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Comment on ‘Operational efficiency in banking: An international perspective’
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Journal of Banking & Finance 21 (1997) 1325 1329
Journalof BANKING & FINANCE
Comment on 'Operational efficiency in banking: An international perspective' Robert DeYoung ~ Office q['the Controller of the Currency, 250 E. Street S. W., Mail Stop 6-5, Washington, DC 20219. USA Received 30 August 1996; accepted 17 March 1997
Abstract
This short comment reinterprets some of the results found by Linda Allen and Anoop Rai in their paper, 'Operational efficiency in banking: An international perspective' (Journal of Banking and Finance 20, 1996, 655-672). J E L classification: G21
Keywords: X-inefficiency; Stochastic cost frontier; Distribution-free model
1. Introduction
In their paper, 'Operational efficiency in banking: An international perspective' Allen and Rai (1996) estimate scale economies, scope economies, and Xinefficiency for an international data set of banks. The authors draw two main conclusions from their results. First, they conclude that 'the prevalence of input X-inefficiencies far outweighs that of output inefficiencies (as measured by economies or diseconomies of scale or scope)'. This finding is a useful contribution to the literature, as it replicates results found until now only in country-specific cost functions. Second, the authors conclude that ~the
I Tel.: +1 202 874 5427; fax: +1 202 874 5394; e-mail: [email protected]. S0378-4266197l$17.00 © 1997 Elsevier Science B.V. All rights reserved.
PIIS0378-4266(97)00028-9
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R. DeYoung l Journal of Banking & Finance 21 (1997) 1325-1329
distribution-free model overestimates the magnitude of X-inefficiencies relative to the stochastic cost frontier approach'. This second conclusion, however, is based on a faulty interpretation of the X-inefficiency estimates from these two approaches. This note provides a more correct interpretation, and shows that the alternative methods employed by Allen and Rai produce quite similar estimates of cost efficiency for some groups of banks.
2. Cost function The authors estimate a total cost function using a combined time series (1988-1992) cross-section (194 banks in 15 countries) data set. The cost function has the following multiplicative form, simplified here to the single inputsingle output case without loss of generality TC = c~op~'y~uv,
(1)
where TC is total expenses, p the price of the input, y the quantity of the output, u a cost inefficiency term, v is a r a n d o m error term, and the subscripts denoting time and bank have been suppressed. 2 Before estimating Eq. (1), the authors take the natural log of both sides and impose a standard translog functional form In TC = ~0 + ~ l n p + + In u + In v.
fll
l n y + ~2 l n p l n p + f12 l n y l n y + 6 l n p l n y (2)
The authors use two separate techniques to estimate the inefficiency term In u: the stochastic cost frontier (SCF) approach (see Jondrow et al., 1982) and the distribution-free (DF) approach (see Berger, 1993). The SCF approach assumes that the In u term is positive for all banks and follows a half-normal distribution, while the D F approach assumes that the In u terms are constant across time for each bank, can be either positive or negative, and are not constrained to follow any particular distribution.
3. Distribution-free approach In this framework, an economically meaningful interpretation of cost inefficiency u requires the raw estimates of In u to be expressed relative to some familiar benchmark. F o r the D F approach, the estimated inefficiency of each
2 In the original article, the authors do not state explicitlythat the two estimates of inefficiency are generated from the same functional form. The approach they take is consistent with this, however, and at any rate using different functional forms would bias any ex post comparisons of the two sets of inefficiencyestimates.
R. DeYoung I Journal of Banking & Finance 21 (1997) 1325-1329
1327
bank is typically compared to the estimated inefficiency of the most cost efficient bank in the sample EFFi = exp(min[ln ut] - In uit) - min[ut]
(3)
nit
Using Allen and Rai's notation, in b/it is the inefficiency term for bank i and min[ln ut] is the smallest value of In uit in the sample. 3 Note that EFF, is a measure of relative cost efficiency, because it equals a maximum of 1.0 for the most efficient bank in the sample (i.e., uit = min[ut]) and is less than 1.0 for all of the other banks in the sample (i.e., uit > min[ut]). Although the authors perform exactly the transformation shown in Eq. (3), they mistakenly interpret the result as being a measure of relative cost inefficiency. This misinterpretation is likely to be the reason for the 'improbable' D F inefficiency estimates that 'range in the vicinity of two-thirds of reported costs' and 'are more than triple' the reported SCF estimates (Allen and Rai, 1996, p. 664). Instead, the authors assert that the large D F inefficiency estimates 'emanate from the model's assumptions that random errors tend to cancel out over time,' and that this tendency is exacerbated by the 'relatively short' 5-year time series component of their data set (Allen and Rai, 1996, p. 664). This assertion may have merit, and it can be directly tested using a method proposed by DeYoung (1997). 4 A rough-and-ready way to undo these misinterpreted results, which Allen and Rai report in their Table 3B, is simply to take their complement INEFFi = 1 - EFFi. This turns the reported numbers into true measures of cost inefficiency. I N E F F has a minimum value of zero for the most efficient bank in the sample and increases with inefficiency (upper bound = 1.0) for the other banks in the sample. Using this correction, it is straightforward to show that the D F averages reported in Table 3B dramatically overstate estimated cost inefficiency. For example, the reported average inefficiency for large banks in Allen and Rai's 'separated' banking nations is 0.673, or about 67% inefficiency, while the corrected average is approximately 1 - 0.673 = 0.327, or about 33°/,+ inefficiency.
