic kernel of the frontier. In this case we estimate: ln TOCkt ¼ f ðy kt ; wkt ; zkt ; hk ; b; dÞ þ vkt þ ukt ;

ð3Þ

where d is an additional vector of parameters in the deterministic kernel accounting for systematic differences across banks due to region, size, and banking type. Thereby, we allow the position of the frontier to be different for various (groups of) banks, but maintain the assumption that the shape of the frontier is identical for all banks, as we do not include interaction terms of dummy variables and other production variables.10 Finally, we account for the possibility that heterogeneity controls affect both the deterministic kernel and the error distribution. For example, a macroeconomic shock in the region may not only render optimal cost higher due to increasing credit defaults among strained bank customers, but could also hinder the banks ability to attain full 8

Here we assume that the omitted variable bias may present itself in the efficiency distribution. 9 A further alternative is to specify the variance component of inefficiency to depend on heterogeneity. But since the economic interpretation of this model is less clear, we do not consider it. 10 Implicitly, we assume that the additional dummy variables remedy an omitted variable bias present in Eq. (1).

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Table 1 Model specifications Specification

Kernel f()

Inefficiency u

1. 2. 3. 4. 5.

f(ykt, wkt, zkt; b) f(ykt, wkt, zkt; b) f(ykt, wkt, zkt; b) f(ykt, wkt, zkt, hk ;b, d) f(ykt, wkt, zkt,hk ; b, d)

ukt ukt ukt ukt ukt

Baseline Truncated Error Kernel General

 N jð0; r2u Þj  N jðl; r2u Þj  N jðl þ dhk ; r2u Þj  N jðl; r2u Þj  N jðl þ dhk ; r2u Þj

efficiency due to sluggish adjustments to a changed business environment. Hence, we consider hk in both the deterministic kernel f ðy kt ; wkt ; zkt ; hkt ; b; dÞ þ vkt þ ukt while accounting simultaneously for systematic deviations in the error by specifying ukt  N j½ðl þ d0 hk Þ; r2u j. This general model allows us to test, first, whether nested reduced forms are sufficient to control for heterogeneity and, second, whether inefficiencies vanish entirely after controlling for other systematic deviations. Note that the reduced forms for the truncated and error models are identical to the benchmark model in Eq. (2). Changes are limited to the assumptions concerning the inefficiency distribution. In contrast, the reduced form of the heterogeneity in kernel specifications require an extension of Eq. (2) with the dummy variables hk for g groups yielding: X d g hgk : ð4Þ ln TOCkt ðw; y; zÞ ¼ ½Eq: ð2Þ þ g

In sum, we estimate cost and profit efficiency for a common sample of German banks with five different specifications (Table 1): (i) the simple baseline specification; (ii) the baseline specification with the mean of the truncated half-normal at l; (iii) the specification with exogenous factors in the distribution of the inefficiency term (i.e., the heterogeneity in error specification); (iv) a specification with exogenous variables in the deterministic kernel; and (v) a general specification accounting for heterogeneity in both error and kernel. 3. Data In selecting our sample of banks, we aim to avoid comparing ‘‘excessively” different banks (Mester, 1993). Our motivation for doing so, is that if heterogeneity already matters in such a sample, the effect is likely to be even stronger in cross-country and/or cross-sector studies. In contrast to Altunbas et al. (2001), we exclude commercial banks due to their focus on wholesale and investment rather than retail banking activities. The savings and cooperative banks in our sample account for approximately 80% of all German banks in terms of numbers and capture more than a third of total assets under management. These banks cater especially to consumers and small and medium sized firms (Agarwal and Elston, 2001), while relying mostly on customer deposits as a source of funds (Koetter et al., 2006). Organizationally, both share a two tier governance struc-

