行政院國家科學委員會專題研究計畫 成果報告
應用一般化共同邊界麥氏生產力指數探討銀行業生產力變
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研究成果報告(精簡版)
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Orea (2002)
(generalized Malmquist productivity index)
O’Donnell et al. (2008) Rao (2006) ; ; ; Abstract
This paper aims to provide new insights of productivity growth with a newly developed generalized metafrontier Malmquist productivity index (gMMPI) for banking industries across 15 West European nations during the period 1993-2006. A key advantage of the proposed method is that it allows for consideration of the role of scale effects. We also broaden our capacity in decomposing various sources of productivity change in the metafrontier context. The empirical results reveal that, on average, the banking industries experienced productivity growth arising from technical efficiency change and scale effects. This confirms that a more competitive and integrated financial market induced by financial deregulations is indeed able to improve banking productivity. Furthermore, the catch-up in scale and in technical change is found to underlie the metafrontier productivity growth. Finally, larger banks and conservative banks tend to grow faster than smaller ones.
Keywords: Metafrontier Malmquist productivity index; Technical efficiency change; Scale effects; Catch-up
II
1.
Introduction ... 1
2.
Literature Review ... 3
3.
Methodology
3.1
Technical Efficiency and Metatechnology Ratios ... 5
3.2
Decomposition of the MMPI ... 7
3.3
The Generalized MMPI ... 10
4.
Econometric Specifications and Data Description
4.1
Econometric Specification ... 13
4.2
Data Description ... 15
5.
Empirical results and analysis ... 16
6.
Conclusion... 20
References ... 22
Figure1~3 ... 28
Table1 ... 29
Table2 ... 30
Table3 ... 31
Table4 ... 32
... 33
1. Introduction
Many previous studies have carried out international comparisons among banks
to estimate a common frontier for banks of different groups (countries) by simply
pooling all observations together. This approach implicitly assumes that banks from
different groups have access to the same technology. This assumption appears to be
strong, as each group has its own cultural tradition, resource endowments, and
political and legal systems, which affect the behaviors and willingness to undertake
new innovations of banks from different groups. Different regulatory environments,
for example, lead to either universal banking countries, such as France and Germany,
or separated banking countries such as Belgium and the US (Allen and Rai, 1996). An
alternative way is to obtain individual production frontiers for each group, which can
be used to measure the group-specific technical efficiency scores. This avoids making
the assumption that all of the groups under consideration share the same technology.
However, the so-derived efficiency scores are not comparable due to the fact that they
are evaluated against different group-specific frontiers, rather than a common frontier.
This motivated Battese et al. (2004) and O’Donnell et al. (2008) to propose a
metafrontier production function, estimated by two steps, which makes it possible for
technical efficiency of banks in different groups to be compared with each other.
Since the conventional Malmquist productivity index (MPI) is defined in the
context of a single group production frontier, we need to extend it under the
framework of metafrontier production function. To this end, Rao (2006) first defined a
metafrontier and then discussed how to derive the metafrontier Malmquist
productivity index (MMPI), implicitly assuming a constant return to scale (CRS)
technology that excludes the existence of scale effect. The MMPI can be used to
2
(MTR) change among various groups. Although their MMPI measure is innovative, it
requires more elaboration to gain further insights into its sources and to get rid of the
presumption of CRS. Indeed, the significance of scale effects will be stressed later.
Recently, Chen and Yang (2011) further introduce the effect of scale efficiency
change into the MMPI formula and the resulting index is called the generalized
MMPI (gMMPI). Following this vein, the current study takes three steps to
decompose productivity change into a variety of sources under the framework of the
metafrontier in an attempt to render more information. First, following Orea (2002),
who developed a parametric approach to decompose a generalized MPI that takes
scale economies into account, we establish a measure of gMMPI in the context of a
metadistance function. Second, the catch-up item in Rao (2006) is further split into
two terms of catch-up in technology (CUT) and potential technological change (PTC)
with an eye on a better description of the relative adjustment speed of the technology
undertaken by a group to that of the potential metafrontier. Third, an output-oriented
group-specific distance function allowing for multiple inputs and multiple outputs is
employed to estimate the technical efficiency for each bank that can be specified as a
function of environmental variables like the one proposed by Battese and Coelli
(1995). In this manner, the variance of the one-sided error term, representing the
technical inefficiency of a bank, is heteroscedastic, as pointed out by Kumbhakar and
Lovell (2000).
The rest of the paper is organized as follows. Section 2 gives an overview of the
literature on the productivity changes and reviews several empirical studies specific to
the banking productivity changes of West European countries. Section 3 shows how to
formulate and decompose the new measure of gMMPI in the context of the
metafrontier function, where contribution of scale economies to productivity change
and variables. Section 5 performs an empirical application using panel data of
commercial banks from 15 West European countries, while Section 6 concludes the
paper.
2. Literature Review
The structure of European banking markets has been experiencing rapid changes
during and after the 1990s, making it particularly suitable for not only comparing
efficiency changes among European banks, but also understanding the determinants
of productivity change. Different frontier approaches based on either parametric or
non-parametric techniques have been carried out in order to evaluate banks’ technical
efficiencies and productivity changes in West Europe and other areas. Berger and
Humphrey (1997), Goddard et al. (2001), and Berger and Mester (2003) offer
excellent reviews on this matter.
