Stochastic-MPI for Measuring Total Factor Productivity Index: With an ...

International Conference on Arts, Economics and Literature (ICAEL'2012) December 14-15, 2012 Singapore

Stochastic-MPI for Measuring Total Factor Productivity Index: With an Illustration of Malaysian Commercial banks Deng, Q.1, Wong, W.P. School of Management, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

Abstract—The conventional malmquist productivity index (MPI) measures the total factor productivity (TFP) relative of a set of decision making units (DMUs) with exact values of inputs and outputs. However, for imprecise data, i.e., mixtures of interval data, the conventional way becomes incapable, the issue of TFP measuring and evaluating for imprecise data is getting widespread attention. This paper suggests a method for introducing a stochastic element in measuring TFP changes with nonparametric data envelopment analysis (DEA) approach of MPI. Based on the concept of general distributions and the application of Monte-Carlo simulation, the Stochastic-MPI generates the interval MPI by using computational power to derive empirical distributions for the TFP measures. The method is applied to the case of Malaysian commercial banking industry.

where

λ ∈ (0,1] and Do t(y,x) ≤ 1 if and only if y ∈ P t ( x).

The value of the distance function is the reciprocal of the Farrell technical efficiency. (Fare et al., 1994a). A distance function which measures the maximal proportional change in output required to make (xt,yt) feasible in relation to the t

technology at t+1 which we call Do ( x

I. INTRODUCTION

T

HIS paper develops a new method in measuring TFP changes with nonparametric DEA approach of MPI for stochastic data. Based on the concept of general distributions and the application of Monte-Carlo simulation, the Stochastic-MPI generates the interval MPI by using computational power to derive empirical distributions for the TFP measures.

II. METHODOLOGY: STOCHASTIC-MPI In a multiple-input and multiple-output production system, where inputs x t are used to produce outputs y t in time period t,t=1,2,. . . ,T. This can be defined as

Mt =

Dot ( x t +1 , y t +1 ) Dot ( x t , y t )

(4)

In order to avoid choosing an arbitrary benchmark, the output-based Malmquist productivity change index is defined as the geometric mean of the above two Malmquist productivity index.

M o(x t +1 ,y t +1 ,x t ,y t ) = [M t • M t +1 ]1 / 2 =  D ot(x t +1 ,y t +1 )  t t t   D o (x ,y ) 

 D ot +1(x t +1 ,y t +1 )  t +1 t t   D (x ,y )    o

1/2

(5)

From the Equation (5) we can see that we need to calculate four distances that is to solve four different linear-programming problems. Assume we have M inputs and N outputs, with S=M+N.

Dot ( x t , y t ) −1 = max λ st M

(1) The TFP index of Malmquist is defined using distance function D o (x,y), where an output distance function is used to consider a maximum proportional expansion of the output, y, given the inputs, x at time t. (Shephard, et al., 1970)

λyit ≤ ∑ λi yi t

Do ( y t , x t ) = inf{λ : y t / λ ∈ P t ( x)}

λi ≥ 0

i = 1,..., M

i =1

S

∑λ x

i = M +1

(2)

i i

t

(6)

≤x

t i

i = M + 1,..., S i = 1,..., S

The computation of Deng, Q. is a PhD student at School of Management, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia; e-mail: dq11_man112@ student. usm.my.

(3)

Dot +1 ( x t +1 , y t +1 ) Dot +1 ( x t , y t )

M t +1 =

P t ( xt ) = { yt : yt can be produced by xt at time t}

t

, y t +1 ) . Equation 3

and Equation4 are the Malmquist productivity index at period t and t+1.

Keywords—Stochastic-MPI, Malmquist index, Monte-Carlo simulation, DEA, Malaysian banking industry

t +1

Dot +1 ( x t +1 , y t +1 ) is exactly like

equation (6), where just need to use t+1 replace t. The other two distances require the information from two periods.

