The Relationship between Size, E�ciency and Locational Factors:a two-stage DEA Analysis of German Cooperative Banks

Maik Dombrowa*

June 12th, 2019

Abstract: This paper extends existing e�ciency literature in regional banking by focus-

ing on the question whether increasing bank size depends on the bank's location. The

empirical approach is to compute bank level e�ciency scores for 785 German coopera-

tives between 2012-2016 using Data Envelopment Analysis and conduct subsequent group

mean regression analyses including several locational factors at NUTS3 level. The re-

sults suggest that optimal bank size ranges between ¤ 100-250 million in total assets.

Also, technical and cost ine�ciency signi�cantly outweigh scale ine�ciency urging bank

managers to focus on internal resource allocation. Banks in urban areas seem to better

translate size increases into scale e�ciency gains. At the same time they are more likely

to operate at optimal scale already. For their rural counterparts the opposite holds. They

seem to have more growth potential but are more likely to fail in reaping economies of scale.

Keywords: Cooperative Bank E�ciency, Size, Location, Data Envelopment Analysis

JEL Codes: C61, D21, D24, G21

*University of Münster, Institute for Cooperative Research, Contact: maik.dombrowa@ifg-muenster.de

1 Introduction

Declining number of customers, persistently low interest rates, stricter equity regulation

and competitive pressure from virtual banks: Regional banks are forced to streamline. The

long-run competitiveness and the provision of banking services especially in economically

underdeveloped areas are at risk. To combat these developments regional banks often

attempt to yield economies of scale and e�ciency gains by merging (Richter et al., 2018;

Barkey, 2018).

However, most researchers agree that there is an optimal size in banking.2 Economies of

scale can be realized up to a certain point, from which on returns to scale start to diminish

as coordination costs within the organization increase disproportionately. Existing stud-

ies in German regional banking provide highly heterogeneous evidence quantifying the

optimal threshold somewhere between ¤ 0.5 - 5 bn (Maurer and Thieÿen, 2016; Maurer,

2015; Tischer, 2011; Radomski, 2008). Locational characteristics, such as the competitive

environment, population or enterprise density, GDP per capita or unemployment rates,

are often neglected. These factors vary signi�cantly across German regions so that policy

implications cannot easily be applied to all regional banks (Hanker, 2007; Maurer, 2015).

This paper takes regional factors into account and investigates whether the success of the

observed predominant strategy of increasing size among German regional banks depends

on the location in which they operate.

The empirical analysis is based on a panel from 2012-2016 with 785 German cooperative

banks.3 In a �rst step, each bank's technical, cost and scale e�ciency relative to the best

practice institutions within the sample is computed using linear optimization methods

(Data Envelopment Analysis - DEA). Four DEA models are speci�ed di�ering in terms of

scale assumptions and the underlying business models. One of the latter de�nes banks as

�nancial intermediaries, the other as producers of �nancial services. In the second step,

DEA e�ciency scores are regressed on a set of locational factors and on their respective

interaction terms with size. This procedure is widely accepted by econometricians (see

McDonald, 2008) and applied researchers (Wutz, 2002; Drake et al., 2006; Hahn, 2007;

Conrad et al., 2014; Reichling and Schulze, 2018).

The results suggest that the relationship between size and e�ciency scores is non-linear.

Whereas a bank's technical ability to transform inputs into outputs is highest if it is small

or large (u-shaped) the reverse relationship holds for pure scale economies (inverted-u-

shaped). Virtually all banks with an ine�cient size are operating at decreasing returns to

scale implying that downsizing improves e�ciency. On average, optimal bank size seems

2For a theoretical foundation see Krasa and Villamil (1992) or for empirical evidence in Germany

Gischer and Richter (2011).3There is another bank type that can be classi�ed as regional bank: However, savings banks are left

out of the analysis. The applied method produces more reliable results the more similar the examined

banks are in terms of their business model.

1

to be approximately ¤ 100-250 million in average total assets and thus signi�cantly lower

than other authors suggest (Maurer and Thieÿen, 2018; Maurer, 2015; Hanker, 2007). In

line with existing studies the results con�rm the explanatory power of locational factors

for deviations in bank e�ciency scores. The e�ects depend on whether banking is modeled

according to the intermediation or the production approach.

The main contribution of this paper is the investigation of interaction e�ects between

size and locational factors. Only Varmaz (2006) provides evidence in this regard arguing

that the existence of high market power promotes the positive e�ect of increasing bank

size. The present results suggest the opposite. Banks in areas with high degrees of

competition, dense and young populations are more likely to translate size increases into

scale e�ciency gains.

The remainder of the paper is organized as follows. Chapter 2 explains the concept

of e�ciency that is used in this paper. Chapter 3 reviews the existing theoretical and

empirical literature. Based on these insights �ve working hypotheses are formulated. In

chapter 4 the methodological approach is presented. Chapter 5 summarizes and discusses

the estimation results. Chapter 6 concludes.

2 Bank E�ciency

The e�ciency term used in this paper originates from production theory (Debreu, 1951;

Farrell, 1957; Färe et al., 1985). In this context e�ciency is de�ned as the ability of a

decision making unit (DMU)4 either to maximize a set of outputs given a level of inputs or

to minimize a set of inputs given a level of outputs. In banking, the latter input-oriented

approach is more commonly applied. Due to �nancial constraints, especially regional

banks are more likely to reduce costs by minimizing inputs rather than to maximize pro�ts

(Lang and Welzel, 1996). Thus, this paper takes on the input-oriented view. E�ciency

can be divided into technical, cost and scale e�ciency (TE, CE and SE). TE describes

how e�ciently DMUs allocate resources within the organization. TE, however, neglects

input prices such that the concept of cost e�ciency is needed to assess the DMU's ability

to minimize actual costs. SE measures if the DMU operates at optimal scale. If it does

not e�ciency can be improved by adjusting its scale of operations.

Figure 1 illustrates TE and SE for a DMU that produces an output y with an input

x. Under the assumption of constant returns to scale (CRS)5 TECRS,G of DMU G is

given by the ratio of the distances ABAG

. G should technologically be able to reduce its

input x by x− x* given the output level 0A. Under the assumption of variable returns to

4The term decision making unit is used in the literature and means the object for which e�ciency is

computed. A DMU might be e. g. a university, an industrial �rm or a bank.5I. e. an increase in inputs leads to a proportional increase in outputs.

2

Figure 1: E�ciency under constant and variable returns to scale

Source: based on Gubelt et al. (2000a, p. 485)

scale (VRS)6 TE of DMU G is given by TEG,VSE = ACAG

. Scale e�ciency can be viewed

as the residual between the two TE scores: SEG = ABAC

. Once it is adjusted by its scale

component TEG,VSE is called "pure" technical e�ciency.

Figure 2: Technical and cost e�ciency

Source: based on Poddig and Varmaz (2005, p. 273)

Figure 2 illustrates CE for a DMU that produces an output y with two inputs x1 and

x2. Assuming that the DMU currently produces the output level y in point D it works

technically ine�ciently because too much of x1 and x2 is used. Reducing the inputs

proportionally to the point A would eliminate technical ine�ciency as the convex isoquant

is reached. The exogenous input price ratio p2p1

implies that point D does not minimize

costs. The DMU should choose its input combination towards point substituting the

relatively expensive input x1 for relatively favorable input x2. In point C technical and

cost e�ciency are reached simultaneously.

6I. e. an increase in inputs leads to a disproportional increase in outputs.

3

3 Theory, Evidence and Hypotheses

In order to shed light on the underlying relationships this section summarizes the most rel-

evant theoretical and empirical contributions focusing on European and more speci�cally

German banking markets. The literature overview is divided into three parts that deal

with the respective relationships separately: (1) bank size and e�ciency, (2) locational

factors and e�ciency and (3) bank size, locational factors and e�ciency.

