Int. J. Production Economics 76 (2002) 177–188
Bias in Malmquist index and cost function productivity
measurement in banking
Ana Lozano-Vivasa,*, David B. Humphreyb
a
Department of Economics, University of Malaga, Plza. El Ejido, 29013 Malaga, Spain
b
Department of Finance, Florida State University, Tallahassee, FL, USA
Received 30 August 2000; accepted 26 March 2001
Abstract
This paper illustrates how many prior studies of productivity growth in the banking industry have been overstated.
This occurs with both DEA (Malmquist index) and parametric (stochastic cost frontier) productivity measurement
approaches. The problem is not due to the technique used but in how it is applied. It is easiest to see in the banking
industry due the nature of the data available. The bias is eliminated when all outputs and inputs are included in the
analysis, ensuring that the balance sheet restriction is met. Although simple in concept, this problem and its solution
have so far been neglected in the literature. r 2002 Elsevier Science B.V. All rights reserved.
Keywords: Productivity; Malmquist index; Frontier cost; Banking industry
1. Introduction
Productivity measurement has a long history,
ranging from changes in output per unit of labor
input to more complex, but more complete,
measures of total factor productivity (TFP). In
this transition, growth accounting measures of
TFP have given way to nonparametric (linear
programming) Malmquist index and parametric
(regression-based) stochastic frontier cost function
TFP estimates. However, many published studies
using the Malmquist index in the banking area
appear to have significantly overstated estimates of
productivity growth. The bias is not due to the
technique used but rather in how it is applied.
*Corresponding author. Tel.: +34-952-131256; fax: +34952-131299.
E-mail addresses: avivas@uma.es (A. Lozano-Vivas),
dhumphr@garnet.acns.fsu.edu (D.B. Humphrey).
This problem in recent productivity studies is
most easily seen in applications of the Malmquist
index to banking industry data but exists in
stochastic frontier cost function applications as
well (including our own work). Fortunately, the
bias is simple to correct and it can be measured,
reduced, or eliminated in future productivity
studies. The problem is not new but has been
consistently overlooked in published work. Our
guess is that it has persisted because no one has
shown how large it can be. While other productivity measurement difficulties remain, the one we
identify is amenable to correction.
In what follows, we explain how the bias arises
in Section 2. Its importance is illustrated by
showing that in many studies, it is the source of
the main part of measured productivity growth.
Due to the special nature of banking data and the
Malmquist index itself, it is relatively simple to
0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 1 ) 0 0 1 6 2 - 1
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
approximate the magnitude of the bias (hence our
focus on this measure). In Section 3, we show that
use of a stochastic frontier cost function to
estimate banking productivity can lead to similar
difficulties. The problem associated with both
techniques is illustrated by applying the Malmquist index and a cost function to the same set of
banking data. For completeness, although it is an
inferior measure, we also show how the growth
accounting approach to productivity measurement
is affected. Our conclusions are in Section 4 and an
appendix contains details of prior studies.
Table 1
Example of included and excluded assets and liabilities in
banking productivity studies
Factor inputs: Labor and physical capital
Assets or outputs
Liabilities or inputs
Included:
Business loans
Consumer loans
Securities
Demand deposits
Included:
Savings deposits
Time deposits
Certificates of deposits (CDs)
Excluded:
Funds sold
Vault cash
Other earning assets
Off-balance sheet
activities
Excluded:
Other liabilities for borrowed money
Short-term purchased funds
Subordinated debt
Equity
2. Bias in productivity measurement
2.1. Where the bias comes from
Single factor measures of productivity are
inferior to total factor measures because they do
not include all the inputs that affect output. In most
industries, it is easy to determine and include all the
inputs that significantly affect output as well as to
include all the outputs being produced. However, in
banking, productivity researchers use labor and
capital inputs but selectively choose only certain
inputs and outputs from balance sheet liabilities
and assets and neglect others. Whatever the choice
of banking balance sheet inputs and outputs, it is
common to measure them in terms of their (real or
nominal) value, along with the number of labor
inputs and the value of physical capital. This is
partly because banks are viewed as intermediating
stocks of the value of deposits into stocks of the
value of loans or security investments. But mostly it
is because data on the number of deposit and loan
accounts (alternative stock measures) or the number of deposit account debits and credits and the
number of new loans made (alternative flow
measures of input and output) are rarely available.
