## 4 Model Implementation and Validation
### Arima Modeling of SIC
The value \(q=3\), which gives the lowest mean AIC among the cross-validations, is finally selected as the best candidate for the ARIMA model. The model is then fitted with the in-sample data to compute the parameters in Equation (11). The size of the in-sample data set will influence the goodness of fit and \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline \(p\)-value & 0.019 \\ test statistic & \(-\)3.208 \\ critical value (1\%) & \(-\)3.446 \\ critical value (5\%) & \(-\)2.868 \\ critical value (10\%) & \(-\)2.570 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Augmented Dickey-Fuller test of seasonally differenced SICs in sub-region No. 2. Figure 10: The SICs in sub-region No. 2 and the decomposed trend and seasonal components. the accuracy of the forecast. To choose a suitable in-sample size (i.e., training set size), we compare the RMSEs between the forecast SICs (obtained from training sets) and the true SICs (in test sets) in Figure 13, in each cross-validation, the model is first fitted by data in the training set. Then, the fitted model forecasts the SICs in the months covered by the test set. The RMSE represents the difference between the forecast and the true SICs in the same period. As the training set size increases, the RMSEs decrease monotonically. For training set sizes larger than 300 months, the RMSE curve shows convergence, following the recommended in-sample size in [30]. Eventually, 432 months are chosen as the in-sample size. Figure 11: The ACF and PACF of the seasonally differenced SIC series in sub-region No. 2. The blue shaded regions represent the range of zero plus/minus two standard errors. When values fall within the shaded region, the ACF and PACF are considered to be zero. The lags of the red dots provide information to determine the orders of the ARIMA model. Figure 12: Schematic of the five folds cross-validations. Figure 13: Convergence study of in-sample sizes. Using the MLE as described in Section 3.2 to solve for the parameters, the ARIMA \((2,0,3)\times(0,1,1,12)\) model is finally expressed as  
\[(1- B^{12})Y_{t}^{\prime}= \tag{15}\] \[\frac{1-0.5592B+0.5290B^{2}+0.3939B^{3}}{1-0.0851B-0.7502B^{2}} \left(1+0.8520B^{12}\right)e_{t}\,\] \[e_{t}\sim N(0,0.0028).\]  
Recalling the limitations of \(\epsilon_{t}\) in Equation (10), it is important to assess the residual of the ARIMA \((2,0,3)\times(0,1,1,12)\) model to ensure its validity. The residual is the difference between the actual SIC values and the values estimated by the model. The quarterly and yearly average residuals are shown in Figure 14, and they oscillate around zero, indicating a white noise pattern. The histogram, Q-Q plot, and autocorrelation function (ACF) of the residuals are plotted in Figure 15. The histogram and Q-Q plot demonstrates that the residuals are almost normal in distribution, and the ACF indicates that the residuals are independent of time, which are the two main assumptions for the residual in the ARIMA model. Hence, the ARIMA \((2,0,3)\times(0,1,1,12)\) model has effectively captured most of the information from the original SIC series and the residual is, in essence, white noise. Since the ARIMA model has demonstrated acceptable goodness-of-fit, we employ the MMSE forecast [28] to project the SIC from 2021 to 2025 (in Figure 16). The red lines depict the input data obtained from reanalysis, while the blue dotted lines serve as a reference for verification. The green lines represent the SIC values estimated via the ARIMA model, and the purple lines indicate the forecasts made by the same model. The red shaded areas  
Figure 14: The quarterly and yearly mean of the residual. Figure 10:0 The residual diagnosis of the proposed ARIMA model. denote the 95% confidence intervals of the forecasts. Prior to 2021, the estimated SIC values are nearly identical to the input data from reanalysis.