## 3 Data and Methods
### The Measure of the Direct and Indirect Effects
#### 3.2.1 The Measure of Direct Effects
\tag{5}\]  
Therefore, the impulse response of \(y_{j}\) to \(y_{1}\) is \(r_{1j}^{(0)},\ r_{1j}^{(1)},\ r_{1j}^{(2)},\ \cdots,\) where \(r_{1j}^{(1)}\) represents the response value of \(y_{j}\) in the first month after the shock of \(y_{1}\). The equation of the response of \(y_{j}\) on time \(q\) after the shock of \(y_{i}\) can be expressed as follows:  
\[r_{ij}^{(q)}=\frac{\partial y_{i,t+q}}{\partial e_{jl}},\ q=0,\ 1,\ 2,\ \cdots,\ N;\ t=1,\ 2,\ \cdots,\ T. \tag{6}\]  
We implement the impulse response function between each pair of price indices and between crude oil and each price index and extract the impulse response values in each month, which can be shown in a matrix \(R^{(q)}\). \[R^{(q)}= \begin{bmatrix}r_{11}^{(q)}&\cdots&\cdots&r_{1j}^{(q)}&\cdots&r_{1n}^ {(q)}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ r_{l1}^{(q)}&\cdots&\cdots&r_{lj}^{(q)}&\cdots&r_{in}^{(q)}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ r_{n1}^{(q)}&\cdots&\cdots&r_{nj}^{(q)}&\cdots&r_{ini}^{(q)}\end{bmatrix}  \tag{7}\]  
Step 5: Measure the direct effects  
Combining the results of Granger causality and impulse response function, we can obtain a new matrix \(GR^{(q)}\) revealing the direct effects among variables at time \(q\) after a shock of a certain variable. \[GR^{(q)}=G\odot R^{(q)}= \begin{bmatrix}G_{11}*r_{11}^{(q)}&\cdots&\cdots&G_{1 j}*r_{1j}^{(q)}&\cdots&G_{1n}*r_{1n}^{(q)}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ G_{i1}*r_{i1}^{(q)}&\cdots&\cdots&G_{ij}*r_{ij}^{(q)}&\cdots&G_{in}*r_{in}^{(q )}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ G_{n1}*r_{n1}^{(q)}&\cdots&\cdots&G_{nj}*r_{nj}^{(q)}&\cdots&G_{nn}*r_{nm}^{(q)} \end{bmatrix}  \tag{8}\]  
where \(\odot\) means the Hadamard product. Thus, if \(G_{ij}*r_{ij}^{(q)}\) does not equal 0, it reveals the direct impact degree of variable \(i\) on variable \(j\) in the \(q\)th month when there is a Granger causality from \(i\) to \(j\). Then, the direct effects of crude oil price shocks on price indices are \(GR_{1s}^{(q)},\ s=2,3,4,\ldots,\ 9\). Therefore, combining the Granger causality test and the impulse response function, we could not only identify the significant transmission relations but also obtain the direct influence degree among variables. Then, we propose a transmission network of oil price shocks to Chinese price levels (TNOP) with variables as nodes and transmission relationships as edges. The response values are the weight of the edges. Therefore, the network can be described as Equation (9). \[TNOP=(V,E,W)=GR^{(q)} \tag{9}\]  
where \(V\) represents the node set, and \(E\) and \(W\) represent the edge set and weight set, respectively. If \(G_{ij}*r_{ij}^{(q)}\) does not equal 0, then there is an edge from variable \(i\) to \(j\). Since the weights of the edges vary over time, we only display the unweighted transmission networks of four oil price shocks to various price indices, as shown in Figure 1. The directional red lines represent the transmission relationships from crude oil price shock to price indices, i.e., the direct impact from crude oil to price indices. It is observed that not all the price indices could be directly affected by crude oil. Furthermore, the transmission relationships among price indices are complicated.