Table 1 illustrates a typical selection of asset
outputs and liability inputs included in productivity models. Excluded assets and liabilities are also
shown.1 These choices are justified by pointing to
similar selections made by previous researchers
1
Table 1, although representative, is only illustrative since
not all productivity studies have chosen to include/exclude the
same balance sheet assets and liabilities.
investigating bank cost/scale efficiencies or by
deciding to include only the ‘‘important’’ balance
sheet categories.
When studying banking cost efficiency or scale
effects, variations in loans have a much larger effect
on costs than do variations in interbank funds sold,
vault cash, or off-balance sheet activities.2 Consequently, loans are included as a banking output
while the other three asset categories are usually
excluded. However, loans, security holdings, funds
sold, and off-balance sheet activities all generate
significant revenues and therefore should be
included as a banking output in revenue or profit
analyses. Our key point is that, unlike cost or profit
analysis, there is no such ‘‘differential importance’’
of any of these liability/asset categories when it
comes to productivity measurement. Since productivity growth should reflect the change in the ratio
of outputs to inputs – and not be biased by
concurrent changes in balance sheet composition –
then all balance sheet inputs and outputs need to
be included.3 Productivity analysis differs from
banking cost, revenue, or profit analyses in that
2
Off-balance sheet (OBS) activities include standby letters of
credit, loan commitments, interest rate derivatives, and foreign
exchange contracts.
3
Including OBS activities as an output is, in our view,
inappropriate in a cost function as these activities cost almost
nothing to produce. However, OBS activities generate substantial revenues (and risk) for a bank and, as a result, should
definitely be included in a revenue or a profit function analysis.
A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
changes in excluded balance sheet inputs or outputs
have the same effect on productivity – not a lesser
or greater effect – as those inputs and outputs
chosen to be included in the analysis.
It is well known that productivity and revenue
of a non-banking firm rises in the upswing of a
business cycle since, with excess capital capacity
and (likely) underemployed labor, output can
expand without a commensurate rise in capital
and (in this case) labor inputs. But banks are
different. There is no ‘‘excess capacity’’ among
balance sheet inputs and outputs (although there
can be for capital and labor). All liability funds
inputs find their way into assets through the
balance sheet constraint and, either directly or
indirectly, earn revenues (which is a common way
to identify outputs). In a profit maximizing bank,
funds sold should be earning the same marginal
risk-adjusted return as do loans or security
holdings but funds sold are almost always an
excluded banking output while loans are included.
Similarly, vault cash is an excluded output even
though cash holdings are an essential component
in providing payment and liquidity services
through branches and ATMs. These and other
depositor services tradeoff with interest costs that
would otherwise be needed to attract funds. In this
sense, they earn implicit revenue or a return valued
by the bank in providing a service flow to
depositors. Thus, all balance sheet assets can be
considered to be outputs since they all are
presumed to earn the same marginal risk-adjusted
return. Similarly, all liabilities are inputs since they
all are presumed to incur the same maturityadjusted interest and/or operating expense.
By excluding various liability and asset categories from the analysis, researchers largely predetermine the size and variation of their
productivity estimates over time and across sizeclasses of banks. The magnitude of this bias can be
inferred by first estimating productivity using the
set of inputs and outputs a researcher may choose
and then re-estimating using all liability and asset
categories as inputs and outputs.4
4
In the interests of tractability and parameter estimation
parsimony, only the major inputs/outputs need to be separately
specified. Those remaining can be aggregated together.
179
Take the case noted above where loans are an
included output and funds sold are an excluded
output. If loan demand expands by $500 million,
banks have a number of alternatives: they can
purchase short-term funds, they can try to attract
more deposits, or they can reduce interbank funds
sold by $500 million. The course chosen will depend
on: the interest cost of short-term purchased funds,
the interest and operating expense of increasing
deposits, and the risk-adjusted return for funds
sold. If more short-term funds are purchased (and if
these funds are an excluded input), then both loans
and measured productivity will be viewed as
expanding without a rise in inputs. If deposits are
increased, then the expansion of loans will be offset
and productivity will not change. Finally, if funds
sold are reduced and shifted into loans, loans will
again be viewed as expanding without any offsetting
rise in inputs and measured productivity will rise.