122

International Conference on Arts, Economics and Literature (ICAEL'2012) December 14-15, 2012 Singapore

Dot ( x t +1 , y t +1 ) −1 = max λ

and get δ *x* = max{ t

st

λy

M

≤ ∑ λi yi

t +1 i

i = 1,..., M

t

(7)

i =1

(11) If we set α = 0.05 , we know the mean value and standard diversion; we can estimate the confidence t

S

t ∑ λi xi ≤ xit +1

interval of x* as:

i = M + 1,..., S

x t* ,l = xˆt* − 1.96δ *t ,x t* ,u = xˆt* + 1.96δ *t (12)

i = M +1

λi ≥ 0

i = 1,..., S

Note in Equation (7), the observations from period t and t+1 are involved. In Equation (6), ( x , y ) ∈ P ( x ), so Do ( x , y ) ≤ 1 , but in Equation t

(7), ( x

t

t +1

t

x*t ,l − xˆ*t x*t ,u − xˆ*t , } − 1.96 1.96

t

t

t

t

, y t +1 ) does not need to belong to P t ( x t ) . t

So the value of Do ( x t +1

t +1

t

, y t +1 ) may be greater than 1. The t

computation of Do ( x , y ) is exactly like equation (6), but the t and t+1 superscripts are transposed. Nevertheless, when some values of the inputs or outputs are uncertain, or we just get the mean or the interval information about the variable. In order to have an accurate result for the Malmquist productivity change index, we add the technical of Monte-Carlo simulations, and run certain replications to get the more accurate index value. Assume we have a productivity system of n decision making units with N inputs and M outputs during the period t, t=1,…, T.

Note that during in the progress of computation, we usually use the sample mean to estimate the true mean, and the sample standard deviation to estimate the true standard deviation if true values are unknown. In the progress of Monte-Carlo simulation, random data is generated following the interval using normal distribution; these generated data will be used to compute the Malmquist productivity change index; a total of n1 replications will be simulated; and n1 index values will be calculated; the confidence interval can then be estimated using Equation (12), with that, more accurate index information can be obtained. The Figure 1 shows the follow chart of the Stochastic-MPI:

t

Ut is the set of uncertain variable, U * is the set of the sample mean of Ut, for an arbitrary

x t ∈ U t ,x t* ∈ U *t ,and x t* ,l ≤ x t ,x t* ≤ x t* ,u , where x t* ,l ,x t* ,u ,is the lower bound, uper bound of x t . Without

loss

of

generality,

we

assume

that

x ~ N ( xˆ , δ ), where xˆ is the true mean of x ,set α as t

t *

x*t *

t *

t

confidence level, then

P ( x*t ,l ≤ x*t ≤ x*t ,u ) = 1 − α

(8)

Base on the knowledge of normal distribution, we have

x − xˆ*t t *

δ *x

t *

P(

~ N (0,1) , then Equation (8) can be replaced as:

x*t ,l − xˆ*t

δ *x

t *

≤

x*t − xˆ*t

δ *x

t *

≤

x*t ,u − xˆ*t

δ *x

t *

) = 1−α

(9)

As we check from the standard normal distribution table,

P(−1 ≤ x ≤ 1) ≈ 0.683, P(−1.96 ≤ x ≤ 1.96) ≈ 0.95 P (−2 ≤ x ≤ 2) ≈ 0.954, P(−3 ≤ x ≤ 3) ≈ 0.997 If we set α = 0.05 , we know the mean value and interval information, we can estimate

δ *x through t *

x*t ,l − xˆ*t

δ *x

t *

= −1.96, or

x*t ,u − xˆ*t

δ *x

t *

= 1.96 (10)

Fig. 1 Flowchart of Stochastic-MPI III.

