Cluster-Adjusted DEA E ciency in the presence of Heterogeneity: An Application to Banking Sector

This paper improves on the issues of extreme data points and heterogeneity found in the linear programming data envelopment analysis (DEA) by presenting a cluster-adjusted DEA model (DEA with cluster approach). This analysis, based on e ciency, determines the number of clusters via Gap statistic and Elbow methods. We use the December quarterly panel data consisting of 122 U.S agricultural banks across 37 states from 2000 to 2017 to estimate the cluster-adjusted DEA model. Empirical results show di erences in the estimated DEA e ciency measures with and without a clustering approach. Furthermore, using nonparametric tests, the results of Ansari-Bradley, Kruskal Wallis, andWilcoxon Rank Sum tests suggest that the cluster-adjusted DEAmodel provides statistically better e ciencymeasures in comparison to the DEAmodel without a clustering approach.

In the presence of extreme data points or wide-spread DMUs and heterogeneity, the DEA model provides misleading and inaccurate results (Po et al., 2009). Using the statistical cluster analysis, this issue of extreme data points and heterogeneity has been addressed (Samoilenko and Osei-Bryson, 2008;Jessop, 2009;and Po et al., 2009). However, de ning clusters based on an input-output frontier provides biased and higher eciency measures due to the localized frontier de ned by the input-output clusters (Meiman et al., 2002 andSaati et al., 2013). Following Shaik et al., (2012), this bias is due to three reasons: 1) The input-output frontier is driven by extreme DMUs. These DMUs force the e ciency to be 1 or closer to 1 suggesting these are the most e cient DMUs; 2) The weights associated with extreme DMUs force them to be 1; and 3) In order to identify clusters based on input-output frontier would require to identify clusters based on output or each individual inputs. Thus, the estimated e ciency accounts for potential clustering in output and inputs, so advantageous in clustering the data using e ciency rather than output-inputs or the associated frontier.
In this paper, we present one possibility to improve the DEA model by developing a cluster-adjusted DEA estimator based on a four-step methodology. The four-steps involve estimation of e ciency measures, cluster analysis based on e ciency, re-estimation of e ciency measures by cluster, and nonparametric statistical tests. First, using the DEA model without a clustering approach, we estimate the e ciency measures of the DMUs by year. Second, based on the estimated e ciency measures, we determine the optimal number of cluster groups. Third, using the statistically identi ed clusters of DMUs, we estimate the cluster-adjusted DEA model by year. Finally, we provide nonparametric tests to evaluate di erences in the DEA e ciency measures estimated with and without a clustering approach using Kolmogorov-Smirnov Statistics, Ansari-Bradley, Kruskal Wallis, and Wilcoxon Rank Sum tests.
Following the introduction in Section 1, the rest of this paper is organized into ve sections. Section 2 presents a brief literature review. Section 3 discusses the theoretical framework of the DEA and clusteradjusted DEA models. Section 4 presents the empirical data and construction of the input and output variables. Section 5 discusses the results and statistical implications. Finally, the summary of our conclusions is presented in Section 6.

