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6. Conclusion
This paper successfully extends the deterministic metafrontier translog cost function to the more general SMF Fourier flexible cost function that has several advantages. Unlike the use of conventional two-step procedure, e.g., Battese et al.
(2004), O’Donnell et al. (2008), and Huang et al. (2011a), which combines the stochastic frontier approach with the mathematical programing technique, the stochastic metafrontier is formulated and applied for estimating the technology gap ratio, instead of the programming technique, such that the first and the second step of the study are compatible. It is therefore of great benefit that the statistical inferences can be drawn, on the basis of a regression model, without relying on simulation or bootstrapping. Furthermore, composed errors can be incorporated to handle random shocks and possible production inefficiency. This avoids the criticism aimed at mathematical programming. Most importantly, our new model specifies the TGR to be a function of some exogenous variables. That is, the group-specific environmental differences are allowed to be included in the second-step estimation procedure. The environments faced with a bank usually impose some restrictions on bank managers, which may limit their ability to optimize the allocation of resources among different uses. This helps characterize the sources of a bank’s production inefficiency, on the one hand, and reduce the problem of heteroskedasticity, on the other hand.
Using the newly developed formulas, this paper devoted to uncovering new evidence on the cost efficiency of Western European banks during the period of 1996 to 2010 and to broaden our capacity for illustrating the usefulness of the proposed estimation approach and comparing with the results of various competing metafrontier models. The empirical application discloses that the TGR and MCE exhibit a gradual upward trend initially during 1996-2000 and then fluctuate with a downward trend reflecting market movements, especially after the subprime crisis of 2007-2010.
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These results show that a single and competitive banking market not only contributes to greater cost efficiencies, but also shrinks the technology gap within the member states of European Union. A representative bank in Western Europe attains a remarkable efficiency improvement after the financial markets become more competitive and integrated due to the creation of a single market in 1993. However, the wave of international economic integration is usually inevitably accompanied by higher degree of risk exposure. The subsequently downward trends reflect this factor during the recession period of global economies after 2000 and are found to have deteriorated after the global financial crisis which started in late 2007.
In seeking further evidence on the role played by the bank’s size, profitability, and risk attitude on relevant efficiency score, several implications can be drawn from the empirical results. First, smaller banks appear to outperform the larger ones in terms of MCE and TGR. On the other words, banks with smaller assets size are able to benefit more from the cost efficiency and advanced technology. Second, higher profitability is connected with greater efficiency. Banks with higher profitability tend to be more cost efficient and to adopt more advanced technology than poor profitable banks. Finally, more conservative banks are related to greater efficiency. Banks with higher values of capital ratio are inclined to enhance cost efficiency and to adopt more advanced technology. This result might be ascribable to the fact that risk-averse bank managers are frequently engaged in monitoring and supervising activities that help reduce the exposure of risks.
As in the case for the other two competing metafrontier models, the choice of methodology seems to influence the estimation results and its associated policy implications derived from the analyses. Not only the parameter estimates considerably differ from each other, but also the so-derived efficiency scores show dissimilar distribution and patterns. The deterministic metafrontier programming techniques
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tends to underestimate the TGR and MCE measures and these estimates exhibit larger variation, probably due to specification error, i.e., the estimates are inclined to be confounded with shocks, and lack of considering environmental variable. Although the SMF92 appear to be better choices due to its ability to incorporate random shocks, it is also apt to underestimate these measures and exhibit larger variation in contrast to the SMF95. Moreover, when we adopt deterministic metafrontier programming techniques to deal with further insights on the effect of assets size, profitability, and risk attitude, there is little evidence to suggest that the widely accepted results.
Therefore, the proposed stochastic metafrontier specification, which includes group-specific environmental variable as determinants of technology gap, should be preferable and consistent with reality.