DATA AND SIMULATIONS

The proposed method was applied on a panel data of 24 commercial banks in Malaysia during the period 2001-2008. The intermediation approach was used whereby output measures comprised of Loans and Advances (LA), Profit (P), while the input measures are Total Assets (TA), Staff (S) and Deposit (D). D is used as the estimate of true mean value of total deposit. D is a dynamic variable; we define Cd as changing

123

International Conference on Arts, Economics and Literature (ICAEL'2012) December 14-15, 2012 Singapore

degree, D* as the real mean deposit in a year. Therefore,

(13) D* ~ [ D(1 − Cd ), D(1 + Cd )] Here we assume Cd=5%, D ~ D , where D is the true

mean of

the total deposit. Following Equation (13), D* ~ [0.95 D,1.05 D] . Table 4 shows the D* during the period of year 2001-2008. The Malmquist productivity change index is then calculated using Equation (3)-(7). We set n 1 =10000 and α = 0.05 . The program is coded using Matlab 7.0. Table 1in Appendix 1 gives the comparison of the initial result and the simulation result. By using the result of TFPC after the Monte-Carlo simulations, the BCB recorded the highest growth in TFPC (296.4%~296.9%) and JPMB records the lowest growth in TFPC with -5.7%~-5.6%. By using the mean value basing on the 95% confidence interval, 12 banks, i. e. , BCB, DBB, BAB, BBB, PBB, UOB, BTB, EBB, HSBC, OCBC, AMB and ABB had positive growth (TFP>1) while 9 banks(i.e., SCBB, BNSB, CitB, CIMB, RHB, MAYB, ABMB, JPMB and HLBB) recorded negative growth (TFP < 1). From the positive growth ones, only 3 are locally owned banks, i.e., EBB, AMB and ABB. The whole commercial banking system in Malaysia during the period 2001-2008 has a positive growth of 15.2%~21.6%. The trend of the whole Malaysia banking system from the period 2001-2008 is in a dynamic changing process, from 2001-2003,2004-2005, it is decreased and from 2003-2004,2005-2007,it is increased. Nevertheless, the whole trend of the total productivity change in banking efficiency is waved decreased by 99.7%~112.5%. A possible reason maybe due to the impact of technology on the banking system productivity. Developments in technology cause a series of opportunities and challenges; management needs to look into the technical (e.g. the updating of hardware for productivity) as well as the soft aspect (e.g. improve managerial skills) to sustain the whole banking productivity system.

growth between the ranges of 15.2%~21.6%, the TFP is in a dynamic changing process in the choused period. 12 banks kept the record of increased in TFP index while 9 banks decreased. It seems that the commercial banks industry has improved in these 8 years, while with the rapid development in information and communication technology(ICT), banks are able to use internet banking, telephone banking, ATM machines and web payment system, which will help to improve the working efficiency. However, banks also will face to a series of challenges in the future, i.e., the rising of labor costs, the intense competition from the same trade or occupation, and the uncertain risk of the economic development. ACKNOWLEDGMENT This study was supported by the Research Grant from Research Creativity and Management Office of Universiti Sains Malaysia (Account No: 1001/PMGT/816191)

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[6] [7]

[8]

IV. CONCLUSION Many MPI studies overstate the level of industry productivity. The conventional way measures the TFP relative of a set of DMUs with exact values of inputs and outputs. However, for stochastic data, the general method becomes incapable. Therefore, we suggest a method of Stochastic-MPI to measure TFP changes with nonparametric data envelopment analysis (DEA) approach of MPI. Based on the concept of general distributions and the application of Monte-Carlo simulation, the Stochastic-MPI generates the interval MPI by using computational power to derive empirical distributions for the TFP measures. This technical also can be widely applied in the other areas which have uncertain values. We apply the algorithm into a balanced panel of Malaysian banks over 2001–2008. This application illustrates the performance of the total commercial banking industry by successfully solving the uncertain data with the statistic theory and simulation technique. The result expresses that the commercial banking system in Malaysia has kept a positive

[9]