Many earlier works have already performed cross-country comparisons of
technical efficiency for the commercial banks of European countries, e.g., Allen and
Rai (1996), Dietsch and Lozano-Vivas (2000), Altunbaş et al. (2001), and Weill
(2004), to mention a few. Differing from the foregoing, Bos and Schmiedel (2007)
and Huang et al. (2010) estimate comparable efficiency measures for European banks
under the framework of the translog and the Fourier flexible metacost functions
respectively. Although the issue of technical efficiency is pivotal, it is static in essence.
That is, it provides no information on whether efficiency is time-invariant during the
sample period. Conversely, a productivity measurement is dynamic, which provides
additional information on whether efficiency and technology have experienced
considerable changes during the sample period.
4
pioneered by Berg et al. (1992). The existing works have recourse to estimate either a
common frontier or individual production frontiers against which the efficiency
measures can be computed and used for calculating the MPI. Berg et al. (1993) and
Murillo-Melchor et al. (2009) adopted non-parametric techniques to estimate a
common distance function and assess the productivity changes of banking industries
across European countries. The former study uses data envelopment analysis (DEA)
to explore the differences in banking efficiency and productivity between Norway,
Sweden and Finland. Evidence is found that the banking industry of Sweden tends to
be the most efficient and has the highest productivity, followed by those of Norway
and Finland. Most of the difference in productivity can be attributed to the efficiency
component. The latter study is devoted to analyzing the differences in bank
productivity growth across 14 major European countries for the post-deregulation
period, namely 1995-2001. Their empirical results show that productivity growth,
which involves technical progress and efficiency losses, exists for the
post-deregulation period.
Another strand of contemporary literature compares productivity change of
banks across European nations by estimating individual production frontiers for each
nation. Chaffai et al. (2001) nonetheless employed the stochastic frontier approach to
estimate and compare productivity differences of banks among French, German,
Italian, and Spanish over 1993-1997, where the productivity difference is divided into
pure technological and environmental effects. The results showed that environmental
conditions are relevant in explaining productivity gaps of banking industries among
these countries. Casu et al. (2004) adopted both non-parametric and parametric
techniques to conduct cross-country comparisons of productivity change for European
banks from France, Germany, Italy, Spain, and United Kingdom with data covering
drove productivity growth during the sample period, rather than technical efficiency
change. Casu and Girardone (2005) also reached similar results based on
non-parametric analysis.
3. Methodology
A production technology can be alternatively represented by a distance function,
which is advantageous if firms hire multiple inputs to produce multiple outputs. This
function does not require the aggregation of outputs, price information, and
behavioral assumption. See, for example, Coelli and Perelman (1999, 2000), Cuesta
and Orea (2002), O’Donell and Coelli (2005), and Jiang et al. (2009). Following
Cuesta and Orea (2002), the present study derives efficiency scores from the
estimation of a stochastic output distance function.
3.1 Technical Efficiency and Metatechnology Ratios
An output distance function describes the maximal degree of expansion of the
output vector for a firm, given an input vector and technology. To define an output
distance function in the context of metafrontier, this study begins with the definition
of production technology at time t which transforms inputs into outputs. Define a
technology as
(
)
{
}
* , : 0; y 0, can produce . t t t t t t t T = x y x ≥ ≥ x y (1) where M t y ∈ ℜ+ and J tx ∈ ℜ+ are the vector of
M
outputs and J inputs, respectively. The output set for any input vector, xt, can be expressed as:( )
{
(
)
}
* *
: ,
t t t t t t
P x = y x y ∈T . (2)
6
function at time t is then defined on the output set as:
(
)
* * ( , ) inf{ : / ( )} t t t t t t D x y y P x λλ
λ
= ∈ (3)where
λ
is the minimum scalar that an output vector can be deflated and still remain producible with a given input vector (Shephard, 1970; Kumbhakar and Lovell, 2000,p. 30). Put differently, the distance function measures the largest feasible radial output
expansion given the current input vector. Consequently, the output metadistance
function is used to quantify the distance of a firm’s actual output from a frontier.