Size and E�ciency

Size is widely acknowledged as one of the main e�ciency drivers in banking. Theoretically,

there are three potential sources for economies of scale in banking. First, with increasing

scale of operation the degree of capacity utilization improves. For example non-operating

�xed costs can be reduced. Second, as opposed to conventional industrial production

processes bank services cannot be stored. Thus, the number of sta� is usually close to the

required number of sta� in times of maximal demand. It is argued that for small banks

it is relatively costly to employ reserve sta�. Large banks perform better at minimizing

idle times. Third, large-scale operations allow a higher degree of division of labor and

specialization. This, in turn, stimulates learning e�ects. On average, large bank employees

become more pro�cient in their respective �elds and thus contribute to the institution's

overall ability to acquire customers and generate pro�ts (Tebroke, 1993, pp. 78-84).

The question whether size has a positive e�ect on e�ciency has been discussed contro-

versially.7 Kammlott and Schiereck (2000) conduct OLS regressions using a broad range of

dependent variables. They �nd signi�cant scale economies for personnel expenses. How-

ever, the overall e�ect of size on income statement indicators reveals that small banks

perform better. Using Stochastic Fourier-�exible cost functions Girardone et al. (2004)

also �nd a negative impact of size on technical e�ciency in Italian banking. In a smiliar

methodological setting Bresler (2007) shows that small German savings banks operate

closer to the technical e�ciency frontier than large banks. Using Fixed E�ects models

and return on equity as dependent variable Gann et al. (2010) support these �ndings. In

contrast, other studies yield positive results. With a two-stage DEA approach Hauner

(2004) �nds that cost e�ciency increases signi�cantly as banks extend their operations.

Also in a two-stage DEA approach Drake et al. (2006) show that the largest institutions of

their sample outperform the smallest. Using a relative size measure, total assets divided

by the number of inhabitants in the area, Conrad et al. (2014) support these �ndings.

In a Fixed E�ects regression analysis Richter (2014) shows that size has a signi�cantly

positive e�ect on the e�ciency of German cooperative banks. As dependent variable he

7Bank size is usually approximated by the amount of total assets. In the following size and total assets

are used synonymously if not stated otherwise.

4

uses the cost-income-ratio8 , which has been criticized as not being scienti�cally valid

(Tischer, 2011; Burger, 2008; Bikker and Bos, 2005).

The studies have in common that they neglect the non-linear relationship between e�-

ciency and bank size. Several DEA studies look at di�erent size classes and �nd a u-shaped

relationship between technical e�ciency and bank size, arguing that small banks enjoy

advantages when it comes to controlling costs whereas big banks are better at recruiting

productive sta�. These authors also consistently show that there is an inverted-u-shaped

link between scale e�ciency and bank size, re�ecting that at some point organizational

costs increase disproportionately (Lang and Welzel, 1996; Gubelt et al., 2000a,b; Wutz,

2000, 2002; Radomski, 2008; Tischer, 2011). Using income statement performance indi-

cators Hanker (2007), Maurer (2015) and Maurer and Thieÿen (2016, 2018) �nd similar

descriptive and OLS results. Only Lang and Welzel (1995), who conduct parametric fron-

tier analyses, argue that for Bavarian cooperative banks an abrupt rise of organizational

costs cannot be observed. The following table 1 summarizes empirical evidence concerning

optimal regional bank size in Germany.

Table 1: Optimal regional bank size in GermanyAuthors Bank type Methodology Optimal Size

Lang and Welzel (1995) Cooperative SFA n/a

Lang and Welzel (1996) Cooperative DEA DM 150-200

Gubelt et al. (2000a,b) Savings DEA DM 300-1000

Wutz (2000, 2002) Cooperative DEA DM 100-150

Hanker (2007) Cooperative Descriptive EUR 500-1000

Radomski (2008) Savings DEA EUR 1200-1500

Tischer (2011) Savings DEA EUR 1000-5000

Maurer (2015) Cooperative OLS EUR 650

Maurer and Thieÿen (2016) Cooperative OLS EUR 571

Maurer and Thieÿen (2018) Cooperative OLS Rural/Urban:

n.a./ EUR 350-900

Notes: Numerical values in millions

Based on the theoretical considerations and the empirical results from table 1 the fol-

lowing hypothesis shall be tested.

H1: The relationship between cooperative bank size and e�ciency is non-linear.

Hanker (2007) and Maurer (2015) acknowledge that above results cannot be translated

into universally valid implications for regional banks as they operate in heterogeneous

regions across Germany. Only Maurer and Thieÿen (2018) incorporate that optimal size

may depend on the location where banks operate. Before turning to the particular in-

8CIR = operating costs / operating income

5

terplay between e�ciency, size and location, the following section discusses the existing

literature on the in�uence of locational factors on regional bank e�ciency.

Location and E�ciency

The �rst bank e�ciency studies to consider regional di�erences are cross-country com-

parisons (Berg et al., 1993; Fecher and Pestieau, 1993). They, however, neglect that

common e�ciency frontiers for banks operating in di�erent socioeconomic regions are

likely to produce biased e�ciency scores. Considering factors, such as GDP per capita,

unemployment rates, population density or the number of competitors, subsequent cross-

country analyses �nd that variations in e�ciency between banks can be, at least partly,

explained by these factors (Dietsch and Lozano-Vivas, 2000; Hauner, 2004; Hahn, 2007,

2008). To this day there are only few authors taking their analyses a step further by ex-

amining a single country and the e�ects of regional factors on bank e�ciency, some down

to the federal (Reichling and Schulze, 2018) or even district level (Wutz, 2002; Bos and

Kool, 2006; Bresler, 2007; Conrad, 2008; Christians and Gärtner, 2014a,b; Conrad et al.,

2014; Christians and Hartl, 2015; Richter et al., 2018; Maurer and Thieÿen, 2016, 2018).

Relevant locational factors can be divided into three categories: demography, economic

wealth and competition (see Riekeberg, 2003; Conrad et al., 2009, p. 455, p. 16). Along

this classi�cation the following paragraphs synthesize existing empirical evidence.

Demography

The most commonly used variables to approximate demographical factors are the num-

ber of inhabitants in a region, its growth rate, the population density measured as the

number of inhabitants per square kilometer or population age measured by the share of

over 65- or 75-year-olds. Theoretically, the more inhabitants in a bank's reach the more

opportunities for scale economies or sta� specialization exist. A concentrated base of

(potential) customers goes along with low distribution costs. Relatively few branches are

required to reach out to customers. This seems particularly relevant in small-scale retail

banking, which is an important pillar for regional banks (Conrad et al., 2014; Dietsch and

Lozano-Vivas, 2000). Population age is supposed to in�uence e�ciency negatively. Older

people are less likely to demand standardized �nancial services. Instead they require

costly personalized counseling. Banks need more personnel to produce a given output in

areas with a high share of old people (Conrad et al., 2014). The empirical results are

mixed depending on the proxy, the methodological approach or the dependent variable

used in the analysis.

Wutz (2002) �nds no signi�cant in�uence of the number of inhabitants on technical

e�ciency using DEA and subsequent OLS regressions for Bavarian cooperative banks.

Bos and Kool (2006) use a parametric approach in the �rst stage. Their second-stage

truncated regression results suggest that an increase in the number of inhabitants reduces

6

cost e�ciency. They argue that in highly populated areas the cost of acquiring customers

is signi�cantly lower. Looking at population density, which is often strongly and pos-

itively correlated with the number of inhabitants, Conrad et al. (2014) �nd a positive

and statistically signi�cant e�ect on DEA technical and revenue e�ciency. Using OLS

regression methods Conrad (2008) did not �nd a signi�cant relationship between popula-

tion density and bank performance. With the same methodological approach Christians

and Gärtner (2014b) show that the population growth rate has a positive e�ect on credit

default rates for German savings and cooperative banks. High credit default rates imply

that banks are not able to screen their customers and may indicate low e�ciency. Using

simple regression analysis and for a similar data set Christians and Gärtner (2014a) �nd a

signi�cantly positive e�ect of the population growth rates on total return on assets. Total

return on assets is a conventional bank performance indicator and expected to be posi-

tively correlated with e�ciency. Other recent studies for Germany have relied on cluster

analyses. The results indicate trends as causal relationships become blurred. For instance

Tischer (2011) and Christians and Hartl (2015) examine DEA scores by regional clusters

and �nd that banks in socioeconomically weak regions perform best. This result may be

misleading because in those regions competition is relatively weak and thus possibly an

omitted e�ciency determinant. Summing up, the following hypothesis can be formulated:

H2: A favorable demographic situation at district level promotes the e�ciency of re-

gional banks.

