Let ${\left\{y_t\right\}}_{t=1}^T$ be an observable return series. The GARCH(1,1)-X model with exogenous variables is constructed as follows:

$\{\begin{array}{ll}{y}_t={\sigma }_t{\epsilon }_t,\hfill & (1\text{a})\hfill \\ Var\left(y_t|{ℱ}_{t-1}\right)={\sigma }_t^2=\omega +\alpha y_{t-1}^2+\beta {\sigma }_{t-1}^2+\gamma x_{t-1}^2\hfill & (1\text{b})\hfill \end{array}$

where $\left\{{\epsilon }_t\right\}$ are independent and identically distributed random error terms with a mean of 0 and a variance of 1; when $t>s$, $y$ and ${\epsilon }_t$ are independent. ${ℱ}_{t-1}$ represents the low-frequency information set at time $t-1$, ${\sigma }_t^2$ is the conditional volatility, $x_{t-1}$ represents the exogenous variable, and $\gamma$ intuitively expresses the impact of the exogenous variable on the volatility. Set the parameters $\omega >0$, $\alpha >0$, $\beta \ge 0$, $\gamma >0$ to ensure the non-negativity of the conditional variance. Since the GARCH(1,1) model is sufficient to characterize the ARCH effect in financial data, this paper uses the GARCH(1,1)-X model as the baseline model to incorporate intraday high-frequency data.

To introduce high-frequency data, we consider extending the daily GARCH model using continuous-time processes. Let the intraday return process be denoted as $y_t\left(u\right)$, representing the logarithmic return of the asset at time $u$ on day $t$. We standardize the time of each trading day by defining each trading day’s time interval as $\left(0,1\right)$ and construct the following model:

$\{\begin{array}{ll}{Y}_t\left(u\right)={\sigma }_t{Z}_t\left(u\right)\hfill & (2\text{a})\hfill \\ {\sigma }_t^2=\omega +\alpha y_{t-1}^2+\beta {\sigma }_{t-1}^2+\gamma x_{t-1}^2\hfill & (2\text{b})\hfill \end{array}$

Where $Z_t(\cdot )$ is a standard independent process. When $u=1$, $Y_t\left(1\right)=y_t$ and $Z_t\left(1\right)={\epsilon }_t$, the model is then a GARCH(1,1)-X model. Distinguished from low-frequency models, here ${\sigma }_t$ is referred to as the scale parameter, and since $Z_t(\cdot )$ is independent, Visser (2011) refers to the above model as a scale model.

Note that model (2a-2b) contains high-frequency data $Y_t\left(u\right)$, $Z_t\left(u\right)$, and low-frequency data $Y_{t-1},x_{t-1}$. Due to the inability to estimate parameters, based on the scale model, Visser (2011) further proposed a function that acts on intraday high-frequency data, referred to as the volatility representation. In general, the volatility representation is positive and satisfies positive homogeneity: $H\left(pY\left(z\right)\right)=pH\left(Y\left(z\right)\right)$. Utilizing this homogeneity, Visser cleverly embeds the volatility representation into low-frequency GARCH models, forming a class of volatility representation models that complete the embedding of high-frequency data. This paper adopts the same idea, embedding the volatility representation into the GARCH-X model.

From Homogeneity, we know:

$H_t\equiv H\left(Y_t\left(u\right)\right)=H\left({\sigma }_t{Z}_t\left(u\right)\right)={\sigma }_{t}H\left(Z_t\left(u\right)\right),$(3)

Let

$z_{H,t}=H\left(Z_t\left(u\right)\right),\mu =E\left(z_{H,t}^2\right),{\epsilon }_t^{\text{*}}=\frac{z_{H,t}}{\sqrt{\mu }}.$(4)

Since $Z_t\left(u\right)$ is an independent and identically distributed sequence, $z_{H,t},{\epsilon }_t^{\text{*}}$ are also independent and identically distributed random variable sequences, and $E\left({\epsilon }_t^{\text{*}}\right)=1$. Combining (2a-4), this paper constructs the following GARCH-X model to represent the embedded intraday return process:

$\begin{array}{l}{H}_t={\sigma }_t{z}_{H,t}={\sigma }_t\sqrt{\mu }{\epsilon }_t^{\text{*}},\\ {\sigma }_t^2=\omega +\alpha y_{t-1}^2+\beta {\sigma }_{t-1}^2+\gamma x_{t-1}^2.\end{array}$(5)

We need to estimate parameter $\theta =\left(\omega ,\alpha ,\beta ,\gamma ,\mu \right)$ in the model, but because there is an unknown redundant parameter $\mu$ in model(5), let ${\sigma }_t^{\text{*}}={\sigma }_t\sqrt{\mu }$, then the model(5) can be transformed to

$\begin{array}{l}{H}_t={\sigma }_t{z}_{H,t}={\sigma }_t^{\text{*}}{\epsilon }_t^{\text{*}}\\ {\sigma }_t^{\text{*}2}={\omega }^{\text{*}}+{\alpha }^{\text{*}}{y}_{t-1}^2+{\beta }^{\text{*}}{\sigma }_{t-1}^2+{\gamma }^{\text{*}}{x}_{t-1}^2\end{array}$(6)

where

${\omega }^{\text{*}}=\omega \mu ,{\alpha }^{\text{*}}=\alpha \mu ,{\beta }^{\text{*}}=\beta ,{\gamma }^{\text{*}}=\gamma \mu .$(7)

The method of maximum likelihood estimation proposed by Visser (2011) allows us to first estimate ${\theta }^{\text{*}}={\left({\omega }^{\text{*}},{\alpha }^{\text{*}},{\beta }^{\text{*}},{\gamma }^{\text{*}}\right)}^{\prime }$ based on (6), then estimate $\mu$, and lastly derive parameter from (7). According to the transformed model (6), the log quasi-likelihood function is expressed as:

$L_T\left({\theta }^{\text{*}}\right)=\underset{t=1}{\overset{T}{{\displaystyle \sum }}}{l}_t\left({\theta }^{\text{*}}\right)=-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(\text{ln}\left({\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)\right)+\frac{H_t^2}{{\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}\right)$(8)

Thus, the quasi-maximum likelihood estimation (QMLE) for ${\theta }^{\text{*}}$ is defined as:

${\hat{\theta }}^{*}=\arg \max L_T\left({\theta }^{*}\right).$(9)

At the same time, we can obtain ${\hat{\sigma }}_t^{\text{*}2}={\hat{\omega }}^{\text{*}}+{\hat{\alpha }}^{\text{*}}{y}_{t-1}^2+{\hat{\beta }}^{\text{*}}{\sigma }_{t-1}^2+{\hat{\gamma }}^{\text{*}}{x}_{t-1}^2$. Next, we only need to obtain the estimate of $\mu$ to get the estimate of $\theta$ based on (7). It is noted that when the volatility representative $H_t=\left|y_t\right|$, model (6) corresponds to a low-frequency GARCH(1,1) model, the ${\hat{\theta }}^{\text{*}}$ derived through (8) and (9) corresponds to the parameter estimation of the model (1a-1b). Denoted as $\tilde{\theta }={\left(\tilde{\omega },\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\right)}^{\tau }$, thus obtaining the corresponding estimate sequence $\left\{{\tilde{\sigma }}_t^{\text{*}2}\right\}$. According to the aforementioned transformation, we find $\mu ={\sigma }_t^{\text{*}2}/{\sigma }_t^2$, thus the estimate for $\mu$ is:

$\hat{\mu }=\frac{1}{T}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\frac{{\hat{\sigma }}_t^{\text{*}2}}{{\tilde{\sigma }}_t^2},$(10)

thus, we can obtain:

