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Dea Model for Estimating the Efficiency of Banks



Tobit model

In 1958, the Tobit regression model was introduced by Tobin. The Tobit model is a linear regression model with dependent variables being dichotomous latent variables, in which there is a certain threshold at which the dichotomous latent variables above or below this certain threshold are lost. In terms of model experimentation, the Tobit model can be presented generally and simply as follows:


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Dea Model for Estimating the Efficiency of Banks

mn

T E o i U ijt i V ijt

ii


TE is the technical efficiency of the research unit in year t estimated by the DEA model;

Uijt is a dummy variable ;


V ijt

is a reflection variable.


The choice of these variables is due to the characteristics of each industry and the requirements of the management agencies.

3.3.1. Selecting variables for the model

From the theoretical foundations of efficiency and efficiency evaluation and of econometric models as presented above, we can summarize some key issues on the basis of selecting variables and models for the topic as follows:

- Efficiency is a concept to evaluate how efficiently resources as inputs are used to produce an output of a specified final product. Determining inputs and outputs in general sectors is complicated but for the banking sector, the problem is much more complicated, because of the difficulties in determining the final products. For example, should bank deposits be considered as inputs or outputs. With two main approaches chosen for input and output variables, the first method considers banking activities as service providers, when deposits are considered as an output, the interest paid to depositors is not included in the total cost of the bank. Therefore, according to this approach, output and input are often taken as units of quantity, number of accounts, number of ATMs, number of transactions, transaction processes, etc. The second approach is the intermediary approach, based on the theory of intermediaries, considering banks as intermediary organizations.


between depositors and borrowers. For the intermediation approach, liabilities are considered as inputs and assets are called outputs.

In previous research topics, many studies have applied the intermediate approach in DEA analysis. Typically, some studies by Fukuyama (1995) used data during the collapse of the bubble economy in the period of 1989 - 1991 to estimate productivity growth and technical efficiency of the banking industry in Japan. Fukuyama used three input variables including Capital, capital sources from customers, labor and two output variables including loans and investments. The results from the model show that the average productivity change indexes in the second period from 1990 to 1991 are lower than in the first period from 1989 to 1990. Also in 1995, Favero and Papi used data of 174 Italian banks in 1991 to estimate technical efficiency, the inputs they used were 04 factors including capital, loan capital, financial capital and labor, the outputs they used were 03 factors investment, lending, non-interest income from lending activities. In 1998, Weizel and Lang evaluated the technology of 1490 German banks using the intermediation approach, according to which the inputs were deposits and the outputs were loans. In 1999, Avkira conducted technical efficiency estimates for commercial banks in Australia over the period 1986 to 1995. Avkira used two input variables, non-interest expenses and interest expenses, and two output variables, non-interest income and interest income. In 2002, similar to the input and output variables studied by Avkira, researcher Sathye estimated the productivity changes of Australian banks during the period 1995 to 1999. In 2001, in Malaysia, the Financial Sector Master plan program was implemented, some researchers tested the efficiency of 10 banks using the DEA model and the Malmquist index after implementing the above FSMP program during the period 2001 - 2005. This study also used the intermediate approach to select variables for the model including 03 inputs of deposits, salary expenses, total assets (excluding loans) and 02 outputs including loans. In 2011, researchers Ke-Chiun Chang, Yu Cao, Chang-liang Lin and Chia-Fu Lu used the DEA model to estimate the efficiency of 151 branches of a Taiwanese commercial bank in 2005. They chose the intermediate approach to select variables for their model, the input variables included: Personnel expenses, Interest fees, incidental expense; the output variables included: net profit, loans. The results showed that there were 26 efficient branches accounting for 17.2% and 125 inefficient branches accounting for 82.8%.


For this research topic, the author will select input and output variables for the analysis model using the intermediate approach. Specifically:

The output variables include:

Interest received ( y 1 ); income from other activities ( y 2 ); total outstanding debt ( y 3 ),

Input variables include:

Salary costs ( x 1 ); total assets ( x 2 ): include fixed assets such as office premises , equipment, technology investment... used by the bank in providing banking services (excluding loans because they are an output variable of this process); all deposits ( x 3 ); labor ( x 4 ).

The unit of measurement for variables y 1, y 2, y 3, x 1, x 2, x 3 is billion Vietnamese Dong (VND) and the unit of measurement for x 4 is person.