Since the distance function provides radial measures of the distance from an
input-output bundle to the boundary of production frontier, the technical efficiency
measure ( *
( , ) t t t
TE x y ) can be defined in terms of an output metadistance function:
* *
0< D x yt( ,t t)=TE x yt( ,t t)≤1 (4) Group k’s output distance function (Dtk( ,x y ) at time t is similarly defined and t t)
inherently greater than or equal to the output metadistance function, namely:
* *
( , ) k( , ) ( , ) k( , )
t t t t t t t t t t t t
D x y ≤D x y ⇒TE x y ≤TE x y (4) An inequality implies that there is gap between group k’s distance function and the
metadistance function. We can formally define the metatechnology ratio ( k t MTR ) at time t to measure how close group k’s frontier is to the metafrontier (O’Donnell et al.,
2008).1 That is: * * ( , ) ( , ) 0 ( , ) 1 ( , ) ( , ) k t t t t t t t t t k k t t t t t t D x y TE x y MTR x y D x y TE x y ≤ = = ≤ (6)
The foregoing provides a useful division of the technical efficiency of a particular
production plan gauged at the metatechnology, relative to that of group k, i.e.,
1
The metatechnology ratio is also referred to as the technology gap ratio (TGR) by Battese et al. (2004)
*
( , ) k( , ) k( , ) t t t t t t t t t
TE x y =TE x y ×MTR x y (7)
3.2 Decomposition of the MMPI
The metafrontier approach has recently been proposed to facilitate comparisons
of performance across different groups. This approach can be further extended to
calculate comparable productivity changes for firms operating under different
technologies. In the context of metafrontier, the MPI can be illustrated under the
framework of an output distance function and so can the MMPI. According to Färe et
al. (1994), the traditional MPI with respect to group k between t and t+1 is written as:
1 2 1 1 1 1 1 , 1 1 1 1 ( , ) ( , ) ( , , , ) ( , ) ( , ) k k k t t t t t t t t t t t t k k t t t t t t D x y D x y MPI x y x y D x y D x y + + + + + + + + + = × 1 2 1 1 1 1 1 1 1 1 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) k k k t t t t t t t t t k k k t t t t t t t t t D x y D x y D x y D x y D x y D x y + + + + + + + + + = × , 1 , 1 k k t t t t
TEC
+TC
+=
×
(8)It can be seen that the MPI is composed of technical efficiency change, TECt tk,+1, and
technical change, TCt tk, 1+. Following this vein, the MMPI between t and t+1 can be similarly formulated as:
1 * * 2 1 1 1 1 1 , 1 1 1 * * 1 ( , ) ( , ) ( , , , ) ( , ) ( , ) t t t t t t t t t t t t t t t t t t D x y D x y MMPI x y x y D x y D x y + + + + + + + + + = × 1 * * * 2 1 1 1 1 1 * * * 1 1 1 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) t t t t t t t t t t t t t t t t t t D x y D x y D x y D x y D x y D x y + + + + + + + + + = × * * , 1 , 1 t t t t
TEC
+TC
+=
×
(9)The first component can be interpreted as the rate at which a firm’s observed output
8
and the second component measures the rate of technical change that shifts the
metafrontier up or down between the two periods. Rao (2006) derives a link between
MMPI and MPI. This paper aims to provide a more complete decomposition of the
MMPI into a variety of sources in order to shed some light on productivity change in
the context of the metafrontier. Our decomposition enables us to integrate a group
specific frontier with the metafrontier. From (9), the MMPI can be re-expressed as:
, 1 t t MMPI + * * * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * * 1 1 1 1 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ( , ) ( , ) k t t t k t t t k t t t t t t k t t t k t t t k t t t t t t t t t k t t t k t t t k t t t k t t t k t t t t t t t t D x y D x y D x y D x y D x y D x y D x y D x y D x y D x y D x y D x y D x y D x D x y D x y + + + + + + + + + + + + + + + + + + + + = × × 1 2 * 1 1 1 1 1 1 1 ( , ) ) ( , ) t t t t k t t t D x y y D x y + + + + + + + 1 * * 2 1 1 1 1 1 1 1 1 1 1 , 1 , 1 * * 1 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) t t t t t t k k k k t t t t t t t t t t t t t t t t k k t t t t t t D x y D x y D x y D x y TEC TC D x y D x y D x y D x y + + + + + + + + + + + + + + = × × × , 1 , 1 , 1 k k k t t t t t t TEC + TC + MTRC + = × × , 1 , 1 k k t t t t MPI + MTRC + = × (10)
Term MTRC can be equivalently written as
1 2 1 1 1 1 1 , 1 1 ( , ) ( , ) ( , ) ( , ) k k k t t t t t t t t k k t t t t t t MTR x y MTR x y MTRC MTR x y MTR x y + + + + + + + = ×
which is the geometric mean of the index of the metatechnology ratio between the two
periods. Rao (2006) refers the inverse of MTRCt tk, 1+ to as the catch-up effect. A value
of MTRCt tk, 1+ greater than unity indicates that group k’s frontier catches up with the metafrontier over time, while a value below unity indicates that group k’s frontier
deviates away from the metafrontier. It is important to note that this term can be split
change (
PTC
): , 1 k t tMTRC
+ 1 2 1 1 1 1 1 1 1 1 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) k k k t t t t t t t t t k k k t t t t t t t t t MTR x y MTR x y MTR x y MTR x y MTR x y MTR x y + + + + + + + + + = × 1 * * 2 1 1 1 1 1 1 1 * * 1 1 1 1 1 1 1 1( ,
)
(
,
)
(
,
)
( ,
)
(
,
)
( ,
)
(
,
)
( ,
)
( ,
)
(
,
)
t t t t t t k k k t t t t t t t t t k t t t t t t t t t k k t t t t t tD x y
D x
y
MTR
x
y
D
x y
D
x
y
D
x y
D
x
y
MTR
x y
D
x y
D
x
y
+ + + + + + + + + + + + + + +
=
×
* , 1 1 1 1 , 1(
,
)
( ,
)
k t t t t t k k t t t t tTC
MTR
x
y
MTR
x y
TC
+ + + + +=
×
1 , 1 1 1 1 * , 1(
,
)
( ,
)
k k t t t t t k t t t t tTC
MTR
x
y
MTR x y
TC
− + + + + +
=
×
, 1 , 1 k k t t t tCUT
+PTC
+=
×
(11)The first term on the right-hand side gauges the relative change of the metatechnology
to group k’s frontier between t and t+1. The gap between the metafrontier and group
k’s frontier is shrinking over time, if CUT exceeds unity, while the reverse is true if CUT falls short of unity. Put differently, the first term explains the change in the
relative technical efficiency of the metafrontier to that of the group frontier between
periods t and t+1. A value of CUT greater than one unveils that the group frontier
catches up with the potential frontier. The second term on the right-hand side is the
ratio of the technical change of the metafrontier to that of group k’s frontier. If the
value of PTCt tk, 1+ is greater than unity, then the metatechnology improves in a faster rate than group k’s frontier, leading to a positive contribution to the MMPI. It follows
that the MMPI can be expressed as:
, 1 t t
10
3.3 The Generalized MMPI
The above decomposition of the MMPI is based on CRS technology, ignoring
entirely the possible scale effects. However, if the production technology exhibits
non-constant returns to scale, this index may not provide an appropriate measure of
productivity change due to its overlooking the potential impact of scale efficiency
change (SEC). To overcome this difficulty, Orea (2002) developed a novel parametric
approach that is able to include scale efficiency change into the conventional MPI.