$\hat{\theta }={\left(\hat{\omega },\hat{\alpha },\hat{\beta },\hat{\gamma }\right)}^{\tau }={\left(\frac{{\hat{\omega }}^{\text{*}}}{\hat{\mu }},\frac{{\hat{\alpha }}^{\text{*}}}{\hat{\mu }},{\hat{\beta }}^{\text{*}},\frac{{\hat{\gamma }}^{\text{*}}}{\hat{\mu }}\right)}^{\tau }.$(11)

Denote ${\theta }_0^{\text{*}}={\left({\omega }_0^{\text{*}},{\alpha }_0^{\text{*}},{\beta }_0^{\text{*}},{\gamma }_0^{\text{*}}\right)}^{\tau }$ as the true value of the corresponding parameter. If we assume:

(A1) The parameter space of ${\theta }^{\text{*}}\in \text{Θ}\subset {\left(0,\infty \right)}^4$, $\text{Θ}$ is compact set;

(A2) $E\left[\text{log}\left({\alpha }_0^{\text{*}}{\epsilon }_t^{\text{*}2}\right)+{\beta }_0^{\text{*}}\right]<0$;

(A3) $E\left({\epsilon }_i^{\text{*}4}\right)<\infty$.

Given that the conditions are met, then

$\sqrt{T}\left({\hat{\theta }}^{\text{*}}-{\theta }_0^{\text{*}}\right)\stackrel{L}{\to }N\left(0,{\text{Σ}}^{\text{*}}\right),T\to \infty ,$(12)

where ${\text{Σ}}^{\text{*}}=A_0^{-1}{B}_0{A}_0^{-1}$, ${\left(A_0\right)}_{i,j}=E\left(\frac{{\partial }^2{l}_t\left({\theta }_0^{\text{*}}\right)}{\partial {\theta }_i^{\text{*}}\partial {\theta }_j^{\text{*}}}\right)$, ${\left(B_0\right)}_{i,j}=E\left(\frac{\partial l_t\left({\theta }_0^{\text{*}}\right)}{\partial {\theta }_i^{\text{*}}}\frac{\partial l_t\left({\theta }_0^{\text{*}}\right)}{\partial {\theta }_j^{\text{*}}}\right)$.

Note: (i) According to the proofs by Visser (2011) and Straumann and Mikosch (2006) , let $v_t\left({\theta }^{\text{*}}\right)={\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)$, then equation (8) can be rewritten as:

$\begin{array}{c}{L}_T\left({\theta }^{*}\right)=\underset{t=1}{\overset{T}{{\displaystyle \sum }}}{l}_t\left({\theta }^{*}\right)=-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(\ln \left(v_t\left({\theta }^{*}\right)\right)+\frac{H_t^2}{v_t\left({\theta }^{*}\right)}\right)\\ =-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(\ln \left(v_t\left({\theta }^{*}\right)\right)+\frac{{\sigma }_t^{*}{\epsilon }_t^{*}}{v_t\left({\theta }^{*}\right)}\right).\end{array}$(13)

Since ${\epsilon }_t^{\text{*}}$ is independent of ${\sigma }_t^{\text{*}}$ and $v_t\left({\theta }^{\text{*}}\right)$, and satisfies $E\left({\epsilon }_t^{\text{*}}\right)=1$, we define $L\left({\theta }^{\text{*}}\right)=E\left(l_0\left({\theta }^{\text{*}}\right)\right)$, then we have:

$\begin{array}{c}L\left({\theta }^{*}\right)=-\frac{1}{2}E\left(\ln \left(v_0\left({\theta }^{*}\right)\right)+\frac{{\sigma }_0^{*}{\epsilon }_0^{*}}{v_0\left({\theta }^{*}\right)}\right)2\left(L\left({\theta }^{*}\right)-L\left({\theta }_0^{*}\right)\right)-1\\ =E\left(\ln \left(\frac{v_0\left({\theta }_0^{*}\right)}{v_0\left({\theta }^{*}\right)}\right)-\frac{v_0\left({\theta }_0^{*}\right)}{v_0\left({\theta }^{*}\right)}\right)\end{array}$(14)