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App pendix A
Figu ure A1. Ima
aging correla77
ation of expplanatory vaariables in AAustria
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Figurre A2. Imag
ging correla78
ation of explanatory varriables in BBelgium
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Figuree A3. Imagiing correlat
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tion of explaanatory variiables in Deenmark
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Figu ure A4. Imag
ging correla80
ation of expplanatory vaariables in FFrance
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Figuree A5. Imagiing correlat
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tion of explaanatory variiables in Geermany
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Figu ure A6. Ima
aging correl82
lation of exxplanatory vvariables in IItaly
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Figure A A7. Imaging
g correlatio83
on of explannatory variabbles in Luxeembourg
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Figu ure A8. Ima
aging correl84
ation of expplanatory vaariables in SSpain
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Figure A9. Imagin
ng correlatio85
on of explannatory variaables in Swiitzerland
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Figure A A10. Imaging
g correlatio86
on of explannatory variabbles United Kingdom
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F Figure A11. Imaging co
orrelation of87
f explanatorry variabless for all sammple countries
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Table B1 Estimates of Fourier series for the sample countries
Austria Belgium Denmark France Germany Italy Luxembourg Spain Switzerland United Kingdom cos 1z 1.197 *** 0.775 *** -0.836 *** 0.470 * 0.584 *** -0.271 *** 0.193 -0.619 ** 0.393 ** 0.051 Notes: Standard errors are given in parentheses. ***, ** and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.
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Table B2 Parameter estimates of the translog cost frontier for the sample countries
Austria Belgium Denmark France Germany Italy Luxembourg Spain Switzerland United Kingdom
constant
2.494 *** 1.515 ** 1.457 *** 1.738 *** 1.014 *** -0.297 0.907 ** 4.237 *** 1.273 *** 1.169 **‧
Notes: Standard errors are given in parentheses. ***, ** and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.‧ 國
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Table B3 Estimates of Fourier series for various competing metafrontier models
Independent Stochastic Metafrontier with
τ
(SMF95)Stochastic Metafrontier
without
τ
(SMF92) Metafrontier (QP) variables Coefficient S.E. Coefficient S.E. Coefficient S.D.cos 1z 0.092 *** 0.009 0.036 *** 0.009 -0.070 0.084 sin 1
z
-0.119 *** 0.006 -0.006 0.008 -0.358 0.060 cos 2z
-0.011 * 0.006 0.061 *** 0.007 -0.146 0.058 sin 2z
0.055 *** 0.007 0.017 ** 0.008 -0.010 0.049 cos 3z
-0.013 * 0.007 -0.086 *** 0.008 -0.029 0.054 sin 3z
0.047 *** 0.005 0.004 0.007 0.235 0.048 cos 2 1z -0.064 *** 0.004 -0.038 *** 0.004 -0.026 0.032 sin 2 1z
0.033 *** 0.003 0.040 *** 0.004 0.046 0.037 cos 2 2z
-0.052 *** 0.003 -0.069 *** 0.003 0.036 0.026 sin 2 2z
0.020 *** 0.003 0.030 *** 0.004 -0.019 0.032 cos 2 3z
0.001 0.004 -0.011 *** 0.004 0.095 0.025 sin 2 3z
-0.026 *** 0.003 -0.013 *** 0.003 -0.016 0.024 cos 1 2z z 0.124 *** 0.004 0.147 *** 0.005 0.019 0.040 sin 1 2z z
-0.031 *** 0.004 -0.019 *** 0.005 -0.165 0.040 cos 1 3z z
-0.025 *** 0.004 -0.055 *** 0.005 -0.064 0.045 sin 1 3z z
-0.088 *** 0.004 -0.052 *** 0.005 -0.088 0.035 cos 2 3z z
0.033 *** 0.004 0.026 *** 0.004 -0.072 0.040 sin 2 3z z
0.117 *** 0.004 0.047 *** 0.005 0.186 0.032Notes: SMF95, stochastic metafrontier with Battese and Coelli (1995) specification; SMF92, stochastic metafrontier with Battese and Coelli (1992) specification; QP, quadratic programming model. The standard errors of the QP estimators are obtained using a bootstrapping method with 1,000 replications.
***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels respectively.
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Table B4 Parameter estimates for various competing metafrontier models (TL)
Independent SMF95 SMF92 QP
variables Coefficient S.E. Coefficient S.E. Coefficient S.D.
constant 1.301*** 0.055 0.906*** 0.098 0.433 0.442
Notes: SMF95, stochastic metafrontier with Battese and Coelli (1995) specification; SMF92, stochastic metafrontier with Battese and Coelli (1992) specification; QP, quadratic programming model. The standard errors of SMF are corrected by sandwich estimators. The standard errors of the QP estimators are obtained from bootstrapping. ***, **, and * indicate statistical significance at the 1%, 5%, and 10%
levels respectively.
