# Influencing Factors Analysis of Crude Oil Futures Price Volatility Based on Mixed-Frequency Data
## 3 Model Construction of Crude Oil Futures Price Fluctuation Based on Mixed-Frequency Data  
### Improved GARCH-MIDAS Volatility Model  
In order to re-examine the relationship between stock market volatility and macroeconomic activities and their volatility, the GARCH-MIDAS model was constructed by introducing the MIDAS method into the GARCH model [34]. Different information may have different effects on financial markets, depending on their short-term or long-term effects. In the GARCH-MIDAS model, the volatility of the high-frequency dependent variable is decomposed into the product of long-term low-frequency volatility components and short-term high-frequency volatility components; the low-frequency volatility component is then explained by the weighted level or weighted volatility of the low-frequency variable. The high-frequency volatility component is described by the traditional GARCH structure. According to the research of [PERSON] and [PERSON], the GARCH-MIDAS model constructed in this paper is expressed as follows:  
\[r_{i,t}=\mu+\sqrt{\tau_{t}\times g_{i,t}}\varepsilon_{i,t},\ \forall i=1,\ldots,N_{t}, \tag{1}\]where \(r_{i,t}\) represents the logarithmic return rate of crude oil futures prices on the \(i\)th day of month \(t\); \(\varepsilon_{i,t}|\Phi_{i-1,t}\)-\(N\)(0,1), and \(\Phi_{i-1,t}\) is the set of information available on the (\(i-1\))th day during \(t\). Volatility has two components: the short-term dynamic component of daily volatility \(g_{i,t}\) and the long-term component of volatility \(\tau_{t}\). \(g_{i,t}\) is related to daily liquidity and other possible short-term factors, and \(\tau_{t}\) is first related to expected future cash flows and future discount rates. Assume that the components \(g_{i,t}\) follow the (day) GARCH (1,1) process:  
\[g_{i,t}=\left(1-\alpha-\beta\right)+\alpha(r_{i-1,t}-\mu)^{2}/\tau_{t}+\beta g _{i-1,t}. \tag{2}\]  
Furthermore, the MIDAS regression method is used to characterize the \(\tau_{t}\) components based on the realized volatility (\(RV\)) of the rate of return:  
\[\tau_{t}=m+\theta\sum\
olimits_{k=1}^{K}\varphi_{k}(\omega_{1},\omega_{2})RV _{t-k}, \tag{3}\]  
\[RV_{t}=\sum\
olimits_{i=1}^{N_{t}}r_{i,t}{}^{2}, \tag{4}\]  
where \(K\) represents the maximum lag order of the low-frequency variable, and \(RV_{t}\) represents the realized volatility in the month \(t\). The weighting function in the equation adopts the Beta function, and the specific form is  
\[\varphi_{k}(\omega)=\frac{\left(k/K\right)^{\omega_{1}-1}(1-k/K)^{\omega_{2}- 1}}{\sum\
olimits_{j=1}^{K}\left(j/K\right)^{\omega_{1}-1}(1-j/K)^{\omega_{2} -1}}. \tag{5}\]  
Equation (1) to (5) together constitute a time-varying conditional variance GARCH-MIDAS model based on realized volatility and parameter space \(\Theta=\{\mu,\alpha,\beta,m,\theta,\omega_{1},\omega_{2}\}\). Compared with the basic GARCH model, this model is equivalent to a new type of component equation. According to the foregoing analysis, the long-term component of crude oil futures price fluctuations has a strong correlation with the fluctuations of its influencing factors. Equation (3) is replaced, and GARCH-MIDAS models based on the level and volatility of their influencing factors are established, respectively. For horizontal effects,  
\[\tau_{t}=m_{l}+\theta_{l}\sum\
olimits_{k=1}^{K_{l}}\varphi_{k}\Big{(}\omega _{1,t},\omega_{2,t}\Big{)}X_{l,t-k^{\prime}}^{mv}, \tag{6}\]  
where \(K_{l}\) is the maximum lag order of the measured level of a certain macroeconomic variable and \(X_{l,t-k}^{mv}\) represents the level value of the macroeconomic variable lagging \(k\) period relative to period \(t\). For the volatility effect, the volatility value of the explanatory variable must first be obtained. This article considers the use of AR(p) autoregressive model to obtain residual items, and the square of residuals is used to measure the volatility of macroeconomic variables.