# Realized Variance (realized volatility)

I'm confused about realized variance. I roughly know the theory around Ito Calculus and quadratic variation and integrated volatility so I understand what realized variance measures (even though as pointed out you don't need Ito Calculus to understand RV). I've read a bunch of papers on the topic of modelling volatility using GARCH(p,q) and understand that conditional heteroskedasticity tends to disappear with higher frequency.

What I am confused about is how realized volatility is used to do out of sample prediction for the volatility. Amongst the many papers, I believe one mentioned using it as a tool for evaluating forecasts from GARCH(1,1). Another mentioned that you can calculate realized volatility series and then use standard ARMA models to do forecasting.

I am writing my thesis and my goal is to model volatility. I have 1-minute bar data. I was going for GARCH but my professor said just go straight to realized vol since I have high frequency data. However, I just don't know if realized vol can be used for modeling. How does one predict $$\sigma_{t+1}$$ using realized vol?

EDIT: To clarify, my question is how can one use realized volatility as a volatility model to do out-of-sample prediction? GARCH(p,q) gives us a model for $$\sigma_{t+1}$$ given data up to t. How is RV a model and how can I do out of sample prediction using RV?

Quotes:

#1

It has become apparent that standard volatility models used for forecasting at the daily level cannot readily accommodate the information in intraday data, and models specified directly for the intraday data generally fail to capture the longer interdaily volatility movements sufficiently well. As a result, standard practice is still to produce forecasts of daily volatility from daily return observations, even when higher-frequency data are available.

#2

However, Diebold (1988) shows that conditional heteroskedasticity tends to disappear with the increasing of the sampling interval.

• Not sure what the exact question is. If it is what realized variance is used for, it's essentially the same answer as what any historical data is used for. with variance there is a direct derivative (variance swap) that has it in the payoff formula. Also, I don't think there is any use or benefit in knowing Ito calculus for understanding variance. A second use is that it's the second moment and useful in describing distributions. May 19, 2021 at 5:59
• In GARCH(1,1) you can calc $\sigma_{t+1}$ based on data up to now. So you can fit the model and predict out-of-sample volatility. How do you do this with realized variance? I just don't get how realized variance is a model and how it can be used to predict out-of-sample volatility. Unless I calculate RV and then fit an ARIMA model and use this to do the prediction.
– s5s
May 19, 2021 at 7:22
• Now, academia has mostly focused on incorporating the additional information procured from high-frequency data by extending existing models and conclusively used them to get better 1-day ahead out-of-sample volatility forecasts. Are your goal to get 1-day ahead out of sample forecasts, or do you want intraday forecasts (eg. 1 minute)? Because the latter is not that popular (within academia) and therefore you're limited to a few models that can do this.
– Pleb
May 19, 2021 at 9:35
• @Pleb Thanks for the reply. In situations like this I always remember a mentor of mine from work who kept reminding me that we are not after a Nobel price here. I'd rather go with the more popular 1-day ahead out of sample forecast using intraday data. It will then allow me to compare this to the GARCH(1,1) 1-day ahead forecast using daily data. My goal here is to deliver a well written thesis on the subject and learn a few things along the way.
– s5s
May 19, 2021 at 14:34

• How can one use realized volatility as a volatility model to do out-of-sample prediction? You extended known models to incorporate additional information procured from high-frequency data. Going from the vanilla GARCH to a realized GARCH model can be done by adding an auxiliary model as an external regressor that captures intraday variation. The realized GARCH model can forecast $$h$$-periods ahead, with $$h$$ being days.

• How is RV a model and how can I do out of sample prediction using RV? RV is not a model but a non-parametric measure of intraday variation. The estimates procured from realized measures have no model-based relationship between days (eg. they lack an auxiliary model, like an ARMA representation, linking them between days), so they cannot be used for $$h$$-period ahead forecasts. That is why you incorporate it into a model: to create a relationship between days that tries to satisfy empirical features of returns and intraday returns (roughly speaking).

## An introduction: From GARCH to realized GARCH

Many of my arguments below, can be found in the paper of Hansen et al. (2012) together with Peter Reinhard Hansens slides on the same topic. Let us briefly consider the vanilla GARCH model. We define the de-meaned returns on the form:

$$$$r_t = \sigma_t z_t, \qquad z_t \overset{iid}{\sim}N(0,1)$$$$

with the GARCH equation being:

$$$$\sigma^2_t = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2.$$$$ Now, we can ask ourselves the question:

• Why can't we use an ordinary GARCH model with intraday volatility? We can, and it works great! The original GARCH model uses noisy squared returns to extract information about the current level of conditional variance. However, noisy squared returns are not suitable under periods of drastic persistent changes in the variance, implying that the updating of the GARCH model is simply too slow to "catch up" to the new level of volatility. This lead to the GARCH-X models, which extend the original GARCH model by adding a realized measure as an external regressor:

$$$$\sigma^2_t = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2 + \gamma x_{t-1},$$$$

where $$x_t$$ is a realized measure of volatility, eg. Realized variance (RV), Realized kernel, etc. In the slides (p. 37) they do a small experiment and observe how many days it takes for the vanilla GARCH model to update to a new level of volatility, as opposed to the GARCH-X model:

Observed above, it is clear that the GARCH-X model is much faster at updating to the new shift in volatility.

NOTE: We utilize both high- and low-frequency information by adding an external regressor instead of replacing noisy squared returns, $$r_{t-1}^2$$, with $$RV_{t-1}$$ and adding no external regressor. The additional information in the former approach gives better estimates (we also have one more degree of freedom).

While the GARCH-X specification improved the time it took for the model to update to the new level of volatility, it did not permit multi-period forecasts due to lacking an auxiliary model on the realized measure. This is where Hansen et al. (2012) comes in.