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Aigner, D., Lovell, C.A.K., & Schmidt, P. (1977). Formulation and Estimation of Stochastic Frontier Production Function Models. Journal of Econometrics ,6, 21–37. Allen, D., & Boobal-Batchelor, V. (2005). The Role Of Post Crisis Bank Mergers In Enhancing Efficiency Gains And Benefits To The Public In The Context Of A Developing Economy:Evidence From Malaysia. Modsim,2005:International Congress Modeling and Simulation:Advances and Applications for Management and Decision Making,2275-2282. Charnes A, Cooper,W.W. & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research,2, 429–444. Drake, L. (2001). Efficiency and productivity change in UK banking. Applied Financial Economics, 11, 557−571. Deng, Q., Wong, W. P., Wooi, H. C., & Xiong, C. M. (2011). An Engineering Method to Measure the Bank Productivity Effect in Malaysia during 2001-2008,Journal of System engineering Procedia,2,1-11. Farrel MJ. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society Series A, 120, 253-281. Fäere, R., S. Grosskopf, & C. A. K. Lovell. (1985). The Measurement of Efficiency and Production.: Kluwer-Nijhoff Publishing Analysis,16, 753-762. Fare, R., Grosskopf, S., Norris, M., & Zhang, Z. (1994). Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries. The American Economic Review,84(1),66-83. Kao, C., & Liu, S.T. (2011). Stochastic Malmquist productivity index, Statistical Concepts and Methods for the Modern World, December 28-30, 2011 Lotfi, F. H. (2006). Application of Malmquist Productivity Index on Interval Data in Education Groups. International Mathematical Forum, 1, 753-762. Meeusen, W., & J. van den Broeck. (1977). Efficiency Estimation for Cobb–Douglas Production Functions with Composed Error. International Economic Review ,18, 435–444. Shephard, R.W. (1970). Theory of Cost and Production Func-tions. Princeton University Press, Princeton. Sufian,F. & Habibullah, M. S. (2010). Does Foreign Banks Entry Fosters Bank Efficiency? Empirical Evidence from Malaysia.Inzinerine Ekonomika Engineering Economics,21(5),464-474. Tortosa-Ausina, E., Grifell-Tatjé, E., Armero, C., & Conesa, D. (2008). Sensitivity analysis of efficiency and Malmquist productivity indices: An application to Spanish savings banks. European Journal of Operational Research, 184(3), 1062-1084. W. D. Doyle, “Magnetization reversal in films with biaxial anisotropy,” in 1987 Proc. INTERMAG Conf., pp. 2.2-1–2.2-6.

International Conference on Arts, Economics and Literature (ICAEL'2012) December 14-15, 2012 Singapore

Appendix 1 Table 1: Comparison of the initial result and the simulation result 2001-2008. (I:Initial result,M:Monte-Carlo result,Interval:Interval result by Monte-Carlo simulations) Period 2001-2002 DMU I

2001-2003

M Interval I

2001-2004

M Interval I

2001-2005

M Interval I

2001-2006

M Interval I

2001-2007

M Interval I

2001-2008

M Interval I

Mean

M Interval I

M Interval

1

1. 043 1. 045 [1. 013,1. 077] 0. 968 0. 969 [0. 922,1. 017] 0. 972 0. 972 [0. 929,1. 015] 1. 121 1. 120 [1. 092,1. 147] 0. 998 0. 998 [0. 980,1. 017] 1. 002 1. 002 [0. 960,1. 043] 0. 911 0. 911 [0. 855,0. 968] 1. 002 1. 002 [0. 964,1. 040]

2

0. 993 0. 993 [0. 982,1. 005] 1. 049 1. 053 [1. 023,1. 083] 0. 978 0. 980 [0. 950,1. 011] 0. 980 0. 978 [0. 967,0. 990] 1. 096 1. 100 [1. 096,1. 104] 0. 897 0. 897 [0. 853,0. 942] 0. 902 0. 903 [0. 855,0. 951] 0. 985 0. 986 [0. 961,1. 012]

3

1. 041 1. 043 [1. 030,1. 055] 0. 934 0. 934 [0. 934,0. 934] 1. 128 1. 128 [1. 126,1. 129] 1. 471 1. 471 [1. 471,1. 471] 0. 684 0. 682 [0. 676,0. 688] 0. 931 0. 931 [0. 920,0. 942] 0. 987 0. 984 [0. 967,1. 002] 1. 025 1. 025 [1. 018,1. 032]

4

0. 993 0. 993 [0. 990,0. 996] 1. 016 1. 014 [0. 986,1. 042] 0. 579 0. 580 [0. 578,0. 582] 1. 383 1. 381 [1. 370,1. 393] 0. 665 0. 664 [0. 661,0. 667] 0. 999 0. 999 [0. 998,0. 999] 0. 946 0. 946 [0. 939,0. 953] 0. 940 0. 940 [0. 932,0. 947]