To those that loans are a more important
category of output than funds sold, we would
agree if the purpose were to measure cost efficiency
or scale economies since funds sold have a de
minimis effect on costs. To buy or sell funds all
that is needed is a small room, a few traders, some
telephones, and a direct or dial-up line to a money
market and/or other banks. However, because of
the way balance sheet inputs and outputs typically
are measured, a $500 million expansion in deposit
inputs that funds $500 million in new loans should
have the same effect on productivity as a $500
million shift in asset composition away from funds
sold to new loans. But this will not occur unless
funds sold and loans are both included in the
analysis as outputs. Similar productivity measurement problems arise when balance sheet liabilities
are excluded from the analysis.5 If researchers do
not want their productivity estimates contaminated by shifts in balance sheet composition over
time or across banks, then they need to effectively
include all liabilities and assets in their analysis.
No one would attempt to measure the productivity
5
For example, in US studies the category ‘‘other liabilities for
borrowed money’’ is often excluded as an input while deposits
are included. Changes in the excluded inputs, however, can
affect the level of loan output which, in turn, affects measured
productivity. If excluded inputs were included in the analysis,
productivity would be measured more accurately.
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
of petroleum refining operations without including
the outputs of gasoline, heating oil, motor oil,
grease, paraffin, etc., together. Otherwise, changes
in output mix due to changes in demand and prices
would yield a biased measure of productivity. Just
as in banking, petroleum refining productivity
growth is based on changes in output per unit of
labor and capital input, not in the amount of raw
petroleum input or in output mix.
2.2. Magnitude of the bias
The Malmquist index approach to productivity
measurement is, in its essentials, an overall index
measure of how the balance sheet outputs have
risen or fallen relative to the
P identifiedPfactor and
balance sheet inputs. Let
Ait and
Ljt represent the summed values of the total balance sheet
assets and total liabilities actually used in a
banking productivity study in year t: Because of
the balance sheet constraint, we know that the
ratio of all assets to all liabilities (plus the value of
equity capital) in year t; TAt =TLt ; has to be equal
to 1.00 in all years (time-series) for all banks
(cross-section). But inPall banking
productivity
P
studies we have seen,
Ait = Ljt a1:00; indicating that some assets or some liabilities (or both)
have been P
excluded
P from the analysis. Consequently, if
Ait = Ljt varies over time or across
banks (or both), this variation will make it appear
that productivity is higher or lower than what it
actually is.6 The balance sheet restriction can be
6
In many cost function applications to banking, the productivity bias described here is less obvious but exists nonetheless. It
is less obvious because cost is the dependent variable and there is
precious little information on either the level or the variation in
the various assets and liability cost shares (not balance sheet
shares). As a result, it is very difficult for researchers to determine
the size and stability of the cost shares of the excluded Ai and Lj
variables. Some cost function studies largely circumvent this
problem by defining the dependent variable, total cost, to be very
close to the sum of only the cost of the included asset and
liability categories in the model. This is not a complete solution
because total cost will include all operating expenses plus the
interest cost on all included liability categories. While interest
costs can be associated with various liability categories,
operating costs are not similarly allocated in the reported data.
As a result, if any excluded asset or liability category contributes
significantly to operating cost, then its exclusion from the model
can bias the productivity results.
P
P
met by either (a) having Ai and Lj include the
value of all assets and the value of all liabilities or
(b) so choosing Ai and Lj such that the values or
the value shares of the excluded assets and
liabilities are equal to each other and constant
over time. In practice, (a) is probably the easiest
to do. This is especially true where liability and
asset mix is significantly affected by the level of
interest rates and the stage of the business cycle so
balance sheet values and/or value shares are not
constant over time. As the business cycle varies
over time and across regions, both time-series
and cross-section productivity estimates can be
affected.
We can approximate the degree of bias by
comparing the Malmquist index measure of
productivity between (say) years 1 P
and 2 with
P the
corresponding
asset/liability
index
ð
A
=
Lj2 Þ=
i2
P
P
ð Ai1 = Lj1 Þ: This comparison across years or
between countries is shown in panel A of Table 2
for productivity studies where deposits are only
treated as an input.7 Panel B of Table 2b makes the
same comparison for studies where some deposits
are treated as an output.8 If there were no bias,
then the value in column 4 representing the ratio of
assets to liabilities used as outputs and non-factor
inputs in each study would always be equal or
close to 1.00. However, comparing the values in
columns 3 and 4 of the table, it is clear that in
almost all cases most of the Malmquist index
productivity increase or decrease can be attributed
not to productivity growth but to changes in the
asset/liability index. A first approximation to a
more accurate productivity estimate is obtained by
subtracting column 4 from 3. This result is shown
in the last column of the table. As seen, this
adjustment markedly lowers the estimate of productivity growth for banking firms, sometimes
7
While there are more Malmquist index banking productivity
studies than those shown in the tables, only those papers which
reported the data needed to compute an asset/liability index
could be included.