### Completing the GARCH-X model: The Realized GARCH model

In Hansen et al. (2012) they extend the GARCH-X model to incorporate intraday return variation, by adding a measurement equation as an external regressor to relate the estimated realized measure (eg. $$RV$$) to the latent volatility process, and moreover include a function that captures asymmetric reaction to shocks (ala. EGARCH type structure), making for a very flexible and rich representation. They provide two specifications called the linear Realized GARCH model and the loglinear version (LRG). I will focus on the linear specification. From the GARCH-X model they extend the $$x_t$$ via the measurement equation given by:

$$$$x_t = \varepsilon + \varphi \sigma_t^2 + \tau(z_t) + u_t,$$$$

where $$u_t \overset{iid}{\sim}D(0,\sigma^2_u)$$ for $$D(\cdot)$$ being a generic distribution (usually Gaussian) and furthermore $$z_t$$ and $$u_t$$ are independent. In their paper they write some important features of the measurement equation,

The inclusion of the realized measure in the model and the fact that $$x_t$$ has an autoregressive moving average (ARMA) representation motivate the name Realized GARCH. A simple yet potent specification of the leverage function is $$\tau(z) = \tau_1 z + \tau_2 (z^2-1)$$, which can generate an asymmetric response in volatility to return shocks.

As you pointed out above, you can use an ARMA representation to explain the realized variance, however, here they combine an ARMA representation, with a GARCH equation. Moreover, as seen above, they define a simple and yet potent specification of the leverage function based on the Hermite polynomials (truncated at the second level). The function is convenient since it ensures $$\mathbb{E}\left[\tau(z)\right]=0$$ whenever $$z$$ is specified with mean zero and unit variance. Based on pure intuition, we see that ceteris paribus, when $$\tau_1 < 0$$, $$x_t$$ will be larger for a negative shock (measured by large negative residuals $$z_t < 0$$), since $$\tau(z)>0$$. For $$\gamma>0$$ this will in turn lead to a pronounced effect in $$\sigma^2_t$$. Their paper contains many additional discussions on the realized GARCH model, however, I have briefly summarized their conclusions below.

Conclusions of their paper:

• The realized GARCH model is parsimoneus, have tractable analysis (can be estimated via QMLE) and allows for multiperiod forecasting.

• From their empirical analysis, they find evidence that the LRG(1,2) model provides the most consistent results and outperforms all other benchmark models. Furthermore they favour the LRG model since it provides less heteroskedasticity in the measurement equation, making QMLE more efficient as opposed to the linear specification (see section 5.5 in the paper).

• They argue that the realized GARCH model is flexible since it can incorporate different realized measures.

## Honorable mentions: Realized EGARCH, HEAVY and HAR

I will briefly discuss some other realized models you can research yourself, without going into details:

• The same authors construct a realized EGARCH model, and is an extended specification of the above realized GARCH model. It can be found in this paper, Hansen et al. (2014).

• Sheppard & Shephard (2009) constructs the HEAVY models, which use realised measures as the basis for multi-period-ahead forecasting of volatility, by having two (or more) latent volatility processes (one for $$\sigma^2_t$$ and another for $$x_t$$) and linking them differently as opposed to the realized GARCH specification.

• Corsi (2009) creates a Heterogeneous Autoregressive (HAR) model, that constructs a reduced-form linear relationship between 3 volatility components being daily, weekly and monthly. The model is extremely parsimoneus and have good forecasting abilities. This is also the reason for its increased popularity. Furthermore, it has been extended to account for different aspects inherent in high-frequency data. Mathematically, the model is based on the premise, that the underlying data generating process follows a pure diffusion, implying that the model will estimate quadratic covariation in presence of jumps. Many authors have further extended the univariate model following the same simplified setup, such as the HARQ and HARQF model of Bollerslev, Patton, & Quaedvlieg, (2016) and the SHAR model of Patton & Sheppard, (2015). I have provided a brief explanation of the SHAR model here, which also gives you an introduction to the structure of a HAR-type model.

### FYI: Linking realized measures to realized models

Below, I have provided a clear and concise meaning of the different terms and how they relate to eachother:

• A realized measure (eg. realized variance): estimates volatility each day and have no model-based relation between two separate days (eg. an auxiliary model like an ARMA representation). A very good paper giving you an overview of different realized measures can be found here.

• HAR with realized measures: Provides a linkage between different days and you can do multiperiod forecasts. Still, it only uses realized volatilities (realized measures).

• Realized GARCH: uses both daily returns and realized volatilities (found from realized measures) to model volatility, $$\sigma_t^2$$, and provide a linkage between different days.

Many of the realized measures and models are implemented in R, either via the rugarch package or the highfrequency package. I hope this provides some insight. I wish you good luck with your thesis.

• Could I double check this is correct: "The estimates are independent between days (The estimate you get today, has no relation to the estimate you got yesterday)". Returns are uncorrelated but not independent. Squared returns are positively autocorrelated. Would this not make RV estimates on different days dependent (because the $r^2_t$ values are dependent). Also, this is a very good explanation - I read quite a few papers (some way over my head) and just couldn't figure out how RV fits into the picture. Thank you so much!
– s5s
May 19, 2021 at 19:34
• Yes, you are correct. "Independence" was used in a vague sense. My point was, that realized measures are just estimating/measuring the volatility at each day by using the highest frequency possible (without suffering from related issues, such as noise and jumps). Therefore, they do not contain any relation between days, since you're not linking them using any model/relationship (eg. like an ARMA representation). I've made an edit. I hope it helps.
– Pleb
May 20, 2021 at 0:04