Chapter 6 Efficiency Ranking Method using SFA and SDEA: Analysis and Discussion

Transcription

1 Chapter 6 Efficiency Ranking Method using SFA and SDEA: Analysis and Discussion 206

2 The proposed approach, in this chapter is based on the theme of integration of SFA and Super efficiency model of DEA (SDEA). This model is called Efficiency Ranking Method by SFA and SDEA (ERM-SSD). It acknowledges stochastic nature of data and recommends the best alternative whose average performance is compared with the best alternative in the set of alternatives. The proposed model is illustrated using a hypothetical data set with two inputs and two outputs and with three inputs and three outputs. The proposed approach is validated using data obtained from PSU banks operating in India. The ranks obtained by the proposed model are compared with the conventional models such as CRS-DEA and Super efficiency DEA models using two different techniques namely: Spearman s rank test and Mean Squared Deviation (MSD). It is hoped that the proposed model is able to consider the case of multiple inputs and multiple outputs in SFA framework and has good amount of discrimination power while ranking the DMUs. 6.1 Introduction Stochastic Frontier analysis (SFA) and Data Envelopment Analysis (DEA) are widely used Multi-Criteria Decision Making (MCDM) tools for performance evaluation and benchmarking (Bazrkar and Khalilpour, 2013; Odeck and Bråthen, 2012; Goncharuk, 2011; Wu et al., 2011; Thoraneenitiyan and Avkiran, 2009; Reinhard et.al, 2000; Coelli and Perelman, 1999). SFA a parametric technique and DEA a non-parametric technique are often used by academicians and practicing managers in complex business situation. For measuring cost and revenue efficiency of the property-casualty insurance companies, Park et al. (2009) proposed a model using SFA. The study considered different types of insurance distribution systems in the U.S. to evaluate the impact of the ownership pattern on the efficiency of these companies. Baltas (2005), applied two-stage SFA model to find the consumer differences in food demand. In this study, SFA was presented as an alternative methodology for analyzing and segmenting store clientele. Vencappa and Thi (2007) applied SFA for decomposing the productivity growth of foreign banks in Czech Republic, Hungary and Poland. Hiebert (2002) applied SFA for estimating cost and operational efficiency of electric generating plants for the period 1988 to Mohamad and Said (2012) used Super efficiency Data Envelopment Analysis (SDEA) model to measure and assess the performance of selected largest listed companies in Malaysia. SDEA model was able to provide distinct ranking to these companies as against those by usual DEA model. It was concluded that that top-ranked companies on the basis of revenue generated are not necessarily top-ranked performers. Chen et al. (2012) used SDEA and DEA to measure 207

3 efficiency of financial and non-financial holding companies in Taiwan. DEA ranked multiple companies as efficient whereas SDEA was able to further rank efficient companies and assign distinct ranks. Yawe (2010) used standard DEA and SDEA models to measure efficiency of hospitals operating in Uganda. Using SDEA model, ranking of the efficient units was possible. Hospitals were further categorized into four groups: strongly super-efficient; super-efficient; efficient and inefficient. Few studies which integrated SFA and DEA to derive benefits from both of them can also be cited. For example, Thoraneenitiyan and Avkiran (2009) used a three stage integrated model using SFA and DEA to measure the impact of restructuring and country-specific factors on the efficiency of post-crisis East Asian banking systems. In the first stage, a non-oriented Slack based (SBM) DEA was used to assess technical efficiency of banks without considering environmental effects. In the second stage, SFA was used for measuring the impact of the environment on bank inefficiency. In stage three, SBM was repeated with adjusted data. The results indicated that domestic mergers play a significant role in developing efficient banks, and restructuring does not lead to more efficient banking systems. Banking system inefficiencies were mostly attributed to country-specific conditions, particularly, high interest rates, concentrated markets and economic development. Azadeh et al., (2009) presented an integrated approach using DEA, Corrected ordinary least squares (COLS), SFA, Principal Component Analysis (PCA) and Numerical Taxonomy (NT) for performance assessment, optimization and policy making of electricity distribution units. This model accounted for both static and dynamic aspects of information on environment due to involvement of SFA. The integrated approach gave an improved ranking methodology and facilitates better optimization of electricity distribution systems. To illustrate the usability and reliability of the proposed algorithm, thirty eight electricity distribution units in Iran were considered, ranked and optimized by the proposed algorithm. Coelli and Perelman (1999) used SFA framework to investigate technical inefficiency in European railways. The objective of the paper was to compare the results obtained from the three alternative methods for estimating multi-output distance functions. Techniques applied were Parametric frontier using Linear Programming (PLP); DEA and Corrected Ordinary Least Squares (COLS). Input-orientated, output-orientated and constant returns to scale (CRS) distance functions were estimated and compared. The results indicated a strong degree of correlation 208

