the temporal behavior of scale economies within a banking firm

Journal of Economics and Finance �9 Volume 20 �9 Number 2 �9 Summer 1996 �9 Pages 33-45

The Temporal Behavior of Scale Economies Within A Banking Firm

by K e n n e t h D a n i e l s a n d D o , a n T i r t i r o ~ l u

A B S T R A C T

Examining the time path of the scale economies has not received much

attention until recently. Moreover, the time path of scale economies for a

given firm has not been studied in the banking literature at all. Examining

scale economies, either from cross-sectional or panel data, based only on

a single estimate ignores the dynamic behavior (both cost- and produc-

tion-wise) of a banking firm as well as of the banking industry. We study

the time-varying scale economies of commercial banking firms. We

employ the Kalman-filter approach in estimating the translog cost func-

tion. The Kalman filter allows the parameters of the translog cost function

and therefore the scale economies to be time dependent and varying. The

estimation results indicate significant variation in inter- and within-firm

scale economies over time for sample banks.

Introduction

Does a banking firm reduce its average cost of production as it expands its output? The answer to this question has been examined empirically by estimating the scale economies for banking firms. 1 These studies use either cross-sectional or panel data and obtain only a single estimate of scale economies. Based on the magnitude of this estimate, a bank (or banking industry) is said to exhibit increasing, constant or decreasing scale economies. Various studies assume different cost functions and estimation techniques, but the findings generally indicate mild scale economies.

The single estimation approach, however, ignores the dynamic behavior (both cost- and production-wise) of a banking firm as well as of the banking industry. Recent papers by Berger and Humphrey (1992), Bauer et al. (1993), and Berger (1993) (BHBB, henceforth) are the only ones that have considered time variation in scale economies in banking. BHBB estimate scale economies cross-sectionally for each sampling period to produce independently time-varying information. Thus, the estimate of scale economy at time t does not assist the estimation of the scale economy at (t+l), contrary to intuition.

Our paper takes the current time period's estimate of scale economy and uses it as an input into next time period's estimate. We examine the time-varying scale economies of each banking firm in our sample - - that is, we examine time variation within a banking firm. We employ a Kalman-filter (KF, henceforth) (Kalman, 1960; Kalman and Bucy, 1961) to estimate the translog cost function. The KF is a time-series, random coefficients technique that allows the parameters of the translog cost function to be stochastic and time dependent. Since scale economies depend on the estimated parameters of the translog cost function, the scale economies would also be time dependent and varying.

This paper is organized as follows. Motivation of the study is explained next while the following sections provide a discussion of the model, data, variable definitions, empirical results and concluding comments along with avenues for future research.

Kenneth Daniels, Virginia Commonwealth University, Department of Finance, Richmond, VA 23284-4000 Do, an ~rtiro~lu, Salisbury State University, School of Business, Salisbury, MD 21801

34 JOURNAL OF ECONOMICS AND FINANCE �9 Volume 20 �9 Number 2 �9 Summer 1996

Motivation of the Study

The standard duality result between a production and its corresponding cost functions has been the pri- mary justification for the empirical literature to estimate a cost function, C(Q,P), with arguments of output, Q, and a vector of input prices, P. When effects of technical change are considered, the cost function is mod- ified to C(Q,P,t), where t is time. Various studies consider the effect of technical change by entering an index of time into the empirical model (e.g., Hunter and Timme 1986, 1991; and Glass and McKillop 1992). However, the time index approach may not provide an accurate picture of technical progress. For example, Bauer et al. (1993, p. 394) note:

"Unfortunately, there is no unique indicator or proxy for measuring the effects of technical change, either for neutral or embodied technical progress. As a consequence, virtually all previous banking studies have chosen to model technical change as a simple time trehd. However, studies of electric utilities have suggested that time'trends may poorly reflect year-to-year variations in technical change when this process is not constant or smoothly increasing or decreasing."

The approach of including a time index in the empirical cost function is acceptable if 1) the technical change is disembodied, 2) various measures of economies have no time pattern, 3) other sources of change are captured by the time-index variable 2, and/or 4) technical change is not latent.

We examine time-varying cost functions and the time-path of scale economies by making the coefficients of the translog cost function stochastic and time-varying) The cost function becomes C(Q,P,B(t)) where B(t) indicates time-dependent parameters. This approach is free from the second, third and fourth criticisms, but is consistent with a disembodied technical change. We avoid the concern about a time-index since time variation is modeled as a stochastic process. We briefly consider these four issues.

The KF model obtains time-varying parameters and therefore can provide time-paths of scale economies. The time-varying measures of economies significantly improves on the constant point estimates of previous studies. Moreover, since the information uncovered at time t is fed into next period's estimation of scale economies, this gives our approach an advantage over independently estimated scale economies over time.

Time-varying economies may result from sources other than technical change such as business cycles, regulatory changes, global competition, changes in the quality and prices of inputs, mergers and acquisi- tions, changes in organizational form, and financial innovations. Making the parameters time-dependent captures the effects of these other sources of change in time. 4

Technological progress is an unobservable (or latent) and a stochastic variable. The time index approach treats it as if it were observable. The KF, by capturing time variation through stochastic changes in the parameter estimates, adopts a latent variable approach. This technique allows the measurement of the time-varying scale economies while controlling for technological progress. Slade (1989) illustrates the use of the intercept term as an approximation of technological progress. We adapt a similar technique which uses the random intercept term of the time-varying cost function as an approximation of the firm's total factor productivity, which is a measurement of the firm's technological progress. We assume that the time-variation effects of technical change under the KF are still disembodied. In other words, a stable relationship exists between inputs, the cost function, and time. 5

The Model

Let C = f (Q, P) be the dual cost function for a typical bank. We follow Hunter and Timme (1991) and Mester (1992) to describe two-output, three-input translog cost function in defining the cost function: 6

C = cx o + Ei ~i lnQi +Em am lnPm + 1/2 Era "~n aro-n lnPm lnPn

+ 1/2 E i Ej [3ij lnQ i lnQj + E i E m (~im lnQi lnPm + �9

ij = L,D m,n = W,K,F

The Temporal Behavior of Scale Economies Within A Banking Firm 35

where outputs are total loans and leases (L) and total produced domestic and foreign deposits (D), and inputs are labor (W), capital (K), and loanable funds (F).

