## 3 Methodology  
### SFA (Stochastic Frontier Analysis)  
SFA expresses the relationship between input and output factors as a production function, and technical efficiency is estimated through the frontier production function, which represents the maximum output versus input. The TE (technical efficiency) of a company indicates where the company's technology level is relative to the efficiency technology level expressed in the form of a frontier production function. The farther the company's technology level is from the frontier production function, the lower the efficiency of the company. The production frontier can be estimated through non-parametric and parametric methods. In this study, the parametric method estimates the production frontier using stochastic frontier analysis. Also, in this study, FRONTIER Version 4.1 provided by Coelli is used for estimation. According to [PERSON] and [PERSON] [59], a stochastic production frontier model is assumed as shown in Equation (1) below to reflect the change in efficiency over time. \[Y_{it}=f(x_{it},\ \beta)e^{V_{it}-U_{it}},\ i=1,2,\ldots,N,\ t=1,\ 2,\ldots,\ T \tag{1}\]  
\(Y_{it}\)  
between groups cannot be performed using traditional SFA. Therefore, in order to compare the efficiency levels of different groups operating under different technical conditions, we use the meta-frontier production function that wraps the production function of all groups [59]. Created by [PERSON], [PERSON], and [PERSON] [60; 61], the meta-frontier production function model is defined by the formula below. \[ \begin{split} Y_{it}^{*}=f(x_{it},\ \beta^{*})=e^{x_{it}\beta^{*}},\ i=1,2, \ldots,N,\ N=\sum\limits_{i=1}^{R}N_{j},\ t=1,\ 2,\ldots,\ T,\ s.t.\ x_{it}\beta^{*}\\ \geq x_{it}\beta_{(j)}\ for\ all\ j\end{split}  \tag{4}\]  
In this case, \(\beta_{(j)}\) is a vector composed of the parameters of the production function of the \(j\)-th group, and \(\beta^{*}\) is an unknown variable vector of the meta-frontier function that satisfies the following equation. As shown in Equation (4) above, the graph of the meta-frontier production function is placed above the graph of the production frontier function of each group for all periods. The meta-frontier production function becomes an encircling form of the frontier function of each group based on the same technology. To simplify, if the function f of Equation (1) is \(e^{X_{it}\beta_{(j)}}\), Equation (1) can be divided as follows. \[Y_{it}=e^{-U_{it(i)}}\times\frac{e^{x_{it}\beta_{(j)}}}{e^{x_{it}\beta^{*}}} \times e^{x_{it}\beta^{*}+V_{it(j)}} \tag{5}\]  
When both sides of Equation (5) are divided by \(e^{x_{it}\beta^{*}+V_{it(j)}}\), the following Equation (6) is derived. \[\frac{Y_{it}}{e^{x_{it}\beta^{*}+V_{it(j)}}}=e^{-U_{it(i)}}\times\frac{e^{x_{it }\beta_{(j)}}}{e^{x_{it}\beta^{*}}} \tag{6}\]  
In Equation (6) above, \(e^{-U_{it(j)}}\) on the right side is the technical efficiency of group \(j\). The statement \(\frac{e^{x_{it}\beta_{(j)}}}{e^{x_{it}\beta^{*}}}\) is the ratio of the j group frontier function to the meta-frontier function, that is called MTR (meta-technology ratio) or TGR (technical gap ratio). TE\({}^{*}\), which represents the technical efficiency of the meta-frontier function, is a value obtained by multiplying TE and MTR, and can be expressed as follows. \[TE_{it}^{*}=\frac{Y_{it}}{e^{x_{it}\beta^{*}+V_{it(j)}}}=TE_{it}\times MTR_{it} \tag{7}\]  
There are two methods to measure the parameters of meta-frontier functions: quadratic programming (QP) and linear programming (LP). QP is a method of minimizing the sum of squares of deviations, and LP is a method of minimizing the sum of absolute values of deviations. According to [PERSON], [PERSON] and [PERSON] [60; 61], QP and LP are defined as Equations (8) and (9) below.