Analogously, we generalized Orea’s approach as suitable for the metafrontier. Assume
that the technology can be described by the transcendental logarithmic (translog)
output-oriented distance function. The (log) index of total factor productivity (TFP)
change is formulated as:
, 1 lnMMPIt t+
(
)
(
)
* * 1 1 1 1 1 1 ln , , 1 ln , , 1 ln 2 ln ln m M t t t t t t t m m m m t t t D y x t D y x t y y y y + + + + = + ∂ + ∂ = + × ∂ ∂ ∑
(
)
(
)
* * 1 1 1 1 1 1 ln , , 1 ln , , 1 ln 2 ln ln n N t t t t t t t n n n n t t t D y x t D y x t x x x x + + + + = + −∂ + −∂ − + × ∂ ∂ ∑
* * 1 * * 1 1 1 1 1 1 1 ln ln 2 2 m n M N m m t n n t t t m t t n m t n t y x y xε
ε
+ε
ε
+ + + = = = + × − − − × ∑
∑
(13) where(
)
* * ln , , ln t t t m t m D y x t yε
=∂ ∂ and(
)
* * ln , , ln t t t n t n D y x t xε
= ∂∂ are output and input
distance elasticities, respectively. Since the translog output distance function belongs
to a quadratic function in ln n t
x , lny , and tm t, Diewert’s (1976) quadratic identity
lemma is applicable to yield:
(
)
(
)
* * 1 1 1 lnDt+ yt+,xt+ ,t+ −1 lnDt y x tt, t,(
)
(
)
* * 1 1 1 1 1 1 ln , , 1 ln , , 1 ln 2 ln ln m M t t t t t t t m m m m t t t D y x t D y x t y y y y + + + + = + ∂ + ∂ = + × ∂ ∂ ∑
(
)
(
)
* * 1 1 1 1 1 1 ln , , 1 ln , , 1 ln 2 ln ln n N t t t t t t t n n n n t t t D y x t D y x t x x x x + + + + = + ∂ + ∂ + + × ∂ ∂ ∑
(
)
(
)
* * 1 1 1 ln , , 1 ln , , 1 2 t t t t t t D y x t D y x t t t + + + ∂ + ∂ + + ∂ ∂ (14)Substituting Eq. (14) into Eq. (12), we obtain:
, 1 lnMMPIt t+ =lnDt*+1
(
yt+1,xt+1,t+ −1)
lnDt*(
y x tt, t,)
(
)
(
)
* * 1 1 1 ln , , 1 ln , , 1 2 t t t t t t D y x t D y x t t t + + + ∂ + ∂ − + ∂ ∂ (15)The first term on the right-hand side measures exactly the change in the
output-oriented measure of Farrell technical efficiency between t and t+1, TECt t*,+1,
and the second term is a measure of technical change, TCt t*, 1+ .
It is a consensus that a TFP index should have four desirable properties: identity,
monotonicity, separability, and proportionality. Although the lnMMPI of (15) satisfies
the first three properties, the proportionality property that requires a homogeneous
condition of +1 in outputs and -1 in inputs may not be fulfilled, since the input
weights do not necessarily sum up to unity. See, for example, Balk (2001) and Orea
(2002). This means that, unless in the case of CRS technology, this index is invalid to
be used as a measure of productivity change. Drawing on ideas recommended by
Denny et al. (1981), Orea (2002) aggregates the growth in inputs using distance
elasticity shares in place of distance elasticities as weights, which warrants the
proportionality property.