As described by the authors, the bank-specific In b/it terms are actually intertemporal averages of the five annual regression residuals (In u + In v) for each bank. The DF assumption of constant intertemporal bank-specific inefficiency implies that this averaging causes the random error components In v to attenuate toward zero, leaving only inefficiency In u in the averaged residuals. 4 There is another potential cause for the large D F inefficiency estimates. In order to minimize the impact of very large random errors, D F practitioners typically truncate the distribution of the In t/it at the 5th and 95th percentiles prior to executing the transformation in Eq. (3). Selecting appropriate truncation levels can significantly affect the magnitudes of the inefficiency estimates (see Berger, 1993; DeYoung and Nolle, 1996), and not truncating at all will virtually guarantee very large estimates of inefficiency. The original article does not indicate the percentiles at which the DF estimates were truncated.
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R. DeYoung I Journal of Banking & Finance 21 (1997) 1325-1329
4. Stochastic cost frontier approach Allen and Rai report the raw SCF estimates of In u in Table 3A, and interpret them as measures of percent cost inefficiency. This is a c o m m o n practice, and is a fairly accurate practice for small values of In u. Large values of In u, however, can understate percent cost inefficiency substantially. Recall that, in the original multiplicative cost function Eq. (1), the cost inefficiency term u is equal to 1 + percent cost inefficiency. Using the identity u = exp(ln u) and rearranging yields the following equation, which can be used to transform the raw SCF estimates of In u into corrected measures of percent cost inefficiency I N E F F i = exp(ln u) - 1.
(4)
I N E F F has a minimum value of zero for the most efficient bank in the sample and increases with inefficiency (no upper bound) for the other banks in the sample. Using this correction, it is easy to show that the SCF averages reported in Table 3A slightly understate estimated cost inefficiency. For example, average inefficiency for large banks in 'separated' banking nations is reported as 0.275, or about 28% inefficiency, while the corrected average is 1 - exp(0.275) = 0.317, or about 32% inefficiency. Note that this is almost exactly the same level of average cost inefficiency generated by the (corrected) D F method for this set of banks.
5. Conclusion The above analysis suggests that Allen and Rai should not reject the D F approach out-of-hand in favor of the SCF approach. Still, direct comparisons between the corrected D F and SCF estimates derived above should be made with caution, because the estimated In u underlying these estimates are based on two different sets of distributional assumptions. A rank order correlation, or some similar test, could be used to reveal whether the D F and SCF approaches yield similar estimates of relative bank efficiency. Finally, it would be interesting to see whether the corrected D F and SCF estimates provide robust or contradictory results in the country-by-country and regression analyses that the authors report in Tables 4 and 5.
References Allen, L., Rai, A., 1996. Operational efficiencyin banking: An international perspective.Journal of Banking and Finance 20, 655-672. Berger, A.N., 1993. Distribution-freeestimates of efficiencyin the US banking industry and tests of the standard distributional assumptions. Journal of Productivity Analysis 4, 261 292.
R. DeYoung I Journal of Banking & Finance 21 (1997) 1325 1329
1329
DeYoung, R., 1997. A diagnostic test for the distribution-free efficiency estimator: An example using US commercial bank data. European Journal of Operational Research 98 (2), 243 249. DeYoung, R., Nolle, D.E., 1996. Foreign-owned banks in the US: Buying market share or earning it? Journal of Money, Credit, and Banking 28, 622 636. Jondrow, J., Lovell, C.A.K., Materov, I., Schmidt, P., 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics 19, 233 238.