Economic wealth

Proxies are vast: GDP and GNI per capita, their respective growth rates, market pen-

etration measured by the relative amount of deposits per bank or the number of cur-

rent accounts, unemployment rates or productivity measured by the GDP relative to the

working population. Theoretically, economic wealth implies a high demand for �nancial

services meaning larger sales volumes and higher pro�t opportunities. Wutz (2002) ar-

gues that regional banks in rich areas can overcompensate ine�ciency of technical nature.

However, easy access to wealthy customers can cause inertia among bank managers who

reduce their e�orts to maintain low cost structures. Also, wealthy customers may demand

relatively specialized �nancial products which require a higher degree of sta� skills and

capacity, possibly leading to higher costs for the bank (Conrad et al., 2014).

Empirical evidence is mixed with a tendency towards a negative relationship. Conrad

et al. (2014) look at GNI per capita, unemployment and deposit penetration and conclude

that for German savings banks economic wealth is strictly negatively related to e�ciency.

Using a two-step DEA approach Hahn (2007) and a two-step parametric frontier approach

Bos and Kool (2006) consider GDP per capita and �nd analogous results for German,

Austrian and Dutch cooperative banks. Using balance sheet and income statement per-

7

formance indicators Maurer (2015) and Maurer and Thieÿen (2016) show descriptively

that a wealthy population implies less pro�t opportunities for banks. This can again,

at least partially, be traced back to the relatively high competition in these areas. In

his doctoral thesis Maurer (2015) thus employs OLS regression techniques to isolate the

respective e�ects. However, he fails to produce statistically signi�cant results for GNI or

GDP per capita, GDP per workforce and GDP growth concluding that economic wealth

does not have an impact on bank performance.

In contrast, some authors �nd evidence in support of a positive e�ect of economic wealth

on bank performance. Conrad (2008) �nds a positive and statistically signi�cant e�ect

of GNI per capita on the sum of interest and provision margins of savings banks in

Germany. These results are supported by Reichling and Schulze (2018), Grigorian and

Manole (2002) and Wutz (2002) who regress DEA e�ciency scores on GNI and GDP per

capita and deposit penetration respectively. Relying on the empirical evidence economic

wealth seems to in�uence bank e�ciency negatively although some of the e�ect is due

to higher degrees of competition that richer, metropolitan areas bring about. Further

work has to be done to disentangle these e�ects from each other. For the time being, the

following hypothesis is to be tested.

H3: A favorable economic situation at district level impairs the e�ciency of cooperative

banks.

Competition

Theoretically, the relationship between competition and bank e�ciency is ambiguous.

The Quiet Life Hypothesis constitutes that competition determines e�ciency. Banks

with high degrees of market power are expected to operate less e�ciently due to the lack

of competitive pressure (Hicks, 1935). A di�erent paradigm, the E�ciency Hypothesis,

implies an inverse causal relationship: E�ciency determines competition. E�cient banks

push ine�cient competitors out of the market reducing overall competition in the market

(Demsetz, 1973).

In banking, empirical evidence in favor of the Quiet Life Hypothesis is rare. Divid-

ing their samples according to the banks' head o�ce locations several authors �nd higher

e�ciency scores in rural or suburban areas relative to their urban or metropolitan counter-

parts for Germany and Austria (Maurer and Thieÿen, 2018, 2016; Maurer, 2015; Hauner,

2004). These results imply market power leaves incumbents with no incentives to improve

e�ciency. Note that only Hauner (2004) �nds statistically signi�cant e�ects whereas the

other studies rely on descriptive results.

Results to support the E�ciency Hypothesis are given by Wutz (2002). He �nds that

an increase in the net interest margin is associated with higher technical e�ciency scores.

The net interest margin is de�ned as the di�erence between average interest rates a bank

8

charges for loans and pays for customer deposits. It is argued that banks can realize high

margins if there is little competition. Grigorian and Manole (2002) �nd that especially

in socioeconomically weak regions a high degree of market power, measured as the bank's

total assets divided by total bank assets within the country, leads to higher e�ciency

scores. In a comparative international study Hahn (2007) �nds similar results. He shows

though that in light of digitization local incumbent banks are increasingly pressurized as

global or direct banks gain market shares in rural areas. Tischer (2011) and Richter et al.

(2018) �nd that German savings banks are more e�cient in peripheral areas and trace

this back to the lack of competition. These studies, however, rely on cluster approaches

and do not properly account for interaction with other locational factors. Christians and

Gärtner (2014b) argue analogously. They �nd higher credit default rates for savings banks

in East than in West Germany. Karmann et al. (2013) provide further evidence in line

with the E�ciency Hypothesis by regressing the Lerner-index9 on the net interest income

of German commercial banks. According to the authors, market power results in a better

intermediation position, in which banks can maximize their interest income by granting

low interest rates on deposits and charging high interest rates on loans.

In total, empirical evidence seems to support the view of the E�ciency Hypothesis im-

plying that too much competition can be harmful to cooperative banking. Accordingly,

the following hypothesis will be tested.

H4: Market power promotes e�ciency of cooperative banks (E�ciency Hypothesis).

Size, Location and E�ciency

In a pioneer paper on the theory of optimal bank size Krasa and Villamil (1992) emphasize

the importance of exogenous factors, such as the industry structure or the macroeconomic

risk in a given region. However, only few and only rather recent studies incorporate this

in their argumentations.

In a Random E�ects regression setting Varmaz (2006) analyses the interaction between

several locational factors and bank size and their e�ects on return on equity of German co-

operative and savings banks. On the one hand, he does not �nd statistical signi�cance for

size and region, which, however, may be due to his imprecise 4-cluster-approach (p. 256,

264). On the other hand, the interaction between market concentration measured by the

Hirshman-Her�ndahl-Index10 and relative bank size has a signi�cantly positive e�ect on

return on equity. In the same regression relative bank size on its own has a statistically

9The Lerner-index describes how high a mark-up suppliers can demand from their customers (Lerner−index = p−GK

p ). It is a common proxy for market power. For a more thorough discussion the reader is

referred to Elzinga and Mills (2011).10The Hirshman-Her�ndahl-Index measures the level of concentration in markets. For details see

Hirschman (1964).

9

signi�cant negative impact. The results can be interpreted as follows: If a bank extends

its scale of operation, return on equity declines. This e�ect is moderated if the bank is

located in an area with little competition (p. 253, 264).

In a DEA analysis Tischer (2011, p. 172) �nds no signi�cant variation in scale e�ciency

scores for German savings banks when controlling for regional clusters. He concludes that

in terms of size banks have adapted similarly well independent of regional conditions.

In a cross-country analysis for European, Japanese and US banks, Hahn (2008) �nds

that banks in socioeconomically weak areas are closer to their optimal size than those in

more prosperous areas.

In a recent study for Germany, Maurer and Thieÿen (2018) explore performance dif-

ferences dividing their sample in urban and rural cooperative banks and size quintiles

respectively. In the years 2005-2011 the biggest 20 % of the rural banks consistently

reach higher pro�t margins than the smaller ones. The authors conclude that rural banks

should be at least as big as they already are. They argue further that optimal size may

be even higher. Due to a lack of big rural banks this could not be investigated (p. 17).