A (log) generalized output-oriented MMPI, lngMMPI, can be defined as:
, 1
12 * * * * 1 1 1 1 * * 1 1 1 1 1 1 1 1 ln ln 2 2 m n n n M N m m t t t t t t m N N n n n m t n t t t n n y x y x ε ε ε ε ε ε + + + + = = + + = = − − = + × − + × − −
∑
∑
∑
∑
(16) where * 1 N n t n ε =−
∑
provides a measure of returns to scale characterizing the output distance function, in which a value of it is greater than (less than or equal to) unityaccording as increasing (decreasing or constant) returns to scale. It is seen that
lngMMPI gauges the growth in outputs net of the growth in inputs and is now a valid
TFP index due to its having the four desirable properties. Using (14), (16) can be
shown to be composed of lnMMPI and the contribution of scale effects, i.e.,
, 1 lngMMPIt t+
(
)
(
)
*1(
1 1)
*(
)
* * 1 1 1 ln , , 1 ln , , 1 ln , , 1 ln , , 2 t t t t t t t t t t t t D y x t D y x t D y x t D y x t t t + + + + + + ∂ + ∂ = + − − + ∂ ∂ * * * 1 * 1 1 * * 1 1 1 1 1 1 1 1 1 ln 2 n n n N N N n t n t t t N t N n n n n n n t t t n n x x ε ε ε ε ε ε + + + = = = + = = + − − × + − − × × ∑
∑
∑
∑
∑
(17)The third term on the right-hand side is the element of scale efficiency change
(SECt t*,+1), which reflects the scale effect under VRS technology and vanishes when CRS technology prevails, such that lngMMPI =lnMMPI. It follows that
, 1 t t
gMMPI +
=
TEC
t t*,+1×
TC
t t*,+1×
SEC
t t*,+1 (18) Nonconstant returns to scale makes a positive contribution to lngMMPI if the scaleelasticity * 1 1 N n t n ε =
−
∑
> , corresponding to increasing returns to scale, and input use expands (ln(
n1/ n)
0 t t x+ x > ), or if * 1 1 N n t n ε =−
∑
< , corresponding to decreasing returns to scale, and input use contracts (ln(
n1/ n)
0t t
x+ x < ). Both cases cause SECt t*,+1 to be greater than unity and reveal that the production scale of the bank is altering toward
the optimal size and enjoying cost savings in terms of a lower long-run average cost.
However, the reverse is true prompting SECt t*,+1 to be less than unity, incurring cost wastes in terms of a greater long-run average cost.
We finally plug (11) into (17) to obtain:
, 1 lngMMPIt t+
(
)
(
)
1(
1 1)
(
)
1 1 1 ln , , 1 ln , , 1 ln , , 1 ln , , 2 k k t t t t t t k k t t t t t t D y x t D y x t D y x t D y x t t t + + + + + + ∂ + ∂ = + − − + ∂ ∂ (
)
(
)
(
(
)
)
(
(
)
)
* * 1 1 1 1 1 1 1 1 1 ln , , 1 ln , , 1 ln , , 1 ln , , 2 ln , , 1 ln , , t t t t t t k k t t t t t t k k t t t t t t D y x t D y x t t t MTR y x t MTR y x t D y x t D y x t t t + + + + + + + + + ∂ + ∂ + ∂ ∂ + + − − ∂ + ∂ + ∂ ∂ * * * 1 * 1 1 * * 1 1 1 1 1 1 1 1 1 ln 2 n n n N N N n t n t t t N t N n n n n n n t t t n n x x ε ε ε ε ε ε + + + = = = + = = + − − × + − − × × ∑
∑
∑
∑
∑
(19)After taking an exponential on both sides, the foregoing can be transformed into:
, 1 t t gMMPI + * , 1 , 1 , 1 , 1 , 1 , 1 , 1 t t k k k k k t t t t t t t t t t k t t
SEC
TEC
TC
CUT
PTC
SEC
SEC
++ + + + +
+
=
×
×
×
×
×
(20)where the last term on the right hand side measures the ratio of scale efficiency
change of the metafrontier to that of the group frontier between two periods
(henceforth, RSEC). It may also be referred to as a catch-up in scale. A value of it
exceeding one implies that the production scale measured on the group frontiers is
correcting toward CRS faster than that measured on the metafrontier.
4. Econometric Specifications and Data Description
4.1 Econometric Specification
14
(=1,…, I ) at time t (=1,…, T) is specified as a standard translog form, including
time trend to deal with technical change, namely:
( )
( )
( )
( )
0 1 1 1 11
ln
ln
ln
ln
ln
2
N M N N k O n n m m jk j n n m j nD
a
a
x
b
y
a
x
x
= = = == +
∑
+
∑
+
∑∑
( ) ( )
( ) ( )
2 0 00 1 1 1 11
1
ln
ln
ln
ln
2
2
M M N M ml m l nm n m m l n mb
y
y
g
x
y
j t
j t
= = = =+
∑∑
+
∑∑
+
+
( )
( )
1 1ln
ln
N M nt n mt m n mt
x
t
y
γ
θ
= =+
∑
+
∑
(21)To estimate the parameters we have to impose the linear homogeneity constraint on
outputs. This leads to normalize the distance function using any one of the M outputs,
say, y1, as the numeraire. After letting u= ln k O
D and appending a statistical noise,
( )
2~ 0, v
v N
σ
, the resulting output distance function is written as:( )
(
)
( )
( )
1 0 1 1 2 1 11
ln
ln
ln
ln
ln
2
N M N N n n m m jn j n n m j ny
a
a
x
b
y
y
a
x
x
= = = =−
=
+
∑
+
∑
+
∑∑
(
) (
)
( ) (
)
2 1 1 1 0 00 2 2 1 21
1
ln
ln
ln
ln
2
2
M M N M ml m l nm n m m l n mb
y
y
y y
g
x
y
y
j t
j t
= = = =+
∑∑
+
∑∑
+
+
( )
(
1)
1 2ln
ln
N M nt n mt m n mt
x
t
y
y
v u
γ
θ
= =+
∑
+
∑
+ +
(22)where term u is a non-negative random variable signifying technical inefficiency and
independent of v. Following Battese and Coelli (1995), the inefficiency term is
allowed to be variant with an array of environmental variables, z, i.e.,
0
u=z
δ
+ ≥W (23)where
δ
is a vector of parameters corresponding to z and W is a normal variate with mean zero and constant variance σ2. Group k’s stochastic frontier model (22) together with (23) can be estimated by the maximum likelihood.Next, parameters of the output metadistance function can be estimated using
linear programming (LP) and quadratic programming (QP) techniques, as proposed
by Battese et al. (2004), who suggest applying either simulations or bootstrapping to
obtain the standard errors for the parameter estimates. Measures * t
TE and MTR tk can be calculated based on these coefficient estimates.