4 between the input and output oriented results for each of the three methods. Authors suggested a method of combining the technical efficiency scores obtained from the three different methods using geometric mean. Ghaderi, et al. (2006), integrated DEA, COLS and Principal Component Analysis (PCA) in a two-stage model. In the first stage technical efficiency of the electricity distribution units operating in Iran were obtained by DEAS and COLS separately. In the second stage, the efficiency scores obtained in the first stage were considered as inputs in the PCA model. The authors claim that the DEA-COLS-PCA model used in this paper provided better ranking of units than would have done by DEA and COLS separately. To assess the impact of regulatory and environmental factors and statistical noise on the efficiency of public transit systems of Italian companies, Margari et al. (2007) used both SFA and DEA approach. The proposed model decomposed DEA inefficiency measures into three components: exogenous effects, managerial inefficiency and stochastic events. The developed measure provided evidence on the determinants of input-specific efficiency differentials across companies. The results also pointed out that managerial skills play a minor role, and emphasized the relevance of regulatory policies aimed at replacing cost-plus subsidization with high-powered incentive contracts as well as improving environmental conditions of public transit networks. In another study by Ta et al. (2008), logistic center location problem was solved by applying DEA and SFA separately and later the ranks obtained by both were compared using Spearman s rank correlation coefficient. Bazrkar and Khalilpour (2013) conducted a comparative study on ranking banks using DEA and SFA approach. The differences between ranks assigned by these two models were tested using Pearson s correlation test. The results showed significant difference between the ranks assigned by two approaches. Moreover, authors found SFA as the superior method of measuring efficiency of the bank as compared to DEA. In order to establish an evaluation system for key discipline scientific research level, Li and Wang (2011), used DEA and SFA models. The objective of this study was to measure scientific research input and output efficiency of fifty eight key disciplines in fifteen universities. The results showed significant difference between input and output efficiencies in terms of numerical sequencing but good consistency in terms of efficiency ranking. Some studies have been carried out to compare the results obtained from these tools. For example, Reinhard, et al. (2000), developed an analytical framework to calculate environmental efficiency in the presence of multiple environmentally detrimental inputs. The model so 209

5 developed enables the aggregation of environmentally detrimental inputs, and it allows the calculation of the environmental efficiency using these inputs. It also indicates which environmentally detrimental input is used most inefficiently, both on individual farms and in the aggregate. In most of the other applications, DEA and SFA were used separately to measure technical efficiency of a DMU and the results obtained were compared using either descriptive statistics like; average or mean efficiency score and standard deviation in the scores or using Spearman s rank test (Odek, 2007; Wadud, 2003; Sharma et al., 1997; Mulwa et al.,2008, etc.). While going through different studies in the literature on the application of DEA and SFA, one may realize that in most of the cases, DEA and SFA have been applied separately and the results obtained through these techniques are compared in the end by various methods. In some of the studies where DEA and SFA have been used in an integrated approach, mostly SFA is used to study the impact of the exogenous variables on the efficiency obtained using DEA. Also, in these studies, mostly one output or an aggregated output with multiple inputs is considered. To the best of our knowledge, there is no study being made which uses an integrated approach using SDEA and SFA. In this study, an attempt has been made to propose an integrated model using SFA and SDEA. Instead of CRS-DEA, Super efficiency DEA (SDEA) is specifically used to improve the discrimination power of the proposed method. Also, the proposed model doesn t require the separation of the exogenous variables in the later stage. Most importantly, the proposed model can accommodate a case of multiple outputs and multiple inputs. This chapter is organized as follows: In the following section 6.2, the need for integrated approach is explained. Section 6.3 presents Spearman s rank test and the MSD approach. These methods are used to compare the ranks obtained using proposed model with those obtained using conventional CRS-DEA and SDEA models. Section 6.4, presents the proposed integrated framework with illustrations using hypothetical data of various input and output combinations. Model is further verified using data from PSU Banks in section 6.5. Analysis and discussion are presented in Section 6.6. Section 6.7 gives summary and conclusions. 210

6 6.2 Need for Integrated Approach In this section, SFA, DEA and SDEA as integrating tools are explained in terms of their model specifications, advantages and disadvantages. This discussion carves the niche for proposed integrated approach Stochastic Frontier Analysis (SFA) Stochastic frontier is a parametric tool for the measurement of technical efficiency of a firm. Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) almost at the same time proposed this model. They proposed a stochastic production function for the cross-sectional data given by ( ), i = 1,, N, (Expression 6.1) Where denote output (or the logarithm of the output) of the i-th firm; is a kx1 vector of functions of actual input quantities used by the i-th firm; β is a vector of parameters to be estimated; and is the composite error term which is further divided into two components defined as:, i = 1,, N, Here s are assumed to be independently and identically distributed (iid) random errors, which have normal distributions with mean zero and unknown variance independent of the random variables which are assumed to account for technical inefficiency in production and are often assumed to be iid truncations (at zero) of the N( µ, ) distribution. In this model, account for random variation of production outside the control of the individual unit or producer. The error term, is composed of two parts: (a) the traditional random error that captures the effect of measurement error, other statistical noise, and random shock due to exogenous (variables which are out of producer s control) variables (if any); and b), one-sided component (as it is iid truncations (at zero) of the N (µ, ) distribution) which captures the effect of inefficiency. 211