8
When deposits are an output, their value is added to the
value of other included asset outputs. As well, to account for
the fact that deposits supply funds to purchase assets with, this
same value of deposits is also added to the value of included
liabilities.
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
Table 2
Malmquist index banking productivity studies: A comparison of productivity and asset/liability index valuesa
Author
(1)
Time period or
Countries covered
(2)
Malmquist Index
productivity
(3)
Asset/liability
index
(4)
Column (3)(4)
1.117
1.115
0.002
(5)
(A) Funding characteristic of deposits
Devaney and Weber [4]
LMDR banks
1990–1993
Non-LMDR banks
1990–1993
1.114
1.116
0.002
Pooled sample
1989–1990
1990–1991
1.314
1.025
1.237
0.973
0.077
0.052
1990–1993
1.115
1.116
0.001
Total sample
1985–1986
1986–1987
1987–1988
1988–1989
1989–1990
1985 –1990
1.013
1.018
1.056
1.043
1.046
1.288
1.012
1.023
1.026
1.038
1.014
1.120
0.001
0.005
0.030
0.005
0.032
0.168
Berg et al. [2]
Finland–Norway
Finland–Sweden
Norway–Sweden
1.120
1.630
1.460
1.033
1.089
1.053
0.087
0.541
0.407
Wheelock and Wilson [3]
1984–1985
1988–1989
1992–1993
Savings banks
1986–1987
1987–1988
1988–1989
1989–1990
1990–1991
1991–1992
1992–1993
Mean: 86/87–92/93
Commercial banks
1986–1987
1987–1988
1988–1989
1989–1990
1990–1991
1991–1992
1992–1993
Mean: 86/87–92/93
0.986
1.018
1.001
0.913
0.987
0.992
0.073
0.031
0.009
1.047
1.061
1.005
0.971
1.043
1.050
1.006
1.026
1.042
1.006
0.995
1.006
1.015
1.017
0.987
1.010
0.005
0.055
0.010
0.035
0.028
0.033
0.019
0.016
1.071
1.092
1.056
0.993
1.008
0.951
0.981
1.021
1.002
1.019
0.999
0.996
1.049
0.999
0.973
1.006
0.069
0.073
0.057
0.003
0.041
0.048
0.008
0.015
1.032
1.152
0.986
1.052
1.193
1.390
0.020
0.041
0.404
1.188
1.125
1.636
1.050
0.812
0.628
0.138
0.313
1.008
Fukayama [5]
Devaney and Weber [6]
(B) Focused on the output characteristic of deposits
Kuussaari [1]
Grifell-Tatj!e and Lovell [7]
Gilbert and Wilson [8]
a
Regional banks
1980–1985
1980–1989
1980–1994
National banks
1980–1985
1980–1989
1980–1994
See the Appendix A for more detail on the comparisons in Panels A and B of Table 2.
182
A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
even changing direction from positive productivity
to negative. The accuracy of this simple approximation is discussed further below. The appendix
contains more detail on the studies used in this
comparison.
3. Illustrating the bias with banking data
3.1. Degree of bias and changes in mean
productivity
We use actual (balanced panel) data on the
Spanish banking system over 1986–1991 to illustrate the importance of the productivity bias.
Three measures of productivity are computed
using (1) a symmetric Malmquist index, (2) a
stochastic frontier (composed error) cost function
and (3) a growth accounting measure.9 As we
show, all three approaches can yieldPbiasedPresults
when
the
P
P asset/liability index ð Ai2 = Lj2 Þ=
ð Ai1 = Lj1 Þa1:00:
To illustrate how the exclusion of various assets
and liabilities can affect the productivity results,
we start by defining our banking inputs and
outputs so that on average 70% of the value of
the aggregate balance sheet of Spanish banks is
covered. This would focus on loan outputs and
deposit and factor inputs. We then successively
expand the coverage of assets and liabilities
until on average 79%, 84%, 98% and finally
100% are covered (which is the no bias situation).