5

0. 950 0. 954 [0. 944,0. 964] 1. 045 1. 038 [1. 015,1. 061] 0. 932 0. 939 [0. 922,0. 956] 1. 199 1. 188 [1. 121,1. 254] 0. 823 0. 824 [0. 794,0. 854] 0. 952 0. 958 [0. 892,1. 025] 0. 932 0. 929 [0. 878,0. 979] 0. 976 0. 976 [0. 938,1. 013]

6

1. 136 1. 135 [1. 121,1. 150] 0. 887 0. 887 [0. 885,0. 890] 0. 849 0. 850 [0. 839,0. 860] 1. 166 1. 166 [1. 153,1. 179] 1. 098 1. 097 [1. 094,1. 101] 0. 918 0. 918 [0. 916,0. 920] 0. 957 0. 957 [0. 957,0. 957] 1. 002 1. 001 [0. 995,1. 008]

7

1. 022 1. 022 [1. 022,1. 022] 1. 070 1. 070 [1. 070,1. 071] 1. 167 1. 160 [1. 151,1. 170] 1. 347 1. 358 [1. 331,1. 384] 0. 724 0. 728 [0. 715,0. 741] 1. 043 1. 023 [0. 962,1. 085] 0. 986 1. 000 [0. 941,1. 058] 1. 051 1. 052 [1. 027,1. 076]

8

1. 067 1. 067 [1. 058,1. 076] 0. 997 0. 993 [0. 979,1. 006] 0. 925 0. 925 [0. 902,0. 949] 0. 872 0. 870 [0. 855,0. 884] 0. 995 0. 997 [0. 960,1. 034] 1. 126 1. 121 [1. 055,1. 187] 0. 911 0. 916 [0. 851,0. 981] 0. 985 0. 984 [0. 951,1. 017]

9

1. 027 1. 027 [1. 018,1. 036] 1. 158 1. 156 [1. 136,1. 175] 0. 956 0. 956 [0. 949,0. 963] 0. 694 0. 694 [0. 694,0. 694] 1. 107 1. 107 [1. 107,1. 108] 1. 161 1. 161 [1. 161,1. 161] 0. 890 0. 890 [0. 888,0. 893] 0. 999 0. 999 [0. 993,1. 004]

10

1. 189 1. 189 [1. 189,1. 189] 0. 991 0. 991 [0. 990,0. 991] 0. 958 0. 958 [0. 957,0. 959] 0. 948 0. 948 [0. 948,0. 948] 1. 051 1. 051 [1. 050,1. 051] 1. 111 1. 112 [1. 111,1. 112] 0. 913 0. 913 [0. 911,0. 914] 1. 023 1. 023 [1. 022,1. 023]

11

1. 017 1. 015 [1. 008,1. 023] 1. 043 1. 039 [1. 005,1. 073] 0. 998 0. 997 [0. 951,1. 044] 1. 037 1. 038 [1. 020,1. 056] 1. 013 1. 012 [1. 002,1. 022] 0. 922 0. 921 [0. 913,0. 930] 1. 019 1. 015 [0. 998,1. 031] 1. 007 1. 005 [0. 985,1. 026]

12

0. 922 0. 922 [0. 915,0. 930] 0. 977 0. 978 [0. 975,0. 980] 0. 958 0. 958 [0. 946,0. 970] 1. 044 1. 044 [1. 033,1. 054] 0. 902 0. 901 [0. 899,0. 903] 1. 000 1. 001 [1. 000,1. 002] 0. 898 0. 898 [0. 898,0. 898] 0. 957 0. 957 [0. 952,0. 963]

13

1. 880 1. 880 [1. 880,1. 880] 0. 450 0. 450 [0. 450,0. 450] 0. 912 0. 911 [0. 879,0. 942] 1. 295 1. 296 [1. 279,1. 313] 0. 864 0. 863 [0. 858,0. 868] 1. 130 1. 129 [1. 107,1. 151] 0. 898 0. 900 [0. 871,0. 930] 1. 061 1. 061 [1. 046,1. 076]