7 Expression 6.1 in log-linear Cobb-Douglas form becomes (Expression 6.2) Estimation of Expression 6.2 by Ordinary Least Square (OLS) method provides consistent estimates of the s., but not of, since E( )= -E( ) 0. Moreover, OLS doesn t provide estimates of producer specific technical efficiency. But OLS does provide a simple test of technical efficiency (TE). If, then = =TE if, then = TE which then is negatively skewed, and there is evidence of technical inefficiency in the data. In the present study, FRONTIER version 4.1 (developed by Coelli, T.J, Centre for Efficiency and Productivity Analysis, University of New England, Australia) to compute TE of a DMU using above model is used Some of the advantages of SFA are Ability to incorporate stochastic data (Bazrkar and Khalilpour, 2013; Odeck and Bråthen, 2012), Ability to test the result using statistical hypothesis testing technique (Reinhard et al., 2000; Coelli and Perelman, 1999), Accounts for statistical noise (which may be beyond the control of the producer) and separates out technical inefficiency (Thoraneenitiyan and Avkiran, 2009; Reinhard et al., 2000; Coelli and Perelman, 1999; etc.) and Compares the performance of each DMU against the average performance of all the units in the group (Ta, et al., 2008). On the other hand, the limitations of this approach are Requirement of a priori specification of the distribution of the inefficiency terms and the functional form of the frontier (Perera and Skully, 2012; Saad and El-Moussawi, 2009; Chen T-Y, 2002), No indication of potential improvement possible in case of inefficient DMU and Inability to handle a case of multiple outputs without developing an aggregate measure of all outputs (Kumbhakar and Lovell, 2000). 212

8 6.2.2 Data Envelopment Analysis (DEA) Data Envelopment Analysis (DEA) is a non-parametric benchmarking tool, based on linear programming technique. It was originally developed by Farrell (1957) and further extended by Charnes, Cooper and Rhodes (1978). Charnes, Cooper and Rhodes (CCR) model measures the relative efficiency of a set of firms that use a variety of inputs to produce a range of outputs under the assumption of constant return to scale (CRS). In economics, return to scale is the term related to the firm s production function. It describes the behavior of the firm in terms of its rate of change in the output/production as a result of change in its input/s. Constant return to scale signifies a production process of a firm where output/production changes are in proportion to the changes in firm s input quantities. As a result the manufacturer is able to scale the inputs and outputs linearly without increasing or decreasing efficiency. An individual unit in this set (of firms) is referred to as DMU. A DMU, for instance, can include hospitals, power plants, universities, schools, banks, bank branches, etc. Performance of a DMU is measured using the concept of efficiency or productivity, which is defined as the ratio of total weighted outputs to total weighted inputs. While measuring the performance, this model captures not only the productivity efficiency of a firm at its actual scale size, but also the inefficiency (Banker, 1984). The best performing unit in the set of DMUs is assigned a score of 100 percent or 1, and the remaining DMUs get a score ranging between 0 and 100 percent, or equivalently between 0 and 1, relative to the score of best performing DMU. DEA forms a linear efficiency frontier which passes through the best performing units within the group whereas all the remaining less efficient units lie off the frontier. The term efficiency used in DEA is the relative efficiency and not the absolute efficiency. Banker, Charnes and Cooper (BCC) in1984, extended the earlier CCR model to BCC model in which, variable return-to-scale (VRS) was introduced. This was achieved by introducing one more constraint in the CCR model which ensures that firms operating at different scales are recognized as efficient. In this case, inefficient firms are compared only with the efficient firms of similar scale. The DEA model developed by Charnes, Cooper and Rhodes with constant return to scale referred to as CRS-DEA is given as under; Let there be N DMUs each with k inputs and m outputs. For the p th DMU under evaluation, the technical efficiency measured by using the CRS-DEA model is given by Maximize 213

9 Subject to ( ) ( ) with (Expression 6.3) where, : Efficiency of the p th DMU value for input criteria i for p th DMU weight of input i value for output criteria j for p th DMU weight of output j : value for input criteria i for n th DMU : value for output criteria j for n th DMU an infinitesimal or non-archimedean constant usually in the order of 10-5 or 10-6 where and here note that n includes p Super Efficiency DEA (SDEA) While ranking the DMUs using DEA model, many a time s several DMUs achieve an efficiency score of one. In such cases, ranking of efficient units is a major challenge faced by the decision maker (DM). In order to overcome this problem, Andersen and Petersen (1993) proposed Super efficiency DEA model called SDEA which is an extension of DEA model. SDEA has been defined either with CRS or VRS assumption. In this study, SDEA model with CRS assumption is considered. This is also a non-parametric method of benchmarking like DEA. The model is given as follows: Subject to 214