4.2 Data Description
This paper compiles unconsolidated accounting statements from the BankScope
database of BVD-IBCA. The sample contains 2573 commercial banks in 15 West
European countries spanning 1993-2006. The unbalanced panel data contain 22627
bank-year observations after excluding all missing observations. According to Dietsch
and Lozano-Vivas (2000), Lozano-Vivas et al. (2001), and Lozano-Vivas et al. (2002),
we consider two micro-level variables consisting of equity to total assets ratio (ETA)
and average return on assets (ROA), as well as three macro-variables including per
capita income (PCI), population density (PD), deposit density (DD), as the
environmental variables that may affect a bank’s technical inefficiency. Variables PCI
and PD are taken from the World Development Indicator (WDI) and DD from the
International Financial Statistics (IFS). We believe that our application to West
European banking is fruitful because of the abundance of quality data available on
banks in this industry.
Based on the intermediation approach, banks are assumed to employ three
inputs – labor (x1), physical capital (x2), and borrowed funds (x3) – to produce three
outputs – loans (y1), investments (y2), and non-interest revenue (y3). As there are
quite a few missing data on the number of employees, we select the total assets net of
16
(2004). Input physical capital is measured by the amount of fixed assets, while input
borrowed funds consist of all deposits and borrowed money. Output non-interest
revenue is used in an attempt to reflect a bank’s degree of product diversification. It
constitutes a crucial source of income for modern universal banks. All of the inputs
and outputs are expressed in millions of real US dollars deflated by the consumer
price index of individual countries with base year 2000.
Table 1 summarizes the descriptive statistics for the aforementioned variables by
countries. It can be seen from these statistics that there exists substantial variation in
these operational characteristics of each sample banks among countries. Banks in
different countries may hire different qualities of inputs to produce heterogeneous
outputs under dissimilar production techniques. Such variations justify the use of
metafrontier model in the study of international comparison.
[Insert Table 1 here] 5. Empirical results and analysis
This study is devoted to decompose the gMMPI into a set of components related
to productivity change using equations (18) and (20). Recall that a value of these
components greater than one indicates a productivity gain, while a value less than one
indicates a productivity loss. As far as the metafrontier is concerned, the left part of
Table 2 presents the rates of change in gMMPI and its individual components per
annum. The average rate of gMMPI is found to be equal to 0.21% per year, which is
consistent with the finding of Casu et al. (2004) and Casu and Girardone (2005) over
the period 1994-2000, Murillo-Melchor et al. (2009) over the period 1995-2001, as
well as Barros et al. (2010) over the period 1996-2003. It is noticeable that this
measure of productivity growth consists of technical efficiency gains (0.06%),
technical progress (0.05%) and the effect of scale improvement (0.10%) toward
change over the sample period. Measure gMMPI varies closely with TEC*
particularly before 2001. SEC* also fluctuates over time, while TC* exhibit a gradual
upward trend initially, accompanied by a slight downward trend.
[Insert Table 2 and Figure 1 here]
We divide the entire sample period into three sub-periods. Evidence is found that
West European banking industries experienced a faster positive productivity growth
during 1993-1998, immediately after the establishment of the single market for
European financial services leading to international economic integration in this area,
than the latter two sub-periods. This productivity gain arose probably from the
regulatory reform in banking, such as financial services restructuring and
consolidation, aiming at increasing competition among financial institutions and
across borders. It is worth mentioning that there exists an increasing positive scale
effect contributing to the productivity growth during the sample period. In fact, this
effect constitutes the main source of the productivity gain, resulted from, e.g., bank
consolidation and the enlargement of banking markets.
