## 3 Modeling of SIC and Time Series Analysis
### Formulation of the ARIMA Model  
The pre-requisite for a time series to be modeled by an ARIMA model is the demonstration of both stationarity and non-randomness. Stationarity greatly simplifies the complexity of the analysis and enhances the accuracy of estimation. Moreover, it helps to mitigate the risk of spurious regression [24]. However, a completely random series, such as white noise, is devoid of any correlations and therefore has no memory or influence on future values. That means that past information has no bearing on future information, and as such, it has no value for further analysis. Hence, before applying the ARIMA model, it is crucial to perform a stationarity test using the Augmented Dickey-Fuller (ADF) hypothesis test [25].  
To represent the ARIMA model, we introduce the lag operator \(B\) first. When a SIC series \(y\) at time \(t\) is multiplied by \(B\) once, it means that time \(t\) has lagged one step. For instance,  
\[\begin{split} y_{t-1}&=B\ y_{t},\\ y_{t-2}&=B^{2}y_{t},\\ &\vdots\\ y_{t-p}&=B^{p}y_{t}.\end{split} \tag{9}\]  
A stationary and non-random time series contains the relevant information extracted by an ARMA model. The information at the current time \(t\) depends on the information in the past, which can be expressed mathematically as  
\[\begin{cases}y_{t}=\underbrace{\varphi_{1}y_{t-1}+\cdots+\varphi_{p}y_{t-p}}_{ \text{auto-regressive (AR)}}+\underbrace{\epsilon_{t}-\vartheta_{1}\epsilon_{t-1}-\cdots-\vartheta _{q}\epsilon_{t-q}}_{\text{moving average (MA)}},\\ \quad=\varphi_{1}By_{t}+\cdots+\varphi_{p}B^{p}y_{t}+\epsilon_{t}-\vartheta_{ 1}B\epsilon_{t}-\cdots-\vartheta_{q}B^{q}\epsilon_{t},\\ \varphi_{p}\
eq 0,\,\vartheta_{q}\
eq 0,\\ \mathbb{E}(\epsilon_{t})=0,\,\forall ar(\epsilon_{t})=\sigma_{e}^{2},\, \mathbb{E}(\epsilon_{t}\epsilon_{s})=0,s\
eq t,\\ \mathbb{E}(y_{s}\epsilon_{t})=0,\,\forall s<t.\end{cases} \tag{10}\]  
Equation (10) implies that  
1. the ARMA model can be written as ARMA \((p,q)\), which indicates that the highest order of the auto-regressive (AR) model is \(p\), and the highest order of the moving average (MA) model is \(q\);
2. the error \(\epsilon_{t}\) is white noise; and
3. the error at the current time is independent of past \(y\).  
If the series is differenced to obtain the stationarity, and the series contains seasonal components, the ARMA model can be extended to an auto-regressive integrated moving average model, which is expressed in a compact form as  
\[\
abla^{d}\
abla^{D}_{S}y_{t}=\frac{\Theta(B)\Theta_{S}(B)}{\Phi(B)\Phi_{S}( B)}\epsilon_{t}, \tag{11}\]  
where  
\[\begin{split}&\
abla^{d}=(1-B)^{d},\qquad\qquad\qquad\qquad \qquad\
abla^{D}_{S}=(1-B^{S})^{D},\\ &\Theta(B)=1-\vartheta_{1}B-\cdots-\vartheta_{q}B^{q},\qquad \qquad\Phi(B)=1-\varphi_{1}B-\cdots-\varphi_{p}B^{p},\\The operator \(\
abla^{d}\) is the ordinary difference component that represents the \(d\)-th differenced time series. The seasonal difference of order \(D\) is defined as \(\
abla^{D}_{S}\), where \(S\) denotes the seasonal period, e.g., \(S=12\) for a yearly seasonal period. \(\Theta(B)\) and \(\Phi(B)\) are the non-seasonal autoregressive and moving average lag polynomials, respectively. The remaining seasonal information is then extracted by the seasonal autoregressive and moving average lag polynomials, \(\Theta_{S}(B)\) and \(\Phi_{S}(B)\), respectively. Equation (11) is denoted as ARIMA \((p,d,q)\times(P,D,Q,S)\) in short.