## 4 Main Results
This simplification is reasonable as (1) time variant \(w_{ij}(t)\) can somehow give an effect similar to spatial order \(h\); and (2) \(w_{ij}(t)\) is always equal to the only non-zero \(w_{ij}^{(h)}(t)\):  
\[w_{ij}(t)=0+0+\ldots+w_{ij}^{(h)}(t)+\ldots+0=w_{ij}^{(h)}(t)\text{ number of zero is }\lambda_{i,k}-1\]  
Considering the combined spatial weight matrix \(W(t)=\sum_{h=0}^{\lambda_{i,k}}W_{i}^{(h)}(t)\), \(Z(t)=\left[z_{1}(t),z_{2}(t),\ldots,z_{N_{i}}(t)\right]^{T}\) is an \(N_{i}\) dimension column vector that includes all neighbor roads within the spatial order to be considered by road \(i\), and \(\Sigma=\left[\sigma_{i}^{2},0,\ldots,0\right]^{T}\), we can obtain a matrix from the LSTAR model according to Equation (5). \[Z(t)=\sum_{k=1}^{p_{i}}\phi_{i,k}W(t-k)Z(t-k)+\Sigma(t) \tag{14}\]  
when the rank of \(W(t)\) is \(N_{i}\), Equation (14) can be rewritten as:  
\[[W(t)]^{-1}[W(t)Z(t)]=\sum_{k=1}^{p_{i}}\phi_{i,k}W(t-k)Z(t-k)+\Sigma(t). \tag{15}\]  
Let \(U(t)=[W(t)Z(t)]\), we obtain:  
\[[W(t)]^{-1}U(t)=\sum_{k=1}^{p_{i}}\phi_{i,k}U(t-k)+\Sigma(t). \tag{16}\]  
\(\left[W_{i}^{(h)}(t)\right]^{-1}\) can be treated as an instantaneous window to \(U(t)\), so \(U(t)\) is stationary in the short term. We have:  
\[\overline{U}(t)=\sum_{k=1}^{p_{i}}\phi_{i,k}U(t-k)+\Sigma(t). \tag{17}\]  
Let \(u_{i}(t)\) be the element of \(U(t)\), then:  
\[\overline{u}_{i}(t)=\sum_{k=1}^{p_{i}}\phi_{i,k}u_{i}(t-k)+\varepsilon_{i}(t). \tag{18}\]  
Pre-multiplying both sides of Equation (18) by \(u_{i}(t-m)\):  
\[\overline{u}_{i}(t)u_{i}(t-m)=\sum_{k=1}^{p_{i}}\phi_{i,k}u_{i}(t-k)u_{i}(t-m )+\varepsilon_{i}(t)u_{i}(t-m). \tag{19}\]  
Taking expected values in both sides, we obtain:  
\[\overline{r}_{i}^{\prime}(m)=\sum_{k=1}^{p_{i}}\phi_{i,k}r_{i}^{\prime}(m-k)+ \sigma_{i}^{2}\delta(m) \tag{20}\]  
where the expected value \(r_{i}^{\prime}(m)=E(u_{i}(t-m)u_{i}(t-k))\), \(\overline{r}_{i}^{\prime}(m)=E(\left[W_{i}^{(h)}(t)\right]^{-1}u_{i}(t)u_{i}(t -m))\), \(E(\varepsilon_{i}(t)u_{i}(t-m))=\sigma_{i}^{2}\delta(m)\). We can then obtain:  
\[\left[ \begin{array}{c}\overline{r}_{i}^{\prime}(0)\\ \overline{r}_{i}^{\prime}(1)\\ \vdots\\ \overline{r}_{i}^{\prime}(p_{i}-1)\end{array} \right]=\left[ \begin{array}{cccc} r_{i}^{\prime}(1)&r_{i}^{\prime}(2)&\cdots&r_{i}^{\prime}(p_{i})\\ r_{i}^{\prime}(0)&r_{i}^{\prime}(1)&\cdots&r_{i}^{\prime}(p_{i}-1)\\ \vdots&\vdots&\vdots&\vdots\\ r_{i}^{\prime}(p_{i}-2)&r_{i}^{\prime}(p_{i}-3)&\cdots&r_{i}^{\prime}(1)\end{array}  \right]\left[ \begin{array}{c}\phi_{i,1}\\ \phi_{i,2}\\ \vdots\\ \phi_{i,p_{i}}\end{array} \right]+\left[ \begin{array}{c}\sigma_{i}^{2}\\ 0\\ \vdots\\ 0\end{array} \right] \tag{21}\]  
\[\text{Let }\Phi_{i}=\left[ \begin{array}{c}\phi_{i,1}\\ \phi_{i,2}\\ \vdots\\ \phi_{i,p_{i}}\end{array} \right],A^{\prime}=\left[ \begin{array}{cccc}r_{i}^{ \prime}(1)&r_{i}^{\prime}(2)&\cdots&r_{i}^{\prime}(p_{i})\\ r_{i}^{\prime}(0)&r_{i}^{\prime}(1)&\cdots&r_{i}^{\prime}(p_{i}-1)\\ \vdots&\vdots&\vdots&\vdots\\ r_{i}^{\prime}(p_{i}-2)&r_{i}^{\prime}(p_{i}-3)&\cdots&r_{i}^{\prime}(1)\end{array}  \right],\overline{R}_{i}^{\prime}=\left[ \begin{array}{c}\overline{r}_{i}^{ \prime}(0)\\ \overline{r}_{i}^{\prime}(1)\\ \vdots\\ \overline{r}_{i}^{\prime}(p_{i}-1)\end{array} \right],\Sigma=\left[ \begin{array} []{c}\sigma_{i}^{2}\\ 0\\ \vdots\\ 0\end{array} \right].\]  
Then the augmented matrix of Equation (21) is:  
\[\overline{A^{\prime}}=\left[ \begin{array}{cccc}r_{i}^{\prime}(1)&r_{i}^{ \prime}(2)&\cdots&r_{i}^{\prime}(p_{i})&\overline{r}_{i}^{\prime}(0)-\sigma_ {i}^{2}\\ r_{i}^{\prime}(0)&r_{i}^{\prime}(1)&\cdots&r_{i}^{\prime}(p_{i}-1)&\overline{r} _{i}^{\prime}(1)\\ \vdots&\vdots&\vdots&\vdots\\ r_{i}^{\prime}(p_{i}-2)&r_{i}^{\prime}(p_{i}-3)&\cdots&r_{i}^{\prime}(1)& \overline{r}_{i}^{\prime}(p_{i}-1)\end{array} \right]=\left[ \begin{array}{ ccc}A^{\prime}&\overline{R}_{i}^{\prime}-\Sigma\end{array}