Journal of Banking & Finance 21 (1997) 1325 1329
Journalof BANKING & FINANCE
Comment on 'Operational efficiency in banking: An international perspective' Robert DeYoung ~ Office q['the Controller of the Currency, 250 E. Street S. W., Mail Stop 6-5, Washington, DC 20219. USA Received 30 August 1996; accepted 17 March 1997
Abstract
This short comment reinterprets some of the results found by Linda Allen and Anoop Rai in their paper, 'Operational efficiency in banking: An international perspective' (Journal of Banking and Finance 20, 1996, 655-672). J E L classification: G21
Keywords: X-inefficiency; Stochastic cost frontier; Distribution-free model
1. Introduction
In their paper, 'Operational efficiency in banking: An international perspective' Allen and Rai (1996) estimate scale economies, scope economies, and Xinefficiency for an international data set of banks. The authors draw two main conclusions from their results. First, they conclude that 'the prevalence of input X-inefficiencies far outweighs that of output inefficiencies (as measured by economies or diseconomies of scale or scope)'. This finding is a useful contribution to the literature, as it replicates results found until now only in country-specific cost functions. Second, the authors conclude that ~the
I Tel.: +1 202 874 5427; fax: +1 202 874 5394; e-mail: [email protected]. S0378-4266197l$17.00 © 1997 Elsevier Science B.V. All rights reserved.
PIIS0378-4266(97)00028-9
1326
R. DeYoung l Journal of Banking & Finance 21 (1997) 1325-1329
distribution-free model overestimates the magnitude of X-inefficiencies relative to the stochastic cost frontier approach'. This second conclusion, however, is based on a faulty interpretation of the X-inefficiency estimates from these two approaches. This note provides a more correct interpretation, and shows that the alternative methods employed by Allen and Rai produce quite similar estimates of cost efficiency for some groups of banks.
2. Cost function The authors estimate a total cost function using a combined time series (1988-1992) cross-section (194 banks in 15 countries) data set. The cost function has the following multiplicative form, simplified here to the single inputsingle output case without loss of generality TC = c~op~'y~uv,
(1)
where TC is total expenses, p the price of the input, y the quantity of the output, u a cost inefficiency term, v is a r a n d o m error term, and the subscripts denoting time and bank have been suppressed. 2 Before estimating Eq. (1), the authors take the natural log of both sides and impose a standard translog functional form In TC = ~0 + ~ l n p + + In u + In v.
fll
l n y + ~2 l n p l n p + f12 l n y l n y + 6 l n p l n y (2)
The authors use two separate techniques to estimate the inefficiency term In u: the stochastic cost frontier (SCF) approach (see Jondrow et al., 1982) and the distribution-free (DF) approach (see Berger, 1993). The SCF approach assumes that the In u term is positive for all banks and follows a half-normal distribution, while the D F approach assumes that the In u terms are constant across time for each bank, can be either positive or negative, and are not constrained to follow any particular distribution.
3. Distribution-free approach In this framework, an economically meaningful interpretation of cost inefficiency u requires the raw estimates of In u to be expressed relative to some familiar benchmark. F o r the D F approach, the estimated inefficiency of each
2 In the original article, the authors do not state explicitlythat the two estimates of inefficiency are generated from the same functional form. The approach they take is consistent with this, however, and at any rate using different functional forms would bias any ex post comparisons of the two sets of inefficiencyestimates.
R. DeYoung I Journal of Banking & Finance 21 (1997) 1325-1329
1327
bank is typically compared to the estimated inefficiency of the most cost efficient bank in the sample EFFi = exp(min[ln ut] - In uit) - min[ut]
(3)
nit
Using Allen and Rai's notation, in b/it is the inefficiency term for bank i and min[ln ut] is the smallest value of In uit in the sample. 3 Note that EFF, is a measure of relative cost efficiency, because it equals a maximum of 1.0 for the most efficient bank in the sample (i.e., uit = min[ut]) and is less than 1.0 for all of the other banks in the sample (i.e., uit > min[ut]). Although the authors perform exactly the transformation shown in Eq. (3), they mistakenly interpret the result as being a measure of relative cost inefficiency. This misinterpretation is likely to be the reason for the 'improbable' D F inefficiency estimates that 'range in the vicinity of two-thirds of reported costs' and 'are more than triple' the reported SCF estimates (Allen and Rai, 1996, p. 664). Instead, the authors assert that the large D F inefficiency estimates 'emanate from the model's assumptions that random errors tend to cancel out over time,' and that this tendency is exacerbated by the 'relatively short' 5-year time series component of their data set (Allen and Rai, 1996, p. 664). This assertion may have merit, and it can be directly tested using a method proposed by DeYoung (1997). 4 A rough-and-ready way to undo these misinterpreted results, which Allen and Rai report in their Table 3B, is simply to take their complement INEFFi = 1 - EFFi. This turns the reported numbers into true measures of cost inefficiency. I N E F F has a minimum value of zero for the most efficient bank in the sample and increases with inefficiency (upper bound = 1.0) for the other banks in the sample. Using this correction, it is straightforward to show that the D F averages reported in Table 3B dramatically overstate estimated cost inefficiency. For example, the reported average inefficiency for large banks in Allen and Rai's 'separated' banking nations is 0.673, or about 67% inefficiency, while the corrected average is approximately 1 - 0.673 = 0.327, or about 33°/,+ inefficiency.
