Tobit Model for Determining Factors Affecting Efficiency

= 1, 2, …, q);

Let z be an N vector of weights where the factors of the vector in number are denoted by z i .

DEA model under constant return to scale (CRS) assumption

With the above assumptions, the author will first calculate technical efficiency according to CRS through the following equation:

i  min  ,

In there:

c , z

N

zx i x i

IP

i  1

where i = 1, 2, …, N; P = 1, 2, …, p;

y zy

N

ii

qiq

i  1

where i = 1, 2, …, N; Q = 1, 2, …, q;

z i  0 for all i

Scale value represents a proportional decrease in all inputs so that 0   1



i

and c

ii

x

is the smallest value of should cp

is the vector of technical efficiency of the factors.

input factors for a commercial bank i. Technical efficiency is achieved at its minimum level.

c

many timesi = 1 or a commercial bank operates at its best when DEA gives

i

i

results​c = 1 and then a commercial bank cannot perform any better with the established observations. In the other case, if c < 1 then one

Commercial banks are operating below their best efficiency.

Illustrated by the diagram, we see that CRS is the OG frontier, a commercial bank located on this frontier is efficient. Suppose there is a branch of the i-th bank located on the right side of the frontier at point E, that is, this bank operates inefficiently. With a given set of observations, these banks can improve the productivity of input factors compared to the most efficient banks located on the OG frontier. According to the diagram, CRS = AB/AE, so the i-th bank can reduce (1 - ) input quantity to achieve optimal efficiency at point B.

Figure 3.4: CRS (OG), VRS (CFC'), NRS (OFC') boundaries

In order to find a best performance benchmark for each commercial bank, DEA has been shown to be superior as shown above. Here, the benchmark for each commercial bank i is constructed from the vector z and its factor values ​​determined when the linear programming problem above is solved. The bank that is not performing at its best will not be included in the best performance benchmark for the bank i , or most of the factors of the vector z will be zero. The factors that are not equal to zero indicate the components of the best performance benchmark. This measure of technical efficiency is often called the measure of total technical efficiency because the residual of total technical efficiency includes all sources of inefficiency, controllable and uncontrollable factors.

control. The estimate of total technical inefficiency includes inefficiencies originating from branch size, poor management, and other unspecified variables.

To determine the optimal size for a commercial bank, two additional DEAs need to be applied, namely VRS and NRS.

DEA model under variable return to scale (Variable return to scale- VRS) assumption

The technical efficiency of a commercial bank i under VRS is solved through the following equation:

i  min  ,

In there:

v , z

N

zx i x i

IP

i  1

where i = 1, 2, …, N; P = 1, 2, …, p;

y zy

N

ii

qiq

i  1

where i = 1, 2, …, N; Q = 1, 2, …, q;

N

z i  1

i  1

z i  0 for all i

is calculated using the formula VRS = AD/AE.

NRS Non-increasing returns to scale

The technical efficiency of a commercial bank i under NRS is solved through the following equation:

i  min  ,

In there

n , z

N

zx i x i

IP

i  1

where i = 1, 2, …, N; P = 1, 2, …, p;

y zy

N

ii

qiq

i  1

where i = 1, 2, …, N; Q = 1, 2, …, q;

N

z i  1

i  1

z i  0 for all i

In the above figure, the OFC' line represents NRS.

SE scale efficiency

The formula for calculating scale efficiency (SE) of a commercial bank i based on the input orientation of the variables will be calculated according to the following formula:

i

i c



s i

v

Scale efficiency is the ratio of the CRS/VRS

In the above approaches, with the results obtained, we do not know whether a commercial bank is operating in the range of increasing or decreasing scale. To solve



i

c

v

to solve this problem, can be determined by looking at the ratio  i

equal to or less than one.

If

i



i

c 1

v

then a commercial bank i is scale efficient then capacity

The productivity of inputs cannot be improved by increasing or decreasing the scale of production. In the figure, the scale efficiency is at point F.

i

i

c 1

If  v

then a commercial bank i is scale inefficient, then



i

v

c

The ratio of output lost due to scale inefficiency can be defined as 1 - the policy intervention that would be more efficient:

 i . has two

sn

- If they are not equal  i  i : then a commercial bank i is scale inefficient due to decreasing returns to scale DRS. Therefore, a commercial bank i can increase input productivity and reduce costs by reducing its size.

sn

- If they are equal  i  i : then an ith commercial bank is scale inefficient due to increasing returns to scale IRS. Therefore, an ith commercial bank can improve input productivity and reduce unit cost by increasing its size.

Through the analysis of the DEA model as above, there are two causes of technical inefficiency of a commercial bank. The first cause is due to pure technical inefficiency, the second cause is due to scale efficiency. If there is no error in determining the input, output and operating environment variables, the first cause of pure technical inefficiency reflects the deviation in management compared to the best performing bank. Thus, the results from the DEA model give us the measures of scale efficiency, pure technical efficiency, and total technical efficiency of each commercial bank, thereby determining the best operating standard in evaluating the performance of a commercial bank.