Given that $\text{ln}\left(x\right)-x\le 1$, with equality if and only if $x=1$, therefore, to maximize $L\left({\theta }^{\text{*}}\right)$, it must be that $v_0\left({\theta }^{\text{*}}\right)=v_0\left({\theta }_0^{\text{*}}\right)$, i.e., ${\theta }^{\text{*}}={\theta }_0^{\text{*}}$. Hence, it follows that $\frac{L_T\left({\theta }^{\text{*}}\right)}{T}$ converges uniformly to $L\left({\theta }^{\text{*}}\right)$, and the parameter ${\theta }^{\text{*}}$ converges almost surely to ${\theta }_0^{\text{*}}$.

(ii) The first-order partial derivative and the second-order partial derivative of $l_t\left({\theta }^{\text{*}}\right)$ respectively are:

$\begin{array}{c}\frac{\partial l_t\left({\theta }^{*}\right)}{\partial \left({\theta }_i^{*}\right)}=\frac{1}{{\sigma }_t^{*2}\left({\theta }^{*}\right)}\frac{\partial {\sigma }_t^{*2}\left({\theta }^{*}\right)}{\partial {\theta }_i}-H_t^2\frac{1}{{\left({\sigma }_t^{*2}\left({\theta }^{*}\right)\right)}^2}\frac{\partial {\sigma }_t^{*2}\left({\theta }^{*}\right)}{\partial {\theta }_i}\\ =\left(1-\frac{H_t^2}{{\sigma }_t^{*2}\left({\theta }^{*}\right)}\right)\frac{1}{{\left({\sigma }_t^{*2}\left({\theta }^{*}\right)\right)}^2}\frac{\partial {\sigma }_t^{*2}\left({\theta }^{*}\right)}{\partial {\theta }_i}\end{array}$(15)

$\begin{array}{c}\frac{{\partial }^2{l}_t\left({\theta }^{\text{*}}\right)}{\partial {\theta }_i^{\text{*}}\partial {\theta }_j^{\text{*}}}=\frac{1}{{\sigma }_t^{\text{*}4}\left({\theta }^{\text{*}}\right)}\left(\frac{2H_t^2}{{\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}-1\right)\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_i}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_j}\\ +\left({\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)-H_t^2\right)\frac{1}{{\sigma }_t^{\text{*}4}\left({\theta }^{\text{*}}\right)}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_i\partial {\theta }_j}\end{array}$(16)

When ${\theta }^{\text{*}}={\theta }_0^{\text{*}}$, we get

$\begin{array}{l}{\left(A_0\right)}_{i,j}=E\left(\frac{\partial l_t\left({\theta }_0^{\text{*}}\right)}{\partial \left({\theta }_i^{\text{*}}\right)\partial \left({\theta }_j^{\text{*}}\right)}\right)=E\left(\frac{1}{{\sigma }_t^{\text{*}4}\left({\theta }_0^{\text{*}}\right)}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }_0^{\text{*}}\right)}{\partial {\theta }_i}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }_0^{\text{*}}\right)}{\partial {\theta }_j}\right)\hfill \end{array}$(17)

Since $H_t={\sigma }_t^{\text{*}}{\epsilon }_t^{\text{*}}$ and ${\sigma }_t^{\text{*}}$ and ${\epsilon }_t^{\text{*}}$ are independent, we have:

$\begin{array}{c}{\left(B_0\right)}_{i,j}=E\left(\frac{\partial l_t\left({\theta }_0^{\text{*}}\right)}{\partial \left({\theta }_i^{\text{*}}\right)}\frac{\partial l_t\left({\theta }_0^{\text{*}}\right)}{\partial \left({\theta }_j^{\text{*}}\right)}\right)\\ =E\left({\epsilon }_t^{\text{*}2}-1\right)E\left(\frac{1}{{\sigma }_t^{\text{*}4}\left({\theta }^{\text{*}}\right)}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_i}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_j}\right)\end{array}$(18)

Thus, the asymptotic variance obtained is ${\text{Σ}}^{\text{*}}=A_0^{-1}{B}_0{A}_0^{-1}$. Since $var\left({\epsilon }_t^{\text{*}}\right)=E\left({\epsilon }_t^{\text{*}2}\right)=1$, we have $var\left({\epsilon }_t^{\text{*}2}\right)=E\left({\left({\epsilon }_t^{\text{*}2}-1\right)}^2\right)$.

