## 3 Data and Methods
### The Measure of the Direct and Indirect Effects  
#### 3.2.1 The Measure of Direct Effects  
The Granger causality test is a common method for determining whether there is causality between two variables. The impulse response function can analyze the dynamic effects (namely the influence degree) of variable changes on the overall system, i.e., how the volatility of one variable is transmitted to other variables over time, thus leading to fluctuations in other variables. We take the Brent crude oil as an example; the measure of the direct effect on each price index includes the following five main steps. Step 1: Building the VAR model  
If the time series is stationary, we construct a bivariate VAR model that pairs the crude oil price and each price index. In a VAR model, all variables are treated as endogenous variables, and it can estimate the dynamical interaction between variables. The mathematical expression of the VAR model is as follows:  
\[y_{t}=\Phi_{1}y_{t-1}+\Phi_{2}y_{t-2}+\cdots+\Phi_{p}y_{t-p}+e_{t},\;t=1,\;2, \cdots,\;T, \tag{1}\]  
where \(y_{t}\) is a k-dimension column vector of an endogenous variable. \(\Phi\) is a coefficient matrix of k \(\times\) k. \(p\) is the lag order, and \(T\) is the simple length. \(e_{t}\) is a \(k\)-dimension white noise vector. The Akaike information criterion is used to determine the value of \(p\). Step 2: Granger causality test  
The Granger causality test is based on precedence and predictability. According to [37], there are nine variables; thus, we carry out \(\frac{9*(9-1)}{2}=36\) Granger causality tests for each pair of variables. The results could be described as a matrix \(G\), shown in Equation (2). \[G= \begin{bmatrix}0&\cdots&\cdots&G_{1j}&\cdots&G_{1n}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ G_{i1}&\cdots&\cdots&G_{ij}&\cdots&G_{in}\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\vdots&\ddots&\vdots\\ G_{n1}&\cdots&\cdots&G_{nj}&\cdots&0\end{bmatrix}  \tag{2}\]  
where \(G_{ij}\) represents the causal relationship (or transmission relationships) from variable \(i\) to variable \(j\), and \(I\
eq j\). \(n\) is the number of variables and equals 9 in this paper. We set the first variable with the crude oil price, i.e., \"1\" represents crude oil, and \"2\" to \"9\" represent price indices. Choosing a suitable significant level is crucial to the result of the Granger causality test, and this paper chooses the significance level of 0.05 according to the existing studies [38; 39; 40]. If there is a significant causal relationship from \(i\) to \(j\), then \(G_{ij}=1\); otherwise, \(G_{ij}=0\). Step 3: Judging whether the VAR model is stationary  
The bivariate VAR model, constructed in step 1, needs to be tested by the AR root. If the reciprocal of the absolute value of all roots in the VAR model is less than 1, the VAR model is stable in the unit circle. If the model is unstable, some results may not be valid (e.g., the standard error of the impulse response function). Step 4: Building the impulse response function  
If the VAR model is stable, we build an impulse response function, which is used to measure the impact of a positive shock of one standard deviation from an endogenous variable on all endogenous variables in the VAR model during the current and future periods. The impulse response function can be expressed as follows:  
First, the VAR model constructed in step 1 can also be expressed as Equation (3). \[y_{t}=(I_{k}+\mathrm{A}_{1}L+\mathrm{A}_{2}L^{2}+\cdots)e_{t},\;t=1,\;2,\cdots,\;T, \tag{3}\]where \(L\) is a lag operator, and \(I_{k}\) is an identity matrix with \(k\)-dimensions. The \(i\)th variable of \(y_{t}\) is \(y_{it}\), which is as follows:  
\[y_{it}=\sum_{j=1}^{k}\Bigl{(}r_{ij}^{(0)}e_{jl}+r_{ij}^{(1)}e_{j(t-1)}+r_{ij}^{( 2)}e_{j(t-2)}+\cdots\Bigr{)},\ t=1,\ 2,\cdots,T. \tag{4}\]  
Supposing that there is a standard deviation to \(y_{1}\) at time 0,  
\[e_{1t}=\left\{ \begin{array}{ll}1,\ t=0\\ 0,\ t=1,\ 2,\ 3,\cdots,\ e_{it}=0;\ \ t=0,1,2,\ \cdots\ and\ i\
eq 1.\end{array} \right.