Daily LGARCH model estimation using high frequency data

1.   Introduction

Volatility clustering is a well-known characteristic of financial time series. Accurately describing volatility is helpful for pricing and risk management of financial assets. Many conditional heteroscedasticity models have been proposed to describe the time varying volatility. Among them, the autoregressive conditional heteroscedasticity (ARCH) model proposed by Engle (1982) and the generalized autoregressive conditional heteroscedasticity (GARCH) model proposed by Bollerslev (1986) have been widely studied, especially in the financial industry. For example, Nelson (1991) applied GARCH model to asset pricing, Zou et al. (2015) used GARCH model to estimate the combination of market investment risk value, and De Davide (2019) used GARCH model to analyze and predict S & P 500 index. In many cases, the impact of assets on the market is asymmetric. That is to say, investors react differently to the same amount of good news and bad news. Therefore, in order to reflect this phenomenon, more and more scholars are involved in the study of asymmetric GARCH model, see, e.g., Hentschel (1995), Pan et al. (2008), Gyamerah (2019) and Linton et al. (2020). In addition, considering the cyclical factors of market fluctuations, many scholars began to study periodic GARCH model, such as Zhao et al. (2016).

As noted by Duffie and Pan (1997), maximum likelihood estimation of the GARCH type model has the potential disadvantage of being overly sensitive to extreme returns. For example, if we consider a market crash, then extreme daily absolute returns may be 10–20 times the normal daily fluctuation, so the quadratic form of GARCH model yields a return effect that is 100–400 times the normal variance, resulting in excessive fluctuation prediction. Therefore, in order to avoid the impact of extreme returns, Xiao and Koenker (2009) proposed a LGARCH model. It has been shown that LGARCH model can produce more robust inferences, compared to the previous GARCH type models. The specific form of LGARCH $(1, 1)$ model is as follows:

where, $\omega > 0, \alpha, \beta\geqslant0$, $\{\varepsilon_{t}\}_{t = 1}^{\infty}$ is an independent identically distributed sequence with mean $0$ and variance $1$, namely, $\{\varepsilon_{t}\}\sim i.i.d(0, 1)$, and $y_{s}$ is independent of $\{\varepsilon_{t}:t\geq1\}$ for $t > s$. Let $\mathcal{F}_t$ be the $\sigma-$ field generated by $\{\varepsilon_t, \ldots, \varepsilon_1, y_0, y_{-1} \ldots \}$. Given $\mathcal{F}_{t-1}$, the conditional mean of $y_{t}$ is $E(y_{t}|\mathcal{F}_{t-1}) = 0$, and the conditional variance of $y_{t}$ is $Var(y_{t}|\mathcal{F}_{t-1}) = E(h_{t}^{2} \varepsilon_{t}^{2}|\mathcal{F}_{t-1}) = h_{t}^{2} E(\varepsilon_{t}^{2}|\mathcal{F}_{t-1}) = h_{t}^{2}$.

With the development of electronic information technology, it is easier to obtain intraday high frequency data in the financial market, and such data usually contain lots of useful information and are valuable in improving model estimation. To achieve a more precise parameter estimator of common GARCH model, Visser (2011) proposed a volatility proxy model, embedding intraday high frequency data into the framework of daily GARCH model. The volatility proxy model not only maintains the parameter structure of daily GARCH model, but also introduces the intraday high frequency data. Available results show that the variance of parameter estimator in GARCH (1, 1) model can be reduced 20 times by selecting an appropriate volatility proxy, which greatly improves the estimation accuracy of model parameters. Many scholars have further extended the results of Visser (2011) to other cases. For example, Wang et al. (2018) proposed a compound quantile regression (CQR) method to estimate the GARCH model based on high frequency data, and proved the asymptotic normality of the estimators without strong moment conditions; Wu et al. (2018) studied the quasi maximum exponential likelihood estimation (QMELE) of non-stationary GARCH (1, 1) model under high frequency data, and obtained the limiting properties under weak moment conditions; Fan et al. (2017) studied the VaR estimation based on periodic GARCH model with high frequency data; Deng et al. (2020) studied the parameter test of GARCH model, where the parameter estimators were obtained from intraday high frequency data, and the corrected likelihood ratio test and Wald test statistics were further investigated.