10 (Expression 6.4) The difference between the standard DEA model and the SDEA model lies in the treatment of the efficient units (Saen, 2008). This model is similar to Banker, Charnes and Cooper (BCC) model, except that in the approach presented by Andersen and Petersen, the DMU under evaluation is excluded from the set, thus allowing the efficient DMUs to increase their inputs proportionally while preserving their efficiency status and achieve an efficiency score above 1. DEA as a MCDM tool has several advantages which are as follows; No need to specify a-priori weights on the input-output criteria (factors). The DEA approach, allows each DMU to choose a set of weights (also called as multipliers) for the input-output criteria that enables it to appear in the best light (George and Rangaraj, 2008; Sufian, 2007; Avkiran, 1999; Al-Faraj et al., 1993; Mester, 1996; Banker, 1984). DEA uses the data to derive an efficiency frontier. This frontier sets the benchmark for less performing units. It is with the reference to this frontier that each DMU is evaluated (Soteriou and Stavrinides, 2002; Ramanathan, 2005; Koster et al, 2009). It s ability to indicate the potential improvement in the performance of an inefficient Decision Making Unit (DMU) (Duffy et al., 2006; Banker and Morey, 1986; Sherman, 1984). DEA, though popular, has few limitations. One of the limitations of DEA is its less discrimination power due to two reasons: a) When the sum of the number of inputs and outputs is large as compared to the total number of DMUs in the sample (Andersen and Petersen, 1993; Zhu, 2001; Saen, 2008). 215

11 b) At times, DEA assigns high efficiency to a DMU due to its very low value of single input or very high value of output, even though that input or output is seen as relatively unimportant (Seiford and Zhu, 1999; Shang and Sueyoshi, 1995). Therefore, the discussion that follows considers an extended version of Data Envelopment Analysis called Super Efficiency Model of DEA (SDEA). SDEA is known for better discriminating power than DEA (Andersen and Petersen, 1993; Balf et al., 2012; Lovell and Rouse, 2003). Some of the advantages of SDEA are Its ability to indicate the potential improvement in the performance of an inefficient Decision Making Unit (DMU) (Lovell and Rouse, 2003; Adler et al., 2002). And most importantly, its ability to rank efficient DMUs. (Andersen and Petersen, 1993; Lovell and Rouse, 2003; Chen et al., 2010). Some of the disadvantages of SDEA are Sometimes, it is likely that a specific set of DMUs are ranked too high (Balf et al., 2012). This model is not unit invariant. This means that the model is deficient in its treatment of the non-zero slacks as its treatment of the slack does not yield a measure that is unit invariant (Cooper et al, 2007). In some cases, DMUs which are rated efficient (efficiency score equal to one) using conventional DEA model do not have feasible solution in SDEA model (Lovell and Rouse, 2003). The model does not take into account the existence of slacks in the inputs and the outputs (Tone, 2002). It can be realized that these models as a stand-alone tool are silent on following issues SDEA and CRS-DEA cannot incorporate stochastic nature of the real data and measure technical efficiency of a DMU after separating out inefficiency and random shock due to exogenous variables (if any), Application of SFA without aggregation of output in case of multiple outputs, CRS-DEA at times cannot provide a tie-breaking procedure and In some cases, DMUs which are rated efficient (efficiency score equal to one) using conventional DEA model do not have feasible solution in SDEA model Sometimes, it is likely that a specific set of DMUs are ranked too high using SDEA 216

12 CRS-DEA and SDEA both being non-parametric in nature does not provide a diagnostic tool to test the validity of the results. Based on above discussion, it is proposed that these issues can be addressed by integrating SFA and DEA. The proposed integrated model will help to a) Incorporate stochastic nature of the real data and measure technical efficiency of a DMU after separating out inefficiency and random shock due to exogenous variables (if any), b) Apply SFA with multiple outputs, c) Provide a tie-breaking procedure and d) Recommend the best alternative whose average performance is evaluated against the best DMU in the sample under study. 6.3 Rank Evaluation Tools In this section, a brief description of two different methods namely Spearman s rank test and Mean Squared Deviation (MSD) is given in brief. These methods are used for comparing ranks obtained by different approaches Spearman s Rank Test The Spearman s correlation coefficient is a measure of the linear association between two variables which are available in ordinal scale. That is, it measures the strength of association between two ranked variables. It is the nonparametric version of the Pearson product-moment correlate ion. Spearman s rank test is used to test the strength of a relationship between ranks assigned by different criteria to the same set of units. In other words, it tests whether there is agreement or disagreement between the ranks obtained by two different techniques. A perfect Spearman correlation of +1 or 1 occurs when each of the variables is a perfect monotone function of the other. The test statistic is given by (Expression 6.5) Where = Spearman s rank correlation coefficient n = the number of items or individuals being ranked = the rank of item i with respect to one variable/criterion = the rank of item i with respect to a second variable/criterion 217