The gMMPI is further decomposed into technical efficiency change, technical
change, and scale efficiency change based on the group frontiers. The results are listed
in the middle of Table 4, while various catch-up effects are shown in the right part of
the table. Recall that MMPIt t,+1=TECt tk,+1×TCt tk,+1×CUTt tk,+1×PTCt tk,+1, which differs from the gMMPI due to the omission of scale effect, i.e., term SEC*. Since the
average rate of productivity growth for MMPI is 0.11% per year, this omission leads
to an underestimation, as compared with the 0.21% of the gMMPI. Among the four
sources, technical efficiency change (SEC) of the group frontiers plays the most
important role, followed by the PTC, while the remaining two sources curb
18
The scale effect on the ground of group frontiers is essential, not only for
capturing scale efficiency change, but also for helping shed light on the effect of
catch-up in scale. The results show that an average bank sustains a slow improvement
of its production scale, 0.02% per annum, toward constant returns to scale during the
sample periods. Most importantly, the ratio of the scale efficiency change of the
metafrontier to that of group frontiers exceeds one, suggesting that there exists
catch-up in scale. That is, the production scale measured on the group frontiers is
adjusting toward CRS faster than that measured on the metafrontier. Finally, before
the advent of European Economic and Monetary Union (EMU) in 1999 Rao’s
catch-up effect (MTRC) is slightly greater than unity and sustains over the following
three years, then falls below unity thereafter, mainly entailed by the regress of CUT.
This reveals that the stimulation effect of the EMU on the banking productivity
appears to last in a short period. Figure 1 depicts the three measures over time. It is
seen that MTRC and CUT are almost fluctuating synchronously, but the curve of PTC
hardly varies with a smooth downward trend.
[Insert Table 3 here]
Table 3 reports the mean values of these measures in a descending order
according to the gMMPI. Banks in nine out of the fifteen countries have positive
measures of the gMMPI, implying that their productivity grows with time evaluated
against the metafrontier, while banks in the remaining countries undergo a
productivity decline. The highest productivity growth is found by Belgium (1.05%),
followed by Sweden (0.99%), and United Kingdom (0.99%), while the lowest rate of
growth occurs in Spain and Portugal (both are found to be equal to -0.56%), followed
by Switzerland (-0.19%), Norway (-0.18%), Italy (-0.16%), and Netherlands (-0.13%).
Our findings are similar to Pastor et al. (1997), Kondeas et al. (2008), and
a closer look at the three elements, all but one (Denmark) and eleven out of fifteen
countries show scale efficiency gain and technical progress, while merely seven out of
fifteen countries enjoy technical efficiency gain. In particular, banks in Spain and
Portugal are found to experience the slowest productivity growth due primarily to
worse technical efficiency change.
It is crucial to note measures TEC, TC, and SEC, shown in the middle columns
of Table 5 are not comparable among countries, because they are computed against
distinct country frontier. The evidence found in the previous paragraph may be
attributed to the trend toward consolidation in response to the intensified competition
among banks within the EMU. According to Goddard et al. (2001), all West European
countries, apart from Portugal, have undergone a decline in the number of banks since
1989. Faced with this new atmosphere, a bank must manage to enhance its
competitiveness by lowering production costs by means of, e.g., merger and
acquisition activities to expand the production scale and adopting innovations in a
prompt manner to stimulate technical progress instantly.
Merger and acquisitions may incur extra costs of adapting to the new
environment and reconsidering business strategies, disappearance of traditional
revenues, and any other unfinished banking reforms. Another explanation related to
the economic environment stresses that the lower the level of economic conditions,
the harder is to engage in banking activities, as noted by Lozano-Vivas et al. (2001,
2002). Our finding confirms this argument in that, on the basis of Table 1, Spain and
Portugal have the lowest levels of per capita income among the sample states, leading
to technical efficiency losses.
We respectively classify the entire sample into eight groups based on total assets
and the ratio of equity to total assets. Table 4 presents the outcomes. According to
20
banks do. Evidence is found to support the establishment of large banks as this form
of banks exhibit swift rates of growth in TC* and SEC*, accompanied by a gradual
regress in TEC* as a consequence of merger and acquisitions. Conversely, the major
contribution to productivity growth of smaller banks stems from TEC*. Figure 2
shows a U-shape for the average gMMPI curve against various size classes. Curve
TEC* decreases as the bank size grows, while the curves of TC* and SEC* rise with
bank sizes. It is also evident that small banks exhibit higher catch-up in technology,
while larger banks exhibit higher catch-up in scale.
[Insert Table 4 and Figure 2 here]
Panel B of Table 4 shows the productivity differences for banks with diverse
attitudes toward risk. Since equity capital provides a buffer against portfolio losses, it
is expected that the higher the capital to assets ratio, the less likely is a bank involving
in insolvency, and vice versa. It is interesting to note that the highest three classes of
banks corresponds to the quickest productivity growth, mainly because they have the
highest average values of TEC*. More conservative banks tend to grow in a rapid rate.
Similarly, TC* improves with ETA for the last four classes, while SEC* varies
irregularly. The outcomes appear to be quite reasonable, since there is incentive for a
risk-averse bank manager to engage in monitoring and supervising activities and
making circumspect decisions for the purpose of avoiding the exposure of excess risks
(Kumbhakar and Wang, 2007). Figure 3 illustrates that MTRC and its two
components have gradual increasing trends, though fluctuate.