Under the KF, equation [1] becomes:

lnCt = %,t + ~'il~i,t lnQi,t + 2m ~m,t lnPm,t + 1/2 ~m 2n ~ lnPm,t lnPn,t

+ 1/2 Ei E i [~ij,t lnQj,t + 2i •m Qim,t lnQi,t InPm,t + �9 2

ij = L,D m,n = W,K,F

The following restrictions are imposed on the translog cost function to ensure that the cost function is linearly homogeneous in factor prices:

For C~r 2mCtmn,t = 1, 2mOtmn,t = 0 C~n), 2mOim,t = O. 3a

Symmetry condition requires that

For (Vt); am.,t = %m.t , 13ij,t = 13ji,t

Scale economies are traditionally defined as:

3b

SEt = (~i-1 31nCt / 31nQt) -1 4a

SEt = (Y-'il3i,t + Y-'i Y--'j 13i.i,t lnQj,t + Y-,i Y-'m | lnPm,t) -1 4b

where equation [4.b] is computed for each time period. If SEt is greater than, equal to, or less than 1, then a bank (or banking industpy) exhibits increasing, constant, or decreasing returns to scale, respectively. All time-varying parameters are modeled as random walks. The parameters are assumed to be random walks because we have no prior knowledge about the appropriate functional form for the random coefficients. In addition, interest costs constitute the bulk of a bank's total costs. There is evidence that interest rates follow a random walk pattern. 7 The starting values for each coefficient are chosen by maximizing the likelihood function. Details of the KF algorithm is provided in the appendix.

Berger, Hanweck, and Humphrey (1987) show that output expansion (or contraction) over time for a banking firm may result from a) growth in deposits and loans at each existing office as the local market grows over time (holding the number of branch offices constant) and b) growth in the number of branch offices which brings with it more deposits and loans. The values of output measures in this study already capture both effects. Adding the expansion of bank branch offices over time in equations [1] and [2] would therefore cause second counting. Hence, we exclude the number of bank branches from the model.

Virtually all bank studies acknowledge the difficulties with respect to defining output and costs and to aggregating output. A number of very recent studies, including Humphrey (1992), Berger and Humphrey (1992), and Hunter and Timme (1991), subscribed to the criterion of value added. We also adopt this model of bank production which incorporates the production of intermediate deposit outputs as well as final loan outputs. Thus, both input and output characteristics of deposits are specified simultaneously. (See Humphrey (1992, pp.ll6-118) for an excellent discussion of the value added approach.)

We aggregate outputs to minimize the degrees of freedom problem. There is a trade off between the lim- ited number of observations and the number of parameters to be estimated. Disaggregating outputs increas- es the number of parameters drastically while the number of observations still remains constant. We believe that equation [1] is a simple and sufficient cost function specification for the purposes of this study.

It should also be noted that by aggregating outputs into the two categories above, we implicitly assume that there is only one cost structure underlying the production of each of the following: all classes of loans and leases and all classes of produced deposits.

36 JOURNAL OF ECONOMICS AND FINANCE * Volume 20 �9 Number 2 �9 Summer 1996

Data

The KF requires time-series data that pertain to individual banking firms. We obtained these data from Bank Compustat, which reports quarterly data for each sample bank and makes it possible to estimate the parameters of the translog cost function and the scale economies on a quarterly basis. The sampling period runs from 1973-Q1 to 1989-Q4 - - 68 quarters of data for each bank. All nominal variables are deflated.by GNP deflator.

We must note that we undertook this study under substantial data constraints. Specifically, we faced two problems. First, Bank Compustat is incomplete over some time periods. Data on number of employees are critical for any cost function analysis. Yet, data for this variable in Bank Compustat files either do not exist or are incomplete. This constraint forced us to survey about 100 banks to collect the information directly from them. Unfortunately, most banks did not respond to our survey. The second difficulty is that some of the Bank Compustat data are of questionable quality. We undertook extensive data checks. Our efforts yield- ed quality time-series data only for ten banks, which are being studied in this paper.

Variable Definitions We define the variables following Hunter and Timme (1991) and Mester (1992):

C = (Salaries and benefits) plus (Occupancy expenses) plus ['(Total Interest Expenses minus Service Charges)]*[Total Loans and Leases/Total Earning Assets]

QL = total loans and leases (includes real estate loans, loans to depository institutions, agricultural production, commercial and industrial loans, personal loans, loans to state and government subdivisions, all other loans, lease financing receivables) minus (unearned income plus allowance for loans and lease losses)

QD = Total domestic and foreign deposits (includes deposits by individuals, partnerships, government, corporations, states and political subdivisions, and all other deposits)

w h e r e

Total Earning Assets = (Investments + Total Loans + Other Earning Assets)

Price of labor (Pw) = Salaries and Benefits divided by the number of employees

Price of Capital (Pr.)= Occupancy expenses divided by bank premises

Price of Deposits (PF) = [(Total interest expense minus service charges)]

divided by total deposits

Empirical Results

The KF provides an estimate for each parameter at every point in time; thus, there are 68 estimates for each of the 21 parameters of equation [2]. We have 21 x 68 = 1,428 estimates for each sample commercial bank. As a result of the abundance of estimates, we have reported the results for the first, median and final time period. 8 While Table 1 reports the estimation results, Figure 1 graphs the time-varying scale economies for the sample banks.


