The leverage and the volatility feedback effect have so far been analyzed to the greatest extent based on discrete time models, using in particular GARCH models. In the continuous time context, Corsi and Renò (2012) recently introduced the leveraged HAR model, stating that when modeling realized volatility, it is necessary to mimic the properties and dynamics of the assumed quadratic variation. The LHAR model for realized variance (which is one out of three proposed specifications) reads as

$$R{V}_{t}=c+{\beta}_{1}R{V}_{t-1}+{\beta}_{2}R{V}_{t-1}^{w}+{\beta}_{3}R{V}_{t-1}^{m}+{\beta}_{3}{r}_{t-1}^{2}{d}_{t-1}+{\epsilon}_{t},$$(1)

where *r*_{t} is the close-to-close return and *d*_{t} is a dummy variable which is one if *r*_{t−1} < 0 and zero otherwise. The weekly (monthly) variance $R{V}_{t}^{w}$ ($R{V}_{t}^{m}$) is calculated as the average of daily variances *RV*_{t} from *t* − 4 (*t* − 21) to *t*. *ε*_{t} is an i.i.d. white noise process. While Corsi and Renò (2012) design the model to deliver optimal forecasts of *RV*_{t+h}, our focus is on learning about the dependence structure of the volatility across the entire distribution of the realized variance and we propose to use the LHAR model as a starting point.

The quantile regression framework introduced by Koenker and Bassett (1978) provides the tools to model the conditional distribution of a random variable. In the present context, the extension to quantile autoregressive (QAR) models of Koenker and Xiao (2006) allows us to characterize the dependence structure of realized volatility across its distribution. We follow Andersen et al. (2003) and treat the realized variance as an observed time series of variance observations which constitute the object of further interest. QAR in general extends the possibilities to describe the features of a time series beyond a standard autoregressive model. The general QAR(*p*) model for a time series variable *y*_{t} is

$${Q}_{{y}_{t}}(\tau \mid {y}_{t-1},\dots ,{y}_{t-p})={\theta}_{0}(\tau )+{\theta}_{1}(\tau ){y}_{t-1}+\dots +{\theta}_{p}(\tau ){y}_{t-p},$$

where *τ* denotes the quantile of interest with *τ* ∈ (0, 1) and *θ*_{i}(*τ*) are quantile specific parameters. The model is particularly useful if the properties of the time series process depend on the level of the time series. As studies by Baur, Dimpfl, and Jung (2012) and Zhu et al. (2016), or Zhu et al. (2015) showed, the dynamic properties of time series are indeed quantile dependent and this also holds for volatility processes (cp. Badshah, 2013).

Combining the LHAR model with the quantile autoregressive estimation approach leads to the central model of our analysis, the leveraged quantile heterogeneous autoregressive (LQHAR) model:

$\begin{array}{rl}{Q}_{R{V}_{t}}(\tau )& ={\theta}_{0}(\tau )+{\theta}_{1}(\tau ){r}_{t-1}^{2}+{\theta}_{2}(\tau ){r}_{t-1}^{2}{d}_{t-1}\\ & \phantom{\rule{1em}{0ex}}+{\theta}_{3}(\tau )R{V}_{t-1}+{\theta}_{4}(\tau )R{V}_{t-1}^{w}+{\theta}_{5}(\tau )R{V}_{t-1}^{m}.\end{array}$(2)

In contrast to the LHAR model in Equation (1), we add the additional regressor ${r}_{t-1}^{2}$. The reason is that it provides additional information related to the entire trading day which is by construction not captured in *RV*_{t−1}. For example, if *r*_{t} is (close to) zero, *RV*_{t} might be high or low, depending on the trading pattern throughout the day. If prices changed heavily and returned to their initial value at the closing of the market, *RV*_{t} will be high.

From Equation 2 it is now possible to deduce the quantile dependent structure of volatility asymmetry and volatility persistence. The asymmetric effect of past returns on RV is directly observed through the estimation of *θ*_{1} and *θ*_{2}. Persistence is slightly more complicated. Of course, the parameter *θ*_{3} is of great interest as it provides a first impression of the immediate persistence. In the long run, however, it is necessary to consider the full dynamics as captured by *θ*_{3}, *θ*_{4}, and *θ*_{5}. In the standard HAR specification of Corsi (2009), the weekly and monthly aggregates replace a longer autoregressive dynamic. In the end, the specification in Equation 1 is a short form of an AR(22):

$${y}_{t}={\alpha}_{1}{y}_{t-1}+{\alpha}_{2}{y}_{t-2}+\dots +{\alpha}_{22}{y}_{t-22}+{\epsilon}_{t}.$$