As described by the authors, the bank-specific In b/it terms are actually intertemporal averages of the five annual regression residuals (In u + In v) for each bank. The DF assumption of constant intertemporal bank-specific inefficiency implies that this averaging causes the random error components In v to attenuate toward zero, leaving only inefficiency In u in the averaged residuals. 4 There is another potential cause for the large D F inefficiency estimates. In order to minimize the impact of very large random errors, D F practitioners typically truncate the distribution of the In t/it at the 5th and 95th percentiles prior to executing the transformation in Eq. (3). Selecting appropriate truncation levels can significantly affect the magnitudes of the inefficiency estimates (see Berger, 1993; DeYoung and Nolle, 1996), and not truncating at all will virtually guarantee very large estimates of inefficiency. The original article does not indicate the percentiles at which the DF estimates were truncated.
1328
R. DeYoung I Journal of Banking & Finance 21 (1997) 1325-1329
4. Stochastic cost frontier approach Allen and Rai report the raw SCF estimates of In u in Table 3A, and interpret them as measures of percent cost inefficiency. This is a c o m m o n practice, and is a fairly accurate practice for small values of In u. Large values of In u, however, can understate percent cost inefficiency substantially. Recall that, in the original multiplicative cost function Eq. (1), the cost inefficiency term u is equal to 1 + percent cost inefficiency. Using the identity u = exp(ln u) and rearranging yields the following equation, which can be used to transform the raw SCF estimates of In u into corrected measures of percent cost inefficiency I N E F F i = exp(ln u) - 1.
(4)
I N E F F has a minimum value of zero for the most efficient bank in the sample and increases with inefficiency (no upper bound) for the other banks in the sample. Using this correction, it is easy to show that the SCF averages reported in Table 3A slightly understate estimated cost inefficiency. For example, average inefficiency for large banks in 'separated' banking nations is reported as 0.275, or about 28% inefficiency, while the corrected average is 1 - exp(0.275) = 0.317, or about 32% inefficiency. Note that this is almost exactly the same level of average cost inefficiency generated by the (corrected) D F method for this set of banks.
5. Conclusion The above analysis suggests that Allen and Rai should not reject the D F approach out-of-hand in favor of the SCF approach. Still, direct comparisons between the corrected D F and SCF estimates derived above should be made with caution, because the estimated In u underlying these estimates are based on two different sets of distributional assumptions. A rank order correlation, or some similar test, could be used to reveal whether the D F and SCF approaches yield similar estimates of relative bank efficiency. Finally, it would be interesting to see whether the corrected D F and SCF estimates provide robust or contradictory results in the country-by-country and regression analyses that the authors report in Tables 4 and 5.
References Allen, L., Rai, A., 1996. Operational efficiencyin banking: An international perspective.Journal of Banking and Finance 20, 655-672. Berger, A.N., 1993. Distribution-freeestimates of efficiencyin the US banking industry and tests of the standard distributional assumptions. Journal of Productivity Analysis 4, 261 292.
R. DeYoung I Journal of Banking & Finance 21 (1997) 1325 1329
1329
DeYoung, R., 1997. A diagnostic test for the distribution-free efficiency estimator: An example using US commercial bank data. European Journal of Operational Research 98 (2), 243 249. DeYoung, R., Nolle, D.E., 1996. Foreign-owned banks in the US: Buying market share or earning it? Journal of Money, Credit, and Banking 28, 622 636. Jondrow, J., Lovell, C.A.K., Materov, I., Schmidt, P., 1982. On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics 19, 233 238.