13 d i = Mean Squared Deviation (MSD) The second method used to compare the ranks assigned by the proposed model and the conventional model is Mean Squared Deviation (MSD). This is calculated by first finding the mean efficiency score for each DMU using each of the models under study. In this pair wise approach, average value of the square of difference of ranks obtained is computed this is called as Mean Squared Deviation (MSD). MSD is computed for all the pairs so formed. MSD closer to the value of zero indicate no (or little) difference between the ranks assigned by two different methods in the pair. 6.4 Proposed Framework In this subsection, the proposed framework is presented. The theme of this approach is as follows: Efficiency Ranking Method using SFA and SDEA (ERM-SSD) Initially, various combinations using each of the outputs and all inputs are considered. Therefore if there are l outputs under study then there will be l such combinations formed called dimensions. For each of these dimensions, technical efficiency for each DMU is computed using SFA. Next, SDEA model is applied with these technical efficiencies obtained from SFA analysis as inputs and dummy output of 1, to get an efficiency score called as SSD-I. Similarly, by considering the technical efficiencies obtained from SFA analysis as outputs and dummy input of 1, SDEA model is applied to get an efficiency score called SSD-O. Finally, efficiency scores obtained through SSD-I and SSD-O are combined together using arithmetic mean to define a value called SSD. Here arithmetic mean is used to combine SSD-I and SSD-O scores so as to ensure equal weightage. The five step procedure is explained as follows: Step 1: Identify n DMUs or alternatives to be evaluated with k inputs and l outputs. Step 2: Define various combinations/dimensions using each output and all inputs. Apply stochastic production function with error component model of SFA to compute technical efficiency of each DMU. The efficiencies thus computed will be for l output 218

14 dimensions. For instance, for two inputs and two outputs there will be two such dimensions, one for each output. Step 3: Use efficiency scores obtained in step 2 as Input Criteria (IC) and output as dummy variable with constant value (say 1) (Dummy Output Criterion, DOC). Apply Super Efficiency DEA with input orientation (SDEA-I) model to find efficiency score for each DMU/alternative. This efficiency score is designated as (SSD-I) i, Where i= 1, 2, 3 n. SDEA with input orientation will help in minimizing inputs (resources, cost etc.) for the given level of outputs. Step 4: Use efficiency scores obtained in step 2 as Output Criteria (OC) and dummy variable with constant value (say 1) as input (Dummy input Criterion (DIC). Apply Super Efficiency DEA with output orientation (SDEA-O) model to find efficiency score for each DMU/alternative. This efficiency score is designated as (SSD-O) i, Where i= 1, 2, 3,n. SDEA with output orientation will help in maximizing outputs (revenue, profit, efficiency etc.) for the given level of inputs. Step 5: Combine SSD-O and SSD-I values obtained in step 3 and 4 using arithmetic mean to get a combined efficiency score (for each DMU). This efficiency score is called SSD score. Use these SSD efficiency scores to rank the DMUs/alternatives in ascending order of magnitude. In this case, arithmetic mean is used to combine two score; SSD-O and SSD-I so as to give equal weightage to both the scores Illustration In this section, an application of the proposed model ERM-SSD is illustrated using hypothetical data set with two different input-output combinations. First a hypothetical dataset using twelve DMUs with two outputs (O1 and O2) and two inputs (I1 and I2) and then a hypothetical data set using eighteen DMUs with three outputs and three inputs is considered for illustration purpose. 219

15 A hypothetical case with two outputs and two inputs Data for a case of twelve DMUs with two outputs (O1 and O2) and two inputs (I1 and I2) presented in Table 6.1. Step 1: Identify DMUs or alternatives to be evaluated with 2 inputs and 2 outputs. Table 6.1: Hypothetical data with 2 outputs and 2 inputs DMU O1 O2 I1 I2 A B C D E F G H I J K L Step 2: First, various combinations/dimensions using single output and multiple inputs are formed and stochastic production function with error component model given by Expression 6.1 is used to compute technical efficiency of each DMU. In this illustration, with two outputs and two inputs, two dimensions are formed. These dimensions are (O1, I1, I2) and (O2, I1, I2). Step 3 and 4: The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute SSDI and SSDO scores using the procedure explained in step 3 and 4 of the proposed framework. The computations are presented in Table