3.3.2. DEA model estimates the efficiency of banks

Although the parametric method is widely applied to studies calculating the efficiency of business operations, industries, etc., the non-parametric method is now also widely applied in cases where production technology is unclear, overcoming the limitations of complex production and distribution technical details. In particular, the non-parametric analysis method DEA has the outstanding advantage of analyzing the efficiency of businesses and industries with many inputs and many outputs, giving evaluation results close to the actual operations of businesses and industries, etc.

In the above section, the thesis presented the models used in the world to measure and evaluate the performance of banks, credit institutions, etc. However, within the framework of the thesis, the author chose the DEA model to evaluate the performance, and from there, based on the Tobit model to evaluate the impact of FDI, GDP and inflation factors on the performance of 23 commercial banks in the study. The reason for choosing the DEA model for evaluation is the advantages of this model. Specifically as follows:

Advantages of the DEA model: First, it allows for efficient analysis in cases where it is difficult to explain the relationship between multiple resources and the results of multiple activities in the production system without requiring the determination of a specific functional form when constructing the production frontier. Second, DEA is capable of analyzing a large number of input and output factors. Third, DEA can also be applied to qualitative variables, so it is used to analyze the efficiency of enterprises operating in many fields. Fourth, it can be used to separately estimate types of production efficiency such as technical efficiency, resource allocation efficiency, cost efficiency and efficiency.


Fifth, the method allows to evaluate the contribution of each input factor and output factor to the overall efficiency (or inefficiency) of the enterprise and to evaluate the level of inefficiency in resource use.

Although this method has some disadvantages such as it only gives the efficiency score as the relative efficiency between units (observation samples) with each other, so if a unit has an efficiency score of 100% and is on the optimal line, it does not mean that it is actually optimal (it is only more optimal than other units in the study). Therefore, DEA is often performed in combination with regression analysis in a 2-stage model (2-stages DEA) or multi-stages DEA to increase the model's persuasiveness.

However, the disadvantages of the DEA model are considered minor compared to the reliability of the results obtained after running the model.

Applying DEA and Tobit models in the thesis:

The efficiency estimation model of the banks considered is a DEA (data envelopment) model with variable returns to scale (VRS). We start by presenting a DEA model with constant returns to scale (CRS) and then extend it to consider variable returns to scale. In the case of technology under constant returns to scale, through linear programming it is possible to establish decision units (DMUs), which in this case are banks, defining an envelope, often called the efficient frontier. This benchmark frontier is a linear combination of the efficient banks in the sample. The set of best practice or frontier observations are those decision units for which there are no other decision units or a linear combination of units for which all outputs are equal to or greater (given a fixed number of inputs - for an output-oriented model) or all inputs are equal to or less (given a fixed number of outputs - for an input-oriented model). The DEA frontier is formed as a piecewise linear combination of these sets of best practice observations, giving us a convex production possibilities set. DEA provides a computational analysis of relative efficiency for multiple input/output situations by evaluating each decision unit and measuring its performance relative to the envelope of best practice units. Units that do not lie on this surface are called inefficient. Thus this method provides a measure of relative efficiency.

Let us briefly describe this corresponding DEA (linear programming) model. We assume that each bank has K inputs and M outputs for every DMU. For the i-th DMU the inputs and outputs are represented by the vectors xi and yi, respectively. For


For each bank (DMU) we want to obtain a measure of the ratio of all outputs

u ij y ij

j

for i i​

on all inputs, such as j , where ui and vi are weight vectors. To choose the

optimal weights, the following problem is proposed:

u ij y ij

max

uij , vij​​


with constraints

u ij y ij

j

v ij y ij

j

j 1

for i i​

j


u ik , v im ≥ 0

i, j = 1,2,…N (1)

k = 1,2,…K m = 1,2,… M

As is known with this representation of the model there are infinitely many solutions. This can be avoided by

v ij y ij

by introducing a constraint j = 1, and obtaining the factorized form of the linear programming problem:

max i j y i j

ij ,Z ij j


with constraints

Z ' i x i 1

i y j Z i ' x j ​​ 0

ik , Z im ≥ 0

i, j = 1,2,…N (2)

k = 1,2,…K m = 1,2,… M


here the vectors ui and vi are replaced by i and Zi. Using the duality property of this linear programming problem, Charnes, Cooper, and Rhodes (1978) derived an equivalent envelope form as:

min i

,


with constraints - yi + Y i ≥ 0

i x i - X i ≥ 0 (3)

in ≥ 0

where is an (N x 1)-dimensional vector; and , a scalar, is the efficiency score for the i-th DMU. The combination (X i, Y i) can be interpreted as a projection of the DMU onto the efficiency frontier, with the constraints interpreted accordingly. Note that i 1, with i = 1 implying a DMU on the efficiency frontier. Because of the smaller number of constraints, this formulaic representation is often used for computation.