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Figure 1: Time-Varying Scale Economies Based on the Kalman Filter Estimations

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Kalman Filter Estimates of Tram-log Cost Function Coefficients.

BANK AMERICA

Parameter FIRST MEDIAN LAST ct0 -0.295 -0.052 0283

(-1.54) (-1.15) (2.8O) ~L 0.484 0.493 0.423

(1.15) (2.28) (1.71) ~'D 0.860 0.893 0.802

(1.98) (2.89) (3.04) aw 0345 0.331 0.227

(1.49) (3.35) (237) ct F 0.500 0.615 0.307

(2.45) (2.43) (2.20) [SLL .0.350 -0.344 -0.337

(-0.09) (-2.8S) (-1.10) [SDD 3.143 3.147 3.172

(0.54) (2.03) (2.96) ~LD -1.636 -1,630 -1.589

(4) .41) (-3.50) (-3.15) Ctw'w 0.796 0.789 0.691

(1.62) (2.60) (2.92) ct~ 0.873 0.829 1.019

(1.96) (3.20) (3.78) r 0.033 -0.027 0.077

(0.07) (-0.~) (0.76) OLW -1.929 -1.924 -1.909

( -1.11) (-.6.01) (-6.46) OL~ 0.635 0,626 0.715

(1.63) (2.22) (229) ODW 2.575 2.576 2.626

(1.10) (2.47) (2.33) OO~ .0.827 -0.816 .0,759

(.0.63) (-2.79) (-2.77)

NOTES: l) 1~6.dr am ia paremb~,~.

Kalman Filter Estimates of Trans-log Cost Function C.ocfficientS. CITICORP

Parameter FIRST MEDIAN o.o --0.297 ..0.096i o.284

(-1.65) (-I.8O) (0.05) ~C 0.434 0 3 9 8 0.445

(1.86) (2.65) 003) 130 0.889 0.820 0.922

(6.50) (622) (6.34) cx w 0.313 0.282 0.372

(0.33) (0.17) (0.08) a F 0.730 0.849 0.774

(6.41) (5.28) (6.59) ~LL -5,524 -5.521 -5.529

(-2.76) (-2.24) (-2.36) ~DD -2.418 -2.418 -2.408

( -4.45) (..4.69) (-4.86) J3LD 3.822 3,824 3.833

(3.74) (5.58) (3.82) aww -0.092 -0.078 -0.121

(-1.81) ( -2 .51) ( -2 .90) aFF 0.067 0.111 -0.032

(0.58) (2.54) (-1.13) Ctw F -0.104 -0.090 -0.184

(-2.06) (-2.62) (-2.63) OLW -0.601 -0..596 -0.617

(-3.42) (-3.99) (-3.22) OLF 1.106 1.091 1.071

(3.89) (3.58) (3.41) Oow 1.173 1.168 1.210

(2.94) (2.83) (2.96) ODF -0,928 -0.955 -0.926

(-4.27) (-3.06) (-2.17)

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38 JOURNAL OF ECONOMICS AND FINANCE * Volume 20 * Number 2 �9 Summer 1996

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Kalman Filter Estimates of Trans-log Cost Function Coefficients.

FIRST MARYLAND Parameter FIRST MEDIAN LAST r O -0.457 qJ.102 0.186

(-Z71) (0.81) (1.20) ~L 0.663 0.5011 0,465

(1.98) (2.03) (1,88) 13 D 0.406 0.371 0.310

(1.10) (1.44) (1.23) a w 0.196 0.152 0,167

(0.70) (0.76) (0.6"0 at: 0.594 0.755 0.702

(2.62) (437) (3.64) !SLL -6.839 -6.796 -6.846

(-1.SS) (-2.70 (-232) [300 -13.501 -13.493 -13.539

(-1.74) (-4.83) (-4.87) [~LO 10.112 10.154 10.045

(1.99) (3.12) (3.83) aww 0,879 0.903 0,873

(0.79) (332) (2.83) aFF 0.037 0.107 0.092

(0.07) (0.47) (0.32) Ctwr- -0.0498 -0.445 -0.483

(-0.80) (-1.71) (-1.70) OLW -1.104 -I.071 -1,113

(-0.66) (-3.94) (-3.67) OLF 0.144 0.099 0,0894

(0.16) (0.40) (0.36) Oow 0330 0.347 0.314

(0.14) (127) (0.99) ODe -0252 -0266 -0241

(-020) (-0.96) (.o.so)

Hot,a: i ) t-t,~ti*t~, s = ht ; = ~

"ABLE i (CONTINUED ['ABLE I ( C O N T I N U E I

Kalman Filter Estimates of Trans-log Cost Function CoefficientS.