Literature Review
Since the seminal works of Koopmans (1951), Farell (1957, and Farrell and Fieldhouse (1962) on e ciency measures, DEA of Charnes et al., (1978), a non-parametric method used for the estimation of production frontiers, can evaluate the performance of DMUs by measuring the relative e ciency using multiple inputs and multiple outputs. This measurement can be either output-oriented or input-oriented. With its application in the banking sector (Aly et al., 1990;Miller and Noulas, 1996;Chen 2002;Casu and Molyneux, 2003;Drake and Hall, 2003;Hauner, 2005;Tao et al., 2013;and Dipayan, 2014), DEA has expanded in dealing with the issues of heterogeneity in DMUs by integrating the DEA model and cluster analysis to alleviate the gaps in DEA modeling (Yang and Kuo, 2003;Paradi et al., 2012;and Maletic et al., 2013).
For example, Thanassoulis (1996) developed a method for simultaneously clustering operating units and determining a di erent set of marginal resource levels. Meiman et al., (2002) applied clustering directly to the results of DEA method with the goal of having multiple references of subsets from the original set of DMUs. Po et al., (2009) employed a piecewise production functions derived from the DEA method to cluster the data with input and output variables. Jessop (2009) used an integer DEA model with both the number and size distribution of groups as objectives and criteria. Samoilenko and Osei-Bryson (2008) presented a three steps methodology: cluster analysis, DEA, and Decision tree. Alirezaee and Sani (2011) presented a new hierarchical process DEA model. Paradi et al., (2012) applied the k-means clustering algorithm with DEA e ciency measures. Amirteimoori and Kordrostami (2014) clustered the operational units then evaluated each unit in its cluster. Tao et al., (2013) presented a hybrid model to conducting performance measurements using DEA and axiomatic fuzzy set clustering. Jahangoshai et al., (2018) integrated dynamic fuzzy C-means and Arti cial Neural Network with a DEA model to solve a multiple criteria optimization problem.
Our paper di ers from the existing literature in four di erent aspects. First, unlike the existing literature such as Tone (2017) that de ned the clusters based on an input-output frontier, this paper addresses the issue of heterogeneity by rst estimating the e ciency measures through the concept of DEA. Second, the optimal number of clusters is identi ed using alternative clustering methods, Gap Statistic and Elbow methods. The results of the optimal number of cluster groups are then compared to the di erent cluster indices discussed in Charrad et al., (2014). Third, the e ciency measures are re-estimated by cluster groups while accounting for the yearly variability. Finally, to test the robustness of the DEA model, nonparametric statistical tests are used to evaluate distributions of the e ciency measures estimated with and without the clustering method.

Theoretical framework
Primal production theory assumes that the relationship between multiple outputs, y = (y , x , ..., y j ) ∈ R J + and inputs, x = (x , x , ..., x i ) ∈ R I + is re ected by the concept of production function. The production function represents the relation between non-allocatable exogenous input vectors, x, used in the production of an endogenous output, y. The production function framework forms the bases in the estimation of the DMU's e ciency using the linear programming DEA model.

. DEA model
The technology that transforms multiple inputs into multiple outputs is represented by input set, L(y). The input set, L(y), satisfying constant returns to scale and strong disposability of input is de ned as: L(y) = {x : y can produce x; x ∈ R I + and y ∈ R J The input set, L(y), denotes the collection of input vectors that yield output vectors. This concept is represented by an input distance function evaluated for any DMU reference production possibility set, T, as: Here, the second expression of equation (2) identi es the linear program used to calculate the distance function, with z being a T × vector of intensity variables. Therefore, z identi es the constant returns-to-scale (CRS) boundaries of the reference set. Under the variable returns-to-scale (VRS), the intensity variable is z=1.
In addition, the scale e ciency measure is computed as the ratio of the e ciency measure estimated under CRS over pure technology estimated under VRS. The e ciency measures estimated by the DEA model (equation 2) forms the basis for a cluster analysis. The optimal number of cluster groups is identi ed using alternative clustering methods, Gap Statistic and Elbow methods. Next, the conceptual framework of the clustering method is presented.

. Estimating the number of clusters based on DEA e ciency measures
Cluster analysis deals with the identi cation of homogenous DMUs with similar patterns. Suppose that we have already estimated the e ciency measures of the DEA model. Let {λ tj }, t = , . . . , T de ne indepen-dent e ciency measures and j = , . . . , p DMU's. Additionally, suppose that we have already clustered the e ciency measures λ tj into k clusters C , . . . , C k with C k denoting the indices of observations in k and n k = |C k |. Thereafter, let d pp be the square Euclidean distance between the two DMUs, p and p , of e ciency measure, λ. There are several di erent indices for choosing the optimal number of clusters, k, in the k-means method, among them we focus on Gap Statistic and Elbow methods.³ By the rule of thumb, we assume that the maximum k value is: kmax = p .

