## 3 Material and Methods
### Spuriousness in the Analysis Results
#### 3.2.2 Autocorrelation
This can be represented as follows [53; 54]:  
\[X_{1i}=\alpha+\beta_{2}X_{2i}+\varepsilon_{i} \tag{5}\]  
From Equation (5), it can be observed that if a variable \(XX_{1i}\) has a _perfect_ relationship with another variable \(X_{2i}\), it will result in the inability to compute the values of the coefficients \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\). This is due to the determinant \(R^{2}=1\), and the values of \(S.E.E.\left(\hat{\beta}_{1}\right)\) and \(S.E.E.\left(\hat{\beta}_{2}\right)\) being equal to infinity. This makes \(\left(\hat{X}X\right)^{-1}\) a singular matrix, which prevents the calculation of the inverse. However, if the correlation is imperfect multicollinearity, the analysis may still proceed, but it will result in biased estimates, leading to potential forecasting errors. Impact of Multicollinearity  
The presence of multicollinearity can have significant impacts on the analysis, as it causes biased estimates, which affects the subsequent use of the model. The effects can be summarized as follows:  
* The correlation, \(r\), will be high, which affects the analysis of the relationships. * Higher estimated values of \(S.E.E. (\hat{\beta}_{1})\) lead to reduced \(t\)-tests, resulting in the potential rejection of \(H_{0}\). If this happens, those variables should be removed. * The analysis cannot properly evaluate the model's convergence toward equilibrium due to the high sensitivity of the variables. Detecting Multicollinearity  
Multicollinearity can be present in almost any model. There are several methods to detect it, with the most commonly used methods being: correlation coefficients, \(r\), and Variance Inflation Factors, VIF, which can be calculated using this formula, \(VIF=\frac{1}{1-R^{2}}\), as follows:  
\[r=\frac{n\sum XY-\sum X\sum Y}{\sqrt{\left[n\sum X^{2}-\left(\sum X\right)^{2 }\right]\left[n\sum Y^{2}-\left(\sum Y\right)^{2}\right]}} \tag{6}\]  
From Equation (6), it can be observed that if the value \(r_{y,x}\) is close to 1, it indicates that the dependent variable \(Y\) is highly correlated with the independent variable \(X\), which benefits the model. However, if the independent variables are highly correlated with each other \(\left(X_{i},\ldots X_{n}\right)\), this will negatively impact the model and increase the likelihood of errors. As \(r_{x_{i},x_{i}}\) approaches 1, the problem of multicollinearity becomes more severe. Therefore, the researcher may need to add additional independent variables to the model or remove certain variables, depending on the appropriateness of the model [46; 47]. Methods for Resolving Multicollinearity  
The approach to resolving multicollinearity depends on the nature of each model. In some cases, researchers may not need to make any changes, as the unbiased properties of the model remain intact. However, in other models, it may be necessary to modify certain independent variables or add additional independent variables to the model. It is essential to consider the type of multicollinearity present. If it is _Perfect Multicollinearity_, this indicates that two or more independent variables in the linear equation model are completely correlated with one another. In such cases, it is necessary to examine and revise the independent variables, as the system will not be able to proceed otherwise. An example equation is provided below:  
\[X_{1i}=\alpha+\beta_{3}X_{3i}+\epsilon_{i} \tag{7}\]  
From Equation (7), if variable \(X_{1i}\) is perfectly correlated with another variable \(X_{3i}\), this results in an inability to calculate the values of \(\hat{\beta}_{1}\) and \(\hat{\beta}_{3}\). Additionally, \(S.E.E. (\hat{\beta}_{1})\) and \(S.E.E. (\hat{\beta}_{2})\) would be equal to infinity, because the \(\left(XX\right)^{-1}\) matrix becomes a singular matrix, making it impossible to calculate its inverse. Therefore, the researcher must revise and reconstruct the model from the very beginning.