Selection of Input-Output Variables in Data Envelopment Analysis - Indian Commercial Banks

Transcription

1 Selectio of Iput-Output Variables i Data Evelopmet Aalysis - Idia Commercial Baks Subramayam T Departmet of Statistics, Christ Uiversity, Bagalore ABSTRACT Data Evelopmet aalysis is a oparametric method used to evaluate the performace of profit ad o-profit orgaizatios. It assumes that the iput ad output variables are kow a priori. I each eviromet, there exists a huge umber of iput ad output variables, ad all the variables will serve as either iput or output variable. If there exist a large umber of iput ad output variables, the discrimiatory power of the DEA will reduce. To overcome this difficulty oe may eed to reduce the iput ad output variables usig appropriate scietific methods. This study proposed a ew stepwise method to reduce the data set with the help of o-parametric tests. The outputs are impressive, ad this proposed method is approximately suitable i reducig the isigificat iput ad output variables ad also tried to miimize the wastage of the iput variables about 2% for the selected study. Keywords: DEA, Evirometal Risk, No-Performig Assets, Commercial Bak 1. Itroductio Productivity ad efficiecy are the two relative key terms to defie the workig eviromet of a orgaizatio or a uit. To measure the relative efficiecy of a orgaizatioal uit the most popular oparametric method i literature is the Data Evelopmet Aalysis (DEA), which was origiated by Chares et al. (1978). DEA is a optimizatio method uses liear programmig techiques to evaluate the relative efficiecy of orgaizatioal uits where multiple iputs ad outputs make compariso difficult. I DEA, the uit which is to be evaluated is kow as Decisio Makig Uit (DMU). The focal idea behid the DEA techology is that the efficiecy of each DMU is evaluated comparig to the other DMUs by assigig the favorable weights to the correspodig iput ad output variables. DEA forms a productio possibility set with the available umber of iput ad output variables ad measures the efficiecy of each DMU. The efficiecy score of all the DMUs lies betwee 0 ad 1. If ay DMU assigs efficiecy score 1, we will call it as efficiet otherwise a iefficiet DMU. The discrimiatig power of DEA will deped o the umber of DMUs ad the available umber of iputs ad outputs respectively. DEA does't explai how to idetify the relevat iput ad output variables i the data aalysis. The efficiecy score of a DMU will deped o the icluded iput ad output variables i the data exploratio. If the umber of iput ad output variables is more, the dimesioality of the productio space will icrease ad proportioally the discrimiatory power of DEA will decrease. The greatest challege i DEA is to idetify the parsimoious model. The major drawback i DEA aalysis is it assumes the iput ad output variables are predefied for evaluatio. The efficiecy of a particular Decisio Makig Uit (DMU) depeds based o the selected iput ad output variables. For example, if we look ito the Idia bakig system, there may be eormous literature o bakig efficiecy but there is o geeral agreemet o the choice of iput ad output variables from the available data set (T.Subramayam et. al, 2008, 2014). The efficiecy of a DMU based o the availability of the data. I regressio aalysis, there may be may well-established methods to idetify the sigificat variables. I DEA, there is o such type of methods to decide the importat variables. The available literature o the selectio of variables is isigificat. Recetly some of the researchers focused o this to stregthe the parsimoious or statistics based methods (Larry Jekis et.al, 2013, Hiroshi Morita et.al, 2009, Niraja et.al, 2011). 51 Subramayam T