. . Gap statistic method
The gap statistic method, rst by developed by Tibshirani et al., (2001) is based on the log standardization of the pooled within cluster sum of square, logW k . The determination of the number of cluster groups is as follows: • Step 1: From the number of clusters, k = , . . . , kmax, compute the pooled within cluster sum of squares around the cluster means, W k , as: where D k = p,p ∈C k d pp is the sum of the pairwise distance for all the points in cluster, k and n k is the number of DMUs in cluster, k. • Step 2: Generate B reference data sets using the uniform distribution and then cluster each data set with a varying number of clusters, k = , . . . , kmax.
Then compute the estimated Gap statistic Step 4: Letw = ( B ) B b= log(W * kb ) and the standard deviation, sd(k), be de ned as: and de ne s k = + B sd(k). • Step 5: Choose the number of clusters as the smallest k such as Gap(k) ≥ Gap(k) − s (k+ ) .

. . Elbow method
One the oldest methods for determining the optimal number of clusters is the Elbow method (Sugar, 1998).
The algorithm can be computed as follows: • Step 1: With a varying number of clusters, k = , . . . , kmax, compute the clustering algorithm using the DEA e ciency scores. • Step 2: For each k, compute the pooled within cluster sum of squares around the cluster means, W k , as: where D k = p,p ∈C k d pp is the sum of the pairwise distance for all the points in cluster, k and n k is the number of DMUs in cluster, k. • Step 3: At some value of k, W k will drop dramatically. Thereafter, it will reach a diminishing return with an increase in k. Therefore, choose k that does not increase much W k .
The optimal number of clusters, k, identi ed by Gap Statistic and Elbow methods forms the basis for the cluster-adjusted DEA model. These techniques use the e ciency measures of the DEA model and the output of the k-means clustering algorithm to form homogenous DMUs. The e ciency of these clusters depends on the change of the within-cluster dispersion, W k . Hence, taking into consideration the prede ned minimum and maximum k values in determining the optimal k value around the DMUs, the cluster of e ciency measures will result in dynamic size of homogenous DMUs.

. Cluster-adjusted DEA model
Once the optimal k value and number of homogenous DMUs are identi ed based on the e ciency measures, the cluster-adjusted DEA model is estimated using equation 3. The technology that transforms inputs into outputs is represented by the cluster-input set, L(y k ), where k is the number of cluster groups. Thus, the cluster-input distance function evaluated for any DMU within each cluster-reference production possibility set, K, is expressed as: where the number of clusters, k, and all other properties of the input distance function remains the same. The empirical application of our method is straightforward. The estimation of the DEA estimator in an e ciency-cluster-e ciency set up is completed as follows:

Empirical data
The agricultural banking sector is a major component of the nancial system. Hence, performance of the agricultural banking sector is critical to the stability and development of the United States' (U.S) economy. One of the most important factors that a ect the agricultural banking performance is the interest rates. An analysis of bank interest rate determinants is crucial to the understanding of the nancial intermediation. Interest rate is the price a borrower pays for the use of money they borrow from a lender or fee paid on borrowed assets (Crowley, 2007). Interest rates determine the pro tability of agricultural banks among other factors (Gardner et al., 2005). Proper interest rate management reduces bank exposure to risk and provides an opportunity to stabilize and improve their net income. According to Flannery (1980), when interest rates rise, banks managers can expect changes on both the asset and liability sides of their balance sheets. Since banks are always faced with the risk of having a low or high interest rate, the application of our methodology to the agricultural banking sector could help managers and regulators make sound banking decisions. Furthermore, in analyzing the interest rates of agricultural banks, we are addressing a signi cant issue in banking -banks are more sensitive to the changes in interest rates.
The Farm Credit Administration (FCA), an institutional part of the United States government provides a uniform call report that contains the nancial data of each Farm Credit System or agricultural banks that must be submitted to FCA quarterly. The data is freely available at: https://www.fca.gov/bank-oversight/callreport-data-for-download. After a post-hoc data cleaning, this paper uses a December quarterly data consisting of 122 agricultural banks across 37 states from 2000 to 2017.⁵ The input and output variables obtained from FCA must be consistent and provide a true re ection of the interest rates.
The output variables were selected to represent the income side of the agricultural banking sector. Thus, the selected two output variables were de ned as the total interest income and the total non-interest income.⁶ The input variables included were the total interest expenses and the total non-interest expenses.⁷ The selection of input and output variables in the DEA model needs careful attention because it may affect the distribution of technical e ciency measures. Since income is output based, the output price index is used to de ate the income. Thus, the deposit service (DS) and loan service (LS) price indexes are used to compute the quantity index (QI) of the output and input variables, respectively. The quantity index of the output and input variables is computed as, QI (output)= aggregate output variable × (100/LS) and QI (input)= input variable × (100/DS), respectively. The aggregate output variable is de ned as a sum of total interest income and total noninterest income. Table 1 presents the summary statistics of inputs, outputs, aggregate output, and price indexes. Since, FCA reports data that contains negative input values, the application of DEA model cautions the researcher to rst convert the negative values into positive by adding a common positive number. Thus, with a negative interest expense, a constant of 9,500 was added to the quantity indices (Table 1).
5 The quarterly data for December would include the previous 3 quarters data. The physical year starts with January 1 and ends with December 31, of each year. 6 The total interest income (RIAD4107) is the sum of 1) Total interest and fee income on loans (RIAD4010); 2) Income from lease nancing receivables (RIAD4065); 3) Interest income on balances due from depository institutions (RIAD4115); 4) Interest and dividend income on: U.S. Treasury securities and U.S. Government agency obligations excluding mortgage-backed securities (RIADB488), Mortgage-backed securities (RIADB489), and all other securities includes securities issued by states and political subdivisions (RIAD4060); 5) Interest income from trading assets (RIAD4069); 6) Interest income on federal funds sold and securities purchased under agreements to resell (RIAD4020); and 7) Other interest income (RIAD4518). The total non-interest income (RIAD4079) is the sum of 1) Income from duciary activities (RIAD4070); 2) Service charges on deposit accounts (RIAD4080); 3) Trading revenue (RIADA220); 4) Fees and commissions from securities brokerage (RIADC886); 5) Investment banking, advisory, and underwriting fees and commissions (RIADC888);6) Fees and commissions from annuity sales (RIADC887); 7) Underwriting income from insurance and reinsurance activities (C386); 8) Income from other insurance activities (RIADC387); 9) Venture capital revenue (RIADB491); 10) Net servicing fees (RIADB492); 11) Net securitization income (RIADB493); 12) Net gains (losses) on sales of loans and leases (RIAD5416); 13) Net gains (losses) on sales of other real estate owned (RIAD5415); 14) Net gains (losses) on sales of other assets (RIADB496); and 15) Other noninterest income (RIADB497). 7 The total interest expense (RIAD4073) is the sum of 1) Interest on deposits in domestic o ces on: Transaction accounts (interestbearing demand deposits, NOW accounts, ATS accounts, and telephone and preauthorized transfer accounts (RIAD4508), Nontransaction accounts with savings deposits (includes MMDA's) (RIAD0093), Non-transaction accounts with Time deposits of $250,000 or less (RIADHK03), Non-transaction accounts with time deposits of more than $250,000 (RIADHK04); 2) Interest on deposits in foreign o ces, edge and agreement subsidiaries, and IBFs (RIAD4172); 3) Expense of federal funds purchased and securities sold under agreements to repurchase (RIAD4180); 4) Interest on trading liabilities and other borrowed money (RIAD4185); and 5) Interest on subordinated notes and debentures (RIAD4200). The total noninterest expense (RIAD4093) is the sum of Noninterest expense on: 1) Salaries and employee bene ts (RIAD4135); 2) Expenses of premises and xed assets (net of rental income) (excluding salaries and employee bene ts and mortgage interest (RIAD4217); 3) Goodwill impairment losses (RIADC216); 4) Amortization expense and impairment losses for other intangible assets (RIADC232); and 5) Other noninterest expense (RIAD4092).

Empirical Results and Discussions
An input-oriented DEA model was adopted because 1) Banks have better control over inputs than outputs and 2) In the presence of negative input values, the output-oriented DEA model becomes infeasible. Using the quantity index of inputs and the aggregate output variables, the DEA model (equation 2) was estimated by year due to the di erences in banks through time under CRS, VRS, and scale assumptions. While accounting for the yearly variability, the optimal number of clusters and the partition of banks into groups were done using the estimated DEA e ciency measures under CRS assumption. In addition, the cluster-adjusted DEA model was estimated using equation 3. All the models were estimated in R language and the nonparametric tests in Statistical Analysis Software.