Looking at the urban institutions, they �nd an inverse relationship. The biggest 20

% perform consistently worse than the other quintiles. Some of this size e�ect can be

explained by the existence of PSD and Sparda banks in urban areas whose business mod-

els are signi�cantly di�erent. When leaving them out of the analysis, the variation in

the pro�t margin between size quintiles of urban banks becomes less pronounced. One

has to critically point out that constructing size classes by quantiles may produce biased

results. In fact, the few very large banks among German cooperatives may distort sample

means inappropriately. Thus, it may be reasonable to classify banks by predetermined

size intervals. Due to a lack of theoretical foundation and empirical evidence the following

explorative hypothesis is formulated.

H5: The magnitude and direction of the e�ect of bank size on e�ciency depends on the

bank's location.

4 Methodology and Data

First Stage: Data Envelopment Analysis

E�ciency scores are computed using the non-parametric approach Data Envelopment

Analysis. It employs optimization methods to construct a piecewise linear e�ciency fron-

tier. This frontier is spanned by the group of e�cient DMUs in the sample. Relative

to these benchmark DMUs e�ciency scores for all other DMUs are derived (Figure 3).

The main advantage of DEA is that no ex ante assumptions about functional forms of

the production process have to be made. At the same time DEA is highly sensitive to

10

potential measurement errors due to outliers.11

Figure 3: Piecewise linear e�ciency frontier in DEA

Source: based on Coelli (1996), p. 8.

The original so called CCR model by Charnes et al. (1978) is based on the assumption

of constant returns to scale. In banking, however, due to competitive and technological

constraints it is rather unrealistic to assume that an increase in inputs is always accompa-

nied by a proportional increase in outputs (Lang and Welzel, 1996). Banker et al. (1984)

extend the CCR model by allowing for variable returns to scale. The extended, so called

BCC model is based on the following linear optimization problem that is to be solved for

each DMU i = {1...N} under consideration. The di�erence between the CCR and BCC

model is that for the latter the convexity condition λ ≥ 0 is introduced:

minλθ

s.t. − yi + Y λ ≥ 0

θxi −Xλ ≥ 0 (1)

λ ≥ 0

eTλi = 1.

The scalar θ is the e�ciency score to be computed for each DMU i. It takes on a

value between 0 and 1 where 1 indicates a perfectly e�cient DMU i∗. yi and xi are the

output and input vectors of DMU i. Y and X are matrices containing the inputs and

outputs of all other DMUs j 6= i. λ is a N × 1 matrix of constants. The optimization

procedure considers one DMU i and contracts the input vector to the e�ciency frontier

up to where it is technologically possible to produce the given output set. Again, this

frontier is determined by the e�cient benchmark DMUs i∗ (θi∗ = 1) within the sample.

In DEA each ine�cient DMU i 6= i∗ has at least two e�cient peers. In Figure 4 DMUs

A, B and C produce an output y with two inputs x1 und x2. A and B do so e�ciently,

i. e. on the frontier, C does not. DEA computes by how much C could reduce its inputs

to reach C', a linear combination of A and B.

11A more thorough discussion of (dis)advantages can be found in Fiorentino et al. (2006), pp. 5-8.

11

Figure 4: E�cient peers in DEA

Source: based on Coelli (1996), S. 17.

Scale e�ciency is then derived by the ratio of the two e�ciency scores under constant

and variable returns to scale:

θSE =θCRSθV RS

.

Beyond technical and scale e�ciency DEA is able to compute cost e�ciency given that

data on input prices is available. Cost e�ciency seems to be practically more relevant

as bank management and other stakeholders make decisions based on cost considerations

rather than purely technical ones.

The computation of DEA cost e�ciency scores is based on the above linear optimization

problem (1) after which the following cost minimization is to be run:

minλpix∗i

s.t. − yi + Y λ ≥ 0

x∗i −Xλ ≥ 0 (2)

λ ≥ 0

eTλi = 1.

pi is a vector of input prices for DMU i and x∗i its cost minimizing set of inputs derived by

the linear optimization problem given pi and output quantities yi (Coelli, 1996, p. 25).

Cost e�ciency can then be de�ned as the ratio of the minimally achievable cost and the

actually accrued costs within the DMUi.

θCE =pix∗i

pixi.

When conducting DEA analyses the choice of inputs and outputs depends on how the pro-

duction process is modeled. In banking two approaches have dominated recent literature.

12

First, banks are supposed to act as �nancial intermediaries mainly transforming deposits

into loans (this intermediation approach was shaped by Sealey and Lindley (1977)). It is

usually applied when the economic signi�ance of banks is examined. Second, banks are

seen as producers of �nancial services (this production approach was shaped by Benston

(1965)). This stance is often taken when the focus is on the business side of banking.

Several scholars have shown that despite using the same data set signi�cantly di�erent

results arise for the two approaches (e. g. Wutz, 2000; Porembski, 2000; Ahn and Le,

2015). The main di�erence between the approaches is that within the intermediation

framework deposits are classi�ed as inputs in the sense that they are transformed into

loans. Whereas for the production approach deposits are considered to be a �nancial ser-

vice provided to customers and hence classi�ed as an output. For the sake of completeness

and robustness, both models are estimated.

Both approaches include labor and capital as input factors. Labor is measured by the

average number of employees and capital by the average amount of total assets in a given

year. Following Wutz (2002) and Conrad et al. (2014) input prices are approximated

by the average salary, average capital costs and average deposit costs (see caption of

table 2). As outputs net commissions, total customer loans divided into three maturity

categories and other earning assets are taken into account. Table 2 presents the inputs,

outputs, input prices and their descriptive statistics with the upper case letters P and

I symbolizing the production and intermediation approach respectively. DEA cannot

process zero values. This problem was solved by substituting the respective values by

in�nitesimally small values (Huguenin, 2012, p. 23).

Table 2: Model overview and descriptive statisticsVariable Description Ø Min Max St. Dev.

xI1 / xP1 No. of Employees 147 2 1,719 162.3

xI2 / xP2 Total Fixed Assets 7.9 0.02 164.3 12.6

xI3 / yP6 Total Deposits 587.5 19.5 12,488 1,312

yI1 / yP1 Net Commissions 3.9 0 538.2 53.4

yI2 / yP2 Customer Loans < 1 year 414.5 0.04 1,068 67.4

yI3 / yP3 Customer Loans 1-5 years 109.3 1.7 2,293 173.6

yI4 / yP4 Customer Loans > 5 years 237.1 2.4 8,088 488.5

yI5/ yP5 Other Earning Assets 3.1 0 65.6 5.4

p1 Average salary (¤) 59,936 27,523 316,667 12,309

p2 Average capital costs 0.89 0.01 11.92 0.66

p3 Average deposit costs 0.009 0.001 0.036 0.005

Source: annual reports; 2012-2016; 785 banks and 3925 bank years, inputs and outputs in mio. ¤ except employees and input prices

De�nitions: p1 = personnel expenses / no. of employees; p2 = other operating expenses / total �xed assets ; p3 = total interest expenses /

total deposits

In order to analyze e�ciency scores depending on the bank's size the data set is divided

13

into ten size classes measured by the amount of total assets, for which descriptive statistics

are summarized in table 3. The intervals were predetermined based on existing studies

that follow a similar approach (Welzel, 1996; Wutz, 2000; Radomski, 2008; Tischer, 2011).

Some studies divide the samples into size quantiles (Maurer and Thieÿen, 2018, 2016).

Considering that bank size is strongly right skewed, i. e. many small and few large banks,

this approach may produce distorted sample means. Others use very few size classes, such

as small, medium and large, providing rather imprecise results (Richter et al., 2018).