In the literature, few studies have been done about introducing the intraday high frequency data to the estimation of daily LGARCH model. However, as mentioned above that LGARCH model is more robust than other GARCH type models. Therefore, it makes sense to introduce the intraday high frequency data to estimate the daily LGARCH model, which is a main contribution of this paper. Another contribution is that the proposed estimation method is adopted to all the parameters of the model and it is not necessary to set $\omega$ in (2) to be 1 as before, see, for example, Visser (2011) and Wang et al. (2018). The rest of this paper is organized as follows. In Section 2, we introduce the volatility proxy model and estimators. In Section 3, we derive the asymptotic results of the model estimator. Simulations and empirical studies are respectively shown in Section 4 and Section 5. We conclude the article in Section 6.

2.   Model and estimation

2.1.   Volatility proxy model

Let $\theta = (\omega, \alpha, \beta){'}$ be the parameter vector for model (1)–(2). In order to introduce intraday high-frequency data, it is necessary to extend the LGARCH model to the volatility proxy model. Denote $Y_{t}(u)$ to be the logarithmic return of an asset at time $u$ on day $t$, where the time of each trading day is standardized to the interval of [0, 1]. From Visser (2011), we firstly consider the following scale model:

here, $0\leq u\leq1$, $h_{t}$ is the volatility of day $t$. Standard process $Z_{t}(u)$ satisfies: when $t\neq s$, $Z_{t}(u)$ is independent of $Z_{s}(u)$ and has the same distribution as $Z_{s}(u)$. When $u = 1$, ($y_{t} \equiv Y_{t}(1)$, $\varepsilon_{t}\equiv Z_{t}(1)$, $EZ_{t}^{2}(1) = 1$), the model (3)–(4) degenerates into model (1)–(2). It is easy to see that the scale model introduces intraday data information $Y_{t}(u)$, and it retains the parameter structure of the daily LGARCH model. Unfortunately, model (3)–(4) can not be directly estimated due to the inconsistent frequency between $h_{t}$ and $Y_{t}(u)$. In order to estimate $\theta$, we further need to construct the intraday high-frequency data into a daily volatility proxy.

The volatility proxy is a daily sequence based on the intraday data. That is, for intraday yield process $Y_{t}(u)$, let $H_{t} \equiv H(Y_{t}(u))$ be a volatility proxy for $Y_{t}(u)$, where $H(\cdot)$ is a given function. Common volatility proxies include realized volatility and intraday price range. Positive homogeneity is an important property of volatility proxy. Namely, for a non-zero constant $\rho$ $(\rho > 0)$, the following equality holds:

Consider a given function $H(\cdot)$ satisfying the positive homogeneity and apply $H(\cdot)$ to equation 3. Then it is obtained that

Define

Because the standard process $Z_{t}(\cdot)$ is independently and identically distributed, hence $\varepsilon_{t}^{*}$ is an i.i.d. sequence with $E\varepsilon_{t}^{*2} = 1$. Then, combined with equations (3)–(7), we have the following volatility proxy model,

In the above model, a redundant parameter $\mu$ appears due to the setting of $\varepsilon_{t}^{*}$, which makes it impossible to directly apply the QMLE method to estimate $\theta$ and $\mu$ simultaneously. Next, we give an indirect approach to estimate the parameters. Let $h_{t}^{*} = h_{t} \sqrt{\mu}$, then (8)–(9) can be rewritten as follows:

where

When volatility proxy $H_{t} = |y_{t}|$, it is easy to have $\mu = 1, h_{t}^{*} = h_{t}$. Therefore, model (1)–(2) is a special case of model (10)–(11). According to (10)–(11), we can use the QMLE method to estimate the $\theta^{*} = (\omega^{*}, \alpha^{*}, \beta^{*})^{'}$. Once an estimator for $\mu$ is given, then we can get the estimation of the parameter $\theta = (\omega, \alpha, \beta)^{'}$ by using (12). The detailed estimation procedures are given in the next section.

2.2.   Parameter estimation

Define $\theta^{*} = (\omega^{*}, \alpha^{*}, \beta^{*})^{'}\in\Theta$, where $\Theta\subseteq \mathcal{R}^{3}$ is a parameter space for model (10)–(11). In addition, suppose that $\theta_{0}^{*} = (\omega_{0}^{*}, \alpha_{0}^{*}, \beta_{0}^{*})^{'}$ is the true value of the parameter $\theta^{*}$, which is an interior point of the parameter space $\Theta$.