FIRST CHICAGO Parameter FIRST MEDIAN LAST r 0 -0.608 -0.0237 0.511

(-2.63) (-332) (1.468) ~u 0.138 0.0567 0.197

(1.36) (2.73) (2.35) t3 n 0.737 0.632 0.724

( 2 . 1 8 ) (3.76) (3.10) etw 0.181 0.124 0.0995

(0.42) (1.48) (0.43) nq: 0.683 0.792 0.771

(3.89) (3.69) (4.30) ~'LL 10.746 10.751 10.748

(0.03) (030) (0.29) [300 5.645 5.652 5.643

(1.12) (2.40) (2.94) ~'LO -7.668 -7.656 -7.670

(-2.57) (-2..2 0 (-2.53) aw'w 1.636 1.640 1.640

(0.46) (0.70) (0.76) ae~ -0.020 0.011 -0.109

(-0.38) (1.45) (-0.63) OtWF "0.055 -0.057 -0,101

(-0.24) (-1.69) (-0.55) OLW .4.964 ,,4.956 -4,972

(-1.62) (-2.28) (-2.54) OLF 0.791 0.777 0.7578

036) (2.18) (2.70 ODw 2.853 2.866 2.830

(1.21) (2.10) (2.S 0 ODF -0.840 -0.838 ..0.913

(-0.95) (-3.12) (-4.53)

N o t ~ : 1) t 4 t t d ~ a ~ in p * ~ m o ~ .

Kalman FIlter Estimates of Trans-log Cost Function Coefficients.

F1R~i'TENNESSEE Parameter FIRST MEDIAN LAST ct o -0399 -0.024 0.022

(-6.90) (-0.45) (0.53) ~L 0.4,85 0.422 0.468

(1.91) (3.41) (235) ~'n 0.906 0.892 0.880

(3.94) (4.51) (4.70) r w 0..303 0.257 0.302

(1.70 (2.aS) (1.82) r 0.589 0.682 0.75

(3.93) (52"0 (5.41) ~LL -1220 -1.216 -1210

(-032) (435) (-531) ~'DO -3.026 -3.027 -3.031

(-0.62) (-2.31) (-3.12) [~LO 4262 4265 4268

(2.09) (35o) (3.78) ctww -0.173 -0.160 ..0.158

(-029) (-124) (-0.71) r -0.108 -0.076 -022.5

(-0.19) (-0.50) (-0.98) aWF 0.175 0.197 0.139

(0.34) (1.59) (0.82) OLW -1.459 -1.460 -1.426

(-2.o4) (-231) (-2.55) OLF 0393 0373 0.415

(137) (2.87) (1.93) ODW 1328 1331 1.3,43

(1.04) (2.34) (2.95) OoF .0259 -0269 -0.218

(-1.31) (-2.16) (-1.10)


Figure 1 clearly demonstrates the basic premise of this paper - - that is, the measure of scale economies (or any other measure of economies for that matter) is dynamic and that obtaining just a single estimate of any measure of economies would not necessarily con- vey an accurate picture in this regard.

It is clear from Figure I that each bank has its own cost/production behavior over time. Bank of America and Citibank have rather flat time paths. Time-paths of other sample banks are dynamic and different. First Chicago, Bank of Boston and National City show rather sharp swings from one time to the next during the entire sample "period. First Maryland, Colorado National, First Tennessee, and especially Norwest exhibit relatively smooth behavior. Finally, US Bancorp exhibits rather sharp swings initially, fol- lowed by a period of smoothly declining and later slightly sharply swinging patterns.

If we define 'within firm variation" as a switch from increasing (decreasing or constant) returns to decreasing (increasing or constant) returns then we observe that all sample banks, with the exception of Citibank, Bank of America, and Colorado National, exhibit within firm variation.

Figure 1 indicates that Citibank and Bank of America consistently exhibit decreasing returns to scale and that Colorado National consistently exhibits increasing returns to scale. 9 Estimates of scale economies for Colorado National decline after the first quarter of 1982. This movement corresponds to merger activities in the first quarters of 1981 and 1982.

The time-paths of First Chicago and Bank of Boston indicate increasing returns to scale except for short time blocks during the sampling period. The volatility of the time path of First Chicago's scale economies is particularly noteworthy. We conducted a Wall Street Journal Index (WSJI) search to understand the reasons for this high volatility. First Chicago evidently had difficulties in attracting deposits in early 1980s. We understand from the WSJI search that while regulators were keeping an eye on this bank during much of the 1980s, it 1) was trying an aggressive strategy to keep bad news from crippling its deposit-gathering strength in 1985, 2) reorganized its corporate banking business to improve perfor- mance in April 1986, 3) merged with Bank of Lansing in July 1986 and agreed to acquire four Chicago area banks in September 1986,

4) was sued for $3.6 billion by Hunt Brothers and countersued them for $440 million for unpaid loans and punitive damages, 5) lost State of Illinois'

['.ABLE I ( C O N T I N U E D

Kalman Filter Estimates of Transqog Cost Function Coefficients.

NORWEST CORPORATION Parameter RRST MEDIAN LAST et o -0.280 -0.097 0.147

(-3.41) (-1.70) (2.24) [SL 0.453 0.269 0.239

(2.92) (3.21) (2-51) [SD 1.039 0.897 0.912

(5.88) (5.42) (5.76) etw 0.078 0.113 0.058

(0.77) (1.35) (1.75) a F 0.746 0.713 0.657

(2.13) (2.83) (2.99) [SLL 0.248 0.333 0.2.53

(2.33) (3.51) (2.01) I~DD 0.324 0.363 0.373

(1.22) (3.72) (2.79) [SLD -0.184 -0.068 -0,122

(-1.18) (-1.76) (-1.49) ~ww 0.0576 0.0725 0.0853

(1.43) (1.99) (1.36) a ~ -0.260 -0.260 -0.251

(1.48) (-2.80) (-1.88) m,V'F 0.091 0.113 0.0887

(-1.78) (-1.24) (1.52) OLW -0.192 -0.196 -0.206

(-1.57) (-2.10) (-2-62) OLF 0.225 0.254 0.292.