Table 3: Descriptive statistics for size classes

Size class mean sd min max #

0 39.6 8.6 13.9 50 167

1 75.5 15 50.5 100 401

2 167 42.6 100 250 1025

3 321 43.2 250 400 681

4 494 60.8 400 600 444

5 688 56.2 601 800 344

6 901 62.4 801 1000 241

7 1490 392 1000 2500 460

8 3570 741 2510 4960 132

9 7240 2390 5010 14000 30

Total 657 1120 13.9 14000 3925

Source: annual reports; 2012-2016; 785 banks and 3925 bank years, values in mio. ¤ except #

Second Stage: Regression Analysis

The two-step approach of this paper has been on the rise in bank e�ciency literature.

FollowingWutz (2002), Banker and Natarajan (2008) and Conrad et al. (2014) �rst-stage

DEA e�ciency scores are regressed on locational factors using pooled OLS models. How-

ever, some authors argue that Tobit regression approaches are more appropriate. They

consider DEA e�ciency scores as generated by a censoring process (Grigorian and Manole,

2002; Bos and Kool, 2006; Reichling and Schulze, 2018). Ho� (2007) compares second-

stage approaches and concludes that OLS performs at least as well as other methods. In

support of OLS, McDonald (2008) argues that DEA e�ciency scores are not censored but

fractional data.

Thus, in order to test for non-linearity of size and to determine the causal e�ect of

locational factors on bank e�ciency scores the following OLS equation is estimated:

θji = β0 + β1SIZEi + β2SIZE2i + β3LOCATIONi + εi (3)

14

θji is bank i′s technical, cost or scale e�ciency score j. SIZE12 is bank i′s amount of

total assets and LOCATION represents a set of locational factors of the district in which

bank i operates.

To explore whether the in�uence of increasing the scale of a bank's operation on e�ciency

depends on the bank's location the following structural break models are estimated:

θji = β0 + β1SIZEi + β2SIZE2i + β3(SIZEi × LOCATIONi) + εi (4)

The term (SIZEi×LOCATIONi) indicates the interaction between size and the respec-

tive locational factor included in the regression. The marginal e�ect of an increase in

bank size now depends on size itself but also on the bank's location:

δθjiδSIZEi

= β1 + 2β2SIZEi + β3LOCATIONi (5)

The following table 4 summarizes all locational and control variables used in second stage

regressions. Economic wealth and demographics indicators are NUTS3, i. e. district

level data. The NUTS classi�cation provides a hierarchical and systematic breakdown

of European economic regions, with NUTS3 being the smallest entity. NUTS2 refers to

federal states and NUTS1 to whole countries. Market power is proxied by each bank's

net interest margin. It is argued that banks with high market power can a�ord to grant

low interest rates on customer deposits while charging relatively high rates on customer

loans (Wutz, 2002, p. 11).

5 Results and Discussion

This chapter follows the two-stage structure of the methodological approach. First, DEA

results, then second-stage regression results are presented.

DEA e�ciency scores

The estimation results are roughly in line with existing studies: On average, scale e�ciency

outweighs technical e�ciency by approximately 10-30 percentage points depending on

the underlying approach. With scale e�ciency scores close to 1 it is suggested that the

12The variables SIZE and SIZE2 were transformed by taking the natural logarithm to achieve a

normal distribution.

15

Table 4: Descriptive statistics for locational factorsLocational factor Proxy Description mean sd min max obs

Market power Net interest margin NIM (%) 2.48 0.38 0.85 4.38 3,685

Economic wealth (1) GDP per capita GDP (1,000 ¤) 34.2 12.7 14.4 133 3,925

Economic wealth (2) Unemployment rate Unemployment (%) 3.94 1.51 2.1 11.4 3,640

Demographics (1) Population density Popdens (ppl/km2) 442 666 669 4701 3,925

Demographics (2) Population Age >65 year olds (%) 20.63 2.33 15.2 30.7 3,925

Control variable

Risk preference Standard deviation of ROA Risk 0.31 0.16 0 1.21 3,925

Data sources: Net interest margin, Risk (annual reports, gathered from �tchconnect); economic wealth (DESTATIS); demographic situation

(EUROSTAT); economic wealth and demographical factor are on NUTS3-level indicators

De�nitions: Net interest margin = (interest income - interest expenses) / average total assets; Population density = no. of inhabitants /

square kilometers for each district, Population age = no. of inhabitants older than 65 / total population; Risk = standard deviation of

return on assets.

predominant source of ine�ciency is of technical nature. This is in line with existing DEA

studies for German cooperative and savings banks (Wutz, 2000, 2002; Gubelt et al., 2000b;

Welzel, 1996; Tischer, 2011). Also, e�ciency scores produced by the intermediation and

the production approach di�er signi�cantly in absolute terms. Wutz (2000) compares the

approaches for an identical data set and argues that an absolute deviation is acceptable

as long as the e�ciency ranks are stable. For instance if for the banks A, B and C it is

true that θIk,A > θIk,B > θIk,C , then θPk,A > θPk,B > θPk,C with k = {VRS, SE, CE} should

hold, too. He conducts a spearman analysis whose results imply low rank correlation

between his estimates. For the data at hand, however, this strong inconsistency cannot

be con�rmed.13

Plotting e�ciency scores against the ten size classes several points can be made (Figure

5, see table 8 in the appendix for detailed results). First, as discussed in the previous para-

graph: Despite the absolute di�erence between production and intermediation e�ciency

scores the similarity in the shape of the curves becomes obvious.

Figure 5: Total, technical and scale e�ciency by size classes

(a) Intermediation Approach (b) Production Approach

13Spearman rank correlation coe�cients, Wutz's (2000) rank coe�cients in parentheses: θIV RS / θPV RS

= 0.62 (0.41), θISE / θPSE = 0.65 (0.58), θICE / θPCE = 0.79 (n.a.).

16

Second, looking at cost and technical e�ciency the intermediation and production ap-

proach show a u-shaped relationship with size. This is in line with theoretical prediction

(Tebroke, 1993) and other empirical studies (Gubelt et al., 2000b; Wutz, 2002; Radom-

ski, 2008; Tischer, 2011). The smallest and largest banks seem to manage the internal

resource allocation more e�ciently than mid-size banks, with the largest banks domi-

nating especially in the production approach. Richter et al. (2018) also �nd a u-shaped

relationship between technical e�ciency and their three size categories. However, their

sample of cooperative banks reveals that the smallest banks perform best. A reason for

the divergence may be the di�erent DEA model speci�cation.

Turning to the di�erence between cost and technical e�ciency the results imply that

banks are more e�cient from a purely technical stance. If banks operate on the technical

e�ciency frontier there is still potential to reduce costs by substituting the relatively

expensive factor for the more favorable one leaving output quantities constant. For the

intermediation approach, this is particularly pronounced. However, it suggests that the

di�erence between technical and cost e�ciency becomes smaller as bank size rises. If

banks are considered �nancial intermediaries, the larger they get the better they are at

substituting inputs towards the cost minimum.

Third, from a scale perspective the intermediation and production approach are largely

consistent. Banks of size class 2 with total assets of ¤ 100-250 million are the most scale

e�cient. Compared to existing studies in table 1 this interval seems quite low. Also, both

approaches reveal the lowest scale e�ciency in the largest size class, supporting the notion

of disproportionately increasing organizational costs when banks become very large. Only

for size class 9 with average total assets of ¤ 14 billion technical and cost e�ciency exceed

scale e�ciency. Note that scale e�ciency only indicates how much banks deviate from

their optimal size. It does not say if banks should grow or shrink to become fully scale

e�cient.

As depicted in �gure 3, DEA results include the information if banks operate in the area

of increasing returns to scale (IRS) or decreasing returns to scale (DRS). If the former

holds banks should increase and if the latter holds banks should decrease the extent of

their operations. Table 9 in the appendix summarizes the number of banks with IRS and

DRS by size classes.14 Overall 60 % of the banks operate at DRS and are thus too big. 27

% are at IRS and thus too small. The remaining 13 % are scale e�cient. Looking at the

size classes in detail, nearly 80 % of the banks in classes 1 and 2 are too small whereas

approximately 95 % of the banks in the classes 7-10 are too big. In terms of size, striving

towards the middle should maximize scale e�ciency.