. E ciency measures
The e ciency measures estimated using the DEA model de ned in equation (2) are used in the cluster analysis. First, the yearly mean DEA e ciency measures range from 0.863 to 0.900 under CRS, 0.933 to 0.953 under VRS and 0.920 to 0.952 under the scale assumptions. The results in mean DEA e ciency measures are validated by the slight uctuation of the standard deviations through the years. Furthermore, the results suggest that during the nancial crisis of 2007-2009, banks were on average e cient. Second, the limitation of a standard formulation of the DEA model is to build a separate linear program for each bank. However, since the data is composed of wide-spread and heterogenous banks, the e ciency measures estimated in equation 2 are biased and inaccurate. That is, the DEA model without a clustering approach fails to de ne the group of banks that are like the banks under evaluation. Henceforth, it may be di cult to interpret the results of Table  2 because of the non-homogeneous banks. Finally, a comparison of the DEA e ciency measures under the scale assumption suggests that the VRS technology is higher than CRS technology. To avoid bias due to scale e ciency, the optimal number of clusters is determined based on the CRS e ciency measures.

Std.dev: Standard deviation. Estimation is based on the input-oriented DEA model.
Existing literature has shown that the analysis of banks' performance and e ciency would be almost impossible if all banks had the same capital structure, o ered the same mix of services, followed identical accounting practices, and were equally a ected by in ation and operated under the same regulatory restrictions (Vittas, 1991). However, with banks exhibiting considerable di erences in their e ciency measures, qualitative problems would undermine the usefulness of these ratios for analytical and policy purposes when not properly accounted for. Figure 1 presents the yearly distribution by year of the DEA e ciency measures under CRS assumption. Unlike the existing literature such as Meiman et al., (2002) and Saati et al., (2013) who clustered the input and output variables, the results in Figure 1 illustrate the presence of cluster groups and the importance of yearly variability of the e ciency measures. Hence, with distinct groups of e ciency measures presented annually in Figure 1, Table 3 summarizes the results of the k-means clustering approach based on the Gap statistic and Elbow methods, and the 30 indices present in the NbClust package. Within each year of Table 3, three cluster groups are su cient to partition the e ciency measures of banks. These results are further validated by the majority rule decision of the 30 indices. In addition, from Table 3, two important results emerge.
First, 12 indices of the NbClust package suggest that within each year, the optimal number of clusters is three. This result is based on the majority vote or ruling of the NbClust. In addition, 5 indices of the NbClust package suggest that four cluster groups are su cient to partition the banks. Second, even though the num-ber of cluster groups is identical from 2000 to 2017, the composition of banks within cluster di ered by year (dynamic). For example, for the rst cluster group, 27 banks were in 2000 whereas 22 banks in 2001. In contrast, for the second cluster group, 11 banks were in 2000 and 17 banks in 2001. After statistically determining the optimal number of clusters, the cluster-adjusted DEA model (equation 3) is re-estimated while accounting for the yearly variability. Overall, most banks experienced an increase in their e ciency measures in the early 2005 and followed a downward trend during the nancial crisis of 2007. Additionally, the results of Table 2 clearly depict that most banks were able to convert inputs to outputs e ectively.

. Cluster-adjusted e ciency measures
Having rst discussed the estimation of the DEA e ciency measures (Table 2), it is important to know the magnitude of the e ciency measures estimated using the cluster-adjusted DEA model. Tables 4, 5, and 6 respectively present the summary statistics of the cluster-adjusted DEA e ciency measures by year within cluster 1, 2, and 3. In comparison to Table 2, the cluster-adjusted DEA e ciency measures in Tables 4, 5, and 6 are higher while accounting for homogenous banks. The higher e ciency measures do not preclude that the banks are performing better. However, this suggests that we are evaluating homogenous banks or DMUs based on similar characteristics identi ed by the cluster analysis of e ciency. Std.dev: Standard deviation. Estimation is based on the input-oriented DEA model. Std.dev: Standard deviation. Estimation is based on the input-oriented DEA model.