The input and output composition for our five
successive levels of coverage, yielding five panel
data sets, is shown in Table 3, along with the
average percent of the balance sheet value of
assets and liabilities that are covered. For each
of the five data sets, we estimate productivity
using the Malmquist index, a stochastic
frontier cost function and a growth accounting
model. To be consistent, all three methods use
the same number and specification of banking
inputs and outputs at each level of asset/liability
9
These three approaches to productivity measurement are
well known. However, they are outlined in our working paper
(available on request).
coverage.10 A corresponding asset/liability index is
also computed to gauge the degree of bias at each
coverage level. The top part of Table 3 shows the
coverage level for all methods and the non-factor
inputs/outputs for the Malmquist index approach.
The bottom part of the table shows the corresponding input price definitions for the cost
function and the cost function specifications used.
Our productivity results averaged over 1986–
1991, are shown in Fig. 1. Over the five levels of
asset/liability coverage, the growth accounting
model provides the highest estimate of Spanish
banking productivity growth. This is followed by
the stochastic frontier cost function with the
Malmquist index giving the lowest average estimate. All the three methods of estimating productivity growth generate lower estimates as the
degree of bias, indicated by the size of the asset/
liability index, falls with greater coverage of assets
and liabilities. When the asset/liability index
equals 1.00 (coverage level 5), then the potential
bias is removed and the Malmquist index suggests
that annual average productivity change was
0.8% over 6-year period. In contrast, the cost
function suggests that productivity change averaged 1.0% a year while the growth accounting
measure (which is inferior to the other two)
estimates productivity change at 0.2% a year.
Regardless of which one of the three methods is
chosen, we see that it can be seriously biased – in
this case overestimated – when some inputs and
outputs are excluded from the analysis. When 30%
of the value of banking inputs/outputs are
excluded from the analysis (coverage level 1 which
focuses only on loans and deposits), estimated
average productivity growth is 5.7% annually with
growth accounting, 5.7% with a cost function and
3.8% with the Malmquist index. When 2.0% of
the value of inputs/outputs are excluded (coverage
level 4) then the same models yield estimated
annual rates of productivity change of 0.2%,
10
For convenience and consistency with many cost function
studies, we specify deposits as an input in the cost function. Our
preferred specification [9], however, would include deposits as
an input (with an average interest rate) and as an output (to
reflect transaction and safekeeping services provided to
depositors which use the majority of capital and labor factor
inputs).
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
Table 3
Banking inputs/outputs and coverage level (Spanish banks, 1986–1991)
Coverage
level
Assets (outputs)
Liabilities (inputs)
Percent covered:
1
A1: Commercial loans
A2: Other loans+loans to other banks
L1: Demand deposits
L2: Savings+time deposits
Labor inputs
Physical capital
56
83
2
Coverage level 1+
A3: Securities
69
88
3
Coverage level 2+
A4: Other assets
Coverage level 3+
A5: Cash and reserves
Coverage level 1+
L3: Borrowed funds +deposits
from other banks
Coverage level 2+
L4: Other liabilitiesa
Coverage level 3+
L5: Equity capitalb
73
95
97
100
100
100
Assets (%)
4
5
Coverage level 4+
A6: Value of physical capital
Coverage level 4
Liabilities (%)
Input prices (P) for the cost function:
P1&2=(Interest cost of total deposits)/(L1+L2)c
P3=(Interest cost of borrowed funds+cost of deposits from other banks)/L3
P4=(Loan loss expense+other costs)/L4
PN=(Personnel expense)/(number of workers)
PK=(Materials cost+building cost+amortization)/A6
Specifications of the cost function:d
1. TC=f(A1, A2, P1&2, PN, PK)
2. TC=f(A1, A2; A3; P1&2; P3, PN, PK)
3. TC=f(A1, A2, A3, A4, P1&2, P3, P4, PN, PK)
4. TC=f(A1, A2, A3, A4, A5, P1&2, P3, P4, PN, PK)
5. TC=f(A1, A2, A3, A4, A5, A6, P1&2, P3, P4, PN, PK)
a
The value of equity capital is excluded here.