14

0. 880 0. 881 [0. 843,0. 919] 0. 555 0. 555 [0. 525,0. 585] 1. 628 1. 628 [1. 516,1. 739] 0. 756 0. 756 [0. 719,0. 793] 1. 420 1. 427 [1. 369,1. 484] 1. 373 1. 371 [1. 355,1. 386] 1. 088 1. 088 [1. 085,1. 092] 1. 100 1. 101 [1. 059,1. 143]

15

22. 299 22.328 [20. 73,23. 927] 0. 384 0. 385 [0. 366,0. 404] 1. 232 1. 233 [1. 161,1. 305] 0. 813 0. 815 [0. 765,0. 864] 0. 966 0. 967 [0. 905,1. 028] 1. 118 1. 117 [1. 073,1. 162] 0. 933 0. 935 [0. 885,0. 984] 3. 964 3. 969 [3. 698,4. 239]

16

1. 060 1. 059 [1. 032,1. 087] 0. 997 0. 999 [0. 964,1. 034] 1. 014 1. 016 [0. 996,1. 036] 0. 772 0. 772 [0. 769,0. 775] 1. 130 1. 129 [1. 126,1. 132] 1. 302 1. 302 [1. 301,1. 303] 0. 915 0. 915 [0. 915,0. 916] 1. 027 1. 027 [1. 015,1. 040]

17

0. 804 0. 804 [0. 768,0. 839] 1. 014 1. 015 [0. 981,1. 050] 0. 179 0. 179 [0. 179,0. 179] 7. 494 7. 494 [7. 494,7. 494] 0. 816 0. 817 [0. 813,0. 820] 0. 929 0. 929 [0. 929,0. 929] 1. 274 1. 274 [1. 271,1. 277] 1. 787 1. 787 [1. 777,1. 798]

18

0. 432 0. 432 [0. 424,0. 440] 0. 846 0. 844 [0. 826,0. 862] 1. 197 1. 198 [1. 167,1. 228] 0. 958 0. 959 [0. 912,1. 005] 1. 124 1. 123 [1. 082,1. 163] 1. 192 1. 194 [1. 144,1. 244] 0. 856 0. 853 [0. 834,0. 872] 0. 944 0. 943 [0. 913,0. 973]

19

1. 018 1. 017 [0. 998,1. 037] 0. 894 0. 893 [0. 872,0. 914] 1. 127 1. 127 [1. 127,1. 127] 1. 003 1. 002 [0. 976,1. 029] 1. 003 1. 002 [0. 983,1. 022] 0. 825 0. 825 [0. 810,0. 839] 1. 126 1. 127 [1. 087,1. 167] 0. 999 0. 999 [0. 979,1. 019]

20

0. 954 0. 954 [0. 952,0. 956] 1. 240 1. 238 [1. 215,1. 260] 0. 961 0. 959 [0. 921,0. 998] 0. 946 0. 946 [0. 913,0. 979] 0. 989 0. 991 [0. 959,1. 023] 1. 122 1. 118 [1. 083,1. 153] 1. 088 1. 089 [1. 053,1. 125] 1. 043 1. 042 [1. 014,1. 071]

21

1. 010 1. 009 [0. 993,1. 025] 1. 110 1. 109 [1. 073,1. 145] 1. 074 1. 071 [1. 040,1. 103] 0. 927 0. 925 [0. 907,0. 943] 1. 010 1. 010 [1. 006,1. 013] 0. 838 0. 838 [0. 838,0. 838] 1. 032 1. 032 [1. 030,1. 033] 1. 000 0. 999 [0. 984,1. 014]

Mean

2. 035 2. 037 [1. 948,2. 125] 0. 935 0. 934 [0. 914,0. 954] 0. 987 0. 987 [0. 951,1. 013] 1. 344 1. 344 [1. 323,1. 364] 0. 975 0. 976 [0. 959,0. 993] 1. 042 1. 041 [1. 018,1. 065] 0. 974 0. 975 [0. 951,0. 999] 1. 185 1. 185 [1. 152-1. 216]

125