ture of centralized head organizations (‘‘Deutscher Sparkassen and Giroverband, DSGV” and ‘‘Bundesverband der deutschen Volksbanken und Raiffeisenbanken, BVR”). Large central cooperative and savings banks (‘‘Landesbanken”) frequently serve as clearing houses and act as providers of interbank financial products. Table 2 provides descriptive statistics for input prices, output quantities, equity, and dependent variables for sample banks gathered from accounting statements reported to the Deutsche Bundesbank between 1993 and 2005. Outputs are interbank loans y 1 , commercial loans y 2 , and securities y 3 . Inputs are fixed assets x1 , labor x2 , and total borrowed funds x3 . For the price of fixed assets w1 we take the ratio of depreciation and other expenditures on fixed assets to fixed assets. The price of labor w2 is calculated as an average wage rate equal to the Euro amount of personnel expenses divided by the number of full time equivalent employees (FTE). The price of borrowed funds w3 is estimated by the ratio of interest expenses to total borrowed funds.11 But even among this relatively homogenous group of banks, some heterogeneity may exist. For example, savings banks operate subject to the principles of regional demarcation (Frankenberger, 2004). Regulations limiting the regional scope of operations imply that regional macroeconomic conditions may affect efficiency systematically different. Further, German savings banks are owned by (local) governments as opposed to the mutual ownership structure of cooperative banks. Hence, the former might have enjoyed funding advantages owed to explicit or implicit government guarantees (Brunner et al., 2004). Additional sources of heterogeneity include alternative deposit insurance schemes in the respective banking sectors. In sum, these banks pursue sufficiently similar business strategies to be comparable from an economic point of view. On the other hand, systematic differences require a more explicit consideration of heterogeneity from an institutional point of view. We aim to capture these differences in a simple manner by specifying a set of dummy variables for different banking groups, regions, and size classes. We distinguish between eight banking groups: two types of savings banks and six types of cooperative banks. Regions are defined as the 16 states (‘‘Bundesla¨nder”) of the Federal Republic of Germany. On the basis of total assets, we allocate banks to four equally distributed size classes in each year.12 To determine if heterogeneity among banks is significant, we conduct a non-parametric rank-sum test (Kruskall and Wallis, 1952). Test statistics are shown in the last three columns of Table 2. All null hypotheses that the respec-

11

Following Maudos et al. (2002), we also excluded extreme outliers at alternative cut-off points. Results are qualitatively similar. 12 We distinguish public savings, independent savings, commercial and rural cooperative banks, Sparda banks, PSD banks (Post-, Spar- und Darlehensvereine), civil servant’s banks, and Raiffeisen banks. Mean total assets in size classes I–IV in millions of Euros are: 51, 142, 345, and 1772, respectively.

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Table 2 Bank production data for German savings and cooperative banks 1993–2005 Variable

y1 y2 y3 w1 w2 w3 z TOC PBT

KW testa

Descriptive statistics Interbank loansb Customer loansb Securitiesb Fixed assetsc Labord Borrowed fundsc Equityb Operating costb Profit before taxb

Mean

SD

Min

Max

States

Groups

Size

55.3 323.0 131.0 13.9 49.1 3.6 24.7 30.1 5.6

158.0 799.0 315.0 6.6 8.0 0.8 57.7 68.3 14.1

0.001 0.762 0.003 5.201 27.69 1.901 0.223 0.261 35.40

4360.0 22,400.0 6570.0 59.2 71.9 5.5 1710.0 1880.0 429.0

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.004 0.000 0.000 0.000 0.000 0.000

Notes: 31,080 observations. a p-Values for Kruskall–Wallis test with 15, 7, and 3 degrees of freedom for states, groups, and size classes. b Measured in millions of Euros. c Measured in percent. d Measured in thousands of Euros.

tively stratified samples are drawn from the same population are rejected.13 Consequently, bank production variables are not the same across banking groups. 4. Results We assess the importance of accounting for heterogeneity arising from alternative model specifications by testing for the preferred specification. Subsequently, we compare efficiency levels, ranks and the tails of the efficiency distribution across specification. The latter helps us shed light on the opportunity cost of not using the preferred specification for society, banks, and bank supervisors. 4.1. Specification Table 3 reports key estimation results for all five specifications of both cost and alternative profit frontiers, respectively.14 Parameter estimates highlight two main results. First, the location parameter l is significantly different from zero for the heterogeneity in error specification and the general specification. This may be explained by the fact that we introduce heterogeneity in the efficiency distribution in these specifications. Second, point estimates of the total error’s inefficiency components cast doubt on the cost truncated and kernel models, respectively. For either specification, both the total variance, r, and the ratio of variance components, k, are insignificant. This result contrasts previous evidence on German bank efficiency (Altunbas et al., 2001; Koetter, 2006). Moreover, parameter estimates of k and r are consistently significant when controlling for heterogeneity in 13 We also conducted independent sample t-tests for east versus west banks as well as cooperative versus savings banks. Results confirmed that differences in means between the two respective sub-samples are significantly different from zero. 14 Complete estimation results are available from the authors upon request.