Now for this general AR(p) process to be stationary, a necessary (albeit not sufficient) condition is that the sum of the autoregressive parameters be lesser than 1 in absolute value, i.e. $\mid \sum _{i=1}^{22}{\alpha}_{i}\mid <1$. In the cascading structure of the HAR, these 22 coefficients are summarized in 3 coefficients which are related to the AR(p) as follows: ${\alpha}_{1}={\beta}_{1}+\frac{1}{5}{\beta}_{2}+\frac{1}{22}{\beta}_{3}$ etc. Basically the HAR imposes the restriction that *α*_{2} to *α*_{5} and *α*_{6} to *α*_{22} are identical. Hence, in the HAR context, the stationarity condition can easily be stated as | *β*_{1} + *β*_{2} + *β*_{3} | < 1.

In the QAR model, Koenker and Xiao (2006) show that a process that is globally stationary might exhibit explosive behavior in the tails of the distribution. For the process to be stationary, the expected value of the QAR parameters needs to be lesser than one in absolute value. In our LQHAR model, the stationarity condition is, thus, that the expected value of the sum (*θ*_{3}(*τ*) + *θ*_{4}(*τ*) + *θ*_{5}(*τ*)) is lesser than one. We can thus sum the coefficients (*θ*_{3}(*τ*) + *θ*_{4}(*τ*) + *θ*_{5}(*τ*)) for each *τ* to obtain a measure for the overall persistence in the LQHAR model. The stationarity condition is satisfied if, in addition, the absolute value of the sum over *τ* is lesser 1.

It should be noted that the model presented in Equation (2) has a structure which is similar to the threshold GARCH model of Glosten, Jagannathan, and Runkle (1993). The heterogeneous terms account for the long persistence which is typically found in realized variance time series, but also a typical feature of the GARCH model where it is captured by the GARCH term. The dependence on lagged squared returns is similar to the ARCH term in the GARCH model (depending on the exact specification of the mean model of course). In order to highlight the gains in information we consider two benchmark models to compare the QAR model to. First, the linear LHAR model of Corsi and Renò (2012) which is parametrized as our QAR model given in Equation (2) and, hence, reads as follows:

$\begin{array}{rl}R{V}_{t}& ={\beta}_{0}+{\beta}_{1}{r}_{t-1}^{2}+{\beta}_{2}{r}_{t-1}^{2}{d}_{t-1}+{\beta}_{3}R{V}_{t-1}\\ & \phantom{\rule{1em}{0ex}}+{\beta}_{3}R{V}_{t-1}+{\beta}_{4}R{V}_{t-1}^{w}+{\beta}_{5}R{V}_{t-1}^{m}+{\epsilon}_{t},\end{array}$(3)

where *ε*_{t} is an i.i.d. white noise process. The second benchmark model is the threshold GARCH model of Glosten, Jagannathan, and Runkle (1993) which is specified as follows:

$${r}_{t}=\mu +\beta {r}_{t-1}+{u}_{t},\phantom{\rule{1em}{0ex}}{u}_{t}\sim N(0,{h}_{t})$$(4)

$${h}_{t}=\omega +\alpha {u}_{t-1}^{2}+\gamma {h}_{t-1}+\delta I({u}_{t-1}<0){u}_{t-1}^{2},$$

where *r*_{t} is the close-to-close return and *I*(⋅) is the indicator function that takes on a value of 1 if *ε*_{t−1} < 0 and 0 otherwise. The main parameter of interest is then *δ* which captures the asymmetric response of the variance *h*_{t} to lagged innovations. We use an AR(1) process for the mean equation in order to account for market imperfections (cp. Campbell & Shiller, 1988).

A drawback of our specification is that we include the correlated variables ${r}_{t}^{2}$ and *RV*_{t} at the same time in our models. The average correlation observed in our data is 0.51. We therefore also estimate a reduced form of Equation (2) where we drop ${r}_{t-1}^{2}$ and replace ${r}_{t-1}^{2}{d}_{t-1}$ by *RV*_{t−1}
*d*_{t−1} (labeled “robustness regression model”). This allows us to test and ensure that the high correlation does not bias our results. On the other hand, we lose the interpretation of the daily shock which is the driving force in the GARCH model, and, thus, direct comparability of these models.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.