## 3 Material and Methods
### Structure of the Path-GMM-Nearest-Neighbor Model  
For this research, a model called Path Analysis--Generalized Method of Moments (GMM) based on a Nearest-Neighbor with Observed Variable model (Path-GMM-Nearest-Neighbor Model) has been defined. The estimation of this model, based on traditional dynamic simultaneous equation models, still has flaws. This is due to the assumption that parameters remain constant across different policy regimes. As a result, small responses to policy changes are not expected and cannot be used for decision making. Therefore, this research has chosen to use GMM, because it is easier to apply to the model structure to draw conclusions about these parameters. This approach is intended to enhance the efficiency of the analysis, as the Path-GMM-Nearest-Neighbor model selects results that are free from spurious outcomes and ensures that the test results meet all the criteria for thorough analysis, as outlined below [8; 11; 15; 57; 58]. From Figure 2, the relationships between latent variables are illustrated, highlighting the causal connections between endogenous and exogenous latent variables under the following hypotheses:  
1. \(X_{1}\) has a direct effect on \(Y_{i}\);
2. \(X_{2}\) directly influences \(Y_{i}\);
3. \(X_{2}\) exerts a direct impact on \(X_{1}\);
4. \(Y_{i}\) directly affects \(X_{2}\);
5. \(X_{2}\) indirectly influences \(Y_{i}\) through \(X_{1}\);
6. \(Y_{i}\) has an indirect effect on \(X_{1}\) via \(X_{2}\). This estimator is based on the Autoregressive Approximation as follows: [59; 60]. \[ \begin{split} Y_{t}=\sum\limits_{j=1}^{\infty}\alpha_{j}(\theta_{0} )Y_{t-j}+\varepsilon_{t}\\ \text{Let}\ \alpha_{j}(\theta_{0})=-(-\theta)^{j},\ \forall\theta,\ j=1,2 \dots\end{split}  \tag{1}\]  
The estimation can be practically performed as follows: [61]  
\[Y_{t}=\sum\limits_{j=1}^{K}\alpha_{j}(\theta_{0})Y_{t-j}+\varepsilon_{Kt} \tag{2}\]  
However, this estimation results in additional errors due to the following equation. \[\varepsilon_{Kt}=\varepsilon_{t}+\sum\limits_{j=K+1}^{\infty}\alpha_{j}( \theta_{0})Y_{t-j}=\varepsilon_{t}+(-1)^{K+1}\theta_{0}^{K+1}\varepsilon_{t- K-1} \tag{3}\]  
Figure 2: The analytical structure of the Path-GMM-Nearest-Neighbor model. Therefore, the model reduced to AR represents the second-order properties of \(Y_{i}\) as a function of variance. Hence, it is necessary to find the variable for \(\theta_{0}\) based on the estimator for \(\alpha_{j}(\theta_{0})\) by determining the vector \(K\)[61; 62]. \[A_{K}(\theta)= \begin{pmatrix}\alpha_{1}(\theta)\\ \vdots\\ \alpha_{K}(\theta)\end{pmatrix} \ \ \forall\theta,\ \alpha_{j}(\theta)=-(-\theta)^{j} \tag{4}\]  
Let \(\hat{A}_{K}\) be the vector \(K\) of the OLS estimator for \((\hat{\alpha}_{1},\ldots,\hat{\alpha}_{K})\). As per derived \(K\), it can thus be defined as follows:  
\[\hat{\theta}_{TK}=\underset{\theta\in\Theta}{\text{argmin}}\big{(}\hat{A}_{K }-A_{K}(\theta)\big{)}^{\prime}V_{TK}\big{(}\hat{A}_{K}-A_{K}(\theta)\big{)} \tag{5}\]  
Let \(\Theta=(-1,+1)\) and \(V_{TK}\) be the matrices with weights \(K\times K\). For the Path-GMM-Nearest-Neighbor model, the Nearest Neighbor estimator is used, which for the simplest non-parameterized estimator, is the Nearest Neighbor \(k-NN\) estimator. The Nearest-Neighbor estimator of conditional expectations is created by averaging the dependent variable values for observations where the conditional variable \(X\) is closest to the estimated value. Let \(\sigma_{XL}\) represent the indicator of the size of component \(l^{th}\) of \(X\) (standard deviation), where \(X\) is in order \(r\), and it is explained that: [61; 62]  
\[\big{\|}X_{i}-X_{j}\big{\|}_{n}=\left\{\sum_{l=1}^{r}\frac{\big{(}X_{il}-X_{jl }\big{)}^{2}}{\hat{\sigma}_{l}}\right\}^{1/2} \tag{6}\]  
From Equation (6), this represents the indicator of the gap between observations \(i\) and \(j\), explaining the nature of the multivariate \(x\). Let us consider the integer \(K,K\leq n\), and  
\[