16 Step 5: Table 6.2: Computations of SSD-I and SSD-O scores (2 outputs and 2 inputs) DMU OC1 OC2 DIC SSD-I DOC IC1 IC2 SSD-O A B C D E F G H I J K L Upon combining SSD-I and SSD-O scores using arithmetic mean, efficiency scores called SSD scores are obtained. These SSD scores are ranked to get the final ranking of DMUs. Ranks are also obtained for each DMU using CRS-DEA and SDEA models. The ranks of these DMU using the proposed model ERM-SSD, CRS-DEA and SDEA are presented in Table 6.3. Table 6.3: Ranks using ERM-SSD, CRS-DEA and SDEA (2 outputs and 2 inputs) DMU ERM-SSD-I Score ERM-SSD-O Score ERM-SSD Score ERM-SSD Rank CRS-DEA Rank SDEA Rank A B C D E F G H I J K L From Table 6.3 it is observed that four DMUs are tied at rank 1, when computed using CRS- DEA approach. However, this tie no more exists in ERM-SSD and SDEA approaches. The proposed approach (ERM SSD) has recommended DMU L as the best DMU while SDEA has given the first rank to DMU E. 221

17 Hypothesis Testing In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA model, Spearman s rank test was carried out. For this purpose, following hypotheses were framed. Hypothesis1 (referred to as Hypothesis 5 in Chapter 3) H0: There is no association/correlation between the ranks of individual DMUs obtained by ERM-SSD and CRS-DEA. H1: There is association/correlation between the ranks of individual DMUs obtained by ERM- SSD and CRS-DEA. Hypothesis 2 (referred to as Hypothesis 6 in Chapter 3) H0: There is no association/correlation between the ranks of individual DMUs obtained by ERM-SSD and SDEA. H1: There is association/correlation between the ranks of individual DMUs obtained by ERM- SSD and SDEA. Hypotheses are tested at 5% level of significance. The results of Spearman s rank test are shown in Table 6.4. Table 6.4: Spearman s Rank test results for 2 output and 2 input data (for all three models) Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank ERM-SSD Rank Correlation Coefficient Sig. (2-tailed) N CRS-DEA Rank Correlation Coefficient ** Sig. (2-tailed) N SDEA Rank Correlation Coefficient ** Sig. (2-tailed) N From Table 6.4 it is seen that there is no significant association/correlation between the ranks assigned by ERM-SSD and CRS-DEA ( with p-value of 0.382) and between ERM-SSD and SDEA ( with p-value of 0.308). This means that there is significant difference between the ranks assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and 2 are accepted and respective alternate hypotheses H1 are rejected. Further, the difference between the ranks assigned by above three approaches is verified by MSD method. The results are shown in Table

18 Table 6.5: MSD results for 2 output and 2 input data (for all three models) DMU ERM-SSD Rank CRS-DEA SDEA (1-2) 2 (1-3) 2 (1) Rank (2) Rank (3) A B C D E F G H I J K L MSD MSD measures the difference between the ranks using average of squared deviation. In case of complete association or agreement between the ranks assigned by any 2 models under study ideally, value of MSD should be 0. From Table 6.5, it can be seen that none of the MSD values are closer to zero. The MSD value ranges from 31.5 between ERM-SSD and SDEA to between ERM-SSD and CRS-DEA. So, this suggests that there is no association between the ranks assigned by ERM-SSD and CRS-DEA and between ERM-SSD and SDEA models A hypothetical case with three outputs and three inputs In order to check the consistency of the proposed model, one more case of hypothetical data with three inputs and three outputs using a set of eighteen DMUs is considered. Similar analysis is carried out for this data set. Step 1: The data required for the analysis is as shown in Table

19 Step 2: Table 6.6: Hypothetical data with 3 outputs and 3 inputs DMU O1 O1 O3 I1 I2 I3 A B C D E F G H I J K L M N O P Q R First, multiple inputs with single output combinations/dimensions are formed and stochastic production function with error component model given by Expression 6.1 is used to compute technical efficiency of each DMU. In this illustration, with three outputs and three inputs, three combinations/dimensions are formed. These dimensions are (O1, I1, I2, I3), (O2, I1, I2, I3) and (O3, I1, I2, I3). Step 3 and 4: The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute SSD-I and SSD-O scores using the procedure explained in step 3 and 4 of the proposed framework. The computations are presented in Table

20 Step 5: Table 6.7: Computations of SSD-I and SSD-O scores (3 outputs and 3 inputs) DMU OC1 OC2 OC3 DIC SSDI DOC IC1 IC2 IC3 SSDO A B c D E F G H I J K L M N O P Q R Arithmetic mean of SSDI and SSDO scores called as SSD are obtained. These SSD scores are ranked to get the final ranking of DMUs using the proposed method ERM-SSD. The ranks obtained are shown in Table 6.8. Ranks obtained for DMUs using CRS-DEA and SDEA models are also shown in Table 6.8. Table 6.8: Ranks using ERM-SSD, CRS-DEA and SDEA (3 outputs and 3 inputs) DMU ERM-SSD-I Score ERM-SSD-O Score ERM-SSD Score ERM-SSD Rank CRS-DEA Rank SDEA Rank A B C D E F G H I J K L M N O P Q R