For regulatory purposes, the value of equity capital exceeds the value of physical capital.
c
Interest costs were not separately available for transaction, savings, and time deposits.
d
TC equals the sum of the separate inputs specified in each coverage level (except for capital and labor operating expenses which are
not separately allocated to the inputs used or outputs produced).
b
Fig. 1. Average productivity by level of data coverage.
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
0.4%, and 0.8%, respectively. In our example,
reducing the bias to zero results in a reduction in
estimated productivity change of 96% (growth
accounting), 118% (cost function), or 121%
(Malmquist index).11 The effects of reducing the
bias in studies of other country’s banking systems
could differ in magnitude, direction, and in terms
of which methodology is most affected.
3.2. Productivity by year and its distribution across
banks
Table 4 shows productivity for all Spanish
banks for each year.12 For simplicity, only coverage levels 1 and 4 are shown.13 There is substantial
year-to-year variation in productivity estimates
when coverage is incomplete – coverage level 1. As
well, for 1987–1988, estimated annual productivity
change ranges from 7.2% (Malmquist Index) to
9.9% (growth accounting). Since the corresponding asset/liability index for that year is 1.076, this
suggests that perhaps as much as 7.6% points of
the productivity growth for 1987–1988 is the result
of not including all assets and liabilities. As well,
since the asset/liability index is not constant, the
year-to-year bias is also not constant and ranges
from 1.026 to 1.076 (with a mean of 1.047).
The level and variation of the productivity
estimates are greatly reduced when coverage is
extended to level 4 (where only 2% of the average
value of assets and liabilities have been excluded).
Here, we show two ways in which the coverage
level can be expanded. The usual way would be to
add new variables to the model which represent
the asset/liability categories that have been excluded. This is the approach outlined in Table 3,
applied in all the figures, and shown in the middle
of Table 4 under the heading ‘‘Adding Variables to
Expand Coverage’’. As seen, the bias is very small
11
Reductions larger than 100% are possible since positive
values at coverage level 1 turn into small negative values at
coverage level 5.
12
The mean values from this exercise were used to plot
changes in average productivity as the coverage level changed in
Fig. 1.
13
Although coverage levels 4 and 5 give very similar
productivity results, coverage level 4 excludes physical capital
as an output since it is already included as a factor input.
at coverage level 4 since the corresponding asset/
liability index is very close to 1.000 – the no bias
situation of coverage level 5. The productivity
estimates for the Malmquist index, the stochastic
frontier cost function, and the growth accounting
models all show a year-to-year variation that
fluctuates between small positive and small negative values. The average result (as indicated
already in Fig. 1) is that annual productivity
growth for Spanish banks over 1986–1991 was
very small – a small decrease or a small increase.
Overall, a conclusion of little or no change in
productivity would be reasonable.
A second approach to show the effects of
expanding asset/liability coverage would be to
simply redefine the first set of new variables added
to the model shown in Table 3 (giving coverage
level 2) to include the additional variables as
coverage is expanded. This reduces the number of
new separate variables added to the model and so
minimizes the tendency of the Malmquist index to
find 100% efficiency merely by specifying additional constraints (variables). The results of this
approach are shown at the bottom of Table 4
under the heading ‘‘Aggregating to Expand Coverage’’. While the year-to-year productivity change
estimates here for coverage level 4 show some
differences from the approach discussed above, the
mean results are quite similar. At this coverage
level, where bias is almost eliminated, average
annual productivity change is 0.2% using the
Malmquist index, 0.4% for the stochastic
frontier cost function and 0.4% for the growth
accounting model. Thus, it would seem that we
obtain quite similar results using the two alternative methods for expanding asset/liability coverage.
