Scaling Realized Variance and Adapting HAR to Intraday Horizons

# Realized Variance and Intraday HAR


# Realized Variance and Intraday HAR












I understand this is quite a simple question, but I'd like to make it rigorous for myself because I've seen it in a few varied contexts and haven't completely gotten it somehow. I'm starting an academic project, so I need to make it concrete. I haven't found any papers which really describe this basic detail, any links would be appreciated.

How do I get the units for realized variance? I.e. after calculating realized variance, how do I interpret that figure? Suppose I plug it into a model (like HAR), how would I transform the RV result to give realized variance predictions for time horizons other than 1 day? (I.e. like something shorter, intraday).

Take this example for realized variance over the last two hours, with returns sampled every 5 minutes ($M=24$), (Suppose abusively that $t-24$ corresponds to $t$ - 2 hours: $$RV_t = \sum_{i=1}^M r_{[t-i, t-i+1)}^2$$

What are the units here? Is this a two-hourly realized variance? How would I convert that to annualized vol? (3.25 * 252?)

Also, I've always seen the HAR model predicting one day ahead realized var. Is it accepted to just put in a different dependent variable, like $RV$ over the 30 minutes or two hours?: $$RV_{t+1} = \beta_0 + \beta_1 RV_t + \beta_2^T \mathbf{x} + \varepsilon_t$$

Again, I understand this is basic, but for some reason I'm getting confused. Any resources or direction to fill in the gaps would be great.

## Answer by R.M. Leushuis (score 1, accepted)

https://quant.stackexchange.com/a/85595

The confusion around units and scaling is common and comes from the fact that realized variance is defined relative to the sampling interval, not to any fixed calendar period.

If you sum 24 squared 5-minute returns over a 2-hour window, you get a 2-hour realized variance denominated in 5-minute return units. To annualize, you need to scale by the number of such windows in a trading year. If your trading day has 6.5 hours (US equities), that is 3.25 two-hour windows per day, so annualized variance is RV × 3.25 × 252, and annualized volatility is the square root of that.

On the HAR question: there is nothing stopping you from using a different dependent variable. The original Corsi (2009) HAR was specified for daily RV, but the logic of aggregating over daily, weekly, and monthly horizons does not depend on the base frequency being one day. You can construct a HAR where the left-hand side is 30-minute or 2-hour RV and the right-hand side uses corresponding multiples of that base frequency. Whether this works well empirically depends on the asset and the noise properties at your chosen frequency. At very high frequencies, microstructure noise becomes a concern and you may need noise-robust estimators like realized kernels.

For a thorough overview of how different RV estimators and forecasting models (including HAR and its extensions, as well as machine learning and deep learning approaches) relate to each other, see Leushuis and Petkov (2026), "Advances in forecasting realized volatility: a review of methodologies," Financial Innovation 12:14 (https://doi.org/10.1186/s40854-025-00809-5). It covers 32 models and discusses the empirical properties each one is designed to capture.

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/85430

Perhaps adding on to @AKdemy's answers, and to answer to your questions specifically,

How do I get the units for realized variance? I.e. after calculating realized variance, how do I interpret that figure?

- It is more intuitive to understand in units of volatility, which is the same as the units of price. For example, if you calculated realized/ex-post volatility using 5 minute returns across the past day, in normal distribution terms that would be the 1-standard deviation volatility in price possible for 5 minutes across a day.

Suppose I plug it into a model (like HAR), how would I transform the RV result to give realized variance predictions for time horizons other than 1 day? (I.e. like something shorter, intraday).

- If you have 5 minute returns, you are essentially calculating variance/volatility for 5 minute intervals. You can then scale it (IID) to longer intervals i.e. 30 minute, days, weeks etc. I don't believe it makes sense to scale shorter i.e. 5 minute to 1 minute, you would need shorter interval returns to estimate shorter interval volatility.

[Some example] What are the units here? Is this a two-hourly realized variance? How would I convert that to annualized vol? (3.25 * 252?)

- This is a two-hour (frequency) realized variance correct. As @AKdemy said, you must multiply it by 12*260 (a day has 12 intervals of 2 hours and 260 business days per year in the FX context).

Also, I've always seen the HAR model predicting one day ahead realized var. Is it accepted to just put in a different dependent variable, like RV over the 30 minutes or two hours?

- I think it comes down to what works and your philosophy towards problem solving. Some people think so long as the independent variable is significant and the adj R^2 is high, it is good. Others think that it is important for the variable to make economic sense, so they disagree.