21 From Table 6.8 one may observe that the proposed approach ERM-SSD assigns rank 1 to only one DMU namely DMU B. Similarly, SDEA assigns rank 1 to only one DMU namely DMU J. On the other hand, CRS-DEA assigns rank 1 to 7 different DMUs. So, in this case also proposed model is able to provide a tie-breaking procedure. This suggests a better discrimination power of the proposed model. Hypothesis Testing In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA model, Spearman s rank test was carried out. For this purpose, same hypotheses as stated above were tested at 5% level of significance. The results are presented in Table 6.9. Table 6.9: Spearman s Rank test results for 3 output and 3 input data (for all three models) Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank ERM-SSD Rank Correlation Coefficient Sig. (2-tailed) N CRS-DEA Rank Correlation Coefficient ** Sig. (2-tailed) N SDEA Rank Correlation Coefficient ** Sig. (2-tailed) N From Table 6.9 it is seen that there is no significant association/correlation between the ranks assigned by ERM-SSD and CRS-DEA ( with p-value of 0.937) and between ERM-SSD and SDEA ( with p-value of 0.848). This means that there is significant difference between the ranks assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and 2 are accepted and respective alternate hypotheses H1 are rejected. Further, the difference between the ranks assigned by above three approaches was checked by MSD method. The results are shown in Table

22 Table 6.10: MSD results for 3 output and 3 input data (for all three models) DMU ERM-SSD Rank (1) CRS-DEA Rank (2) SDEA Rank (3) (1-2) 2 (1-3) 2 A B C D E F G H I J K L M N O P Q R MSD As seen from Table 6.10, in this case as well it is seen that none of the MSD values is zero. The MSD value ranges from between ERM-SSD and SDEA to 68.5 between ERM- SSD and CRS-DEA. So, this suggests that there is no association between the ranks assigned by ERM-SSD and CRS-DEA and between ERM-SSD and SDEA models. 6.5 An Application of the Proposed Approach: PSU Banks In this section, an application of the proposed framework using data on Public Sector Unit (PSU) Banks operating in India is presented. For the purpose of analysis, all 26 PSU Banks are selected. These twenty-six PSUs control more than ninety percent of all deposits, assets and credits of the Indian banking sector ( The parameters on which data are collected are as follows: Net Profit (output) Total Income (output) Operating Expenses (input) and Total Assets (input) 227

23 Data are obtained for the financial year from the official website of Indian Bank Association ( Step 1: Select 26 PSU Banks operating in India in the sample. Data are presented in Table Step 2: Table 6.11: PSU Bank data with 2 outputs and 2 inputs 228 (Fig. are in Crores) Sr. No. Banks Net Profit Total Income Operating expenses Total Assets 1 Allahabad Bank Andhra Bank Bank of Baroda Bank of India Bank of Maharashtra Canara Bank Central Bank of India Corporation Bank Dena Bank Indian Bank Indian Overseas Bank Oriental Bank of Commerce Punjab & Sind Bank Punjab National Bank Syndicate Bank UCO Bank Union Bank of India United Bank of India Vijaya Bank State Bank of India (SBI) State Bank of Bikaner & Jaipur State Bank of Hyderabad State Bank of Mysore State Bank of Patiala State Bank of Travancore IDBI Ltd Source: Multiple inputs with single output combinations/dimensions are formed and stochastic production function with error component model given by Expression 6.1 is used to compute technical efficiency of each DMU. Here, with two outputs and two inputs, two dimensions are formed. These dimensions are (O1, I1, I2) and (O2, I1, I2).

24 Step 3 and 4: The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute SSD-I and SSD-O scores using the procedure explained in step 3 and 4 of the proposed framework. The computations are presented in Table Table 6.12: Computations of SSD-I and SSD-O scores (PSU Banks) Sr. No. PSU Bank OC1 OC2 DIC SSD-I DOC IC1 IC2 SSD-O 1 Allahabad Bank Andhra Bank Bank of Baroda Bank of India Bank of Maharashtra Canara Bank Central Bank of India Corporation Bank Dena Bank Indian Bank Indian Overseas Bank Oriental Bank of Commerce Punjab & Sind Bank Punjab National Bank Syndicate Bank UCO Bank Union Bank of India United Bank of India Vijaya Bank State Bank of India (SBI) State Bank of Bikaner & Jaipur State Bank of Hyderabad State Bank of Mysore State Bank of Patiala State Bank of Travancore IDBI Ltd Step 5: Upon combining SSDI and SSDO scores using arithmetic mean, efficiency scores called SSD scores are obtained. These SSD scores are ranked to get the final ranking of DMUs. The ranks obtained are shown in Table Table 6.13 also shows ranks obtained for DMUs using CRS- DEA and SDEA models. 229