Following the convention in the literature, see Visser (2011), we consider the quasi conditional log-likelihood function (apart from a constant term).

Then, the QMLE of parameter $\theta^{*}$ can be defined as follows:

According to (12), to further estimate $\theta$, we need to know $\mu$. After $\hat{\theta}^*$ is estimated, the fitting sequence $\{ \hat{h}_t^{*} \}$ is obtained from (11). It is already known that the absolute value of return $|y_t|$ can also be regarded as a special volatility proxy ($H_t = |y_t|$, $\varepsilon_t^* = |\varepsilon_t|$). When $H_t = |y_t|$, the estimated parameter obtained by the likelihood function (13) is actually the estimator of $\theta$ in the LGARCH (1, 1) model (1)–(2), which only uses the information of daily data $\{y_t\}$ and no intraday high frequency data is introduced. When $H_t = |y_t|$, the estimated value of $\theta$ is denoted as $\tilde{\theta} = (\tilde{\omega}, \tilde{\alpha}, \tilde{\beta})'$ and the corresponding fitting series for $h_t$ is denoted as $\{\tilde{h}_t\}$. According to $h_{t}^* = h_t \sqrt{\mu}$ or $\mu = h_t^{*2} / h_t^2$, we can get an estimator of $\mu$ as followed:

Finally, the parameter estimators of LGARCH model with high-frequency information, denoted by $\hat{\theta} = (\hat{\omega}, \hat{\alpha}, \hat{\beta})'$, are given by

Different from the usual estimator $\tilde{\theta} = (\tilde{\omega}, \tilde{\alpha}, \tilde{\beta})'$ corresponding to $H_t = |y_t|$, the estimator $\hat{\theta} = (\hat{\omega}, \hat{\alpha}, \hat{\beta})'$ generally contains intraday high frequency data information and hence is expected to have a better performance.

3.   Asymptotic theory

Before stating the asymptotic results for $\hat{\theta}^{*}$, we firstly make the following assumptions.

A1 The parameter space $\Theta$ is compact, and $\theta_{0}^{*}$ is an interior point of $\Theta$.

A2 The i.i.d. random sequence $\{\varepsilon_{t}^{*}\}$ satisfies $E(\varepsilon_{t}^{*4}) < \infty$, and there exists a positive and continuous probability density function almost everywhere.

A3 The series $\{H_{t}, h_{t}^{*}\}$ generated from model (10)–(11) are strictly stationary and geometrically ergodic for the considered parameter space $\Theta$.

Based on assumptions (A1)–(A3), adopting similar arguments to Visser (2011), it is not difficult to prove the asymptotic distribution of $\hat{\theta}^{*}$,

where $\Sigma^{*} = \Omega_{I}^{-1} \Omega_{S} \Omega_{I}^{-1}$,

To obtain the asymptotic variance of $\hat{\theta}^{*}$, we give the partial derivatives of the likelihood function $l_{t}(\theta^{*})$ with respect to $\theta^{*}$ as follows:

Further,

In terms of (7), $E\varepsilon_{t}^{*2} = 1$, $E\left(\varepsilon_{t}^{*2}-1\right) ^{2} = Var(\varepsilon_{t}^{*2})$. Therefore,

where,

According to (11), it can be obtained that:

Further,

Let $s_{\omega^*}^2$, $s_{\alpha^*}^2$, $s_{\beta^*}^2$ be the asymptotic variances of $\omega^*$, $\alpha^*$, $\beta^*$ respectively. Based on the asymptotic property of $\hat{\theta}^*$, we can get the following conclusion provided the parameter $\mu$ is given:

In practice, it is important to choose a proper volatility proxy $H_{t}$ for parameter estimation. According to (24), it can be seen that the impact of $H_{t}$ to the asymptotic variance is based on $E$ $(\varepsilon_{t}^{*4})$ which equals ${E(z_{H, t}^{4})}/{[E(z_{H, t}^{2})]^{2}}$ from (7). Consequently, it is expected to choose a volatility proxy $H_{t}$ with small value for $E(\varepsilon_{t}^{*4})$. From (6)–(7), similar to Liang et al. (2021), we can obtain

