“How efficient are public sector banks in India? A non-parametric approach”

This study examines the efficiency of Public Sector Banks (PSBs) in India using Data Envelopment Analysis (DEA). Analysis is carried out on a sample of 19 PSBs that are existed during the study period from 2005 to 2018. There are two different aspects deliberated, namely technical efficiency of PSBs and the growth in their productivity. Input variables envisaged for the study are deposits, borrowings, fixed assets, and the number of employees. Loans and advances along with investments act as output variables to measure technical efficiency and productivity. The results indicate that the technical efficiency of PSBs ranges between 97% and 100%. Corporation Bank, Indian Bank, and Oriental Bank of Commerce outperformed their peers with 100% technical efficiency. Productivity growth among the sampled banks during the study period stood between 0.8% and 20%. However, Corporation Bank, Indian Bank, and Oriental Bank of Commerce registered 9.1%, 5.4% and 6.4% productivity growth, re-spectively. The results reveal that PSBs are working hard to optimize resource utiliza- tion. Researchers around the world can use DEA as a tool to measure the efficiency of banks with different input and output variables related to financial, marketing and managerial performance.


INTRODUCTION
The economic development of any country depends on the financial sector, especially the banking sector. The role of financial intermediation in economic growth has been a widely recognized aspect of empirical research. Finance can stimulate the main drivers of growth, efficiency, and productivity of an economy (Yusifzade & Mammadova, 2015). The Indian banking sector is one of the healthiest performers in terms of competitiveness, growth, efficiency, profitability, and soundness in the world banking industry (Kumar Mishra, 2017). Banks' efficiency depends on a diversified banking system that attracts savings and channelizes them into productive investments to generate income.
Economists like Schumpeter have accepted the indispensable role of the financial system in the development of the economy. He characterized their significance as follows: "He [the banker] stands between those who wish to form new combinations and the possessors of productive means. The banker is a phenomenon of development" (Schumpeter, 1934). However, the banker can achieve development only through the process of efficient and effective financial intermediation. Simultaneously, the banking sector focuses on improving the quality of assets, appropriate capital, and an expectation of higher returns. Three eminent researchers representing The World Bank (Yeyati), Ministry of Finance Chile (Alejandro Micco) and Debt and Development Finance Branch, UNCTAD (Ugo Panizza), respectively,

Evolution of data envelopment analysis and its application
In 1957, US economist Michael Farrell published a critical paper titled "The Measurement of Productive Efficiency," considering single-output/ single-input to measure the technical efficiency of decision-making units. Charnes, Cooper and Rhodes improved this method with multiple-output/multiple-input in the year 1978. However, Farrell's seminal work has undergone various improvements, mainly categorized as three schools: Afriat School, Charnes School and Shephard School, as mentioned by Thompson et al. (1993). Allen N. Berger, a senior economist (Federal Reserve System), and David B. Humphrey, an eminent scholar in Banking at Florida State University, investigated efficiency issues in commercial banking in 1992 and concluded that technical efficiency progress was an outcome of a firm's technology (Berger & Humphrey, 1992). Five years later, the same researchers carried out an extensive review of international literature and found that frontier efficiency measurement techniques like "Stochastic Frontier Analysis, Data Envelopment Analysis and Thick Frontier Analysis" were extensively used (Berger et al., 1997). There was a shift from non-parametric methods used to estimate the production frontier to parametric tests. Lovell and Schmidt (1998) investigated and presented a comparative view of these approaches. In 2008, Wade and Larry explored the thirty years of research work using DEA considering four essential models used to measure efficiency, approaches to integrate restrictions on multipliers, considerations about the status of variables and data variation modeling (Lovell & Schmidt, 1998 The results revealed that the sustainable banks were more efficient than non-sustainable banks. Since maintaining efficiency in operations and utilizing resources (assets and liabilities) are a continuous requirement for creating sustainable operations, academicians and researchers worldwide are particularly interested in questions of efficiency. Previous research has focused on measuring banks' efficiency based on ownership patterns (public, private and foreign), size, country-specific, and operations during events like financial crises. This study aims to assess the technical efficiency of PSBs in India with the variables relating to the financial intermediation process. Perhaps the methodology adopted (DEA and MPI are worldwide accepted tools to measure efficiency and productivity) in the study is applicable to measure the efficiency of any bank across the globe. Understanding current levels of efficiency will enable banks, government, and policymakers in enhancing the future-readiness of banks. PSBs or state-owned banks in emerging nations hold 70% of the market share. Hence, this study is of direct relevance to further the understanding of the industry and its functioning.

