USING DATA ENVELOPMENT ANALYSIS (DEA) TO FORECAST BANK PERFORMANCE

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1 USING DATA ENVELOPMENT ANALYSIS (DEA) TO FORECAST BANK PERFORMANCE Ronald K. Klimberg, Saint Joseph s University, Haub School of Business, Philadelphia, PA, 19131, , klimberg@sju.edu Kenneth D. Lawrence, School of Management, NJIT, Newark, NJ 07102, (973) , carpetfour@yahoo.com Tanya Lal,, Saint Joseph s University, Haub School of Business, Philadelphia, PA, 19131, , tl361386@sju.edu ABSTRACT Forecasting is an important tool used by businesses to plan and evaluate their operations. One of the most commonly used techniques for forecasting is regression analysis. Often forecasts are produced for a set of comparable units which could be individuals, groups, departments or companies that perform similar activities such as a set of banks, a group of mangers and so on. We apply a methodology that includes a new variable, the comparable unit s DEA relative efficiency, into the regression analysis. The results of applying this methodology to the performance of commercial banks will be presented. (Keywords: Forecasting, Data Envelopment Analysis, Regression) INTRODUCTION Quantitative forecasting models, even rather sophisticated models, are easier to develop and use today as result of our improving computer technology. These quantitative forecasting techniques use historical data to predict the future. Most quantitative forecasting techniques can be categorized into either time series approaches or causal models. Time series forecasting techniques are forecasting techniques that only use the time series data itself and not any other data to build the forecasting models. These time series approaches isolate and measure the impact of the trend, seasonal, and cyclical time series components. Causal models use a set of predictor/independent variables, possibly also including the time series components, that are believe to influence the forecasted variable. One of the most popular causal model approach is regression analysis. Regression techniques employ the statistical method of least squares to establish a statistical relationship between the forecasted variable and the set of predictor/independent variables. Many forecasting situations involve producing forecasts for comparable units. A comparable unit could be an individual, group of individuals, a department, a company, and so on. Each comparable unit should be performing similar set of tasks. When applying regression analysis, the established statistical relationship is an average relationship using one set of weights assigned to the predictor/independent variables. However, when regression is applied to a set of comparable units the relative weight/importance of each of the predictor/independent variables

2 will most likely vary from comparable unit to comparable unit. For example, if advertising is an independent variable, one comparable unit might emphasize advertising more (or less) than other comparable units. Either way is not necessarily better nor worse, it is just how that particular comparable unit emphasizes advertising. As a result, in some cases, the regression model could provide forecast estimates that are too high or too low. In this paper, we will apply and extend some of our recent previous work, Klimberg et al., [3, 4], in which we introduced a methodology that incorporates into the regression forecasting analysis a new variable that captures the unique weighting of each comparable unit. This new variable is the relative efficiency of each comparable unit that is generated by a non-parametric technique called data envelopment analysis (DEA). In the next section, we provide a brief introduction to DEA. Subsequently, we discuss the methodology and present the results of applying our methodology to a data set of commercial banks. Finally, the conclusions and future extensions are discussed. m vixik = 1 i = 1 DATA ENVELOPMENT ANALYSIS (DEA) DEA utilizes linear programming to produce measures of the relative efficiency of comparable units that employ multiple inputs and outputs. DEA takes into account multiple inputs and outputs to produce a single aggregate measure of relative efficiency for each comparable unit. The technique can analyze these multiple inputs and outputs in their natural physical units without reducing or transforming them into some common measurement such as dollars. The Charnes, Cooper and Rhodes (CCR) DEA model [1] is a linear program that compares the ratio of weighted outputs to weighed inputs, i.e., efficiency, for each comparable unit. The efficiency of the k th comparable unit (i.e., E k ) is obtained by solving the following linear formulation: t MAX Ek = u ryrk r = 1 s.t. t m u ryrj - vixij 0 j = 1,..., n r = 1 i = 1 u r, vi ε r,i where: Parameters Y rj = amount of the rth output for the jth comparable unit; X ij = amount of the ith input for the jth comparable unit; t = the number of outputs; m = the number of inputs, and; n = the number of comparable units;, = is a small infinitesimal value;

