## 3 Materials and Methods
### Methods
#### 3.3.2 GARCH Family Models  
In the analysis and research of time series data, most of the early regression analysis models usually assume that the series variance is constant; i.e., the fluctuation of the time series will not change with time. However, for the prediction problem of the time series of the time-varying volatility, the parameter estimations obtained under the above assumptions are not valid, and the relevant significance tests of the model parameters cannot be performed at this time. GARCH-family models effectively solve these problems and were initially used to analyze financial time series. Its essence is the historical volatility information as a condition and a change in autoregressive form to describe fluctuations. It can better solve the heteroskedasticity, volatility aggregation, leverage effect, and asymmetric effect of the series, so it can better grasp and fit the characteristics of the volatility change of the series. In 1986, [PERSON] [45] further developed the GARCH model based on the original autoregressive conditional heteroskedasticity model (ARCH). When dealing with garlic-price series, this model is able to describe the effect of a prior period's external information on the current period's price fluctuations and the effect of a prior period's price fluctuations on the sharpness of current price fluctuations through the parameters. The GARCH model can solve the heteroscedasticity and volatility aggregation of garlic-price series. The specific process of the GARCH model is  
\[y_{t}=\mu_{t}+\sigma_{t}\eta_{t} \tag{7}\]  
\[\varepsilon_{t}=\sigma_{t}\eta_{t} \tag{8}\]  
\[\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\varepsilon_{t-1}^{2}+\cdots+\alpha_{q} \varepsilon_{t-q}^{2}+\beta_{1}\sigma_{t-1}^{2}+\cdots+\beta_{p}\sigma_{t-p} ^{2} \tag{9}\]  
where \(\eta_{t}\) is independently and identically distributed and \(\eta_{t}\sim N(0,1)\); \(\sigma_{t}^{2}\) is the conditional variance; \(\alpha_{0}\) is the constant; \(\alpha_{i}\) is the ARCH term coefficient; \(\beta_{i}\) is the GARCH term coefficient; and \(\varepsilon_{t}\) is the random disturbance term. As the research progressed, it was found that when the price series is affected by positive and negative shocks the same degree, the impact varies. Generally, most of the positive shocks have less impact on the time series, whereas the negative ones are more powerful and will have a more significant impact on the price series. Such fluctuations are difficult to explain due to the limitations of the GARCH model. Therefore, to further describe this asymmetry, some scholars have started to propose asymmetric GARCH models. In 1991, [PERSON] [46] proposed the EGARCH model, which effectively overcomes the drawbacks of the GARCH model. It does not require every coefficient in the variance equation to be nonnegative. By reducing the constraints, the solution process becomes simpler and more flexible. It can describe the leverage effect of the garlic-price series to examine the effect of the nature of information in the market on the degree of volatility. The mean equation of this model is the same as that of the GARCH, and its standard conditional variance equation is  
\[ln(\sigma_{t}^{2})=\alpha_{0}+\beta ln(\sigma_{t-1}^{2})+\alpha\frac{| \varepsilon_{t-1}|}{\sigma_{t-1}}+\gamma\frac{\varepsilon_{t-1}}{\sigma_{t-1}} \tag{10}\]  
where \(\gamma\) is the leverage coefficient, and the presence of leverage can be determined by a hypothesis test of \(\gamma<0\). If \(\gamma\
eq 0\), it implies that there is asymmetry in the impacts on the shock. In 1993, [PERSON] et al. [47] proposed a PGARCH model for solving asymmetric effects, and the standard conditional variance equation for this model is  
\[\sigma_{t}^{\delta}=\alpha_{0}+\beta\sigma_{t-1}^{\delta}+\alpha(|\varepsilon _{t-1}|-\gamma\varepsilon_{t-1})^{\delta} \tag{11}\]where \(\delta>0\); when \(i=1,2,\cdots,r\), \(|\gamma_{i}|\leqslant 1\); when \(i>r\), \(\gamma_{i}=0\), \(r\leqslant p\).