As noted earlier in panels A and B of Table 2, a
first approximation to an unbiased value of
productivity change using the Malmquist index
was obtained by subtracting an asset/liability
index from a computed, but biased, Malmquist
index. This simple procedure is quite accurate
when compared to the more involved method of
actually including all or almost all assets and
liabilities and directly computing an unbiased
Malmquist index. Table 5 shows the result of computing a (biased) Malmquist index for different
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Table 4
Productivity estimates by year (Spanish banks, 1986–1991)a
Year
Malmquist
index (%)
Coverage level 1
1986–1987
1987–1988
1988–1989
1989–1990
1990–1991
Mean
Cost function
(%)
Asset/liability
index
5.1
8.7
5.5
4.0
5.1
5.7
5.8
9.9
5.8
2.6
4.2
5.7
1.055
1.076
1.041
1.026
1.037
1.047
Coverage level 4: Adding variables to expand coverage
1986–1987
2.8
1.0
1987–1988
0.1
1.6
1988–1989
1.4
1.3
1989–1990
0.5
1.1
1990–1991
0.7
0.1
Mean
0.8
0.4
0.2
0.8
0.3
0.5
0.9
0.2
1.001
1.003
1.002
0.999
0.998
1.000
Coverage level 4: Aggregating to expand coverage
1986–1987
1.8
1987–1988
0.8
1988–1989
0.1
1989–1990
0.1
1990–1991
0.2
Mean
0.2
1.8
3.0
0.5
1.6
1.5
0.4
1.001
1.003
1.002
0.999
0.998
1.000
a
2.4
7.2
2.4
2.2
4.6
3.8
Growth
accounting (%)
0.7
1.3
0.1
1.0
3.0
0.4
All values have been rounded off.
Table 5
Accuracy of approximating Malmquist index productivity bias. Malmquist index productivity minus asset/liability index by coverage
level and year
Coverage level
1986–1987
1987–1988
1988–1989
1989–1990
1990–1991
Mean
Value with
no biasa
1
2
3
4
5
3.1%
2.7%
3.0%
2.9%
2.7%
0.4%
0.4%
0.5%
0.4%
0.3%
1.7%
1.1%
1.8%
1.6%
1.5%
0.4%
0.7%
0.9%
0.4%
0.8%
0.9%
0.3%
1.5%
0.9%
1.2%
1.0%
0.9%
0.9%
0.9%
0.8%
0.8%
0.8%
0.8%
0.8%
0.8%
a
Value of average Malmquist index productivity at coverage level 5 (also equals the same value at coverage level 4 in Table 4).
coverage levels and years for Spanish banks and
subtracting from it the appropriate asset/liability
index for each coverage level and year. As seen in
the table, the results of this procedure (a) yield
very similar values for different coverage levels for
the same year and for the mean across years
(indicating that the approximation method gives
consistent results across coverage levels) and (b)
the mean value of this approximation method is
very close to the value obtained when the unbiased
Malmquist index is computed by including all
assets and liabilities in the computation process
(which is the ‘‘no bias’’ situation of 0.8% annual
average productivity growth).14
14
Such a strong result is not obtained with the cost function.
This is likely due to the fact that the cost function relies on total
cost and input prices and is not dual to the Malmquist index
‘‘production function’’ specification used here.
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
Fig. 2. Malmquist index (coverage levels 1, 2 and 5).
Fig. 3. Cost function (coverage levels 1, 2 and 5).
Lastly, we show how the Malmquist index and
stochastic frontier cost function mean productivity
estimates vary across the 52 banks in the data set.
In Figs. 2 and 3, we adopt the approach of adding
variables to expand coverage. Fig. 2 shows that the
distribution of productivity change by bank from
the Malmquist index varies from around 4% per
year to a negative 8% per year even when all bias
has been removed (coverage level 5). As the
coverage level is increased from 1 to 2, finally to
level 5, the distribution essentially shifts just down
in a parallel fashion so that about half of the banks
A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
experienced negative productivity change while
half experienced a positive productivity increase
(balancing out at 0.8 overall). Fig. 3 shows the
same information for productivity change estimates from the stochastic frontier cost function.
While this distribution is somewhat flatter than
that for the Malmquist index, the same general
parallel downward shift is evident as more and
more of the bias is removed. Although not shown
in Table 4, average overall productivity change
when all the bias is removed is 1.0% per year.
4. Conclusion
Many Malmquist index studies overstate the
level of banking industry productivity. A similar
problem can exist with a stochastic frontier cost
function or a growth accounting approach. The
bias is not due to the technique used but rather in
how it is applied. The bias is easiest to see in
banking industry studies due the nature of the data
available and used to represent balance sheet
outputs and inputs. The problem is simple to
correct and can easily be measured, reduced, or
eliminated in future analyses. The problem has
been consistently overlooked and likely has
persisted because no one has shown how large
the bias has been.
Although other productivity measurement difficulties remain, the problem we identify is eliminated when all outputs and inputs are included in
the analysis. This differs from common practice in
the banking productivity literature where only a
subset of balance sheet inputs and outputs are
included. Importantly, productivity analysis differs
from banking cost, revenue or profit analyses in
that changes in excluded balance sheet inputs or
outputs have the same effect on productivity – not
a lesser or greater effect – as those inputs and
outputs chosen to be included in the analysis.