25 Table 6.13: Ranks using ERM-SSD, CRS-DEA and SDEA (PSU Banks) Sr. No. PSU Bank SSD-I SSD-O SSD ERM-SSD Rank CRS-DEA Rank SDEA Rank 1 Allahabad Bank Andhra Bank Bank of Baroda Bank of India Bank of Maharashtra Canara Bank Central Bank of India Corporation Bank Dena Bank Indian Bank Indian Overseas Bank Oriental Bank of Commerce Punjab & Sind Bank Punjab National Bank Syndicate Bank UCO Bank Union Bank of India United Bank of India Vijaya Bank State Bank of India (SBI) State Bank of Bikaner & Jaipur State Bank of Hyderabad State Bank of Mysore State Bank of Patiala State Bank of Travancore IDBI Ltd As seen from Table 6.13, Indian Overseas Bank emerges as the best bank according to ERM- SSD, whereas Bank of Baroda is the top performing bank according to SDEA. However, in this case, CRS-DEA has assigned first rank to eight different banks showing poor discrimination power. So, in this case as well, the discrimination power of the proposed model ERM-DT is as good as that of SDEA. Hypothesis Testing In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA model, Spearman s rank test was carried out. The results are given in Table

26 Table 6.14: Spearman s Rank test results for PSU Bank data (for all three models) Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank ERM-SSD Rank Correlation Coefficient Sig. (2-tailed) N CRS-DEA Rank Correlation Coefficient ** Sig. (2-tailed) N SDEA Rank Correlation Coefficient ** Sig. (2-tailed) N From Table 6.14 it is seen that there is no significant correlation between the ranks assigned by ERM-SSD and CRS-DEA (0.022 with p-value of 0.916) and between ERM-SSD and SDEA (0.030 with p-value of 0.883). This means that there is significant difference between the ranks assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and 2 are accepted and respective alternate hypotheses H1 are rejected. Further, the agreement between the ranks assigned by above four approaches is checked by MSD method. The results are shown in Table

27 Table 6.15: MSD results for PSU Bank data (for all three models) Sr. No. Banks ERM-SSD CRS-DEA SDEA (1-2) 2 (1-3) 2 Rank (1) Rank (2) Rank (3) 1 Allahabad Bank Andhra Bank Bank of Baroda Bank of India Bank of Maharashtra Canara Bank Central Bank of India Corporation Bank Dena Bank Indian Bank Indian Overseas Bank Oriental Bank of Commerce Punjab & Sind Bank Punjab National Bank Syndicate Bank UCO Bank Union Bank of India United Bank of India Vijaya Bank State Bank of India (SBI) State Bank of Bikaner & Jaipur State Bank of Hyderabad State Bank of Mysore State Bank of Patiala State Bank of Travancore IDBI Ltd MSD From Table 6.15 it is seen that MSD values for all pairs are different from zero. The MSD value for ERM-SSD and CRS-DEA is whereas that for the pair ERM-SSD and SDEA is This means that the ranks assigned by different methods are different. So, this suggests that there is no association between the ranks assigned by ERM-SSD and CRS-DEA and between ERM-SSD and SDEA models. 232

28 6.6 Analysis and Discussion From the analysis done for three different cases above, one may acknowledge few common points which are important to notice. For all three cases analyzed above one may notice that the proposed model ERM-SSD has given consistent results and has provided a tie-breaking procedure. It can also be noted that in the proposed approach, the best DMU has emerged out after comparing its average performance with the performance of the best DMU in the sample. Although data under study consisted of multiple outputs, the proposed approach is able to compute efficiency scores in SFA framework without aggregating the outputs. The proposed model acknowledge the stochastic nature of the data by decomposing the error term into two parts: (a) the traditional random error that captures the effect of measurement error, other statistical noise, and random error; and b), one-sided component (as it is iid truncations (at zero) of the N (µ, ) distribution) which captures the effect of inefficiency. Thus, the technical efficiency obtained for each DMU in Step 2 is more real in the sense that this efficiency is obtained after separating out the inefficiency and the effect of random shock due to exogenous variables (if any). Also, for various input-output combinations and bank data analyzed, it can be generalized that there is no association between the ranks obtained by the proposed model ERM-SSD, CRS-DEA model and SDEA model. 6.7 Conclusions SFA is a parametric tool for measuring technical efficiency of a DMU. It benchmarks the performance of a DMU against the average performance of the group of DMUs under study. SDEA is a non-parametric tool which is an extension of conventional DEA model. It is been developed to overcome some of the limitations of DEA such as poor discrimination power of conventional DEA model in ranking efficient DMUs. In this chapter, an integrated approach using SFA and SDEA is proposed which overcomes some of the shortcomings of the individual integrating tools (in this case SFA and SDEA) and tries to make best use of their individual strengths. The proposed model ERM-SSD has number of advantages which are discussed below. 233