(3.50) (2.86) (2.18) ODW 0.152 0.151 0.143

(3.87) (1.58) (-1.18) ODF -0.739 -0.717 -0.695

(-2.45) (-2.69) (-2.44)

No~: 1) t 4tathd~ a~ ie p*~eab,r

F A B L E I I C O N T I N U E I

Kalman Filter Estimates o f Trans*[og Cost Function Coefficients.

NORWEST CORPORATION Parameter FIRST MEDIAN LAST

-0.280 -0.097 0.147 (-3,41) (-1.70) (2.24)

[st. 0.453 0.269 0..239 (2.92) (3.21) (2.51)

[50 1.039 0.897 0.912 (5.88) (5.42) (5.76)

C~w 0.078 0.113 0.058 (0.77) (1.35) (1.75)

cq: 0.746 0.713 0.657 (2.13) (2.83) (2.99)

~LL 0.248 0.333 0.253 (2.33) (3.51) (2.01)

[SDD 0.324 0.363 0.373 (1.22) (3.72) (2.79)

[st:~ - 0 . ~ -0.o68 -0.122 (-1.18) (-1.76) (-I.49)

C~ww 0.0576 0,0725 0.0853 (1.43) (1.99) (1.36)

ct~ -0.260 -0.260 -02.51 (1.48) (-2.80) (-1.88)

aWF 0.091 0.113 0.0887 (-1.78) (-1-24) (1.52)

OLW -0.192 -0.196 -0.206 (-1.57) (-2.10) (-2-62)

OLv 0.225 0.254 0.292 (3.50) (2.86) (2.18)

Oow 0.152 0.151 0.143 (3.87) (1.58) (-1.18)

ODF -0.739 -0.717 -0.695 (-2.45) (-2.69) (.2.44)

Neea*: I) t..ua dsam a~ il


deposits of $227 million, in January 1987, as a protest to the bank's 19.8% credit card interest rate, which the state treasurer described as 'legal loan sharking' and 6) extended and suffered from risky loans to Brazil during most of 1980s.

Bank of Boston shows a sharp declining pattern after 1987. The timing of this pattern is consistent with the timing of banking problems which caused major shake ups in most New England banks.

While National City has a trend of moving from decreasing returns to increasing returns between 1973 and 1985, it generally presents a pattern of inci'easing returns to scale after 1985. We identify from the WSJI that it had a reorganization in May 1974; that it acquired three different banks between May 1975 and August 1976; that it acquired another bank in the first quarter of 1982; that it purchased 18 branches of a financially troubled financial institution in August 1988; that it acquired First Kentucky National Corporation in July 1988; and that it completed a capital restructuring program which was started in 1986.

First Tennessee and Norwest exhibit a consistently declining pattern after 1980, yet their patterns are different before 1981. While evidence for First Tennessee point to increasing and constant returns and volatility in scale economies during much of 1973-1980 period, evidence for Norwest suggests a smoothly increasing pattern moving from decreasing to increasing returns during the same time period. We observe from our WSJI search that First Tennessee was involved in a number of merger activities during 1973, 1974 and in 1982. The WSJI indicates that Norwest acquired two banks in 1980 and 1981, and that Standard and Poor's Corporation put the bank on its weekly CreditWatch list as under special surveil- lance for a possible credit rating change.

US Bancorp maintains an increasing returns to scale pattern, except briefly between late 1973 and early 1975, until around 1979. It starts on a declining pattern, leading to a switch from increasing and constant returns to decreasing returns, in late 1979 until 1987. The bank goes back to a state of increasing returns to scale within the final years of our sampling period. Although US Bancorp's time path is indeed quite interesting, we could not find much information in the WSJI on this bank.

Finally, First Maryland moves consistently from decreasing returns towards constant returns to scale. We observe an upward jump in scale economies in late 1983. This corresponds to merger activity.

Obtaining a single estimate of scale economies for the banking industry does not provide a complete pic-

FABLE I {CONTINUEI

Kalmaa Filter Estimates o f Trans-log Cost Function Coefficients.

BANK OF BOSTON Parameter FIRST MEDIAN LAST % -0.505 -0.081 0.22.84

(..4.40) (-1.82) (2.84) ~L 0.492 0.368 0.479

(1.70) (3.43) (2.25) [3 v 0.428 0385 0.426

(1.41) (3.57) (236) a w 0.280 0.217 0.156

(1.22.4) (231) (2.86) a F 0.789 0.848 0.674

(4.37) (5.73) (4.34) 13LL 0.772 0.793 0.790

(0.47) (3.70) (2.79) l~Va 3.545 3.545 3.554

(1.93) (2.69) (1.67) [31. o -1.303 -1.288 -1.281

(-4.58) (-5.98) (-.4.84) aww 1.160 1.17 1.198

(0.85) (4.83) (428) err 0.2.85 0318 0..239

(0.71) (0.75) (0.99) awl: -0.449 -0.428 -0.455

(-0.81) (-2.67) (-1.7"2) OLW -0.562 -0.526 -0.520

(-0.53) (4..51) (-2.00) e l f 0.722 0.'F26 0.743

(1.08) (3.29) (3.05) ODW -0.485 -0.471 -0.466

(-0.36) (-3.97) (-1.67) ODF -0.564 -0.563 -0.554

(-2.66) (-2.76) (-2.06)

No~ I) bm fisl/~ s~ ie ~h~es.

:ABI,E I iCONTINUED

Kalman Filter Estimates of Tran~log Cost Function Coefficients.