Model specification for data envelopment analysis
Production process can be described as a procedure that can turn a set of resources into attractive outcomes by production units. During this process, efficiency is used to determine how well a production unit is performing in using its resources to produce the outcomes. DEA provides a comprehensive analysis of relative efficiencies of multiple input and multiple output situations by evaluating each DMU and measuring its performance in relation to an envelopment surface composed of other DMUs. Those DMUs forming the efficiency reference set are known as the peer group for the inefficient units (Yang, 2009 In contrast, the latter refers to exploiting the scale of the economy by operating at a point where the production frontier exhibits CRS. The DMUs are homogeneous units whose performance is to be measured (Sekhri, 2011). For this study, DMUs are commercial banks. Technical efficiency score is the total weighted sum of input divided by the total weighted sum of output divided by the ratio of weighted inputs. The efficiency of a bank can be measured as to how efficiently it utilizes its inputs.
The following linear programming equations represent the input-oriented BCC model with VRS assumption: Thus: where u and v are weights for the outputs (y 1 ... y n ) and the inputs (x 1 ... x n ), respectively. The models with CRS to scale are known as the CCR model as it is proposed by Charnes et al. (1978) to estimate the input-oriented technical efficiency of the Korean banking sector.
The CCR model can be formulated as follows:

Model specification for the Malmquist productivity index
Malmquist productivity index is used to measure the changes in firms/banks' efficiency over some period of time. Productivity indices are resultant of production frontier models. Total Factor Productivity (TFP) has various components to provide a clear understanding of Technical Change and Efficiency Change (Fare et al., 1994). Shifts in the production frontier are measured by technical changes, whereas efficiency change measures shifts in the frontier position of a production unit or bank. In 1992, various researchers from Norway representing banking (Atle Berg, Eilev S. Jansen) and the education sector (Finn R. Forsund) measured the average productivity growth, frontier growth and the spread of growth rates in the banking industry using MPI. They concluded that there were very negligible productivity growth at the frontier and noticeable improvement in most banks' relative efficiency (Berg et al., 1992). Borrowing MPI concepts from the earlier researchers, Economics professor Suleyman Desgirmen in collaboration with a professor from the Business Education department, Benli Yasemin Keskin, attempted to apply DEA based Malmquist Total Factor Productivity Index on Turkish banks (TB). They found that with the advantage of technology, foreign banks outperformed other groups. Further, they specified that the financial crisis had brought a setback in TB's technical efficiency (Benli & Degirmen, 2013).
Economics professors from Australia studied the changes in productivity of financial institutions in Botswana during 2009 and found that both decrease and improvement in productivity were technological regress outcomes. Considering the above research that has used MPI to measure productivity, this study intends to apply the same with the following description of the methodology.
Malmquist productivity Index is defined using distance functions. Suppose the function that describes the technology of production is given as F(X,Y) = 0, where X = (x 1 , x 2 , x 3 , ..., x m ) is the input vector and Y = (y 1 , y 2 , ..., y s ) is the output vector. Caves et al. (1982) provide an alternative interpretation of production technology using the concept of 'distance function'. They defined the output distance function as where μ Y is the minimum equiproportional change in the output vector. The distance function measures the maximum proportional change in output required to place (X,Y) on the efficiency frontier. If the evaluated production unit is efficient, D 0 (X,Y) = 1, otherwise D 0 (X,Y) < 1. A distance function may also be computed with input orientation, with reference to technology in a certain period and with CRS or VRS specification. Let tion with period t technology and with CRS and VRS specifications, respectively. The output distance function is the maximum equiproportional increase in output for a given input. This is the output oriented (Farrell, 1957) technical efficiency. Therefore, the distance function can be determined using the DEA method. Caves et al. (1982) as cited in (Galagedera & Edirisuriya, 2005) define the output-based MPI to compare the performance of a production unit in time period t and t + 1 with reference to period t technology as The output-based productivity index measures the maximum level of outputs that can be produced using a given input vector and a given production technology relative to the observed levels of outputs (Coeli et al., 1998 as cited in Galagedera & Edirisuriya, 2005).
Alternatively, output based MPI can be defined with reference to t + 1 technology as M 0 > 1 indicates higher productivity in period t than in period t + 1.
where the component outside the square brackets is the change in the output oriented measure of technical efficiency between periods t and t + 1. The other component in equation (6) captures the shift in technology (technological change Index -TCI) between the two periods t and t + 1. The output based Malmquist total productivity change index is the geometric mean of output-based Malmquist productivity indices with reference to period t and period t + 1 technology. The ratio outside the square brackets in equation (6) is often referred to as the boundary shift component. The catch-up term compares the closeness of the production unit in each period to that period's efficient frontier, whereas the boundary shift term represents productivity gain/loss by the industry and not necessarily by the production unit itself. When the boundary shift is equal to 1, the industry has on average registered no productivity gain or productivity loss between period t and t + 1 (Thanassoulis, 2001 as cited in Galagedera & Edirisuriya, 2005).
When the efficient production is characterized by variable returns-to-scale, the change in the productivity of a production unit may be impacted by changes in scale size. In that case, the component outside the square brackets in equation (6) can be decomposed into a pure technical catch up component and a scale efficiency catch-up component. Pure technical efficiency catch-up and scale efficiency catch-up are orientation dependent. Now for a given production unit, the indices that capture changes in period t+1 relative to period to period t are given by: Total factor productivity change index (TFPCI) The indices of MPI are less than one, one and greater than one. One is subtracted from the index to estimate the growth, and then the value is multiplied with 100 to get the growth rate. If the index is more than one, the bank is said to be with growth in TFP, and less than one indicates negative growth. However, each operation of a bank is supported by technical, technological, management and scale/size efficiencies to have better productivity. Collectively a positive change in all these aspects will lead to growth in total factor productivity (TFPCI). This study used MPI to measure the PSB's growth in TFP during the study period.