Let $G{\left({\theta }_0^{\text{*}}\right)}_{i,j}=E\left(\frac{1}{{\sigma }_t^{\text{*}4}\left({\theta }^{\text{*}}\right)}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_i}\frac{\partial {\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)}{\partial {\theta }_j}\right)$, then the asymptotic variance of ${\hat{\theta }}^{\text{*}}$ is

${\text{Σ}}^{\text{*}}=var\left({\epsilon }_t^{\text{*}2}\right)G^{-1}\left({\theta }_0^{\text{*}}\right)$

It is not difficult to see that the elements of the matrix $G\left({\theta }^{\text{*}}\right)$ and ${\sigma }_t^{\text{*}2}\left({\theta }^{\text{*}}\right)$ are both related to the parameters ${\theta }^{\text{*}}=\left({\omega }^{\text{*}},{\alpha }^{\text{*}},{\beta }^{\text{*}},{\gamma }^{\text{*}}\right)$ and the first-order partial derivatives of $l_t\left({\theta }^{\text{*}}\right)$. From this, we can conclude that:

$\begin{array}{c}{\sigma }_t^{\text{*}2}={\omega }^{\text{*}}+{\alpha }^{\text{*}}{y}_{t-1}^2+{\beta }^{\text{*}}{\sigma }_{t-1}^{\text{*}2}+{\gamma }^{\text{*}}{x}_{t-1}^2\\ =\frac{{\omega }^{\text{*}}}{1-\beta }+{\alpha }^{\text{*}}\underset{i=0}{\overset{\infty }{{\displaystyle \sum }}}{\left({\beta }^{\text{*}}\right)}^i{y}_{t-i-1}^2+{\gamma }^{\text{*}}\underset{i=0}{\overset{\infty }{{\displaystyle \sum }}}{\left({\beta }^{\text{*}}\right)}^i{x}_{t-i-1}^2,\end{array}$(19)

and

$\begin{array}{l}\frac{\partial {\sigma }_t^{\text{*}2}}{\partial {\omega }^{\text{*}}}=\frac{1}{1-\beta },\frac{\partial {\sigma }_t^{\text{*}2}}{\partial {\alpha }^{\text{*}}}=\underset{i=0}{\overset{\infty }{{\displaystyle \sum }}}{\left({\beta }^{\text{*}}\right)}^{i-1}{y}_{t-i}^2,\\ \frac{\partial {\sigma }_t^{\text{*}2}}{\partial {\beta }^{\text{*}}}=\underset{i=0}{\overset{\infty }{{\displaystyle \sum }}}{\left({\beta }^{\text{*}}\right)}^{i-1}{\sigma }_{t-i}^2,\frac{\partial {\sigma }_t^{\text{*}2}}{\partial {\gamma }^{\text{*}}}=\underset{i=0}{\overset{\infty }{{\displaystyle \sum }}}{\left({\beta }^{\text{*}}\right)}^{i-1}{x}_{t-i}^2.\end{array}$(20)

Let the asymptotic variances of the estimates $\hat{\omega },{\hat{\alpha }}^{\text{*}},{\hat{\beta }}^{\text{*}},{\hat{\gamma }}^{\text{*}}$ be ${\sigma }_{\omega }^{\text{*}2},{\sigma }_{\alpha }^{\text{*}2},{\sigma }_{\beta }^{\text{*}2},{\sigma }_{\gamma }^{\text{*}2}$ respectively. Based on the asymptotic properties of ${\hat{\theta }}^{\text{*}}$, the following results are obtained:

$\begin{array}{l}\sqrt{T}\left(\hat{\omega }-{\omega }_0\right)\stackrel{L}{\to }N\left(0,{\sigma }_{\omega }^{\text{*}2}/{\mu }^2\right)\\ \sqrt{T}\left(\hat{\alpha }-{\alpha }_0\right)\stackrel{L}{\to }N\left(0,{\sigma }_{\alpha }^{\text{*}2}/{\mu }^2\right)\\ \sqrt{T}\left(\hat{\beta }-{\beta }_0\right)\stackrel{L}{\to }N\left(0,{\sigma }_{\beta }^{\text{*}2}/{\mu }^2\right)\\ \sqrt{T}\left(\hat{\gamma }-{\gamma }_0\right)\stackrel{L}{\to }N\left(0,{\sigma }_{\gamma }^{\text{*}2}/{\mu }^2\right)\end{array}$(21)

We can see from the estimates that the choice of volatility representation affects the efficiency of parameter estimates. Therefore, we need to provide criteria for selecting the volatility representation. Based on the criterion of minimizing the asymptotic variance of the estimates, we proposed the optimal volatility representation selection criterion function for the general quasi-maximum likelihood estimation:

$MH=\frac{E\left(H_t^4\right)}{{\left(E{H}_t^2\right)}^2}.$(22)

Li and Zhang (2021) prove that the smaller the value of $MH$, the smaller the asymptotic variance of the parameter estimates, leading to more accurate estimates. This paper also seeks to select the optimal volatility representation by finding the minimum value of $MH$.

To demonstrate that the high-frequency GARCH-X model constructed in this paper has better predictive performance, we establish a high-frequency GARCH-X model using the CSI 300 Index as the research object, with the CSI 500 Index selected as the exogenous variable. The data covers a period from January 3, 2019 to July 15, 2021, totaling 615 trading days, with a sampling frequency of one minute. All data are sourced from the Tongdaxin Financial Terminal.

The CSI 300 Index is selected in this paper because it is one of the earliest indices launched in China, comprising the 300 most representative securities with large scale and good liquidity from the Shanghai Stock Exchange and the Shenzhen Stock Exchange. This index serves as a price revelation function that characterizes fluctuations in stock prices and is a key indicator that best reflects the overall trend of the CSI market. The CSI 500 Index, on the other hand, is composed of the top 500 stocks in terms of total market capitalization after excluding the constituent stocks of the CSI 300 Index and the top 300 stocks in total market capitalization among all A-shares. This index is an important indicator reflecting the overall performance of small-cap stocks. There is a certain correlation between these two indices. Therefore, this paper studies the CSI 500 Index as an exogenous variable together with the CSI 300 Index.

The opening hours of China’s stock exchanges are from 9:30 AM to 11:30 AM and from 1:00 PM to 3:00 PM, totaling 4 hours or 240 minutes. Therefore, all price sequence data consist of 1-minute closing prices, generating 240 data points per day. Data at a 5-minute frequency yields 48 data points per day, while data at a 10-minute frequency produces 24 data points per day, and data at a 30-minute frequency results in 8 data points per day.

The daily logarithmic rate of returns is calculated as follows:

$Y_t\left(u\right)=\left[\text{ln}g\left(P_t\left(u\right)\right)-\text{log}\left(P_{t-1}\left(1\right)\right)\right]\ast 100.$

Based on the calculation method for realized volatility presented in (23, we use high-frequency data at 1-minute, 5-minute, 10-minute, and 30-minute intervals to calculate the realized volatility of intraday returns, denoted as RV1, RV5, RV10, and RV30, respectively. Figure 1 presents the time series plots of the four realized volatilities for the CSI 300 Index. As can be seen from Figure 1 , there is no apparent periodicity in the high-frequency data of the CSI 300 Index. According to the ADF test, these four realized volatilities are stationary series (with a p-value of 0.03 for RV1 and 0.01 for the others). Therefore, predictions can be made based on these series.