2 Some of the authors argued that the umber of iput ad output variables must be ot more tha oe-third of the DMUs (Friedma ad Siuay-Ster, 1998). This paper raises a simple statistical method, but based o robust statistical techiques such as parametric ad o-parametric methods. The researchers workig o DEA with little mathematical kowledge are also able to uderstad this approach. 1.2 Objectives of the Study: To idetify the isigificat iput ad output variables To test whether there is ay sigificat differeces betwee the full ad reduced model efficiecy scores To reduce the iput losses usig reduced model 2. Review o variable selectio methods DEA itself does't provide ay guidace i selectio of iput ad output variables ad this selectio left to the user's directio. But, there may be umerous studies o selectio of the variables i DEA. Some authors argued that correlatio aalysis, regressio aalysis ad Priciple Compoet Aalysis (PCA) are useful i selectig iput ad output variables. Larry Jekis et.al, (2003) preseted a multivariate statistical approach to idetify the sigificat variables with least loss of iformatio. This paper discussed the correlatio amog the iput ad output variables. To iclude the iput ad output variable correlatio aalysis utilized as a tool ad if the iput ad output variables have more correlatio that variables are icluded i the model. But, to idetify which variables are appropriate to iclude i the model the variace of the variables cosidered ito the evaluatio. JM Wager et.al, (2007) preseted a method usig the stepwise selectio of iput ad output variables. This paper focused o the least differece of the average efficiecy scores. But, there is o cut poit to stop the variables to be icluded or excluded from the data exploratio. Hiroshi Morita et.al, (2009) discussed a method call desig of experimets. They have selected output variables usig the 2-level fractioal factorial desigs. The test statistic used to idetify the distace betwee two variables is Welch statistic. Niraja et.al, (2011) outlied variable selectio techiques i DEA. This paper discussed four mostly used methods to variable specificatio. These are ECM, priciple compoet aalysis, regressio based test ad test based o bootstrappig methods. The ECM is the best method for low correlatio of variables. 3. Basic CCR ad BCC DEA Models: 3.1 CCR Model: Chares, Cooper ad Rhodes (1978) proposed a liear programmig techique to measure the relative efficiecy of decisio makig uits (DMUs) i a competitive eviromet whe multiple iputs are comprised to produce the multiple outputs. Suppose we have -decisio makig uits (DMUs) with m-iputs ad s-outputs. The DMU j j=1,2,,, is to be evaluated uder ivestigatio with iput ad output vectors X j =(x 1j, x 2j,, x mj ) ad Y j =(y 1j, y 2j,, y sj ) respectively. Where X j > 0 ad Y j > 0. The CCR model to evaluate the relative efficiecy of DMU k is uder costat returs to scale is give by: θ CCR 52 Subramayam T Miλ : j1 λ j x ij λx i0 ; j1 λ u j rj u r0 ; λ j 0,i 1,2,, m; r 1,2,,s

3 3.2 BCC Model: Baker, Chares, ad Cooper (1984) corrected the scale differeces by itroducig a additioal costrait ito the CCR model. The BCC model to evaluate the relative efficiecy of DMUs uder variable returs to scale is give by: BCC Miλ : λ jx ij λx i0 ; λ ju rj u r0; λ j 1, λ j 0; i 1,2,, m; r 1,2,,s j1 j1 This BCC model is kow as the evelopmet problem sice the productio possibility set evelops all the observatios tightly ad hece the ame Data Evelopmet Aalysis. The Scale efficiecy is calculated by usig the ratio CCR. BCC 4 Reductio of Iput/output set i DEA The geeral procedure to evaluate the efficiecy of DMUs is to idetify the suitable iput ad output variables a prior. Sice, the iput ad output variables differ from oe researcher to aother the efficiecy of a DMU will also chage from oe researcher to aother researcher, ad there is o geeral agreemet o the efficiecy of DMUs. To reduce this discrepacy this paper proposed a geeral method to select the sigificat iput ad output variables from the available data set. The earlier literature has show that i geeral there is a high correlatio betwee the iput ad output variables ad the correlatio ad regressio aalysis o more suitable to reduce or idetify the sigificat variables from the data set. The followig stepwise procedure is useful to idetify the sigificat variables or reduce the variables i the data set. 4.1 Stepwise Procedure Assume that we have -decisio makig uits with m (i=1, 2, m) iput variables ad s (j=1, 2,,s) output variables from the available data set. To reduce the umber of iput ad output variables we proposed the followig stepwise procedure. Step1: Ru the full model with costat retur to scale ad store i E. Step2: Drop oe variable at a time ad ru the DEA model. Store the efficiecy values i the set E ij, i=1,2,,i, j=1,2,,o Step3: Use oparametric test for testig the sigificace of the dropped variables. Observe the percetage chage i sigificace value. Step4: If the percetage chage i sigificace value is greater tha 20% retai the variable otherwise exclude the variable from the data exploratio. Step5: If more tha oe variable has the sigificat value less tha 20%, use mea efficiecy chage ad remove the variable with least sigificat chage. Step6: Ru the DEA model with ew set of variables ad repeat the steps 1-5. Step7: Repeat the procedure util all variables percetage chage i sigificace value is greater tha 20%. At least oe iput ad output variable is required to ru the fial model. 5 Empirical Study: Idia Commercial Baks Idia bakig system is a secure ad stable idustry comparig to most of the developig coutries. I Idia, the baks were workig i a heterogeeous eviromet whose maagemet policies, importace to urba ad rural areas are extremely differet amog the maagemets. All the researchers assumed that the baks were workig uder the same frotier to evaluate their efficiecy (T. Subramayam et.al, 2008, 2012). If a bak is i a efficiet eviromet, oe may thik to prefer that bak comparig to other baks. 53 Subramayam T