All in all, however, the results support the view that putting strategic focus on scale

economies is questionable. Improving the use of input factors in a purely technical way as

14Only intermediation approach results are displayed as they largely coincide with the production

approach.

17

well as in terms of costs seems more promising. The exception is the group of the largest

banks whose scale ine�ciency is higher than their cost or technical ine�ciency.

Size, location and e�ciency

So far results have been shown at an aggregate level. Before relating e�ciency to the

interplay between size and locational factors, the relationship between e�ciency and those

factors is investigated. Table 11 in the appendix gives an overview of correlations between

DEA scores, locational factors and balance sheet performance indicators. Interestingly,

technical and cost e�ciency scores are correlated with nearly all locational factors and

size for both the intermediation and the production approach.

Size is positively correlated with cost and technical e�ciency. High population densi-

ties and low shares of people older than 65 are associated with high technical and cost

e�ciency scores, which is in line with theoretical predictions and H2. Unemployment is

positively but weakly correlated with cost e�ciency. The correlations are stronger for

technical e�ciency. Surprisingly, for the intermediation approach it is negative and for

the production approach positive. GDP per capita is consistently positively linked to

cost and technical e�ciency supporting theoretical considerations and H3. For market

power (NIM) only the correlation with technical e�ciency in the intermediation approach

supports the E�ciency Hypothesis. The corresponding cost e�ciency and both scores of

the production approach are negatively correlated supporting the Quiet Life Hypothesis.

Looking at scale e�ciency, the production and intermediation approach coincide. The

coe�cients imply that banks with a high degree of market power, in an area with low

GDP per capita, high unemployment rates, low population and low creditable population

operate closest to their optimal size. This seems to be in line with the demonstrations

of Varmaz (2006), Hahn (2008) and Maurer and Thieÿen (2018) who consistently argue

that banks in socioeconomically weak areas operate closer to their optimal size.

To test these descriptive insights multivariate analyses are conducted. The �ve-year

panel contains a signi�cant number of missing values. Also, most variables do not show

much variation over time such that time series analysis does not seem appropriate. For

these reasons group means for all 785 banks and all variables are computed. The Breusch-

Pagan / Cook-Weisberg test for heteroskedasticity could not be rejected for all model spec-

i�cation. Consequently, robust standard errors are applied. As robustness checks, Tobit

regression with upper truncation at 1 are conducted. The results are largely congruent

and thus not reported here. Table 5 summarizes the regressions results.

The estimation results provide compelling and consistent evidence con�rming H1. The

relationship between bank size and e�ciency is non-linear. SIZE shows a negative and

SIZE2 a positive sign con�rming the u-shaped relationship already depicted in �gure 5

for technical and cost e�ciency (p < 0.01). When calculating the marginal e�ect of an

increase in size on e�ciency scores one can rearrange equation (5) to obtain the minimum.

18

Table 5: In�uence of locational factors on technical, cost and scale e�ciency

Technical E�ciency Cost E�ciency Scale E�ciency

θV RSI θV RSP θCEI θCEP θSEI θSEP

SIZE -0.71∗∗∗ -1.14∗∗∗ -0.75∗∗∗ -1.08∗∗∗ 0.44∗∗∗ 0.47∗∗∗

SIZE2 0.019∗∗∗ 0.030∗∗∗ 0.020∗∗∗ 0.028∗∗∗ -0.012∗∗∗ -0.013∗∗∗

NIM 0.088∗∗∗ -0.033∗ 0.00093 -0.033∗ -0.0018∗ -0.012∗

GDP -0.012 -0.022 -0.012 -0.017 -0.0089∗ -0.0075

Unemployment 0.00077 0.020∗∗∗ 0.011∗∗∗ 0.017∗∗∗ -0.0041∗∗∗ -0.0012

Popdens -0.00077 0.00089 -0.00020 0.000017 0.000054∗ 0.00012∗∗

Age -0.0081∗∗∗ -0.0051∗∗ -0.012∗∗∗ -0.0065∗∗ -0.0024∗∗∗ -0.0049∗∗∗

Risk 0.079∗∗∗ 0.092∗∗ 0.12∗∗∗ 0.095∗∗∗ 0.022∗∗∗ 0.0072

constant 7.75∗∗∗ 11.7∗∗∗ 7.99∗∗∗ 11.2∗∗∗ -3.09∗∗∗ -3.20∗∗∗

N 785 785 785 785 785 785

adj. R2 0.195 0.216 0.147 0.164 0.386 0.330

Notes: Group mean regressions for 2011 - 2015; N = number of groups = number of banks; Huber/White standard errors applied;

signi�cance levels: ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01; SIZE and SIZE2 were transformed into their natural logarithms to achieve a

normal distribution.

For technical e�ciency it is reached at ¤ 367 or 299 million for the intermediation and

production approach respectively. Banks of this size have the most potential to increase

e�ciency by improving internal resource allocation. The same holds for cost e�ciency

and banks with total assets amounting to ¤ 245 or 205 million.

Turning to scale e�ciency the relationship reverses. The coe�cient of SIZE is positive

and of SIZE2 negative (p < 0.01) supporting the notion of the existence of an optimal

size in regional banking. Again computing the marginal e�ect of increasing bank size the

maximum can be found at ¤ 195 or 50 million. These values are quite consistent with

the descriptive results of the previous section and much lower than what was found by

other recent studies. See �gure 5 that reveals the highest scale e�ciency scores for banks

between ¤ 100-250 million and in contrast see table 1 with optimal values only starting

at ¤ 500 million.

Adjustments in size seem to inevitably go along with opposing e�ects on overall ef-

�ciency. This trade-o� between technical and cost on the one and scale e�ciency on

the other hand is to be harmonized. With technical and cost ine�ciencies considerably

outweighing scale ine�ciency the dominant strategy of bank management should be sus-

tainable improvements of internal resource allocation.

Market power:

Looking at the locational factors the regression outcomes for cost and technical e�ciency

partly depend on whether the intermediation or production approach is assumed. The

19

production approach supports the Quiet Life Hypothesis. It predicts that market power

(NIM) reduces technical and cost e�ciency (p < 0.1). Conrad et al. (2014) �nd simi-

lar results for German savings banks. The intermediation approach indicates a positive

impact of market power on technical and cost e�ciency in line with the E�ciency Hy-

pothesis and the results of Wutz (2002), Grigorian and Manole (2002) and Hahn (2007).

Though only the coe�cient for technical e�ciency is statistically signi�cant (p < 0.01).

Recall that in the production approach deposits are outputs and in the intermediation

approach inputs. Thus, high e�ciency scores are associated with low deposit levels in the

intermediation and high deposit levels in the production approach. Market power seems

to negatively in�uence the production of �nancial services but to positively in�uence a

cooperative bank's ability to intermediate between lenders and borrowers.

The relationship between market power and scale e�ciency appears to be negative

(p < 0.1) implying that banks with high market shares in rather rural areas are less scale

e�cient than those in urban areas with higher competition. This result contrasts Hahn

(2007, 2008) who argues diametrically.

Economic wealth:

Concerning economic wealth of the region the results support the notion of a negative

but insigni�cant relationship. GDP, GDP per worker and their growth rates were tested

inconclusively. For GDP per capita some consistency was found: The coe�cients are

negative in all speci�cations but only weakly statistically signi�cant. For unemployment

there is more consistency in the results. High unemployment rates seem to be associated

with high technical e�ciency scores independent of the underlying approach (p < 0.01).

This could be explained by the assumption that in areas with an economically weak

population the demand for consultation-intensive �nancial services is rather low (Conrad

et al., 2014). All in all, the results suggest a negative association of economic wealth and

technical or cost e�ciency.