COLORADO NATIONAL. Parameter FIRST MEDIAN LAST ot o -0376 -0.1103 0.160

(-335) (~ (2.19) 0.22.96 0.185 0.22.9

(135) (1.74) (1.70) 13 o 0.905 0.806 0.907

(4.01) (5.51) (5.31) a w 0.315 0269 0.267

(2-12) (2.74) (1.95) r F 0.527 0.631 0.567

(3.42) (4.90) (4.31) ~ 2.709 2.735 2.680

(0.51) (2.12) (2.45) 13no 3.335 3.355 3312

(0.52) (2.84) (2.43) l$t.D -3.22.06 -3.161 -3.261

(-0.56) (-2.75) (-2.51) Ctww -0338 -0315 -0.333

(-1.65) (-2.81) (-138) ctrr -0.051 0.045 0.104

(-0.12) (0.41) (0.52) Ctw F 0.115 0.089 0.109

(0.41) (0.77) (0..56) OLW 0317 0.341 0.242

(1.38) (2.15) (2.18) O ~ 0.150 0.115 0.115

(1.13) (2.63) (2.23) ODW 0.316 0.345 0.246

(1.31) (2.95) (2.75) ODF -0.617 -0.637 -0.672

(-1.45) (-6.14) (-539)

l) J-mfisfio arc io ~ mlm~i.


ture of banking efficiency. For example, Noulas, Ray and Miller (1990) using a 1986 dataset estimate that mild scale economies exist for large U.S. commercial banks but decreasing returns to scale are found for the largest U.S. commercial banks. Our sample, is select- ed from Bank Compustat which also reports data for the largest commercial banks in the United States. In our opinion, it is incomplete to generically describe all large banks as exhibiting decreasing returns to scale. Clearly, Citicorp and Bank America can be described as banks that exhibit decreasing returns to scale. On the otherhand, Colorado National does not fit the generic decreasing returns to scale label given to large U.S. commercial banks. The remainder of our sample exhibits 'within firm variation' therefore one should inspect the time-series properties of the economies of scale before defining the returns to scale of the banking industry.

I 'ABLE I ( C O N T I N U E D

C o n c l u s i o n a n d S o m e I d e a s f o r F u t u r e R e s e a r c h

The time path of scale economies has received lit- tle attention until recently. Moreover, the time path of scale economies for a given firm has not received attention in the banking literature. Examining scale economies, either from cross-sectional or panel data, based on a single estimate ignores the dynamic nature of the behavior (both cost- and production-wise) of a banking firm as well as of the banking industry. We study time-varying scale economies for ten commercial banking firms by employing the Kalman-filter in estimating the translog cost function. The Kalman filter allows the parameters and therefore the scale economies of the translog cost function to be time dependent.

Our results indicate significant variation in inter- and within-firm scale economies over time. Such variation in economies of scale makes it incomplete to portray the U.S. banking industry as exhibiting mild scale economies.

In addition, we think that the KF approach has a lot of potential for further research. The following list appear be potential research ideas:

1. Slade's (1989) work is particularly important in that it applies the KF in a production/cost function framework. Measurement of TFP in banking is an issue of current interest (see Humphrey, 1992). Since technological progress is a latent variable and TFP is a concept with a time dimension, the KF should provide a reliable methodology in examining TFP.

Kalrnan Filter Estimates of Trans-log Cost Function CoefficientS.

U.S. BANCORP Parameter FIRST MEDIAN LAST % *0.325 *0.055 0.209

(-1.867) (..0.647) (0.982) ~L 0.617 0.568 0.497

(2.904) (2.114) (3.88) [3 D 0.597 0.533 0.513

(2.41) (3.90) (3.17) c~ w 0.515 0.416 0.389

(2323) (3.349) (1.725) ~F 0.612 0.672 0.658

(1.85) (1.71) (1.69) ~l.L 8,429 8.438 8.399

(7.639) (10.289) (7.798) I~DD 16.427 16.435 16.416

(2.952) (2.442) (2.090) IBt. D -11.729 -11.711 -11.770

(-5.084) (-2.S86) (-3.364) ttv,~ -0.813 *0.793 *0.799

(-2.913) (-3.667) (-4.315) ct~ 1.001 1.0288 1.003

(3.88) (3.96) (3.73) CtwF -0.138 -0.122 -0,158

(-S.038) (-10.007) (-7.839) OLW 1,012 1,043 1.009

(7.859) (8.006) (8.413) O ~ -1.851 -1.847 -1,859

(-1.66) (-2.036) (-2.82) Ovw -1.188 -1.159 -1.195

(-1.44) (-2.36) (-2.75) ODe 3.272 3.275 3.245

(4.75) (4.19) (4.75)

No~s: l) t-Jt~dat/.~ am in ~ml~d:~.

"ABLE I ( C O N T I N U E D

Kalman Filter Estimates of Trans-log Cost Function Coefficienls.