Table 3 Estimation results Specification

LL

Cost model Baseline Truncated Error Kernel General

21,403 22,114 27,463 28,179 29,817

Alternative profit model Baseline 22,251 Truncated 19,337 Error 17,974 Kernel 17,887 General 16,726

l

k

r

32.455 0.524*** 21.463 1.442***

1.826*** 18.301 1.614*** 16.348 3.871***

0.174 *** 1.639 0.143*** 1.207 0.282***

164.580 24.379*** 169.240 1.539***

3.864*** 37.721** 16.542*** 42.126** 5.049***

0.835*** 8.691*** 3.939*** 8.751** 1.147***

Notes: 31,080 observations; LL = Log Likelihood; r ¼ ðr2v þ r2u Þ1=2 and k ¼ ru =rv . ***,**,* indicate significant at the 1,5,10 percent level.

the error and general specification. Together, these findings cast doubt on the adequacy of the truncated and kernel specifications to estimate the efficiency of German cooperative and savings banks. For the profit model, differences are less pronounced, and the parameter estimates of k and r are consistently significant. In Table 4, we present log-likelihood ratio tests. The heterogeneity in error specification and the heterogeneity in kernel specifications cannot be compared directly against each other. However, all specifications are nested in the general model with heterogeneity in both kernel and error. We first test the error and kernel specifications against the general specification. For the former, the hypothesis is that d ¼ 0 in the error distribution, while for the latter the same restriction applies to the deterministic kernel. For both cost and profit frontiers, we clearly reject the absence of simultaneous effects of controls on both the technology and the ability to achieve full efficiency. Next, we test if the kernel and error specifications reduce to the truncated specification. We reject both hypotheses as well. Hence explicitly accounting for sample heterogeneity

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Table 4 Specification tests Specification Unrestricted

Restricted

General General Error Kernel Truncated

Error Kernel Truncated Truncated Baseline

J

LLR statistic

Critical v2

Cost

Alternative profit

1% level

25 25 25 25 1

4708 3277 10,699 12,130 1425

2494 2321 2727 2900 5828

44.31 44.31 44.31 44.31 6.63

Notes: Log likelihood ratio (LLR) tests with J restrictions.

is important. Finally, we also reject the possibility that the specifications truncated at l reduce to the half-normal model. Despite insignificant point estimates of the respective cost and profit efficiency location parameters, the fit of the model improves sufficiently to reject exclusion of this parameter. This suggests that the level of efficiency depends also on additional factors different from those suggested by the theoretical intermediation approach model alone, but requires a more differentiated specification.15 In sum, we find that sample heterogeneity significantly influences stochastic cost and profit frontier estimates and therefore should be included in (bank) efficiency studies. Based on log-likelihood ratio tests the preferred specification includes controls in both the deterministic kernel and the error distribution. Hence, already simple dummy variables affect both optimal costs and the ability to achieve the latter differently. The truncated specifications (with or without hk in the deterministic kernel) suffer from some difficulties when estimating location parameters of the inefficiency distribution. In the next section, we compare the effect this has on the estimated foregone profits (PE) and additional costs (CE), compared to our preferred, general specification. 4.2. Efficiency levels Knowledge of the level of (in)efficiency is important, both to banks themselves and from a welfare point of view. Decreasing cost efficiency, ceteris paribus, constitutes a welfare loss. Likewise, banks’ foregone profits increase as profit efficiency decreases. Correct estimation of efficiency levels (absolutely and relatively) thus is important as well. In Table 5, we provide descriptive statistics for cost efficiency (CE) and profit efficiency (PE) scores. We first discuss the scores that result from regular estimations, before we turn to our bootstrapped results. The first important result in Table 5 is that CE is higher than PE. This difference between the means of CE and PE varies depending on specifications, and ranges between 11 (for the heteroge15 We also reject the absence of time trend variables capturing technical change. In fact, partial derivatives of the frontiers yield some technical progress regarding the cost functions of German banks similar to the magnitudes reported in Altunbas et al. (1999). Since we focus here on the effects of heterogeneity on efficiency scores, we do not discuss these results here.