Figure 2 and Figure 3 present the realized volatility series plot and Q-Q plot of the exogenous variable, the CSI 500 Index, respectively. From Figure 2 , it can be observed that the exogenous variable does not exhibit any apparent periodicity. According to the ADF test, these four sequences are stationary (with a p-value of 0.03 for RV1 and 0.01 for the others), and they do not follow a normal distribution.

According to the aforementioned estimation method of the GARCH-X model under high-frequency data, we respectively give the parameter estimation results of the model and the corresponding MH estimates under different volatility representations, as shown in Table 3 .

It can be seen from Table 3 that among the four frequencies, the minimum MH value corresponds to the frequency of 1 minute. As the frequency decreases, the MH value increases, corresponding to the minimum $Var{\left({\epsilon }_t^{\text{*}}\right)}^2$ of the 1 minute. As the frequency decreases, the $Var{\left({\epsilon }_t^{\text{*}}\right)}^2$ value increases, and the MH value of all frequencies is smaller than that of the daily frequency. Therefore, it is appropriate to select the 1-minute volatility representation RV1 as the optimal volatility representation, and its corresponding estimation result as the final parameter estimation result.

Therefore, the results of the low-frequency GARCH(1,1)-X model fitting the CSI 300 index and selecting the 500 volatility index as exogenous variables are as follows:

$\begin{array}{l}{y}_t={\sigma }_t{\epsilon }_t\\ {\hat{\sigma }}_t^2=0.0187+0.1211y_{t-1}^2+0.8007{\hat{\sigma }}_{t-1}^2+0.0314x_{t-1}^2\end{array}$(24)

The fitting result of the GARCH(1,1)-X model with realized volatility in 1 minute is as follows:

$\begin{array}{l}{y}_t={\sigma }_t{\epsilon }_t\\ {\hat{\sigma }}_t^2=0.0044+0.2477y_{t-1}^2+0.5402{\hat{\sigma }}_{t-1}^2+0.14x_{t-1}^2\end{array}$(25)

Based on the obtained fitting results, the volatility estimates based on daily and 1-minute realized volatilities are shown in Figure 4 . It is evident from the figure that the trends of these two estimates are nearly identical.

In today’s financial markets, there is a strong emphasis on financial risk management. Value at Risk (VaR) stands out as an effective measurement method in risk management for financial markets, quantifying the potential maximum loss that a portfolio may face within a certain holding period and at a given confidence level. It has become a widely used metric.

This paper applies the estimation results of high-frequency data using the GARCH(1,1)-X model, namely (25), to the VaR estimation and forecasting of the CSI 300 Index. The VaR estimation process based on high-frequency data is as follows: Firstly, the parameters $\theta ={\left(\omega ,\alpha ,\beta ,\gamma \right)}^{\tau }$ of the GARCH(1,1)-X model are estimated as $\hat{\theta }=\left(0.0044,0.2477,0.5402,0.14\right)$ through model fitting using high-frequency data. Next, by substituting the estimated values $\hat{\theta }$ and daily logarithmic returns into the conditional variance formula of Equation (25) in the GARCH(1,1)-X model, we obtain 514 estimated conditional variances, denoted as ${\hat{h}}_t$. Let ${\epsilon }_t=\frac{H_t}{\sqrt{{\hat{h}}_t}}$, resulting in a sequence of ${\epsilon }_t$. Setting the quantile $p=0.1$, we take the 0.1 quantile of ${\epsilon }_t$ as ${\hat{\epsilon }}_{0.1}$, finally, we set ${\hat{q}}_t={\hat{\epsilon }}_{0.1}\sqrt{{\hat{h}}_t}$, which represents the VaR estimate of $H_t$ on day $t$ at the $p=0.1$ level at time $t-1$.