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App pendix C
Figure C1
Figure C
. Trends in
C2. Trends i
93
CE, TGR, a
in CE, TGR
and MCE (S
R, and MCE
SMF92-FF)
E (QP-FF) )
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Figure C3
Figure C
. Trends in
C4. Trends i
94
CE, TGR, a
in CE, TGR
and MCE (S
R, and MCE
SMF92-TL)
(QP-TL) )
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Table C1 Summary statistics of relevant efficiency scores over time (SMF92-FF)
Period CE TGR MCE
Mean S.D. Mean S.D. Mean S.D.
1996 0.887 0.070 0.821 0.105 0.727 0.103 1997 0.891 0.071 0.822 0.106 0.732 0.104 1998 0.897 0.063 0.824 0.106 0.738 0.104 1999 0.900 0.062 0.823 0.107 0.739 0.105 2000 0.904 0.063 0.828 0.106 0.747 0.106 2001 0.893 0.075 0.831 0.105 0.742 0.109 2002 0.894 0.081 0.828 0.104 0.741 0.115 2003 0.896 0.085 0.827 0.107 0.740 0.119 2004 0.899 0.093 0.829 0.118 0.745 0.113 2005 0.893 0.081 0.844 0.105 0.753 0.111 2006 0.894 0.083 0.846 0.104 0.756 0.113 2007 0.890 0.095 0.848 0.112 0.754 0.106 2008 0.880 0.087 0.850 0.099 0.747 0.106 2009 0.878 0.088 0.846 0.104 0.741 0.110 2010 0.871 0.096 0.850 0.100 0.738 0.109 96-98 0.892 0.068 0.822 0.105 0.732 0.104 99-01 0.899 0.067 0.827 0.106 0.743 0.107 02-04 0.896 0.081 0.828 0.106 0.742 0.116 05-07 0.893 0.080 0.846 0.104 0.754 0.110 08-10 0.877 0.090 0.848 0.101 0.742 0.108 Average 0.891 0.077 0.834 0.105 0.742 0.109
Note: SMF92, stochastic metafrontier with Battese and Coelli (1992) specification.
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Table C2 Summary statistics of relevant efficiency scores over time (QP-FF)
Period CE TGR MCE
Mean S.D. Mean S.D. Mean S.D.
1996 0.887 0.070 0.617 0.168 0.545 0.150 1997 0.891 0.071 0.627 0.174 0.557 0.157 1998 0.897 0.063 0.642 0.177 0.575 0.163 1999 0.900 0.062 0.660 0.170 0.594 0.159 2000 0.904 0.063 0.656 0.173 0.592 0.162 2001 0.893 0.075 0.665 0.172 0.594 0.161 2002 0.894 0.081 0.683 0.170 0.611 0.164 2003 0.896 0.085 0.686 0.171 0.616 0.166 2004 0.899 0.093 0.689 0.189 0.620 0.167 2005 0.893 0.081 0.702 0.172 0.627 0.163 2006 0.894 0.083 0.690 0.173 0.618 0.166 2007 0.890 0.095 0.672 0.191 0.598 0.164 2008 0.880 0.087 0.654 0.177 0.575 0.165 2009 0.878 0.088 0.654 0.183 0.574 0.167 2010 0.871 0.096 0.657 0.177 0.571 0.160 96-98 0.892 0.068 0.628 0.173 0.559 0.157 99-01 0.899 0.067 0.660 0.172 0.593 0.161 02-04 0.896 0.081 0.686 0.172 0.616 0.166 05-07 0.893 0.080 0.688 0.173 0.615 0.165 08-10 0.877 0.090 0.655 0.179 0.573 0.164 Average 0.891 0.077 0.662 0.175 0.590 0.164
Note: QP,quadratic programming model.
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Table C3 Summary statistics of relevant efficiency scores over time (SMF92-TL)
Period CE TGR MCE
Mean S.D. Mean S.D. Mean S.D.