neity in error specification) and 29 (for the heterogeneity in kernel specification) percentage points. Consequently, banks incur substantially more slack in terms of foregone profits than in terms of unrealized cost savings. A second important observation is that accounting for heterogeneity does not automatically increase efficiency estimates. For example, controlling for different banking groups in terms of business group, region and size in the distribution of the inefficiency component in total error yields mean CE estimates that are 8 percentage points lower compared to the baseline model. In turn, mean PE improves approximately by the same magnitude. This result differs from Valverde et al. (2007), who report that inefficiency vanishes almost entirely after adding further control variables. However, their study specifies technical and external controls in the deterministic kernel of the frontier only. Our results suggest that it matters substantially whether such factors influence optimal costs (profits), banks’ abilities to achieve optimal costs (profits) given technology, or both. This is corroborated by the results for both mean CE and PE according to the heterogeneity in kernel specification. In fact, this is the only approach which yields univocally enhanced efficiency measures in both the cost and the profit dimension by approximately two percentage points. Recall that point estimates of bank-specific inefficiency are based on the conditional expectation of uk given ek . Therefore, reducing primarily total estimation error by adding increasingly many controls is likely to reduce the level of estimated inefficiency. However, when there is reason to assume that some factors also affect the relative position of the bank in the inefficiency distribution, adding controls in both kernel and error may increase the bank’s relative inefficiency since this depends on both the bank’s estimated position in the inefficiency and total error distribution. This is confirmed by our results for the error and general specification. We also conduct a bootstrapping analysis along the lines of Atkinson and Wilson (1995) and Koetter (2006). Bootstrapped standard errors are reported together with confidence bands for all five specifications in Table 5.16 Regarding cost efficiency, the precision of the general model mimics that of less involved models. Standard errors are of a similar magnitude. More importantly, CE scores generated with the specification accounting for heterogeneity in both kernel and error differ significantly from measures obtained from nested specification. Hence,

16 We draw j ¼ 1; . . . ; 1000 bootstrap samples with replacement of the original size N, i.e. 31,080 observations. For each draw j, we estimate efficiency EFFj relative to each of the five cost (CE) and profit (PE) frontiers, respectively. Mean statistics obtained with the original sample are denoted EFFobs . Standardo errors are calculated according to n 1=2 P c ¼ 1 PJ ðEFF  EFF Þ2 , where EFF ¼ J1 Jj¼1 CEj : Lower SE j j¼1 J 1 c and upper bounds as bounds are calculated as ½EFFobs  t1a=2;k1 SE c where t1a=2;k1 is the ð1  a=2Þth quantile of the ½EFFobs þ t1a=2;k1 SE,

t-distribution with k  1 degrees of freedom.

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Table 5 Efficiency levels Specification

Regular

Bootstrapped

Mean

SD

Min

Max

Mean

SEa

LBb

UBc

0.890 0.828 0.813 0.912 0.848

0.057 0.076 0.074 0.044 0.066

0.650 0.590 0.667 0.693 0.677

0.988 0.986 0.987 0.990 0.989

0.894 0.828 0.810 0.912 0.850

0.005 0.002 0.010 0.004 0.005

0.881 0.823 0.793 0.903 0.839

0.899 0.832 0.833 0.921 0.858

Alternative profit model Baseline 0.610 Truncated 0.693 Error 0.694 Kernel 0.622 General 0.709

0.168 0.178 0.179 0.164 0.179

0.303 0.257 0.253 0.346 0.274

0.967 0.964 0.963 0.971 0.965

0.610 0.693 0.689 0.622 0.702

0.002 0.002 0.002 0.002 0.010

0.605 0.689 0.690 0.618 0.689

0.614 0.696 0.699 0.626 0.728

Cost model Baseline Truncated Error Kernel General

Notes: 31,080 observations; 1000 bootstrapped samples with replacement. a Standard errors after bootstrapping. b Lower bound. c Upper bound.

accounting for heterogeneity is crucial to obtain unbiased cost efficiency estimates. Regarding profit efficiency scores, we find that the general model estimates are only statistically significantly different from the specifications that either fail to account for heterogeneity entirely or restrict dummies to enter the kernel only. In contrast, specifications allocating such controls to the error yield no significant differences. Thus, accounting for heterogeneity is necessary, but affects especially banks’ abilities to attain profit maximizing production plans rather than reducing optimal profits. In sum, efficiency levels differ significantly when controlling for heterogeneity with alternative specifications. Adding controls implies neither an automatic elimination of inefficiency nor an identical direction of changes of cost and profit efficiency measures across specifications. Thus, frontier models are quite sensitive to the way in which we account for sample heterogeneity. This sensitivity matters. For example, compared to the mean efficiency scores resulting from our preferred, general specification, relying on the heterogeneity in kernel specification decreases the estimate of potential cost savings by approximately 6%, whereas estimated potential cost savings drop with 4% if we rely on the heterogeneity in error specification. Likewise, foregone profits are 8% higher if we consider the heterogeneity in kernel specification, instead of the, preferred, general specification. For both the profit and the cost model, use of either the kernel or the baseline specification results in the largest bias in mean efficiency estimates (compared to the preferred, general specification). 4.3. Efficiency ranks The foregone profits and additional potential cost savings that result from efficiency level differentials are important on average. But for banks, comparing yourself to your