Interestingly, the e�ect of unemployment on scale e�ciency is negative. Both approaches

predict a negative relationship suggesting that banks in areas with high unemployment

rates are more likely to operate at an ine�cient scale, which is again in contrast to the

�ndings of Hahn (2007, 2008).

Demographics:

Turning to demographical factors the results for Popdens do not reveal any statistically

signi�cant relationship with technical and cost e�ciency scores. These results back up

the studies of Wutz (2002) and Conrad (2008).

A clearer relationship is found for scale e�ciency, independent of the underlying ap-

proach. The higher the population density in a given region the more likely banks operate

at their optimal size, again, contrasting the results of Hahn (2007, 2008). The results at

20

hand may indicate that economies of scale are easier to reap in urban/metropolitan areas.

As expected the share of people that are older than 65 (AGE) has a negative and highly

statistically signi�cant impact on bank e�ciency in line with Conrad et al. (2014) and

Maurer (2015). This relationship holds throughout all model speci�cations. Older people

demand speci�c rather than standardized �nancial products and make use of personal

counseling rather than modern communication or distribution channels, such as online

banking (Conrad et al., 2014, p. 544).

The control variable Risk measures the risk preference of each bank and is approximated

by the mean standard deviation of annual return on assets. The coe�cient consistently

show a positive impact on e�ciency, which is highly statistically signi�cant (p < 0.01,

except for scale e�ciency in the production approach).

The e�ect of increasing bank size depending on its location

In order to investigate whether the e�ect of adjusting bank size on e�ciency depends

on locational factors respective interaction terms were included in the regression. The

estimation results are summarized in table 6.15

Table 6: Interaction between size, locational factors and their e�ect on e�ciency

Technical E�ciency Cost E�ciency Scale E�ciency

θV RSI θV RS

P θCEI θCE

P θSEI θSE

P

...

SIZE_NIM 0.014 0.014 0.017 0.0078 -0.018∗∗∗ -0.045∗∗∗

SIZE_Unemployment 0.0033 -0.0028 0.00018 0.000085 -0.0011 0.0020

SIZE_Popdens 0.00051∗∗ 0.0012∗∗∗

SIZE_Age 0.0022 0.0037 0.0056∗∗ 0.0026 -0.0017∗ -0.0015

...

N 785 785 785 785 785 785

adj. R2 0.199 0.217 0.148 0.164 0.435 0.426

Notes: Group mean regressions for 2011 - 2015; N = number of groups = number of banks; Huber/White standard errors applied;

signi�cance levels: ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01; SIZE and SIZE2 were transformed into their natural logarithms to achieve a

normal distribution

Looking at technical and cost e�ciency, the estimation results do not provide much

useful and robust evidence for relationship. Only the interaction between SIZE and Age

shows a signi�cant e�ect on cost e�ciency in the intermediation approach implying that an

extension of a bank's scale of operation is more likely to be translated into cost e�ciency

gains if the population age in the district is relatively high. Diametrically, size increases

are more likely to be translated into scale e�ciency gains in regions with a younger pop-

ulation. The same goes for regions with dense populations and much competition (i. e.

15GDP was entirely left out due to the lack of signi�cance in the speci�cations without interaction terms.

Similarly, population density was not considered in the regressions for technical and cost e�ciency. For

the sake of a clear overview other variables are not displayed. The relevant factors do not change.

21

low market power = low NIM). Summing up, it seems that scale e�ciency gains are more

likely if the bank operates in a socioeconomically strong region (high population density,

low population age and little market power).

Limitations

Some econometric problems inherent in the present approach should be noted. First, due

to availability problems important internal bank e�ciency drivers, such as management

skills or sta� quali�cations, were not included. This is con�rmed by the Ramsey RESET

test for omitted variables for half of the model speci�cations. Second, theoretical consid-

erations imply that some relationships between the variables used are interdependent. On

the regional level, it is plausible to assume that not only economic wealth in�uences bank

e�ciency. The existence of e�cient banks providing local �rms with �nancial resources

may lead to a prosperous economy. Similar arguments can be made for competition in

banking. Competitive pressure can induce e�ciency. However, e�cient banks may also

squeeze ine�cient banks out of business determining the overall competition in the mar-

ket. Both omitted variable bias and reversed causality cause the independent variables to

be correlated with the error term. The OLS estimators become biased and inconsistent.

Another problem with the usage of NUTS3 regional data may occur. Some banks in

the sample do not strictly operate within their rural district. If, for instance, a bank is

headquartered in a weak district but has branches in a strong district the impact of the

local conditions in the strong district are not accounted for in the analysis. Thus, the

true impact of locational factors on bank e�ciency is over- or underestimated.

6 Conclusion

The aim of this paper was to investigate whether there is an optimal size in cooperative

banking, whether bank e�ciency depends on locational factors and whether the e�ect of

increasing bank size on bank e�ciency is a�ected by locational factors. The methodolog-

ical approach was divided into two steps. First, e�ciency scores were computed using the

non-parametric optimization approach Data Envelopment Analysis. Second, e�ciency

scores of 785 German cooperative banks for the years 2012-2016 were regressed on size

measures, locational factors and corresponding interaction terms.

The �ndings indicate that German cooperative banks bene�t from scaling their opera-

tions up to a certain threshold providing further evidence for the existence of an optimal

size of cooperative banks. Cooperatives with total assets of ¤ 100-250 seem to exploit

scale economies most e�ciently. This is signi�cantly lower than existing studies sug-

gest (see table 1). Most banks operate in the area of decreasing returns to scale. This

strengthens recent scienti�c evidence recommending bank managers to review expansion

strategies critically. Maurer and Thieÿen (2016) even discuss that demerging may be a

22

viable option. Owing to the lack of evidence, such strategies cannot be backed up with

data. This may be a starting point for further investigation.

In line with several scholars the results con�rm that locational factors at the district

level do have an impact on bank e�ciency (Wutz, 2002; Hauner, 2004; Drake et al., 2006;

Richter et al., 2018). The second-step regression results for technical and cost e�ciency

depend on whether banks are assumed to be �nancial intermediaries transforming deposits

into loans or producers of �nancial services. Some results contrast each other directly

giving reasonable grounds to further examine the di�erence between the two approaches.

For the time being, the focus is on those results that were found to be relatively consistent.

Economic wealth seems to play an insigni�cant role. GDP, GDP per capita, per worker

and their growth rates do not reveal any statistical signi�cance. Only unemployment is

consistently and positively associated with technical and cost e�ciency con�rming the

majority of other empirical studies discussed in section 3 of this paper.

Concerning demographics, the �ndings point out that population density does not have

an impact on technical and cost e�ciency. This contrasts the theoretical prediction that

dense populations imply high demand potential and thus low distribution cost. Population

age, however, shows the expected link: The older the inhabitants in a region the less

technical and cost e�cient the bank.

The results for the link between locational factors and scale e�ciency directly contrast

the �ndings of Hahn (2008, 2007) who claims that banks in socioeconomically weak areas

are closer to their optimal size. In this paper high degrees of market power, high unem-

ployment rates, low population density and age are associated with lower scale e�ciency

scores. This supports the view of Maurer and Thieÿen (2018). They argue that small

banks in rural areas perform worse than big banks in rural areas concluding that among

the small ones there is much potential for growth.

Looking at the interaction of increasing bank size and locational factors, however, the

results also suggest that size increases are less likely to be accompanied by scale e�ciency

gains in rural areas. This gives rise to the assumption that despite high potentials for

economies of scale rural banks struggle to actually reap them. This may be due to the

lack of a broad base of young, �nancially well endowed population.

This study paves the way for further research. Besides the econometric pitfalls discussed

in the previous section several extensions and improvements can be made. First, the

compilation of a data base including competitor branches on NUTS3 level would be a

better proxy for competition than the net interest margin. Second, the overall framework

under consideration can be extended by balance sheet performance indicators, such as

return on equity or assets, in order to emphasize and validate the practical relevance

of bank e�ciency studies. Providing empirical evidence for a positive causal e�ect of

bank e�ciency on bank performance would equip decision makers in banking with a more

comprehensive set of action parameters.