1996 0.874 0.080 0.775 0.107 0.675 0.102
1997 0.875 0.082 0.776 0.110 0.677 0.105
1998 0.878 0.077 0.777 0.109 0.681 0.106
1999 0.878 0.079 0.777 0.110 0.681 0.105
2000 0.883 0.080 0.782 0.107 0.689 0.107
2001 0.872 0.089 0.787 0.104 0.684 0.107
2002 0.874 0.092 0.783 0.105 0.683 0.112
2003 0.878 0.096 0.779 0.106 0.682 0.115
2004 0.877 0.139 0.782 0.128 0.686 0.112
2005 0.869 0.099 0.803 0.108 0.696 0.113
2006 0.871 0.099 0.805 0.109 0.699 0.114
2007 0.868 0.133 0.806 0.126 0.698 0.110
2008 0.854 0.106 0.811 0.105 0.689 0.111
2009 0.855 0.102 0.806 0.109 0.686 0.113
2010 0.849 0.105 0.807 0.105 0.683 0.109
96-98 0.875 0.080 0.776 0.109 0.678 0.104
99-01 0.877 0.082 0.782 0.108 0.685 0.106
02-04 0.876 0.092 0.781 0.105 0.684 0.113
05-07 0.869 0.097 0.804 0.108 0.697 0.112
08-10 0.853 0.104 0.808 0.107 0.686 0.111 Average 0.871 0.091 0.790 0.108 0.686 0.109
Note: SMF92, stochastic metafrontier with Battese and Coelli (1992) specification.
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Table C4 Summary statistics of relevant efficiency scores over time (QP-TL)
Period CE TGR MCE
Mean S.D. Mean S.D. Mean S.D.
1996 0.874 0.080 0.572 0.152 0.498 0.135
1997 0.875 0.082 0.585 0.161 0.510 0.142
1998 0.878 0.077 0.592 0.167 0.519 0.148
1999 0.878 0.079 0.615 0.160 0.538 0.140
2000 0.883 0.080 0.610 0.164 0.536 0.145
2001 0.872 0.089 0.626 0.163 0.544 0.146
2002 0.874 0.092 0.641 0.157 0.560 0.147
2003 0.878 0.096 0.643 0.164 0.564 0.153
2004 0.877 0.195 0.656 0.187 0.576 0.152
2005 0.869 0.099 0.677 0.161 0.587 0.149
2006 0.871 0.099 0.662 0.168 0.576 0.156
2007 0.868 0.200 0.644 0.192 0.558 0.151
2008 0.854 0.106 0.638 0.164 0.542 0.149
2009 0.855 0.102 0.651 0.180 0.554 0.161
2010 0.849 0.105 0.656 0.172 0.555 0.152
96-98 0.875 0.080 0.583 0.160 0.509 0.142
99-01 0.877 0.082 0.617 0.163 0.539 0.144
02-04 0.876 0.092 0.647 0.161 0.567 0.151
05-07 0.869 0.097 0.662 0.166 0.574 0.152
08-10 0.853 0.104 0.648 0.172 0.550 0.154 Average 0.871 0.091 0.629 0.167 0.546 0.150
Note: QP, quadratic programming model.