Here, $c = E(h_{t}^{4}) / [E(h_{t}^{2})]^{2}$ is a positive constant and hence ${E(z_{H, t}^{4})}/{[E(z_{H, t}^{2})]^{2}}$ is proportional to ${EH_{t}^{4}}/{(EH_{t}^{2})^{2}}$. Let

Then we have smaller $MH$ $\longleftrightarrow$ smaller $E(z_{H, t}^{4})/[E(z_{H, t}^{2})]^{2}$ $\longleftrightarrow$ smaller $Var(\varepsilon_{t}^{*2})$.

From the above, among several candidates, one can choose the volatility proxy $H_{t}$ according to its $MH$ value. The optimal $H_{t}$ should have the smallest $MH$ value.

4.   Simulations

In this section, we carry out Monte Carlo experiments to assess the finite-sample performance of the proposed estimators. In order to simulate the process of (3)–(4), we refer to Visser's (2011) example to simulate the intraday standard stochastic process $Z_t(u)$, which is produced by the following stochastic difference equations:

here, $B_t^{(1)}$ and $B_t^{(2)}$ are two uncorrelated Brownian motions, $Z_t (0) = 0$, and $\Gamma_t(0)$ is randomly generated from the stationary distribution $N$ $(\mu_\Gamma, \sigma_{\Gamma}^2)$. The time interval [0, 1] within the day is equally divided into 240 cells to correspond to the frequency of 1 min in a real trading day. The settings for parameters in (33)–(34) are $\delta = 1/2, \; \sigma_\Gamma = 1/4$ and $\mu_\Gamma = -1/16$. To further generate the stochastic process $Y_t(u)$, the parameters in equation (4) are set in the following two cases: $\theta_0 = (0.1, 0.4, 0.2)'$ and $\theta_0 = (0.1, 0.3, 0.5)'$.

We consider realized volatility ($RV$) as the volatility proxies $H_{t}$ in (8) under different frequencies, namely, 1-minute (RV1), 5-minute (RV5), 10-minute (RV10), 15-minute (RV15) and 30-minute (RV30). For RV1, the formula is given by:

where, the value of $Y_t(u_{0})$ is replaced by $Y_t(0) = 0$. Other volatility proxies can be computed similarly. For comparison, we also consider the case $H_t = |y_t|$ where the estimator is reduced to the usual estimator which only uses the daily data. The sample sizes are $T = 500, 1000$ and $1500$ and the replication time is set to be 1000. For each $H_t$, its MH value in (32) is estimated as:

The mean of 1000 estimated MH values is taken as the final estimated value for MH and is used to judge whether the volatility proxy is optimal.

Tables 1 and 2 summarize the empirical biases (Bias), empirical standard deviations (SD), asymptotic standard deviations (AD) and the value of MH ($\widehat{\rm MH}$) for $\hat{\theta} = (\hat{\omega}, \hat{\alpha}, \hat{\beta})'$. It can be seen that the Bias for each case is generally small. SD and AD of all parameters are very close and decrease when the sample size becomes large, which is consistent with the asymptotic results. Compared to the case $H_t = |y_t|$, the SD and AD of the parameters estimated by $RV$ are significantly smaller, which implies that introducing the intraday high frequency data can effectively improve the precision of the estimator.

By comparing the MH values, it can be found that in the simulation examples considered, the optimal order of different volatility proxies is: $RV1_t > RV5_t > RV10_t > RV15_t > RV30_t > |y_t|$. Namely the estimator under $RV1_t$ shows the best performance. The simulation results show that the volatility proxies model with intraday high frequency data have a good effect on parameter estimation, which is helpful to improve the estimation accuracy of LGARCH model.

5.   Empirical studies

In this section, model (3)–(4) is applied to study a real data set. We analyze SSE 50 Index (Shanghai Stock Exchange 50 Index) return series over the period April 28, 2008–July 24, 2013. The data set consists of the closing prices of the SSE 50 Index at one minute intervals, with a total of 1273 trading days and 240 observations per day. Denote the price sequence as $\{P_t (u), t\in[0, 1273], u\in[0, 1]\}$. Then the intraday return at $u$ time on day $t$ can be calculated as

For demonstration, we plot $Y_t(1)$ in Figure 1.

