# Governance and Dynamic Efficiency with Network Structure in the Brazilian Natural Gas Utilities
## 2 Materials and Methods
### Econometric Model  
We used an econometric model to identify the determinants of the efficiency of companies in the natural gas sector in Brazil. This step was conducted for a second-stage analysis, using the overall efficiency results obtained through the DNSBM model, with the estimated efficiency score as the dependent variable. Stage 2 uses regression to relate efficiency scores to factors considered to influence efficiency. As the efficiency measure estimated by the DEA methodology, which represents the dependent variable, assumes values ranging from 0 to 1, and even though these are not truncated values, they are not freely distributed between \(-\infty\) and \(+\infty\), since the data vary within a certain range [45]. Despite this, the efficiency scores are not censored values; consequently, the Tobit model is an inadequate estimator [46]. Although it is common to use the second stage in efficiency analyses using DEA, there is still an intense debate about which regression model is most appropriate [47]. Amid this discussion, one can highlight the maximum likelihood estimation of a truncated regression model [48; 49] but this model is more suitable for efficiency analyses in their subdivisions, and the estimation of models by ordinary least squares [46]. It was found that various methods have been applied in studies, such as multiple regression and panel analysis with fixed and variable effects. However, no method specifically applied to the natural gas sector using the DNSBM model was identified. Thus, we opted to use a panel data model, as in [50]. The choice of panel data was due to lower collinearity between variables and a higher degree of freedom. This approach offers greater efficiency in estimation [51]. The estimation of the panel data model was performed with the aid of Stata/SE(r) 15-Statistic/Data Analysis software (version 15.1), and the model is represented by the following equation:  
\[eff_{it}=\alpha_{i}+\beta X_{it}^{j}+\varepsilon_{it} \tag{5}\]  
where the index \(i=i\)-th indicates the observations and \(t=t\)-th indicates the time period, \(\mathrm{eff_{it}}\) is the dependent variable for the unit i in the period t; \(\alpha_{i}\) is constant; \(X_{it}^{j}\) represents the value of the \(j=j\)-th independent variable for unit I in the period t; and \(\varepsilon_{it}\) is the error term. In the model of this study, independent variables were selected according to their presence in other studies available in the literature. Table 2 shows the variables selected for this research. The business model that distributors possess does not allow for real competition [52]. This is due to the high cost of the infrastructure required for distribution. As a result, it is impossible to have more than one independent service provider available to consumers in a geographic region (although, with the offer of open access regulation, this condition is somewhat corrected). Regulation must implement compliance mechanisms aimed at encouraging good efficiency and correcting poor performance [45, 53]. Thus, our approach to the performance of distributors for the second stage is based on agency theory. To measure the quality of internal governance in pipeline gas distributors, we calculated a corporate governance index based on the institutional and legal design of each distributor's regulatory system, where decisions are made. It is important to highlight that governance is defined by the laws, processes, and procedures that determine whether the actions and parameters regulated by state agencies make regulatory decisions, resources, and information available to society. According to GNGI [56], a comprehensive assessment of regulatory effectiveness must consider both governance and regulatory substance.