However, the above approach is simplistic because it assumes constant returns to scale. The constant returns to scale assumption is only appropriate when all banks operate at an optimal scale. Factors that may cause banks to not operate at an optimal scale include imperfect competition, leverage concerns, and certain requirements. The fact that banks face varying returns to scale has been documented empirically by McAllister & McManus (1993), Wheelock and Wilson (1997), and others. This phenomenon led Banker, Charnes, and Cooper (1984) to suggest an extension of the DEA model under the constant returns to scale assumption to account for variable returns to scale (VRS) by adding a convex constraint N1' = 1 to Problem 3 above (where N1 is an (N x 1)-dimensional vector of 1s). This condition ensures that an inefficient bank is “benchmarked” against banks of similar size. Therefore, the VRS technique covers the data more closely than the CRS technique, and thus the VRS technical efficiency scores are greater than or equal to the CRS technical efficiency scores. The advantages of the VRS model outweigh the increase in computational power required to solve the model, which allows the VRS to gain more popularity than the CRS approach (Fried, Lovell and Schmidt (1993), Coelli, Rao and Battese (1998), Cooper, Seiford and Tone (2000)) Berger, Leusner and Mingo, (1997).

The model with additional constraints has the form:

min i

,


with constraints - yi + Y i ≥ 0

ixi - Xi ≥ 0 (4)

N 1 ' = 1

in ≥ 0

Types of efficiency: technical efficiency (TE), allocative efficiency (AE) and cost efficiency (CE) in the DEA model.

In 1957, Farell introduced the first efficiency measures based on the studies of Kopmans and Debeu (1951) to measure the efficiency of a unit with (in this study, we consider that unit to be a commercial bank) multiple inputs. Accordingly, he assumed that the efficiency of a commercial bank includes Technical efficiency (TE) and Allcative efficiency (AE), these two efficiencies reflect the ability to use inputs in optimal proportions. When combining these two components, we will have Cost efficiency (CE).

To illustrate his idea, Farell assumes that with constant returns to scale , a commercial bank uses two inputs x 1 , x 2 to produce one output

y. The AA' line is the total efficiency isotopes of a commercial bank.



Figure 3.3: TE, AE

Looking at Figure 3.3, if a commercial bank uses the inputs at point E to produce one unit of output, then the technical inefficiency of that bank is segment DE, which is the amount of input that a commercial bank can add.


decrease without decreasing output. The percentage by which all inputs can be reduced is DE/0E. The technical efficiency of a commercial bank is measured by the ratio TE = 0D/0E. A commercial bank is completely technically efficient when TE = 1.

To calculate Allcative efficiency, we can use the isocost curve BB'. Then the AE of a commercial bank operating at E is calculated as follows: AE = 0C/0D. The distance CD represents the reduction in inputs for a commercial bank to operate at allocative efficiency instead of operating at technical efficiency C'.

Cost efficiency (CE) is calculated as the ratio CE = 0C/0E. The CE segment is also considered as the reduction in input costs and we can see that CE = AExTE.

Efficiency of scale

In 1972, Afriat proposed the first model based on ideas in neoclassical production theory, such as consistency, restricted forms, recovery and extrapolation without maintenance and the hypothesis of functional forms. This method was applied to time series data and has been used in many studies to assess technical efficiency. In 1985, Fare et al introduced a non-parametric method for calculating efficiency for firms, extending Farell's approach by relaxing the limiting assumptions of constant returns to scale and strong utilisation of inputs, which were the main criticisms of Farell's method. Farell showed that the input efficiency of a firm does not necessarily imply the output efficiency for that firm. Technical efficiency, resource allocation efficiency, and other output efficiencies cannot be derived from corresponding efficiency measures, and vice versa, because output and input efficiency focus on different aspects of production. Therefore, it is important to determine the type of efficiency being assessed. Input-oriented technical efficiency can be defined as a firm's ability to produce as much output as possible, given a given level of inputs and given technology.

With the above guidelines, to obtain a separate estimate of technical efficiency for a commercial bank, the author will apply input-oriented technical efficiency measures. This measure must satisfy three scale behaviors, namely: constant returns to scale (CRS), non-increasing returns to scale (NRS), and variable returns to scale (VRS). The model I use is derived from Charnes et al. (1978) and later extended by Banker et al. (1981).

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