. Nonparametric tests of di erence in e ciency with and without the clustering approach
This subsection focuses on the statistical comparison of the DEA e ciency measures estimated with and without a clustering approach. The statistical comparison involves the following: 1) compute the magnitude of di erences in e ciency measures with and without a clustering approach, and 2) conduct statistical tests to evaluate the signi cance of di erences in e ciency measures with and without a clustering approach. The rst analysis, accomplished by computing the change in the e ciency measures, is de ned as: whereμ c it andμ it are respectively the e ciency measures estimated with and without a clustering approach, and i represents the individual bank, and t represents the time.
To evaluate di erences in e ciency measures estimated with and without a clustering approach, we investigate the feasibility of the parametric approach to statistically test the signi cance level ofδ it . Figures  2, 3, and 4 respectively show the distributions ofδ it for the CRS, VRS, and scale e ciency measures. The visual representation shows whether the distributions are bell-shaped and provide indication about their respective skewness. These results suggest that the use of parametric tests, i.e., normality assumptions (normality and equal variances) for the pooledδ of CRS, VRS, and scale e ciency measures are not valid.

Ansari-Bradley Test
Ho: The di erence in population dispersion, σ, of the DEA e ciency measures obtained with a clustering approach,μ c it , and without a clustering approach,μ it , is equal to zero. That is: σμc it − σμ it =0. Ha: Population dispersion parameters, σ, of the DEA e ciency measures obtained with a clustering approach,μ c it , is greater than the population dispersion parameters, σ, of the DEA e ciency measures obtained without a clustering approach,μ it . That is: σμc it − σμ it > .  ** denotes the signi cance at a 1 percent level and * denotes the signi cance at a 5 percent level.  ** denotes the signi cance at a 1 percent level and * denotes the signi cance at a 5 percent level.

Challenges and Conclusions
This paper, addressing the issues associated with extreme data points and heterogeneity found in the linear programming data envelopment analysis (DEA) model, presents an alternative cluster-adjusted DEA model. However, unlike existing literature that de nes the clusters based on inputs-outputs, we de ne the clusters based on the DEA e ciency measures. The number of clusters based on the DEA e ciency measures is statistically determined using Gap statistic and Elbow methods. We use the December quarterly panel data consisting of 122 U.S agricultural banks across 37 states from 2000 to 2017 to estimate the cluster-adjusted DEA model. The proposed cluster-adjusted DEA model involves 4 stages (or steps). First, we estimate the e ciency measures using linear programming DEA model. Second, based on the estimated e ciency measures, the optimal number of cluster groups is determined using the Gap statistic and Elbow methods. These results are further validated by the 30 indices of the NbClust package. Accordingly, the majority rule of the clustering indices concluded that the optimal number of clusters is three groups. Furthermore, these results were supported by the distribution of DEA e ciency measures under the CRS assumption ( Figure 1).
Third, using the statistically identi ed clusters of banks, we estimate the cluster-adjusted DEA model while accounting for the yearly variability. Finally, in the evaluation of di erences in the e ciency measures estimated with the DEA and cluster-adjusted DEA models, the nonparametric tests of Kolmogorov-Smirnov Statistics, Kruskal-Wallis, Wilcoxon Rank Sum, and Ansari-Bradley are conducted. These tests were conducted to compare the distributions, medians, and dispersions of the DEA and cluster-adjusted DEA e ciency estimators. Our results provide evidence that the deterministic DEA model does not guarantee accurate e ciency measures in the presence of non-homogeneous banks or DMUs. However, there are limitations that future researchers could study to improve the discriminatory power of the cluster-adjusted DEA results. For example, future research could incorporate banks merger and acquisition in order to achieve optimal economies of scale. Compared to the current framework, the results of the e ciency measures could vary regarding the type of mergers and the number of yearly mergers. It could be great to further incorporate the nancial crisis as a dummy and study its implication of the DEA cluster e ciency measures. Research could also focus on classifying the total assets using the FCA's classi cation requirement and comparing the statistical properties with the clustering approach of DEA e ciency measures.
Financial Support: This research was supported by the Center for Agricultural Policy and Trade Studies, North Dakota State University [N/A].

Con ict of Interests Statement:
The authors have no con icts of interest to disclose.