4 To calculate the efficiecy of a commercial bak usig DEA models first we eed to idetify the possible umber of iput ad output variables from the available data set. I this study, we select the followig variables as the iput ad output variables. Iput Variables: 1. Number of employees 2. Fixed assets Output Variables: 1. Deposits 2. Ivestmets 3. Advaces 4. Iterest icome 5. Other icome 6. Results ad aalysis 6.1 Stepwise Method The preset data relatig to 26 public sector baks is workig uder govermet sector eviromet. This study attempted to reduce the possible umber of output variables from the data exploratio. From Table1, we ca observe that there is a high correlatio amog the iput ad output variables. There exists high correlatio amog all the variables. I geeral, DEA deals with the highly correlated variables i ature. I the give data, there exist strog correlatio amog the output variables, but techically they are ot correlated i ature. All the variables are idepedet i ature ad are all the products of the baks. The biggest problem with correlatio aalysis is that which variables ca omit with least loss of iformatio. The proposed stepwise method applied to idetify the isigificat variable from the available data set. The iput-orieted DEA model used to calculate the efficiecy of commercial baks. First, the iput-orieted DEA model performed with 2-iputs ad 5-output variables. By droppig oe variable at a time we got the sigificat differeces. I stage1 (from table2) the variable Ivestmet has the lease sigificat differece ad the correspodig chage i sigificat level is 0%. This variable does t have ay sigificat impact o the overall efficiecy ad amog all the variables this variable is a appropriate oe to exclude from the data exploratio. I stage2 (from table3) with 2-iput ad 4-output variables, the overall efficiecy score seems to be Droppig each of the output variables separately results i chages i average efficiecy scores that seem fairly substatial. The variable Iterest Icome' has the least sigificat differece with 0% chage i sigificat level. Of course, the appropriate variable to elimiate at this stage is Iterest Icome'. I stage3 (from table4) with 2-iput ad 3-output variables the overall efficiecy score is By droppig each of the output variables, we observe that all the variables have miimum 1% chage i mea efficiecy score. The chage i sigificat level is approximately more tha 20% i all the variables. Based o the cotributio of the chages i mea efficiecy ad sigificat level all the three variables amely, Other Icome, Advaces, ad Deposits were selected as sigificat output variables for the efficiecy evaluatio. 6.2 Statistical sigificace The sigificace chage at 5% level compared usig Wilcoxo Matched Pair siged rak test. This test examied the sigificat of full ad reduced models i each stage. The CCR ad Scale efficiecy chages are ot statistically sigificat, ad there is some sigificat effect i BCC scores. CCR BCC Scale Full Model Reduced Model Sig * Subramayam T