peers may be at least as important. Therefore, we also compare the ranks of the efficiency scores that result from each of our specifications. To start with, Table 6 reports p-values from a non-parametric test between efficiency scores from the five alternative cost and profit specifications. The null hypothesis that efficiency scores are equal is rejected for any combination of cost and profit specifications. However, while a number of comparative efficiency studies test the equality of efficiency scores in similar ways, we caution that most approaches may be prone to rejecting equality in large samples. Potentially, a few fairly extreme differences suffice to reject the null hypothesis. Therefore, we also present additional evidence below. In Table 7, we list rank order correlations for all five specifications. The table further improves our understanding of the effect of heterogeneity on efficiency scores. First, as is common in the literature, the rank correlation between CE and PE is low for all specifications. This might indicate that banks focus on distinctively different managerial measures regarding either the enhancement to generate profits by increasing revenues versus efforts to eliminate slack on the cost side, for example product innovations versus cost-cutting programs.17 Note, however, that some of the frequently reported low correlation between CE and PE also appears to be owed to the failure to account for heterogeneity given the increase in correlation coefficients from 9.4 percentage points between baseline models to 24.9 percentage points in the preferred general model. Second, in general PE scores for different specifications are more highly correlated than CE scores. For example, 17

An in-depth investigation of the determinants of the low correlation between CE and PE might thus require additional detailed information on ‘soft’ factors in a case study setting, such as for example strategic programs launched in a bank, or corporate culture. We deem the issue out of the scope of the present paper and reserve it for further research.

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Table 6 Wilcoxon matched-pairs sign-rank test on the equality of efficiency scores Model

Cost model

Baseline Truncated Error Kernel

Alternative profit model

Truncated

Error

Kernel

General

Truncated

Error

Kernel

General

0.000

0.000 0.000

0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000

0.000 0.000

0.000 0.000 0.000

0.000 0.000 0.000 0.000

Notes: p-Values for the null hypothesis of equal efficiency scores.

Table 7 Rank order correlations Cost model Baseline CE Truncated CE Error CE Kernel CE General PE Baseline PE Truncated PE Error PE Kernel PE General Notes:

*

Alternative profit model Truncated

Error

Kernel

General

Baseline

Truncated

Error

Kernel

0.937* 0.815* 0.848* 0.099* 0.107* 0.104* 0.178* 0.216*

0.775* 0.865* 0.121* 0.131* 0.149* 0.182* 0.250*

0.918* 0.189* 0.197* 0.191* 0.191* 0.187*

0.181* 0.193* 0.216* 0.202* 0.249*

0.998* 0.983* 0.948* 0.919*

0.989* 0.948* 0.927*

0.936* 0.944*

0.965*

*

0.997 0.934* 0.816* 0.845* 0.094* 0.103* 0.101* 0.174* 0.212*

Indicates significant pairwise rank-order correlation at the 1% level (31,080 observations).

for the cost model, choosing the kernel specification rather than the error specification results in a 5% increase in rank order correlation. For the profit model, rank order correlation in that case only increases by 2%. For a peer group analysis, this may constitute an important finding. In general, the ‘‘loss” of not using the general, preferred specification is higher for the cost model than it is for the profit model. In Table 8, we further explore the tails of the efficiency distribution by looking at the re-rankings of best and worst performers, compared to the general specification. For example, 0.25% of the banks are ranked eight deciles higher in the baseline cost specification than they are in the general cost specification, whereas only 0.02% of the banks are ranked eight deciles lower. In fact, most re-rankings for the cost model result in banks being upgraded compared to the general specification. In contrast, re-rankings for the alternative profit model are more sparse. This confirms our observations that the ‘‘loss” of not using the general, preferred specification is higher for the cost model than it is for the profit model. 4.4. Efficiency tails As we have seen so far, bank efficiency levels differ significantly across specifications, and banks may be considered best or worst performers, depending on the specification that is being used to estimated their efficiency. Bank efficiency scores, however, are not just interesting for banks themselves. For quite some time now, bank supervisors have paired their increasing interest in financial stabil-