3.3.3. Model to determine factors affecting efficiency (Tobit)

Although it is possible to evaluate the relative efficiency of DMUs along the frontiers using linear programming problems, and to make recommendations for improving the efficiency of the unit based on the estimated efficiency results, this may not help us to detect the factors that limit efficiency. Therefore, we will use the model that allows analysis to find out the influencing factors. Since the value of efficiency from the estimated models is the dependent variable. If the Ordinary Least Squares (OLS) method is used to estimate unknown parameters, it will lead to biased and unstable estimates. Therefore, the author considers the Tobit regression model, which is based on the principle of maximum likelihood estimation, to obtain stable estimates of the parameters. The Tobit regression model was proposed by Tobin in 1958, and since then many scholars have continuously developed and improved the model. This regression model belongs to the category of econometric models with restricted or truncated dependent variables, and the essence is that the important explanatory variables can take real observations but the dependent variable can only be observed in a limited way, and the standard model is as follows:

iiii

TE * X

 , 

N (0, 2 )

(5)

Here, i denotes the ith DMU, ​​TEi* is the hidden variable (i.e., unobservable), Xi is the K × 1-dimensional matrix of the independent variables.  is the random error with distribution N(0,  2). The restricted sample value yi is

 TE * iff TE *  0

TE i

  ii

i

 T if TE *  0

TEi is the estimated efficiency from model (4). To explain the leakage of Tobit models, we define:

Output set from the boundary model: TE=  TEi: TEi is the solution of problem (4)}

TEi = α0 + α1.GDP + α2.INF + α3.FDI + α4.ROE + α5.ROA + α6.K/L + α7.SIZE + Ω

TE i : technical efficiency or scale efficiency of commercial bank i ; GDP : impact of GDP on commercial bank i ;

INF: impact of inflation on commercial bank i ; FDI: impact of FDI on commercial bank i;

Ω is the error.

And the input includes the variables of the vector defined as follows: Variables belonging to technology and business size:

Capital equipment level per labor:

KL i

Ki​

L i

Variables belonging to corporate finance:

ROA is the return on total assets ratio, measured by profit after tax over total assets (TTS), meaning

ROA tax collection

TTS

ROA is an indicator measuring the efficiency of using total assets of the bank. The profit margin on total assets shows how much profit after tax is generated from one dong of assets. The higher the ROA result, the better the ability to generate income from assets of the bank, the better the business performance and vice versa. According to international practice, the ROA index ≥ 1% shows that the bank is highly effective in using total assets (Tran Tho Dat and Le Thanh Tam, 2016). According to CAMEL, the bank is most effective when the ROA ratio ≥ 1.5% (Rozzani and Rahman, 2013).

ROE is the return on equity ratio, it is measured by profit after tax on equity (equity) which means

ROE tax collection

vcsh

ROE is an indicator measuring the efficiency of a bank's use of equity. The return on equity ratio reflects the bank's ability to earn net profit on 1 dong of equity. A high return on equity is considered good, demonstrating the bank's ability to generate a lot of profit on equity, or in other words, demonstrating the optimization of the bank's use of equity. According to international practice (Moody's Investors Service: MIS-Moody's Investors Service),

ROE ≥ 12%-15% is considered good. In Vietnam: ROE is considered good if it is between 14% - 17% (Tran Tho Dat and Le Thanh Tam, 2016). According to CAMEL standards, banks are most effective when the ROE index is ≥ 22% (Rozzani and Rahman, 2013). The higher the ROE index, the higher the net profit that the bank's shareholders receive.

(vi). E/A self-financing ratio, it is measured by equity (equity) over total assets (TTS), which means:

E / A  vcsh

TTS (Self-funding ratio)

Malmquist Index

Caves et al. (1982) developed the Malmquist (output-based) idea into a productivity index. The authors used the concept of a gap function, although they did not establish a link with Farrel-type efficiency measures (see Farrel, 1957). Fare et al. (1992) developed a productivity index using the geometric mean of two Malmquist productivity indices of technical change and changes in technical efficiency. Furthermore, Fare et al. (1994) used data envelopment analysis to measure Malmquist productivity indices as the ratio of the values ​​of the output gap functions to a reference technology with constant returns to scale (CRS). Many studies on the Malmquist index have followed the study of Fare et al. (1994) such as Grifell-tatje and Lovell (1996), Ray and Desli (1997), Ray (1999), Wheelock and Wilson (1999), Bukh et al. (1995), Rebelo, J., and V. Mendes. (2000).

The thesis also uses an advanced decomposition of the Malmquist index developed in Fare et al. (1994). It can be expressed as follows

 d t (x

, y )

d t  1 (x

, y )  1/ 2

0

m 0  (y t  1 , x t  1 , y t , x t ) 0 t  1 t  10 t  1 t  1 

0

 d t (x t

, y t )

d t  1 (x t

, y t ) 

(6)



This Malmquist index is decomposed into two factors: one factor only for technological progress and the other factor only for changes in technical change, which can be interpreted as a “catch-up” effect. We represent it as follows

d t  1 (x

, y )  d t (x

, y )

d t (x

, y )

 1/ 2

m 0  (y t  1 , x t  1 , y t , x t ) 

0 t  1

t

t  1 t 0

t  1

t  1 0

t

tt 

d 0 (x t , y t )

d  1 (x t , yes

) d  1 (x , y )

0 1

t  1

0 tt 

(7)

The first factor on the left-hand side can be further decomposed into two factors. The changes

The efficiency change is decomposed into two components: one component represents the efficiency change.

pure result and the other component is the change in scale. We can also represent the above formula as follows

d t  1 (x

, y )

d t  1 (x

, y )

 d t (x , y )

d t  1 (x

, y ) 

0

0

0

0 t  1 t  1 CRS0 t  1 t  1 VRS0 tt VRS0 t  1 t  1 CRS

0

d t (x t

, y t

) CRS

d t (x t

, y t )

VRS

 d t (x t

, y t )

CRS

d t  1 (x

t  1 , y

t  1

) VRS (8)

All this content is presented in chapter 4.