5 I CCR eviromet, the major chages occur i Cetral Bak of Idia (1.42%), State Bak of Hyderabad (5.08%) ad State Bak of Travacore (3.99%) respectively. I BCC eviromet, huge chages occur i some of the bak efficiecies amely, Caara Bak(17.35%), Cetral Bak of Idia(8.10%), Pujab Natioal Bak (13.61%), Uio Bak of Idia (11.43%) ad State Bak of Hyderabad (1.28%). There is o chage i the umber of efficiet DMUs before ad after the stepwise method. The major chage i scale efficiecy scores occurs i Caara Bak (15.25%), Cetra Bak of Idia (11.93%), Pujab Natioal Bak (13.18%) ad Uio Bak of Idia (13.22%) respectively. The compariso betwee full ad reduced model reveals that there is 1% chage i the average efficiecy scores i CCR eviromet ad 2% average efficiecy chage i BCC eviromet. The reduced model tried to miimize the wastage of the iput about 2% comparig to the full model. 7. Coclusios: I this paper, a stepwise method was developed to reduce the iput waste by reducig the isigificat output variables from the data. This study depeds o the mea efficiecy chage ad sigificat chages. This method is useful for researchers with little statistical kowledge. The Wilcoxo Siged Rak test used to compare the chages are statistically sigificat or ot betwee full ad reduced models. The CCR model utilized as the base for performig the stepwise method. This study fixed two iput variables amely, the umber of employees ad fixed assets as iput variables ad tried to reduce the output set; this leads to miimize the iput losses. I CCR eviromet, there is o sigificat chage betwee full ad reduced model, but there is a statistically sigificat differece betwee the full ad reduced models i BCC eviromet. The reduced model miimized o a average 2% iput loss i each of the DMU. This paper tried to develop a ew stepwise method with the help of the existig literature i DEA model. This model will provide some cofidece amog the researchers i reducig the isigificat i iput ad output variables. 8. Refereces: 1. Baker, R. D., Chares, A., & Cooper, W. W. (1984). Some models for estimatig techical ad scale iefficiecies i data evelopmet aalysis. Maagemet sciece, 30(9), Chares. A., Cooper. W.W., ad Rhodes, E. 1978, Measurig the efficiecy of decisio makig uits, Europea Joural of Operatioal Research 2(2): Friedma, L., & Siuay-Ster, Z. (1998). Combiig rakig scales ad selectig variables i the DEA cotext: The case of idustrial braches. Computers & Operatios Research, 25(9), Jekis, L., & Aderso, M. (2003). A multivariate statistical approach to reducig the umber of variables i data evelopmet aalysis. Europea Joural of Operatioal Research, 147(1), Subramayam T

6 5. Morita, H., & Avkira, N. K. (2009). SELECTING INPUTS AND OUTPUTS IN DATA ENVELOPMENT ANALYSIS BY DESIGNING STATISTICAL EXPERIMENTS (< Special Issue> Operatios Research for Performace Evaluatio). Joural of the Operatios Research Society of Japa, 52(2), Nataraja, N. R., & Johso, A. L. (2011). Guidelies for usig variable selectio techiques i data evelopmet aalysis. Europea Joural of Operatioal Research, 215(3), Subramayam, T., & Reddy, C. S. (2008). Measurig the risk efficiecy i Idia commercial bakig-a DEA approach. East-West Joural of Ecoomics ad Busiess, 11(1-2), Reddy, C. S., & Subramayam, T. (2011). Data Evelopmet Aalysis Models to Measure Risk Efficiecy: Idia Commercial Baks. IUP Joural of Applied Ecoomics, 10(4), Subramayam, T. (2013). TECHNICAL AND RISK EFFICIENCY EVALUATION OF INDIAN COMMERCIAL BANKS USING DEA MODELS. Iteratioal Joural of Iformatio, Busiess ad Maagemet, 5(3), Wager, J. M., & Shimshak, D. G. (2007). Stepwise selectio of variables i data evelopmet aalysis: Procedures ad maagerial perspectives. Europea joural of operatioal research, 180(1), Table1: Correlatio: Employees Employees 1 Fixed Assets Appedix Deposits Fixed Assets Deposits Ivestmets Advaces Ivestmets Advaces Iterest Icome Iterest Icome Other Icome Other Icome Table2:Stage1 Overall Mea Efficiecy (2- iputs, 5-outputs) Efficiet Baks Mea Efficiecy Variables Dropped mea efficiecy Sig. Sig. level (%) Other Icome Iterest Icome Advaces Ivestmets Deposits Subramayam T

7 Table3: Stage2 Step1: Overall Mea Efficiecy (2I, 4O) Variables Dropped Efficiet Baks Mea Efficiecy mea efficiecy Sig. Sig. level (%) Other Icome Iterest Icome Advaces Deposits Table4:Stage3 Step2: Overall Mea Efficiecy (2I, 3O) Variables Dropped Efficiet Baks Mea Efficiecy mea efficiecy Sig. Sig. level (%) Other Icome Advaces Deposits Subramayam T