ity (Oosterloo and de Haan, 2004; Oosterloo et al., 2007) with the practice of using efficiency scores for bank rating purposes (King et al., 2006).18 Mostly, of course, supervisors are interested in the tails of the efficiency distribution, in particular the lower tail, with banks that are potentially hazardous to the health of the financial system. A number of studies provide evidence that efficiency scores (as a proxy for managerial capabilities) are an important determinant of bank distress (Cole and Gunther, 1995; Wheelock and Wilson, 1995, 2000; DeYoung, 2003). The increasing use of these measures to estimate the risk of distress by supervisory authorities, therefore suggests that alternative specifications to obtain these proxies matter for more than mere methodological reasons. We obtain data on distress events among German cooperative and savings banks collected by the Bundesbank to assess the discriminatory power of the five efficiency specifications to discern troubled and healthy banks. Distress is defined according to the official taxonomy employed by supervisory authorities (the Bundesbank and the Federal Financial Supervisory Authority) and includes five events. First, compulsory notifications of supervisory authorities required by the banking act about events that may jeopardize the existence of the institution or impair its development. Second, losses amounting to 25% of liable capital. Third, a decline of operating profits exceeding 25%.

18 King et al. (2006) survey the genesis of methods to predict bank distress with hazard rate models.

J.W.B. Bos et al. / European Journal of Operational Research 195 (2009) 251–261

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Table 8 Re-rankings of best and worst performers Specification

Deciles re-ranked 9 8 7 6 6 7 8 9

Cost model

Alternative profit model

Baseline

Truncated

Error

Kernel

Baseline

Truncated

Error

Kernel

0.01 0.02 0.06 0.22 0.90 0.70 0.25 0.02

0.01 0.02 0.05 0.19 0.91 0.68 0.22 0.02

0.00 0.02 0.03 0.06 0.88 1.17 0.73 0.04

0.00 0.00 0.00 0.04 0.00 0.00 0.00 0.00

0.01 0.02 0.12 0.50 0.07 0.04 0.02 0.01

0.01 0.00 0.07 0.42 0.06 0.05 0.02 0.01

0.00 0.00 0.04 0.34 0.04 0.05 0.02 0.01

0.00 0.00 0.03 0.20 0.00 0.00 0.00 0.00

Notes: 31,080 observations. Re-rankings relative to the general specification. Negative values indicate downgrades. Frequencies per re-ranking decile in percentages.

Fourth, capital preservation measures and, fifth, restructuring mergers.19 Table 9 depicts the estimated coefficient of efficiency measures across specifications from a logit estimation on distressed events.20 We focus on the changes of respective managerial skill measures’ discriminatory power. To that end, we follow Hosmer and Lemeshow (2000) and use the Area Under the Receiver Operating Characteristics curve (AUR). It indicates how well a covariate predicts true events for a range of alternative probability cut-off levels beyond which an observation is assigned to the (predicted) event group. Both cost and profit efficiency scores exhibit a good discriminatory power as demonstrated by AUR-values ranging between 62.9% and 68%.21 The almost always significant coefficients imply that lower operational slack in either the cost minimizing or profit generating abilities reduce the probability of distress (PD). More importantly, the results from logit estimations specifying only respective efficiency measures (columns labeled No) highlight that CE and PE obtained from the general frontier specification (General) exhibit the highest discriminatory power. We test if this is a mere reflection of the additionally included information embedded in the heterogeneity dummies. Therefore, we also estimate a logit model that conditions bank distress on efficiency and a vector of banking group, size, and regional dummies in the columns labeled Yes. Estimated PD reductions due to a given increase in cost efficiency decline and the discriminatory power of hazard models improves across all specifications. In line with Porath (2006) and common sense, this suggests that other 19

Note that the latter two incidents are administered by banking pillars’ head organizations and not the supervisory authorities. For a more indepth discussion of bank supervision in Germany and the distress taxonomy see Koetter et al. (2007) and Kick and Koetter (2007). 20 Data on distressed events are only available until 2004. Thus, the sample is reduced to 29,760 bank-year observations. 21 This compares to AUR-value of 58.8% for capitalization and 69.8% for return on equity, two important predictors of bank distress in virtually any failure study.