NATIONAL CITY Parameter FIRST MEDIAN LAST o. o -0.239 -0.011 0.274

(2.901) (1.964) (3.729) f'L 0.790 0.561 0.573

(9.246) (5.260) (4.197) I~D 0.645 0.468 0.474

(6,094) (4.143) (3.383) ct w 0,129 0.165 0.107

(2.883) (2.671) (4.326) al: 0.677 0.799 0.684

(15.346) (9.044) (7.740) 13Lt" 3,021 3.136 3.051

(7.552) (10.115) (16.404) ~ov 7.460 7.517 7.487

(4.804) (5.681) (5.029) [~Lo -4.857 -4.694 -4.810

(-0.40) (*0.63) (*0.75) aww 0.508 0.517 0.473

(4.975) (4.253) (6.368) atr 0.352 0.383 0.337

(3.767) (2.229) (3.623) CtwF -0.405 *0.395 *0.466

(-1.882) (-2.556) (-3.334) OLW 1.598 1.609 1.587

(4.335) (4.316) (4.910) OLV -1,612 -1.656 -1.707

(-1.750) (-2.034) (-3.851) Ovw -2.407 -2.409 -2.399

(-3.033) (-4.459) (-3.422) OoF 2.490 2.455 2.424

(2.220) (2.050) (2.599)

42 JOURNAL OF ECONOMICS AND FINANCE �9 Volume20. Number2. Summer 1996

2. The time-paths of economies of scope can be modeled and studied under the KF. 3. What are the factors that influence the behavior of scale economies over time? This question can be

examined empirically, provided that consistent and reliable time-series data for factors, which may explain the variation in time-varying scale economies, exist.

BIBLIOGRAPHY

Bauer, RW., A.N. Berger, and D. Humphrey, 1993, Efficiency and Productivity Growth in U.S. Banking, in: H.O. Fried, C.A.K. Lovell and S.S. Schmidt, eds., The Measurement of Productive Efficiency: Techniques and Applications (Oxford University Press), 386-413.

Bell, EW. and N.B. Murphy, 1968, Costs in Commercial Banking: A Quantitative Analysis of Bank Behavior and Its Relation to Bank Regulation, Research Report No. 41, (Boston: Federal Reserve Bank of Boston).

Benston, G.J., 1965, Branch Banking and Economies of Scale, Journal of Finance, 20, 312-31.

Benston, G.J., 1972, Economies of Scale of Financial Institutions, Journal of Money, Credit, and Banking, 4, 312-41.

Benston, G.J., G.A. Hanweck and D.B. Humphrey, 1982, Scale Economies in Banking: A Restructuring and Reassessment, Journal of Money, Credit, and Banking, 14, 435-56.

Berger, A.N., G.A. Hanweck and D.B. Humphrey, 1987, Competitive Viability of in

Banking: Scale, Scope, and Product Mix Economies, Journal of Monetary Economics, 16, 501-520.

Berger, A.N., and D.B. Humphrey, 1992, Measurement and Efficiency Issues in Commercial Banking, in: Zvi Griliches, ed., Output Measurement in the Service Sectors, National Bureau of Economic Research, (University of Chicago Press, Chicago, IL), 245-279.

Berger, A.N., 1993, Distribution-Free Estimates of Efficiency in the U.S. Banking Industry and Tests of the Standard Distributional Assumptions, Journal of Productivity Analysis, 4, 261-292.

Cebenoyan, S.A., 1988, Multiproduct Cost Functions and Scale Economies, Financial Review, 23, 499-512.

Chambers, R.G., 1988, Applied Production Analysis: A Dual Approach, (Cambridge University Press).

Clark, J.A., 1984, Estimation of Economies of Scale in Banking Using a Generalized Functional Form, Journal of Money, Credit, and Banking, 16, 53-67.

Evanoff, D.D. and P.R. Israilevich, July/August 1991, Productive Efficiency in Banking, Economic Perspectives, Federal Reserve Bank of Chicago, 11-32.

Ferrier, G.D. and C.A.K. Lovell, 1990, Measuring Cost Efficiency in Banking: Econometric and Linear Programming Evidence, Journal of Econometrics, 46(2), 229-245.

Fields, J., N.B. Murphy and D. Tlrhro_lu, 1993, An International Comparison of Scale Economies in Banking: Evidence from Turkey, Journal of Financial Services Research, 7(2), 111-125.

Glass, J.C. and D.G. McKillop, 1992, An Empirical Analysis of Scale and Scope Economies and Technical Change in an Irish Multiproduct Banking Firm, Joumal of Banking and Finance, 16, 423-437.

Humphrey, D.B., 1992, Flow versus Stock Indicators of Banking Output: Effects on Productivity and Scale Economy Measurement, Journal of Financial Services Research, 6, 115-135.

Hunter, W.C. and S.G. Timme, 1986, Technical Change, Organizational Form, and the Structure of Bank Production, Journal of Money, Credit, and Banking, 18, 152-166.

Hunter, W.C. and S.G. Timme, 1991, Technological Change in Large U.S. Commercial Banks, Journal of Business, 44(3), 339-362.

The Efficiency of Financial Institutions, April 1993, Journal of Banking and Finance (Special Issue), vol. 17, No.s 2-3.

Kalman, R.E., 1960, A New Approach to Linear Filtering and Prediction Problem, Transactions o ASME, Series D: Journal of Basic Engineering, 82, 3545.

Kalman, R.E. and R.S. Buoy, 1961, New Results in Linear Filtering and Prediction Theory, Journal of Basic Engineering, 83, 95-108.

The Temporal Behavior of Scale Economies Wtthin A Banking Firm 43

Meinhold, R.J. and N.Z. Singpurwalla, 1983, Understanding Kalman Filter, American Statistician, 37(2), 123-127.

Mester, L.J., 1992, Traditional and Non-traditional Banking: An Information-Theoretic Approach, Journal of Banking and Finance, 16, 545-566.

Noulas, A.G., S.C. Ray and S.M. Miller, 1990, Returns to Scale and Input Substitution for Large U.S. Banks, Journal of Money, Credit, and Banking, 22, 94-108.

