## 4 Regression Analysis  
### Estimation Model  
In the last subsection, we examined the relationship between green purchasing of products/services in certain item categories and green purchasing policy or practices. However, other external factors such as the size of municipalities may also affect the decision regarding whether to adopt a GPP. For example, a municipality that has a larger budget can more easily adopt a GPP and implement green purchasing. In this case, it is difficult to distinguish between the effect of GPP adoption and that of the size of the municipality. Therefore, it is necessary to conduct a multivariate regression analysis to reveal the impact of GPP adoption on both the implementation of green purchasing and the municipality's measurement of its own green purchasing rate. The purpose of this section is to identify the treatment effects of GPP adoption on both the implementation of green purchasing and the municipality's measurement of its own green purchasing rate while controlling for effects of other factors such as municipality size. Let _Implementation_\({}^{*}_{i}\) and _Measurement_\({}^{*}_{i}\) be municipality \(i\)'s net benefits from the implementation of green purchasing and the measurement of its own green purchasing rate, respectively. In addition, we assume that these variables are explained by GPP adoption and other explanatory variables such as the size of each municipality. We specify the models as follows:  
\[\textit{Implementation}^{*}_{i}=\beta_{1}GPP_{i}+\gamma_{1}X_{i}+\varepsilon_{1i} \tag{1}\]  
\[\textit{Measurement}^{*}_{i}=\beta_{2}GPP_{i}+\gamma_{2}X_{i}+\varepsilon_{2i} \tag{2}\]  
where \(GPP_{i}\) is an indicator variable for GPP adoption, \(X_{i}\) is a vector of control variables, and \(\varepsilon_{1i}\) and \(\varepsilon_{2i}\) are idiosyncratic errors. In addition, \(\beta_{1}\), \(\beta_{2}\), \(\gamma_{1}\), and \(\gamma_{2}\) are parameters to be estimated. Figure 6: Share of municipalities measuring their green purchasing rate. However, we cannot observe both _Implementation\({}_{i}^{*}\)_ and _Measurement\({}_{i}^{*}\)_. Instead, we can observe whether the municipality implements green purchasing (_Implementation\({}_{i}\)_) and whether the municipality measures its own green purchasing rate (_Measurement\({}_{i}\)_). Hence, we assume that _Implementation\({}_{i}\)_ takes the value one if _Implementation\({}_{i}^{*}\geq 0\)_, otherwise zero. Thus, the municipality implements green purchasing if it evaluates its net benefit to be greater than or equal to zero. _Measurement\({}_{i}\)_ and _Measurement\({}_{i}^{*}\)_ are defined similarly. Our variable of interest is _GPP\({}_{i}\)_, which takes the value one if a municipality \(i\) adopts a GPP. This variable is expected to identify the impacts of GPP adoption on whether a municipality implements GPP and/or measures its own green purchasing rate. Therefore, we expect _GPP\({}_{i}\)_ to have positive signs. A control variable vector, \(X_{i}\), includes the natural log of the standard financial scale of municipalities and populations. The natural log of the standard financial scale is expected to capture the effects of the financial constraints of the municipality and the municipality size. Financial constraints and a small size can be barriers to green purchasing [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. Therefore, the coefficient of this variable is expected to be positive. We include the natural log of population as a proxy for the effects of the size of the municipality. Finally, we include prefecture dummies to control for time-constant unobservable factors such as the environmental consciousness of the prefecture to which municipality \(i\) belongs. We assume that \(\varepsilon_{1i}\) and \(\varepsilon_{2i}\) are distributed by standard logistic distributions, respectively. Therefore, we apply the logit model to estimate the parameters.