3 Decision Variables u r = the weight assigned to the rth output, and; v i = the weight assigned to the ith input. The CCR DEA formulation determines objectively the set of weights, u r and v i, that maximizes the efficiency of the kth comparable unit, E k. The constraints require the efficiency of each comparable unit, including the kth comparable unit, not to exceed 1, and the weights, u r and v i, must be positive. A similar DEA formulation must be solved for each comparable unit. A comparable unit is considered relatively inefficient (i.e., E k <1) if it is possible to increase its outputs without increasing inputs or decrease its inputs without decreasing outputs. A comparable unit identified as being efficient (i.e., E k =1) does not necessarily imply absolute efficiency. It is only relatively efficient as compared to the other comparable units that are being considered. These efficiency ratings allow decision-makers to identify which comparable units are in need of improvement and to what degree. Each efficiency score measures the relative efficiency of the comparable unit. These efficiency scores can be use to evaluate performance of the comparable units and provide benchmarks. Nevertheless, besides each efficiency score being comprised of a different set of inputs and outputs values, each comparable unit s efficiency score includes a unique set of weights. The DEA process attempts to find objectively the set of weights which will maximize a comparable unit's efficiency. Therefore, the DEA model has selected the best possible set of weights for each comparable unit. The variation of these weights from comparable unit to comparable unit allows each comparable unit to have their own unique freedom to emphasize the importance of each of these input and output variables in their own way. How well they do this is measure by the efficiency score. Since the Charnes, et al. s 1978 paper, there have been thousands of theoretical contributions and practical applications in various fields using DEA. DEA has been applied to many diverse areas such as: health care, military operations, criminal courts, university departments, banks, electric utilities mining operations, and manufacturing productivity, [2, 5, 6]. REGRESSION FORECASTING METHODOLOGY Our regression forecasting methodology is designed to be applied to a historical data set of multiple inputs and outputs variables from a set of comparable units, [3, 4]. Additionally, one output variable is assumed to be the principal/critical variable that will be needed to be forecasted, e.g., sales, production, or demand. Since the data set we studied has a relatively small number of inputs and outputs we adjust our procedure and eliminate the initial stepwise regression. As a result, the first step is to run a DEA for each comparable unit. We use these efficiency scores as surrogate measures of the unique emphasis of the variables and of performance. Using a principal/critical output variable as the regression dependent variable, all the input variables plus the DEA efficiency score as regression independent variables, we run the regression. This regression model with the DEA efficiency variable should be superior, i.e., should have a significantly lower standard error of the mean and increase R 2, to the regression model without the DEA efficiency score variable.

4 EXAMPLE Seiford and Zhu, [7], applied DEA to the 55 U.S. commercial banks that appeared in the Fortune 1000 list in April The DEA input variables were the number of employees, assets, and stockholder s equity; and the DEA output variables were revenue and profit. The selection of these variables were based on Fortune s original choice of factors for performance characterization, [7]. We retrieved the same Fortune 1000 list of U.S. commercial banks from 2003 to We ran similar DEA models, i.e., same input and output variables as Seiford and Zhu, for 2003 to Table 1 below lists the frequency distribution of the DEA efficiency scores for these years. As shown in Table 1, these efficiency scores are rather dispersed. Table 1. Frequency distribution of the DEA efficiency scores for each year. Using the DEA efficiency scores as an input and revenue as our primary output variable, we ran regression models for 2004 to The basic regression equation used was: Revenue(t) = Employees(t-1) + Assets(t-1) + Equity(t-1) + DEA(t-1); where t = 2004, 2005, 2006; (we refer to this model as w/dea). Additionally we ran the same regression without the DEA efficiency score variable (we refer to this model as NoDEA). Tables 2 and 3 summarize the regression models results with R 2 values and standard errors. The w/dea models were consistently better than the NoDEA models. In terms of R 2 values, the NoDEA models in each year had extremely high R 2 values. The w/dea models only slightly increase the R 2 values; averaging only.1% improvement. Table 2. The regression R 2 values for each year and for the two models.

5 Table 3. The regression standard errors for each year and for the two models. The standard error values for the w/dea models, in Table 3, had a more significant improvement; averaging 8.19% decrease in the standard errors. Table 4 summarizes the residual results by displaying the maximum and minimum residual for each model. In each case, the w/dea regression models performed better than the NoDEA regression models. Table 4. Residual analysis for each year and for the two models. CONCLUSIONS In this paper, we applied a new regression forecasting methodology to forecasting comparable units. This approach included in the regression analysis a surrogate measure of the unique weighting of the variables and of performance. This new variable is the relative efficiency of each comparable unit that is generated by DEA. The results of applying this new regression forecasting methodology including a DEA efficiency variable to a data set demonstrated that this may provide a promising rich approach to forecasting comparable units. We plan to perform further testing with other data sets, some with more comparable units and more years of data. REFERENCES [1] Charnes A., Cooper, W.W. and E. Rhodes, Measuring Efficiency of Decision Making Units, European Journal of Operational Research. 2, pp , [2] Klimberg, R.K. and D. Kern, "Understanding Data Envelopment Analysis (DEA)," Boston University School of Management Working Paper, pp , [3] Klimberg, R. K., Lawrence, K. D., and S.M. Lawrence, Improved Performance Evaluation of Comparable Units with Data Envelopment Analysis (DEA), Advances in Business and Management Forecasting, Volume 5, Elsevier Ltd., pp , 2008.

6 [4] Klimberg, R. K., Lawrence, K. D., and S.M. Lawrence, Forecasting Sales of Comparable Units with Data Envelopment Analysis (DEA), Advances in Business and Management Forecasting, Volume 4, JAI Press/North Holland, pp , [5] Seiford, L.M., Data Envelopment Analysis: The Evaluation of the State of the Art ( , The Journal of Productivity Analysis, 9, pp ,1996. [6] Seiford, L. M. and R. M. Thrall, Recent developments in DEA: the mathematical programming approach to frontier analysis." Journal of Econometric, 46, pp. 7-38, [7] Seiford, L. M. and J. Zhu, Profitability and Marketability of the Top 55 U.S. Commercial Banks, Management Science, Vol. 45, No. 9, Sept. 1999, pp