23

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7 Appendix

Table 7: Mean DEA e�ciency scores between 2012-2016

2012 2013 2014 2015 2016

θV RSI 0.873 0.869 0.854 0.852 0.853

θSEI 0.969 0.970 0.947 0.955 0.962

θCEI 0.753 0.727 0.703 0.672 0.668

θV RSP 0.587 0.567 0.592 0.581 0.607

θSEP 0.863 0.935 0.886 0.911 0.910

θCEP 0.502 0.459 0.563 0.549 0.567

Observations 736 734 728 741 746

Table 8: Mean pooled DEA e�ciency scores by size classes (2012-2016)

Size Class 0 1 2 3 4 5 6 7 8 9

θV RSI 0.932 0.884 0.834 0.844 0.847 0.856 0.878 0.885 0.928 0.940

θSEI 0.918 0.967 0.989 0.977 0.961 0.948 0.943 0.922 0.887 0.874

θCEI 0.749 0.693 0.653 0.678 0.712 0.716 0.752 0.777 0.825 0.878

θV RSP 0.689 0.566 0.518 0.545 0.606 0.601 0.625 0.673 0.777 0.873

θSEP 0.848 0.932 0.949 0.918 0.886 0.864 0.855 0.840 0.832 0.794

θCEP 0.636 0.516 0.472 0.493 0.542 0.535 0.553 0.593 0.665 0.816

SIZE 39.6 75.4 167.5 320.6 493.5 688.1 902.7 1491 3574 7274

Observations 143 373 1042 653 400 313 161 436 108 56

Note: Size = total assets in million Euros

Table 9: Number of banks by their scale position (Intermediation Approach)

0 1 2 3 4 5 6 7 8 9

Size optimum 36 64 176 118 33 22 5 19 14 10

Increasing returns to scale 107 301 520 55 3 0 0 0 0 0

Decreasing returns to scale 0 12 345 484 364 295 160 417 94 47

Total 143 377 1041 657 400 317 165 436 108 57

29

Table10:Pairwisecorrelationcoe�

cients

θV

RS

IθSE

IθC

EI

θV

RS

PθSE

PθC

EP

ROA

CIR

ROE

SIZE

NIM

GDP

Unemployment

Popdens

Age

Risk

θV

RS

I1

θSE

I-0.134∗∗∗

1

θC

EI

0.776∗∗∗

-0.143∗∗∗

1

θV

RS

P0.628∗∗∗

-0.288∗∗∗

0.881∗∗∗

1

θSE

P-0.146∗∗∗

0.714∗∗∗

-0.252∗∗∗

-0.372∗∗∗

1

θC

EP

0.633∗∗∗

-0.194∗∗∗

0.900∗∗∗

0.966∗∗∗

-0.281∗∗∗

1

ROA

0.187∗∗∗

0.207∗∗∗

0.110∗∗

-0.00333

0.119∗∗∗

0.0248

1

CIR

-0.0792∗

-0.149∗∗∗

-0.217∗∗∗

-0.102∗∗

-0.178∗∗∗

-0.145∗∗∗

-0.113∗∗

1

ROE

0.0985∗∗

0.174∗∗∗

0.0943∗∗

-0.00402

0.110∗∗

0.0130

0.877∗∗∗

-0.102∗∗

1

SIZE

0.0714∗

-0.406∗∗∗

0.260∗∗∗

0.296∗∗∗

-0.471∗∗∗

0.224∗∗∗

-0.0931∗∗

0.0772∗

0.00854

1

NIM

0.158∗∗∗

0.204∗∗∗

-0.132∗∗∗

-0.210∗∗∗

0.165∗∗∗

-0.182∗∗∗

0.180∗∗∗

-0.0840∗

-0.0140

-0.430∗∗∗

1

GDP

0.0651

-0.146∗∗∗

0.112∗∗

0.132∗∗∗

-0.105∗∗

0.116∗∗

0.0206

-0.145∗∗∗

0.0947∗∗

0.303∗∗∗

-0.279∗∗∗

1

Unemployment

-0.0820∗

-0.197∗∗∗

-0.00980

0.125∗∗∗

-0.117∗∗

0.0787∗

-0.268∗∗∗

0.0628

-0.292∗∗∗

0.0511

0.111∗∗

-0.240∗∗∗

1

Popdens

0.0499

-0.243∗∗∗

0.112∗∗

0.248∗∗∗

-0.169∗∗∗

0.193∗∗∗

-0.0891∗

-0.107∗∗

-0.00142

0.401∗∗∗

-0.305∗∗∗

0.582∗∗∗

0.0512

1

Age

-0.184∗∗∗

-0.112∗∗

-0.183∗∗∗

-0.0699

-0.122∗∗∗

-0.0984∗∗

-0.246∗∗∗

0.173∗∗∗

-0.284∗∗∗

-0.0788∗

0.222∗∗∗

-0.412∗∗∗

0.496∗∗∗

-0.191∗∗∗

1

Risk

0.187∗∗∗

0.207∗∗∗

0.110∗∗

-0.00333

0.119∗∗∗

0.0248

1-0.113∗∗

0.877∗∗∗

-0.0931∗∗

0.180∗∗∗

0.0206

-0.268∗∗∗

-0.0891∗

-0.246∗∗∗

1

N785

Notes:

Test

signi�cancelevels:

∗p<

0.05),

∗∗p<

0.01,∗∗∗p<

0.001,De�ntions:

ROA=

Return

onAssets,CIR

=Cost

IncomeRatio,Unemployment=

Unemploymentrate,Popdens=

Populationsdensity,Age=

Share

of

>65year-olds,Risk=

Standard

DeviationofReturn

onAssets.

30

Table 11: Tobit: In�uence of locational factors on technical, cost and scale e�ciency

(1) (2) (3) (4) (5) (6)

x_vrs_i x_vrs_p x_ce_i x_ce_p x_se_i x_se_p

main

lnsize -0.87∗∗∗ -1.26∗∗∗ -0.85∗∗∗ -1.15∗∗∗ 0.44∗∗∗ 0.47∗∗∗

lnsize2 0.023∗∗∗ 0.033∗∗∗ 0.022∗∗∗ 0.030∗∗∗ -0.012∗∗∗ -0.013∗∗∗

nim 0.095∗∗∗ -0.036∗ 0.0041 -0.033∗ -0.00054 -0.012∗

lngdp -0.013 -0.026 -0.013 -0.018 -0.0099∗ -0.0077

unemplrates 0.0017 0.021∗∗∗ 0.012∗∗∗ 0.017∗∗∗ -0.0040∗∗∗ -0.0012

popdens -0.0000086 0.000010 -0.000019∗ 0.00000034 0.0000062∗∗ 0.000012∗∗∗

age -0.0089∗∗∗ -0.0055∗ -0.012∗∗∗ -0.0067∗∗ -0.0025∗∗∗ -0.0049∗∗∗

se_roa 0.088∗∗∗ 0.095∗∗∗ 0.12∗∗∗ 0.095∗∗∗ 0.023∗∗∗ 0.0070

_cons 9.28∗∗∗ 13.0∗∗∗ 8.97∗∗∗ 11.9∗∗∗ -3.04∗∗∗ -3.17∗∗∗

/

var(e.x_vrs_i) 0.0086∗∗∗

var(e.x_vrs_p) 0.020∗∗∗

var(e.x_ce_i) 0.019∗∗∗

var(e.x_ce_p) 0.020∗∗∗

var(e.x_se_i) 0.0013∗∗∗

var(e.x_se_p) 0.0031∗∗∗

N 785 785 785 785 785 785

pseudo R2 -0.183 -0.379 -0.215 -0.239 -0.151 -0.160

∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01

31