For comparison and forecasting purposes, the first 514 trading days of data are used as the training set to estimate model parameters and VaR estimates. Then, a rolling forecasting method is employed to calculate the VaR forecast values for the subsequent 101 samples (forecasting set). The VaR estimation results for the training set and the VaR forecasting results for the forecasting set are shown in Figure 5 .

The left graph in Figure 5 shows the estimated results of VaR for the training set with a significance level of 0.1. It can be seen that there are over twenty instances where the predicted returns are greater than the actual returns during this period, while most of the time, the predicted returns are less than the actual returns. According to theory, there are 514 trading days in the training set, and 10% of the sample points, which amounts to an average of 52 trading days, should fall outside the VaR. This indicates that the prediction results for the training set are quite satisfactory. The right graph in Figure 5 presents the prediction results for the forecasting set. The forecasting set consists of 101 trading days, and with a 0.1 quantile set, ideally, an average of 10 sample points fall outside the VaR. In the results of this forecasting set, there are 10 sample points that fall outside the VaR, which suggests that the prediction results for the forecasting set are also quite satisfactory. Overall, it can be concluded that the high-frequency GARCH model with exogenous variables has good predictive power for VaR.

To assess whether the high-frequency GARCH(1,1)-X model with exogenous variables can better reflect the estimation and prediction of the CSI 300 Index, the Mean Absolute Error (MAE) is introduced as a criterion for evaluating the VaR prediction results:

$\text{MAE}=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left|{\hat{y}}_i-y_i\right|.$

To test whether the high-frequency GARCH(1,1)-X model with exogenous variables can better reflect the estimation and prediction of the CSI 300 Index, referring to Iqbal and Mukherjee’s (2012) test on the VaR prediction results of training set. Let $N^{\text{*}}={\displaystyle {\sum }_{t=1}^N{I}_t}$, $\hat{p}=\frac{N^{\text{*}}}{N}$, where $I_t=I\left(H_t<{\hat{q}}_t\right)$, with $N$ being the total number of samples. Let $N_{i,j}$ be the count of points $t$ such that $I_{t-1}=j$ and $I_t=i$ for $2\le t\le N$, where $i,j=0,1$, also, let ${\hat{\tau }}_{i,j}=\frac{N_{i,j}}{N_{i,0}+N_{i,1}}$, $\hat{\tau }=\frac{N_{0,1}+N_{1,1}}{N}$. Define:

$L{R}_{uc}=2\left\{\text{log}\left[\frac{{\left(1-\hat{p}\right)}^{N-N^{\text{*}}}{\hat{p}}^{N^{\text{*}}}}{{\left(1-p\right)}^{N-N^{\text{*}}}{p}^{N^{\text{*}}}}\right]\right\},$

$L{R}_{ind}=2\left\{\text{log}\left[\frac{{\left(1-{\hat{\tau }}_{0,1}\right)}^{N_{0,0}}{\hat{\tau }}_{0,1}^{N_{0,1}}{\left(1-{\hat{\tau }}_{1,1}\right)}^{N_{1,0}}{\hat{\tau }}_{1,1}^{N_{1,1}}}{\left(1-{\hat{\tau }}^{N_{0,0}+N_{1,0}}\right){\hat{\tau }}^{N_{0,1}+N_{1,1}}}\right]\right\},$

then the statistic $L{R}_{cc}=L{R}_{uc}+L{R}_{ind}$ is asymptotically distributed as ${\chi }^2$.

Comparing the low-frequency GARCH model, high-frequency GARCH model, and high-frequency GARCH model with exogenous variables, the results are shown in Table 4 .

As can be seen from Table 4 , in the training set, the LRcc value of the high-frequency GARCH model with exogenous variables is the smallest, followed by the GARCH model with high-frequency data, and the LRcc value of the GARCH model with low-frequency data is the largest. In the prediction set, the ordering of the MAE value is the same as the LRcc value ordering of the training set. This indicates that the prediction results of the high-frequency GARCH model with exogenous variables closer to the real value and that the prediction of the high-frequency GARCH model with exogenous variables is better and more effective.