DATA AND RESULTS
This section describes the data used for the analysis, its rationale and the findings of empirical investigation. This paper used secondary data of 19 banks collected from the RBI website. It considers the banks that existed during the study period (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018) GST (2018) are few of the reforms initiated during the study period. Since these initiatives are taken to mark an improvement in the efficiency of banks, it is appropriate to measure the efficiency and productivity of PSBs. Reviews indicate that DEA can be applied to measure the relative efficiency of public sector banks; also, MPI can be used on the panel data to analyze the productivity and its growth across the period. Hence, DEA and MPI are chosen as a tool to measure efficiency and productivity. Accordingly, variables are chosen to assess the public sector banks' technical efficiency and productivity. Commercial banks' primary operation is to accept deposits and provide loans and advances, which is fundamental for financial intermediation. Hence, deposits, borrowings, number of employees and fixed assets are used as inputs; loans and advances, as well as investment, are used as outputs to carry out the analysis. A concise explanation about the results and its interpretations is displayed in tabular form. Table 2 exhibits various levels of efficiency and productivity scores as an outcome of DEA.  Table 3 expresses the individual bank-wise TE score of PSBs during the study period. Table 1 shows that two banks, namely, Corporation Bank and Oriental Bank of Commerce, have emerged as 100% efficient banks throughout the study period. The Corporation Bank and Oriental Bank of Commerce are the references set for the remaining PSBs in India. This finding reveals that these two banks managed and utilized the resources well and had an optimum scale of operations. Another fascinating point to note is that the Corporation Bank and Oriental Bank of Commerce have not wasted their resources during the financial intermediation process. The best practices of these banks form the benchmark for the remaining banks.
Though Indian Bank captured a mean TE score of 1, it had less than 100% (99.3%) during 2011. The TE score of the remaining 17 banks is relatively lower (less than one), indicating scope for improvement in resource utilization and man-     Optimum resource allocation will facilitate the bank's resources to boost financial intermediation by extending banking facilities to more beneficiaries. Also, increased financial intermediation will positively affect the output level of goods and services in the economy.
The standard deviation reveals that dispersion among the banks in a year has been very moderate. This means that the TE scores achieved among all the banks have a very negligible deviation. Low product differentiation in the banking sector can contribute to negligible deviation among the TE scores of banks. All the banks, irrespective of their size and spread, cater to analogous segments and terrestrial populations. Strong product differentiation to satisfy the demands of heterogeneous customers may improve efficiency.
Having grasped TE and TIE of PSBs, it is essential to identify the growth or productivity changes during the study period. For this purpose, MPI is being used to trace the growth in productivity. As per MPI, Total Factor Productivity (TFP) is the outcome of better resource utilization. This study evidenced the relationship between efficiency and productivity. Out of 19 banks, 17 banks recorded productivity growth though they had less than 100% TE score. With the current status, only two banks with 100% TE score and the remaining banks with 97% to 99%, 17 banks registered with an increase in productivity. Though 89% of the sample banks (17 banks) were with less than 100% technical efficiency, they all could register productivity growth. This means that PSBs have done the effective channelization of resources in the process of financial intermediation. PSBs ensure a perfect link between the income earners with surplus and market players with the deficit. Through this process, PSBs impact the level of productivity in the economy.

CONCLUSION
This study focused on investigating the levels of technical efficiency and productivity of PSBs.
The study period of 2005-2018 (14 years) witnessed tremendous changes in banking regulation. However, the DEA results show that individual banks could achieve an overall 97% to 100% technical efficiency, leaving 3% as inefficiency. This means that PSBs are using more resources than required, also they can achieve the present output level with lesser input. MPI exhibits productivity growth ranging between 0.8% and 20.8%. PSBs with 97% to 100% TE could register productivity growth (17 banks out of 19 banks). This proves that efficiency and productivity are interrelated.
Perhaps it can be reaffirmed that PSBs have taken all efforts to be the best in the financial intermediation process with optimum use of their resources. The individual banks with less than 100% efficiency should refer to their peers, namely Corporation Bank and Oriental Bank of Commerce, that have 100% TE to improve their performance.
This study considers two economic effects to conclude. Namely, PSBs are government-controlled and preferred banks. Their long established existence in the economy and government regulation makes them work under a fixed framework to maintain efficiency, leading to better productivity. On the other hand, PSBs face a massive challenge by catering to the priority sector's requirement at a nominal rate. PSBs operating under these two regimes have proven that they excel in their operations. However, if they can optimally allocate their resources, that will bring better efficiency and productivity.
Academicians and researchers can adapt the considered research methodology to carry out further research in different regions. Besides, researchers can extend the scope of the study to any ownership structure, size, period and variables. The findings can also be used by policymakers to understand banks' efficiency and productivity level in India to establish additional policies. Moreover, DEA's output can be augmented for a second-stage analysis to scrutinize the significance of the input and output variables in determining bank efficiency.