## 2 The Proposed Model
### FLR-FSWGARCH Model
#### 2.1.1 Parameter Estimation Using Fuzzy Linear Regression  
The first step of the FLR-FSWGARCH is the parameter estimation using FLR. The FLR contains three components; the first component concerns the dependent variable, the second concerns the independent variables, and the third concerns the parameter coefficients. To select the independent variables, a fuzzy sliding window algorithm was developed. In the fuzzy sliding window algorithm, the minimum and maximum values of the dataset are determined. Then, the dataset undergoes a fuzzification process in which the crisp values of the data are transformed into triangular fuzzy numbers (TFNs). A matrix size of \(12\times 1\) for the current year (\(\gamma\)) and a matrix size of \(24\times 1\) for the previous year (\(PY\)), which consists of 24 months from the previous two years, are constructed, as adopted from the study by [PERSON] and [PERSON] [29]. The matrix \(PY\) is used to construct 12 sliding window slices. The sliding windows are labeled \(SW_{i}\) and have the same size as the matrix \(Y\). The partitions of the sliding windows \(SW_{i}\) and the matrix \(Y\) are given in Equations (1) and (2), respectively:  
\[SW_{i}=\Big{\{}X_{n-i},X_{n-(i-1)},X_{n-(i-2)},X_{n-(i-3)},\ldots\Big{\}}, \tag{1}\]  
\[Y_{i}=\{X_{n-11},X_{n-10},X_{n-9},\ldots,X_{n}\}, \tag{2}\]  
where \(X_{n-i}\) (\(i=1,2,3,\ldots,12\)) is the previous observation of the first element in matrix \(Y\). Both matrices \(SW_{i}\) and \(Y\) are in the TFN form where \(SW_{i}=(x_{1},x_{2},\)\(x_{3})\) and \(Y=(y_{1},\)\(y_{2},\)\(y_{3})\). Then, the Euclidean distance, \(Edi\), for every 12 fuzzified sliding windows is obtained by using Equation (3):  
\[Y_{j}-SW_{j}=(y_{1}-x_{3},y_{2}-x_{2},y_{3}-x_{1})=(z_{1},z_{2},z_{3}) \tag{3}\]  
where \(j=1,2,3,\ldots,12\) is the row of the elements in both the \(Y\) and \(SW\) matrices. Next, the Euclidean distance computed using Equation (3) is further defuzzified using Equation (4):  
\[Edi_{i,j}=\frac{z_{1}+2(z_{2})+z_{3}}{4}. \tag{4}\]  
The mean Euclidean distance is calculated using Equation (5):  
\[Edi_{i}=\sqrt{\sum_{j=1}^{12}\big{(}Edi_{i,j}\big{)}^{2}}. \tag{5}\]  
The three windows that produce the lowest mean Euclidean distance are selected to be the independent variables of the FLR. Now, the dependent variable is the TFN of the actual data, defined as \(\widetilde{Y}\), while the three selected windows act as the independent variable of the regression. This study uses multivariate FLR [36], as shown in Equation (6):  
\[\widetilde{Y}=\widetilde{\beta}X_{1}+\widetilde{\alpha}X_{2}+\widetilde{ \gamma}X_{3}. \tag{6}\]  
The TFN of the actual data is expressed as Equation (7):  
\[\widetilde{Y}=([1-\mu]e_{i,L},y_{i},[1-\mu]e_{i,R}), \tag{7}\]  
where \(\mu\) is the membership function, \(y_{i}\) is the mode of the TFN, and \(e_{i,L}\) and \(e_{i,R}\) are the width of the left and right values with the mode of the TFN, respectively. In this study, we considered the dependent variables to be symmetrical TFNs given as \(\widetilde{Y}_{i}\); thus, \(e_{i,L}=e_{i,R}\), as shown in Figure 2.  
Then, matrix algebra is employed to estimate the parameters. The FLR can be expressed in the following matrix form:  
\[\widetilde{Y}=\widetilde{\delta}X, \tag{8}\]  
where  
\[\widetilde{Y}=\left[ \begin{array}{ccc}(1-\mu)e_{1,L}&y_{1}&(1-\mu)e_{1,R}\\ (1-\mu)e_{2,L}&y_{2}&(1-\mu)e_{2,R}\\ \vdots&\vdots&\vdots\\ (1-\mu)e_{n,L}&y_{n}&(1-\mu)e_{n,R}\end{array} \right], \tag{9}\]  
\[X=\left[ \begin{array}{ccc}x_{11}&x_{12}&x_{13}\\ x_{21}&x_{22}&x_{23}\\ \vdots&\vdots&\vdots\\ x_{n1}&x_{n2}&x_{n3}\end{array} \right], \tag{10}\]  
\[\widetilde{\delta}=\left[ \begin{array}{c}\widetilde{\beta}\\ \widetilde{\alpha}\\ \widetilde{\gamma}\end{array} \right]=\left[ \begin{array}{ccc}(1-\mu)\beta_{L }&\beta_{2}&(1-\mu)\beta_{R}\\ (1-\mu)\alpha_{L}&\alpha_{2}&(1-\mu)\alpha_{R}\\ (1-\mu)\gamma_{L}&\gamma_{2}&(1-\mu)\gamma_{R}\end{array} \right].