## 2 Materials and Methods
### Causality Analysis
#### 2.1.1 Model Specifications  
As said earlier, this study developed two separate models for the two categories of solid waste emissions: CH\({}_{4}\) emissions from municipal solid waste (MSW) and CH\({}_{4}\) emissions from industrial waste (ISW). For each category of solid waste emissions, a range of independent or explanatory variables were identified. Model 1 considered CH\({}_{4}\) emissions from municipal solid waste (MSW) as the dependent variable and GDP per capita (GDPPC), urban population (UPOP), FDI inflows (FDI), older population number (OP), and literacy rate (LR) as independent or explanatory variables. The independent or explanatory variables for CH\({}_{4}\) emissions from industrial waste (ISW) include GDPPC, UPOP, FDI, and unemployment rate, and their relationships are presented in Model 2.  
Model 1 and model 2 are specified in Equations (1) and (2), respectively.  
\[\text{(MSW)}_{\text{t}}=\alpha_{1}+\beta_{1}\text{ GDPC}_{\text{t}}+\beta_{2}\text{ UPOP}_{\text{t}}+\beta_{3}\text{ FDI}_{\text{t}}+\beta_{4}\text{ OP}_{\text{t}}+\beta_{5}\text{ LR}_{\text{t}}+\varepsilon_{1\text{t}} \tag{1}\]  
\[\text{(ISW)}_{\text{t}}=\alpha_{2}+\beta_{6}\text{ GDPC}_{\text{t}}+\beta_{7}\text{ UPOP}_{\text{t}}+\beta_{8}\text{ FDI}_{\text{t}}+\beta_{9}\text{ UR}_{\text{t}}+\varepsilon_{2\text{t}} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data (Symbol)** & **Description** & **Period** & **Unit** & **Source** \\ \hline CH\({}_{4}\) emissions from municipal solid waste (MSW) & This includes CH\({}_{4}\) emissions & & & \\ \hline CH\({}_{4}\) emissions from & This includes CH\({}_{4}\) emissions & & & \\ \hline CH\({}_{4}\) emissions from & & & & \\ \hline CH\({}_{4}\) emissions from & & & & \\ \hline industrial solid waste (ISW) & This includes CH\({}_{4}\) emissions & & & \\ \hline Older population (OP) & This presents the total number of & & & \\ \hline \multirow{2}{*}{Literacy rate (LR)} & This shows the percentage of people & & & \\  & who can read and write in & & & \\  & Saudi Arabia & & & \\ \hline \multirow{3}{*}{Unemployment rate (UR)} & This shows the share of Saudi & & & \\  & Arabia’s labor force who are & & & \\ \cline{1-1}  & unemployed but seeking & & & \\ \cline{1-1}  & employment opportunities & & & \\ \hline \multirow{3}{*}{Foreign direct investment & This presents the monetary value of & & & \\ \cline{1-1}  & inward direct investments made by & & & \\ \cline{1-1}  & inflows (FDI) & non-resident investors in & & \\ \cline{1-1}  & Saudi Arabia & & & \\ \hline Urban population (UPOP) & The total number of population & & & \\ \cline{1-1}  & living in urban areas & & & \\ \hline \multirow{3}{*}{GDP per capita (GDPPC)} & The monetary value of all domestic & & & \\ \cline{1-1}  & products produced in a year in Saudi & & & \\ \cline{1-1}  & Arabia, divided by the & & & \\ \cline{1-1}  & country’s population & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Data used in causality analysis.  
where \(\alpha_{1}\) and \(\alpha_{2}\): The intercepts; t: The year; \(\beta_{1}\) to \(\beta_{9}\): Coefficients for independent or explanatory variables; and \(\varepsilon_{1}\) and \(\varepsilon_{2}\): Constant error terms.  
The examination of stationarity and non-stationarity of time series data was the initial step for developing a VECM, and this required a logarithmic form of the proposed models (Equations (1) and (2)). Equations (3) and (4) represent the logarithmic form of Equations (1) and (2), respectively.  
\[\text{Ln (MSW)}_{t}=\text{ln ($\alpha_{1}$)}+\beta_{1}\ \text{ln (GDPPC}_{t})+\beta_{2}\ \text{ln (UPOP}_{t})+\beta_{3}\ \text{ln (FDI}_{t})+\beta_{4}\ \text{ln (OP}_{t})+\beta_{5}\ \text{ln (LR}_{t})+\text{ln }\varepsilon_{1t} \tag{3}\]  
\[\text{Ln (ISW)}_{t}=\text{ln ($\alpha_{2}$)}+\beta_{6}\ \text{ln (GDPPC}_{t})+\beta_{7}\ \text{ln (UPOP}_{t})+\beta_{8}\ \text{ln (FDI}_{t})+\beta_{9}\ \text{ln (UR}_{t})+\text{ln }\varepsilon_{2t} \tag{4}\]