To estimate the model, we choose realized volatility as the volatility proxy based on different intraday sampling frequency: 1-minute, 5-minute, 10-minute, 15-minute and 30-minute. Accordingly, the realized volatility of different frequencies are recorded as $RV1$, $RV5$, $RV10$, $RV15$ and $RV30$. Similar to the simulation part, for comparison, we also consider the volatility proxy $H_{t} = |y_{t}|$ which corresponds to the estimator without using high frequency data. Figure 2 shows the time series plots of different volatility proxies.

The LGARCH (1, 1) model estimation results among different $H_t$ are given in Table 3. The AD in the table means the asymptotic variance of the estimated parameter which is calculated according to the formula (17) and (30).

It can be observed from Table 3 that the $\widehat{\rm MH}$ values of realized volatility are generally much smaller than that of $|y_t|$, where $RV1_t$ is the smallest and the corresponding $\widehat{Var}(\varepsilon_t^{*2})$ is also the smallest, which is consistent with the previous theoretical results. Compared with the estimation results of $RV1_t$ and $|y_t|$, the asymptotic variance of $\hat{\omega}$, $\hat{\alpha}$ and $\hat{\beta}$ corresponding to $RV1_t$ are obviously smaller.

From Table 3, when $H_t = |y_t|$, the estimated LGARCH (1, 1) model is

When $H_t = RV1_t$, the fitting model is

To further compare the estimation effect between $|y_t|$ and $RV1_t$, the 95% confidence intervals of parameter estimators are calculated based on the AD in Table 3. Let $\hat{\theta}_L = (\hat{\omega}_L, \hat{\alpha}_L, \hat{\beta}_L)'$ and $\hat{\theta}_U = (\hat{\omega}_U, \hat{\alpha}_U, \hat{\beta}_U)'$ be the lower bound and the upper bound respectively. Then we can calculate the upper and lower bounds of $h_{t}$ as follows:

For comparison, we plot the computed $h_t$, $h_{Lt}$ and $h_{Ut}$ based on equations (38) and (39) in Figure 3. It can be seen from Figure 3 that the estimated $h_t$ from two models are basically close. However, the interval $[h_{Lt}, h_{Ut}]$ (circle) of model (39) is significantly narrower than that of model (38) (triangle).

It is of some interest to compare the performance between $RV{{1}_{t}}$, $RV{{5}_{t}}$, $RV{{10}_{t}}$, $RV{{15}_{t}}$ and $RV{{30}_{t}}$. Due to limitations of space, we only show the comparisons between $RV{{1}_{t}}$ and $RV{{30}_{t}}$. We draw the 95% confidence intervals of ${{h}_{t}}$ calculated by the $RV{{1}_{t}}$ and $RV{{30}_{t}}$ in Figure 4. We can find that the confidence interval under $RV{{1}_{t}}$ is narrower than that under $RV{{30}_{t}}$, which means that the model estimation effect is better with $RV{{1}_{t}}$.

To sum, from the estimation results of Table 3 and the plots in Figure 3, it is shown that introducing the intraday high frequency data can help to improve the LGARCH model estimation for the considered data. Hence the proposed approach is of certain practical value.

6.   Conclusions

With the motivation that to obtain more precise estimators for the well-known daily LGARCH model, this paper studies how to introduce intraday high frequency data into the model estimation. Based on the existing volatility proxy idea, the estimation methods of all parameters of the model and the corresponding asymptotic properties are given.

The selection criteria of the optimal volatility proxy is also proposed. The simulation results show that the parameter estimators perform well under finite samples. The empirical study based on SSE 50 index shows that using the high frequency data can significantly improve the estimation accuracy of daily frequency LGARCH model, compared to the usual method which only uses the daily data.

Acknowledgments

The work is partially supported by National Natural Science Foundation of China 11731015, 11701116, Natural Science Foundation of Guangdong Province 192 2018A030310068, Innovative Team Project of Ordinary Universities in Guangdong Province 2020WCXTD018, Science and Technology Program of GuangZhou 202102020368 and Guangzhou University Research Funding YG2020029, YH202108.

Conflict of interest

The authors declare no conflict of interest.