Is there a HAR that deals with the leverage effect?

Question

The EGARCH is a special GARCH model that treats the leverage effect of the volatility. The HARV does not make a distinction between negative and positive returns. Is there a special HARV that deals with the leverage effect? If not, is there a reason why the leverage effect is perhaps unimportant for HARV?

Answer 1

There exists a modification of the HAR model that accounts for leverage effect (á la GJR-GARCH) in a high-frequency setting.

The semi-variance HAR model, termed the SHAR model of Patton and Sheppard (2015), decomposes the first volatility component in the original HAR model into realized semi-variances, and hence, tries to deal with the leverage effect in a high-frequency setting, by observing how past negative returns might impact future volatility. For clarity, let us define the volatility components of the HAR model as:

\begin{equation} RV_{t-1}^{(day)} = RV_{t-1}^{(n)}, \qquad RV_{t-1}^{(week)} = \frac{1}{5}\sum_{k=1}^{5}RV_{t-k}^{(n)}, \qquad RV_{t-1}^{(month)} = \frac{1}{22}\sum_{k=1}^{22}RV_{t-k}^{(n)}, \end{equation} assuming 22 days in a month, and $n$ being the amount of intraday data. Now splitting up the first volatility component into a positive and negative part: \begin{equation} RV_{t-1}^{(day)} = RV_{t-1}^+ + RV_{t-1}^-, \qquad RV_{t}^+=\sum_{i=1}^n r_{t,i}^2 1_{\{r_{t,i}>0\}}, \qquad RV_{t}^-=\sum_{i=1}^n r_{t,i}^2 1_{\{r_{t,i}<0\}}. \end{equation} Remember that the above decomposition is only true in a univariate setting. Then, the SHAR model can be described as, \begin{equation} RV_t = \phi_0 + \phi_1^+ RV_{t-1}^+ + \phi_1^- RV_{t-1}^- +\phi_2 RV_{t-1}^{(week)} + \phi_3 RV_{t-1}^{(month)} + u_t, \end{equation} where we would expect that $\phi_1^+ = \phi_1^- = \phi_1$ if no additional information was added from the decomposition. If we expect negative returns to have a larger impact on future volatility, then $\phi_1^+ < \phi_1^-$ (which is what they find in the empirical analysis). They further try to include a signed jump-term in the above regression, $\phi_J \cdot \Delta J_t^2$ with $\Delta J_t^2 = RV_t^+ - RV_t^-$, to explore the role that signed jumps play in future variance. If $\phi_J<0$, then days dominated by negative jumps lead to higher future volatility, while days with positive jumps lead to lower future volatility ($\phi_J<0$ is also observed from the empirical analysis).

While I definitely encourage you to read the paper, I will highlight some key findings in the above paper, which might give you a more nuanced understanding:

• Main conclusion 1: For a set of 105 individual stocks, they find that negative semi-variance has a larger and more significant impact on future volatility than positive realized semi-variance, and disentangling the effects of these two components significantly improves forecasts of future volatility. This is empirical evidence of the leverage effect in a high-frequency setting.

• Main conclusion 2: They also find that signed jump variation is important for predicting future volatility, with volatility attributable to negative jumps leading to significantly higher future volatility, whereas positive jumps leads to lower volatility.

• The authors further construct a small study on whether the negative semi-variance provide any improved benefits in contrast to a realized version of a simple leverage effect variable, $\nu RV_t{1}_{\{r_t<0\}}$. They conclude that, the negative semi-variance captures the asymmetric impact of past negative returns on future volatility better, than the usual method of using an indicator function for the sign on the lagged daily returns. Replacing the realized leverage effect variable with the non-realized version (found in the GJR-GARCH model) make the results even more significant.

• It is argued that the decomposition for the weekly and monthly variances have a less pronounced effect, and therefore it is (often) excluded in order to adhere to the principle of parsimony.

I hope this provides some help.