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App pendix D
Figur
Figure
e D1. Trend
e D2. Trend
ds in CE, TG
s in CE, TG
99
GR, and MC
GR, and MC
CE by size c
CE by ROA
class (SMF9
class (SMF
92-FF)
F92-FF)
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Figure
Figu
e D3. Trend
ure D4. Tre
ds in CE, TG
ends in CE,
100
GR, and MC
TGR, and M
CE by ETA
MCE by siz
class (SMF
ze class (QP
F92-FF)
P-FF)
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Figu
Figu
ure D5. Tren
ure D6. Tre
nds in CE, T
nds in CE, T
101
TGR, and M
TGR, and M
MCE by RO
MCE by ETA
OA class (QP
A class (QP P-FF)
P-FF)
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Figure
Figure
e D7. Trend
e D8. Trend
ds in CE, TG
s in CE, TG
102
GR, and MC
GR, and MC
CE by size c
CE by ROA
class (SMF9
class (SMF
92-TL)
F92-TL)
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Figure
Figu
e D9. Trend
ure D10. Tre
ds in CE, TG
ends in CE,
103
GR, and MC
TGR, and
CE by ETA
MCE by siz
class (SMF
ze class (QP
F92-TL)
P-TL)
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Figur
Figu
re D11. Tre
re D12. Tre
nds in CE, T
ends in CE,
104
TGR, and M
TGR, and M
MCE by RO
MCE by ET
OA class (QP
TA class (QP P-TL)
P-TL)
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Table D1 Summary statistics of relevant efficiency scores by asset sizes, return on
assets and the ratio of equity to total assets (SMF92)Class Obs. CE TGR MCE
Panel A. Total Assets
SIZE1 below 100 758 0.90208 0.85441 0.76963
SIZE2 100-200 1,100 0.89761 0.86174 0.77316
SIZE3 200-400 1,368 0.89494 0.85552 0.76529
SIZE4 400-800 1,403 0.89718 0.84892 0.76135
SIZE5 800-2,000 1,700 0.89564 0.84050 0.75207
SIZE6 2,000-5,000 1,179 0.88448 0.82319 0.72671 SIZE7 5,000-15,000 793 0.87694 0.78166 0.68281 SIZE8 above 15,000 888 0.87490 0.77242 0.67181
Panel B. ROA (%)
ROA1 below 0 879 0.86279 0.84296 0.72624
ROA2 0-0.15 870 0.88802 0.81757 0.72431
ROA3 0.15-0.3 1,148 0.89121 0.79648 0.70883
ROA4 0.3-0.5 1,581 0.89828 0.80957 0.72619
ROA5 0.5-1 2,029 0.88942 0.83909 0.74507
ETA2 3-4 720 0.88546 0.77337 0.68289
ETA3 4-6 1,539 0.88866 0.81575 0.72358
ETA4 6-8 1,585 0.89083 0.82777 0.73603
ETA5 8-12 1,795 0.88451 0.84815 0.74976
ETA6 12-20 1,514 0.89170 0.88169 0.78530
ETA7 20-30 591 0.90662 0.88209 0.80007
ETA8 above 30 641 0.93018 0.86681 0.80562
Notes: SMF92, stochastic metafrontier with Battese and Coelli (1992) specification. The values of total assets are measured in millions of US dollars to save space.
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Table D2 Summary statistics of relevant efficiency scores by asset sizes, return on
assets and the ratio of equity to total assets (QP)Class Obs. CE TGR MCE
Panel A. Total Assets
SIZE1 below 100 758 0.90208 0.59463 0.53523
SIZE2 100-200 1,100 0.89761 0.67565 0.60661
SIZE3 200-400 1,368 0.89494 0.69265 0.62070
SIZE4 400-800 1,403 0.89718 0.70076 0.62935
SIZE5 800-2,000 1,700 0.89564 0.68779 0.61614
SIZE6 2,000-5,000 1,179 0.88448 0.66445 0.58630 SIZE7 5,000-15,000 793 0.87694 0.61247 0.53432 SIZE8 above 15,000 888 0.87490 0.58767 0.51188
Panel B. ROA (%)
ROA1 below 0 879 0.86279 0.64634 0.55684
ROA2 0-0.15 870 0.88802 0.62631 0.55480
ROA3 0.15-0.3 1,148 0.89121 0.62665 0.55806
ROA4 0.3-0.5 1,581 0.89828 0.64239 0.57647
ROA5 0.5-1 2,029 0.88942 0.68551 0.60908
ETA2 3-4 720 0.88546 0.61407 0.54386
ETA3 4-6 1,539 0.88866 0.66274 0.58823
ETA4 6-8 1,585 0.89083 0.67896 0.60367
ETA5 8-12 1,795 0.88451 0.69052 0.61134
ETA6 12-20 1,514 0.89170 0.70450 0.62816
ETA7 20-30 591 0.90662 0.68606 0.62312
ETA8 above 30 641 0.93018 0.62752 0.58414
Notes: QP, quadratic programming model. The values of total assets are measured in millions of US dollars to save space.