[Insert Figure 3 here] 6. Conclusion
Using the newly developed formulae, this paper aims to gain further insights on
the productivity growth of West European banking industries during the period
productivity growth. Most importantly, these components can provide useful
information to managers, industry consultants, and regulators to assess performance
and to adjust business strategy and enforce new regulation policy. The empirical
application reveals that a representative bank in West Europe sustains a positive
productivity growth during the sample period when the financial markets become
more competitive and integrated after the creation of a single market for financial
services in 1993. The productivity gains are chiefly stimulated by scale efficiency
change, justifying the significance of the scale effect in the evaluation of a bank’s
productivity change. Overlooking the role played by the scale effect is likely to result
in an underestimation for the measure of productivity change.
Banks in nine out of the fifteen countries are confronted with productivity
growth and scale efficiency gains and technical progress prevail in the vast majority
of the sample states. However, the data failed to identify the prevalence of positive
technical efficiency change in most of the countries under consideration. Larger banks
seemed to grow faster than smaller ones, due to technological advance and the
enhancement of production scale, rather than efficiency change. Finally, a bank with a
higher value of ETA is inclined to grow faster than a bank with a lower value of ETA.
This may be ascribable to the fact that risk-averse bank managers are frequently
engaging in monitoring and supervising activities that help reduce the exposure of
22
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Figure 1. Trends in gMMPI and MTRC components
Figure 2. Trends in gMMPI
Figure 3. Trends in gMMPI and MTRC components by ETA class
28
Trends in gMMPI and MTRC components
Trends in gMMPI and MTRC components by asset size class
Trends in gMMPI and MTRC components by ETA class Trends in gMMPI and MTRC components
and MTRC components by asset size class
Table 1. Descriptive statistics
AUS BEL DNK FIN FRA DEU ITA LUX NLD NOR PRT ESP SWE CHE GBR Number of banks 146 77 97 11 280 876 106 138 22 113 17 49 101 498 42 Number of observations 1235 632 1132 81 2496 9640 690 1279 121 557 109 342 609 3435 269
Labor (total assets net 2613 14068 2652 12975 13494 4934 7689 5576 1629 3588 8406 7294 6845 3868 1783 fixed assets) (8788 (48850 (16737 (16059 (64075 (39664 (17619 (9682 (2076 (12398 (12102 (18369 (22281 (45107 (2580) Physical capital 21 69 22 91 51 31 124 18 4 26 135 131 32 31 13 (43) (195) (97) (150) (243) (97) (272) (45) (9) (82) (189) (356) (175) (259) (34) Borrowed funds 2430 12707 2216 11556 11070 4504 6620 4772 1460 3223 7643 6550 5655 3188 1366 (8192 (43001 (13600 (14398 (48257 (33413 (15209 (8227 (1859 (11066 (10890 (16553 (18257 (36506 (1884) Loans 2080 8706 1234 8763 8880 3459 5605 3927 1038 2987 6244 5245 4635 2526 1238 (7022 (30089 (6944) (11540 (37557 (24202 (13200 (7002 (1063 (10162 (8919) (12680 (14783 (26426 (1675) Investments 441 4429 1182 3109 3112 1175 1503 1442 481 408 1377 1582 1589 896 328 (1642 (14601 (7975) (3648) (17383 (10396 (3235) (3252 (1425 (1542) (1997) (4673) (5721) (12744 (813) Non-interest revenue 16 47 18 83 76 29 74 30 6 25 48 58 54 44 41 (43) (145) (91) (119) (321) (229) (157) (49) (14) (87) (90) (145) (162) (404) (106) Equity over total asset 8.42 6.23 13.67 5.69 8.23 6.08 12.69 5.90 9.10 9.43 8.37 9.81 12.64 12.19 15.30 (9.67) (5.20) (5.31) (3.31) (10.47) (7.09) (12.22) (7.19) (6.73) (3.70) (4.96) (10.70) (6.26) (14.16) (10.33 Return on assets 0.54 0.49 1.40 0.34 0.32 0.31 0.36 0.61 0.46 0.91 0.67 -0.15 1.02 0.89 0.51
(0.27) (0.35) (0.44) (0.52) (0.50) (0.09) (0.47) (0.16) (0.27) (0.24) (0.50) (1.50) (0.37) (0.29) (0.66) Per capita income 3.16 3.07 3.36 3.09 3.07 3.11 2.91 3.75 3.12 3.64 2.30 2.57 3.34 3.52 3.19 (0.07) (0.08) (0.07) (0.13) (0.07) (0.06) (0.05) (0.14) (0.09) (0.09) (0.09) (0.09) (0.08) (0.04) (0.10) Population density 4.57 5.82 4.82 2.73 4.67 5.44 5.24 5.12 5.94 2.64 4.70 4.37 2.99 5.16 5.49 (0.01) (0.01) (0.01) (0.01) (0.02) (0.00) (0.00) (0.05) (0.02) (0.02) (0.01) (0.02) (0.01) (0.01) (0.01) Deposit density 7.74 8.93 7.77 5.32 7.51 8.53 7.65 10.24 9.14 5.85 7.04 6.77 5.66 9.18 8.79 (0.17) (0.22) (0.20) (0.18) (0.18) (0.25) (0.14) (0.26) (0.28) (0.29) (0.11) (0.22) (0.25) (0.20) (0.46) Notes: All inputs and outputs are expressed in millions of real US dollars with base year 2000. Standard deviations are in parentheses.
