## 2 Technology Forecasting Methodology
### Efficiency and Technology Change Analysis Using Malmquist Index  
The Malmquist Productivity Index (MPI) measures the productivity changes along with time variations and can be decomposed into changes in efficiency and technology with DEA, such as a nonparametric approach. Productivity decomposition into technical change and efficiency catch-up necessitates the use of a contemporaneous version of the data and the time variants of technology in the study period. The MPI can be expressed in terms of distance function (E) as Equations (6) and (7) using the observations at time t and t + 1 [40]. \[\text{MPI}_{\text{I}}^{\text{t}}=\frac{\text{E}_{\text{I}}^{\text{t}}(\text{x}^{ \text{t+1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{I}}^{\text{t}}(\text{x}^{ \text{t}},\text{y}^{\text{t}})} \tag{6}\]  
\[\text{MPI}_{\text{I}}^{\text{t+1}}=\frac{\text{E}_{\text{I}}^{\text{t+1}}( \text{x}^{\text{t+1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{I}}^{\text{t+1}}( \text{x}^{\text{t}},\text{y}^{\text{t}})} \tag{7}\]  
where I denotes the orientation of MPI model. The geometric mean of two MPI in Equations (6) and (7) gives Equation (8). \[\text{MPI}_{\text{I}}^{\text{G}}=(\text{MPI}_{\text{I}}^{\text{t}}\text{MPI}_{ \text{I}}^{\text{t+1}})^{\frac{1}{2}}=[(\frac{\text{E}_{\text{I}}^{\text{t}}( \text{x}^{\text{t+1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{I}}^{\text{t}}( \text{x}^{\text{t}},\text{y}^{\text{t}})})\cdot(\frac{\text{E}_{\text{I}}^{ \text{t+1}}(\text{x}^{\text{t+1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{I} }^{\text{t+1}}(\text{x}^{\text{t+1}},\text{y}^{\text{t+1}})})]^{1/2} \tag{8}\]  
The input oriented geometric mean of MPI can be decomposed using the concept of input oriented technical change (TECHCH) and input oriented efficiency change (EFFCH) as given in Equation (9). \[\text{MPI}_{\text{I}}^{\text{G}}=\text{EFFCH}_{\text{I}}\text{-TECHCH}_{\text{I }}^{\text{G}}=(\frac{\text{E}_{\text{I}}^{\text{t+1}}(\text{x}^{\text{t+1}}, \text{y}^{\text{t+1}})}{\text{E}_{\text{I}}^{\text{t}}(\text{x}^{\text{t}}, \text{y}^{\text{t}})})\cdot[(\frac{\text{E}_{\text{I}}^{\text{t}}(\text{x}^{ \text{t}},\text{y}^{\text{t}})}{\text{E}_{\text{I}}^{\text{t+1}}(\text{x}^{ \text{t}},\text{y}^{\text{t}})})\cdot(\frac{\text{E}_{\text{I}}^{\text{t}}( \text{x}^{\text{t+1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{I}}^{\text{t+1}} (\text{x}^{\text{t+1}},\text{y}^{\text{t+1}})})]^{1/2} \tag{9}\]  
The first and second terms represent the efficiency change and the technology change, respectively. MPI given by Equations (8) and (9) can be defined using DEA, such as distance function. That is, the components of MPI can be driven from the estimation of distance functions defined on a frontier technology. [PERSON], [PERSON], [PERSON], and [PERSON] [40] provided the formal derivation of MPI and it is the most popular method among the various methods that have been developed to estimate a production technology [41]. By utilizing both CRS and VRS DEA frontiers to estimate the distance functions in Equation (9), the technical efficiency can be decomposed into scale efficiency and pure technical efficiency components. A scale efficiency change (SECH) is given in Equation (10). \[\text{SECH}=[(\frac{\text{E}_{\text{Vrs}}^{\text{t+1}}(\text{x}^{\text{t+1}},\text{y}^{\text{t+1}})/\text{E}_{\text{crs}}^{\text{t+1}}(\text{x}^{\text{t+ 1}},\text{y}^{\text{t+1}})}{\text{E}_{\text{Vrs}}^{\text{t}}(\text{x}^{\text{t }},\text{y}^{\text{t}})/\text{E}_{\text{crs}}^{\text{t+1}}(\text{x}^{\text{t }},\text{y}^{\text{t}})})\cdot(\frac{\text{E}_{\text{Vrs}}^{\text{t}}(\text{x} ^{\text{t+1}},\text{y}^{\text{t+1}})/\text{E}_{\text{crs}}^{\text{t}}(\text{x }^{\text{t}},\text{y}^{\text{t}})}{\text{E}_{\text{trs}}^{\text{t}}(\text{x} ^{\text{t}},\text{y}^{\text{t}})/\text{E}_{\text{crs}}^{\text{t}}(\text{x}^{ \text{t}},\text{y}^{\text{t}})})]^{1/2} \tag{10}\]  
Additionally, a pure efficiency change (PECH) is given in Equation (11).