Unlike cost or profit analysis, all liability/asset
categories affect productivity measurement equally
(if their balance sheet values are equal) and so need
to be included to obtain a more accurate estimate
of productivity.
In the text, we explain in more detail how the
bias arises in banking studies and show that the
187
majority of measured productivity in many existing studies is due to this problem. We also apply
three methods of measuring productivity (Malmquist index, stochastic frontier cost function, and
growth accounting) to a balanced panel of Spanish
banks over 1986–1991. This application illustrates
how mean productivity change estimates are
reduced as more and more of the bias is
eliminated.
Sophisticated linear programming and/or
econometric techniques can not, of course, overcome specification problems. If researchers wish to
have their productivity results taken seriously by
policy makers, it is necessary to reduce those
biases that can be reduced and note those which
remain. While the bias we identify is ‘‘obvious’’ ex
post, it clearly has not been obvious enough ex
ante in many banking productivity studies to date.
Acknowledgements
We wish to thank Jesus T. Pastor, participants
of the Workshop in Efficiency and Productivity
(Denmark, November 1999) and two anonymous
referees for helpful comments on an earlier version
of this paper. A. Lozano-Vivas is grateful for the
financial support from the CICYT, PB98-1408
(Ministerio de Educacio! n y Ciencia, Spain).
Appendix A. Outputs, inputs, and asset/liability
ratio for panels A and B of Table 2
1. [1]: Outputs are short-term loans to nonbanks, long-term loans to non-banks, checking
accounts by the public, other deposits by the
public, number of branches, and other earning
assets. Inputs are number of personnel, operating
costs, and book value of machinery and equipment.
Asset/liability ratio: Assets=short-term loans to
non-banks+long-term loans to non-banks+other
earning assets+book value of machinery and
equipment. Liabilities=checking accounts by the
public+other deposits by the public.
2. [2]: Outputs are total deposits from other than
financial institutions and total loans to other than
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A. Lozano-Vivas, D.B. Humphrey / Int. J. Production Economics 76 (2002) 177–188
financial institutions. Inputs are man-hours per
year, book values of machinery and equipment,
and non-labor-non-capital operating expenses.
Asset/liability ratio: Assets=total loans to other
than financial institutions+book value of machinery and equipment. Liabilities=total deposits
from other than financial institutions.
3. [3]: Outputs are real estate loans, commercial
and industrial loans, consumer loans, all other
loans, and total demand deposits. Inputs are fulltime equivalent employees, book value of premises
and fixed assets, and purchased funds.
Asset/liability ratio: Assets=real estate loans
+commercial and industrial loans+consumer
loans+all other loans+book value of premises
and fixed assets. Liabilities=demand deposits
+purchased funds.
4. [4]: Outputs are securities, real estate loans,
personal loans, and commercial loans. Inputs
are number of employees, value of premises and
fixed assets (including capitalized leases), and
deposits.
Asset/liability ratio: Assets=securities+real estate loans+personal loans+commercial loans+
value of premises and fixed assets. Liabilities=deposits.
5. [5]: Outputs are revenue from loans and
revenue from business investment activities. Inputs
are number of full-time employees, the value of
premises and real estate, and liabilities including
deposits.
Asset/liability ratio: Assets=revenue from
loans+revenue from business investment activities. Liabilities=liabilities including deposits.
6. [6]: Outputs are securities, real estate loans,
personal loans, and commercial loans. Inputs are
number of employees, the value of premises and
fixed assets (including capitalized leases), and
deposits.
Asset/liability ratio: Assets=securities+real estate loans+personal loans+commercial loans+
value of premises and fixed assets (including
capitalized leases). Liabilities=deposits.
7. [7]: Outputs are loans, checking deposits, and
savings deposits. Inputs are number of employees
and operating expenses.
Asset/liability ratio: Assets=loans. Liabilities=checking deposits+savings deposits.
8. [8]: Outputs are demand deposits, loans with
domestic currency, and loans by trust account.
Inputs are number of employees, value of physical
capital, and purchased funds.
Asset/liability ratio: Assets=loans with domestic currency+loans by trust account. Liabilities=
demand deposits+purchased funds.
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