Table 9 Logit estimation of probabilities of distress Model

Controls

Cost

Alternative profit

No

Yes

No

Yes

Baseline

b SE AUR

4.998 [0.655]*** 63.9%

2.955 [0.739]*** 65.7%

0.778 [0.233]*** 63.4%

0.962 [0.219]*** 63.7%

Truncations

b SE AUR

4.423 [0.524]*** 64.1%

2.609 [0.577]*** 66.0%

0.709 [0.212]*** 63.4%

1.021 [0.195]*** 64.1%

Error

b SE AUR

2.789 [1.027]*** 62.9%

0.606 [1.240] 63.4%

0.698 [0.214]*** 63.4%

1.011 [0.197]*** 64.1%

Kernel

b SE AUR

3.643 [0.528]*** 63.6%

1.966 [0.586]*** 65.4%

0.874 [0.212]*** 63.7%

1.144 [0.197]*** 64.4%

General

b SE AUR

5.81 [0.488]*** 65.5%

3.831 [0.615]*** 68.0%

1.106 [0.216]*** 64.1%

1.447 [0.197]*** 64.4%

Notes: 29,760 observations. Robust standard errors in brackets. Significant at 1%. The sample spans the years from 1993 to 2004. The limited dependent variable denotes distress (1) or no distress (0). We use a logistic regression of the form PDk ¼ ebX k =ð1 þ ebX k Þ, where X is the efficiency score of bank k and b is a parameter to estimate. The discriminatory power is assessed by the Area Under the Receiver Operating Characteristics curve (AUR) as detailed in Hosmer and Lemeshow (2000). ***

factors, such as regional location, contain information on bank risk, too. However, our focus is here not on fitting an optimal hazard rate model of bank distress but to assess the relative merit of efficiency measures accounting for heterogeneity. In fact, the bottom panel in Table 9 confirms that with or without accounting directly for dummies in the logit estimation, our general specification has the highest discriminatory power. As was the case with rank order correlations, choosing an alternative specification matters more for the cost model than it does for the profit model. But the difference with other specifications is not very high. Put differently, whereas efficiency levels and ranks are

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significantly altered when using a different specification, for supervisors interested in predicting distress, the cost from not using the general specification is not very high. In any specification, quite a number of distressed banks are in the lower tail of the efficiency distribution. 5. Conclusion In this study, we examine the effects of heterogeneity on bank efficiency scores. We compare different specifications of a stochastic frontier cost and profit model with a baseline specification. After conducting a specification test, we subsequently discuss the effect of accounting for heterogeneity on efficiency levels, ranks and the tails of the efficiency distribution. As a result, we find a preferred specification, and discuss the various costs of using another specification. For a sample of German cooperative and savings banks we find that banking type, regional location, and size group controls affect both optimal costs and profits as well as banks’ abilities to operate efficiently. Between 1993 and 2005, cost and profit efficiency levels differ significantly between a general specification that accounts for systematic differences across banks in both the deterministic kernel and the inefficiency distribution, and alternative, nested specifications. The mean profit efficiency level appears to be underestimated when not using the preferred specification, whereas the mean cost efficiency level can be either under- or overestimated. Our main conclusion is therefore that accounting for heterogeneity matters. Moreover, it matters not only whether we account for systematic differences across banks but also how. Specifically, we find evidence that both optimal costs and profits as well as the distribution of inefficiency should be allowed to depend on bank traits. The resulting differences in efficiency scores are important for more than only methodological reasons. We report three findings. First, our discussion of differences in efficiency levels shows that different ways of accounting for heterogeneity result in estimates of foregone profits and additional costs that are significantly different from what we infer from our general specification. Second, our ranking analysis shows that banks are significantly re-ranked when their efficiency is estimated with a specification other than the preferred, general specification. Third, when we focus on the tails of the efficiency distribution, we find that the general specification gives the most reliable estimates of the probability of distress, although the differences with others specifications are not as large as what may be expected based on our other findings. We find that efficiency scores generated with a general specification accounting for heterogeneity exhibits the highest discriminatory power to discern troubled from healthy banks. Furthermore, a bootstrapping exercise confirms that especially cost efficiency scores vary significantly depending on how to account for alternative banking group indicators.

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