Slade, M., 1989, Modeling Stochastic and Cyclical Components of Technical Change,

APPENDIX The Kalman-Filter Methodology This section presents the main features of the Kalman filter. Consider a dynamic system represented by

the stochastic linear vector equations: Yt = BtXt + et [A.1]

and Bt§ 1 = mtB t + vt, t = 0, 1, 2, ..., N-l,

where Xt : an exogenous state vector, Yt : the observation vector, e t : a random noise vector, v t : a random noise vector, B t : a random coefficient vector, A t : a given design vector.

The subscript t refers to discrete instants of time.

tics:

[A.21

The following notation will be adopted for this system: E(.) : the expected value, cov(.) : the variance-covariance matrix, Yt : an ordered sequence of the observation vectors Yt, superscript T : the transpose of a matrix,

: equality by definition. The vectors et and vt are assumed to be independent random vectors having the following staffs-

E(et) = 0, [A.31

E(vt) = 0, [A.4]

cov(etej'r) = Qt, [A.5]

cov(vtvjr ) = Rt, [A.6]

Cov(etviT ) = 0, [A.7] for t, j = 0, 1, 2 , . . . , N-1.

The initial state vector B 0 is also assumed to be a random vector with a given mean and covariance. An important and interesting feature of the model is that each time-varying coefficient is not observed, or measured directly. However, the stochastic formulation of the model allows an estimate of each time-varying coefficient by the conditional expectation, E(Bt~Yt) , which is the minimum-variance estimator of B t given yr. In this sense it is the optimal estimator and the KF allows us to obtain each coefficient and the scale economies recursively.

Intuitively, the KF can be interpreted in two ways (see Meinhold and Singpurwalla, 1983): 1. The KF is an updating procedure that consists of forming a preliminary guess about the state of

nature and then adding a correction guess to it, the correction being how well the guess has performed in predicting the next observation,


2. It could be viewed as the evolution of a series of regression functions of estimates on the resid- uals at times 0,1,2,...,t, each having a potentially different intercept and regression coefficient; the evolution stems from a learning process involving all the data.

Now, let

Bt" _ E(Bayt), [A.8]

S t _ cov(Bt~Yt), [A.9]

Pt _ c~ �9 [A.10] Denote the optimal prediction of B t when Yk is available by Bk. 1.

With these definitions, the KF algorithm consist of the following relationships:

Pk+l = mkSkAk T + Qk, [A.11]

Sk+l = Pk " PkXkT(XkPkXk + Rk)'lXkPk, [A.12]

Kk+ll = Sk+lXk+l' i~R-lk+l, [A.13]

Bk+ 1 = AkBk* , [A.14]

Bk+l* = Bk+ 1 + Kk+l(Yk+ 1 - Bk+lXk+l). [A.15] The KF algorithm begins with the forward recursions in equation [A.14]. This computes the one-step

forecast, Bk.1, which is an update at each interval of the previous estimated value. This update is then used in equation [A.15] to estimate the optimal estimator which is a weighted average of Bk+ I and the error that one makes in predicting Yk+l-

Estimating the Model: The model to be estimated is represented by equations [2] and [A-16] - [A.20]:

~i,t = I~i,t-1 + Ui,t [A.16]

otto, t = e%,t. 1 + urn, t [A.17]

amn,t = etmn,t. 1 + Umn,t [A.18]

13ij,t = 131j,t. 1 + uij,t [A.19]

0im,t---- 0im,t. I q- 0im,t [A.20]

where m=n=W,K,F and i=j=L,D.

Equation [2] represents the observation equation and equations [A.16] - [A.20] represent the state equations. Note that each time-varying coefficient is assumed to follow a random walk process. Also each random disturbance term is assumed to follow independent normal distributions. The inclusion of the restrictions in equation [3.a] and [3.b] allows the above model to estimate the scale economies in a recur- sive manner.

NOTES

1. The literature includes Benston (1965, 1972), Bell and Murphy (1968), Benston, Hanweck and Humphrey (1982), Clark (1984), Hunter and Timme (1986), Berger, Hanweck, and Humphrey (1987), Cebenoyan (1988), Evanoff and lsrailevich (1991), Noulas, Ray and Miller (1990), Ferrier and Lovell (1990), Glass and McKillop (1992), Mester (1992), Fields, Murphy, and Tmlro_lu (1993). Also, the Journal of Banking and Finance published a special issue in 1993 that considered various banking issues including scale economies.

2. Chambers (1988, Chapter 6) explains that an embodied technical change will require a change in the input bundle with new inputs as well as in the production function. Disembodied technical change does not require a change in the input bundle but influences the production and the cost functions over time.

The Temporal Behavior of Scale Economies ~thin A Banking Firm 45

3. Berger (1993) notes that scale economies is a local concept and therefore may not provide an accurate picture of global scale efficiencies. He examines this issue with data for a large sample of banks over a long period of time and reports that scale economies do not differ significantly from scale efficiencies.

4. Although Hunter and Timme (1986) assume that time index coefficient captures solely technical change, they (1991, p.342) clearly state that the time index approach is influenced by regulation, financial innovation, and so on and that there is no way to disaggregate the technical change from these other sources.

5. This approach is, however, consistent with factor-augmenting technical change where the usage of an input may decrease or increase as a result of learning process or input substitution due to changes in input productivities over time. Factor-augmenting technical change does not require new inputs and new functional forms over time.

6. Hunter and Timme (1991, p.349) choose aggregated output measures since they primarily want to examine technological change and its interaction with bank size. We follow Hunter and Timme since we want to illustrate the KF method.

7. We thank Anoop Rai for this point.

8. Other estimation results can be obtained from the authors upon request.

9. We thank an anonymous referee for this point.

the temporal behavior of scale economies within a banking firm

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