The HAR-RV Forecasting Model

Realized volatility — computed from high-frequency intraday returns rather than estimated from a model — has a stubborn statistical quirk: it's highly persistent, but not in the clean, simple way a basic AR(1) model captures. It behaves as though shocks to volatility decay slowly across many different timescales at once, a property called "long memory." Full long-memory models (like ARFIMA) can capture this properly but are mathematically heavy and finicky to estimate. The HAR-RV model, introduced by Corsi, sidesteps this: it's just an ordinary linear regression, but with a clever trick — it includes realized volatility averaged over three different horizons as separate predictors, and this simple combination approximates long-memory behavior astonishingly well in practice.

An analogy: predicting tomorrow's mood from three kinds of memory

Imagine predicting someone's mood tomorrow using three pieces of information: how they felt today, how they've felt on average this past week, and how they've felt on average this past month. Today's mood alone misses slower-building trends (a rough month wearing someone down); the monthly average alone misses today's specific spike (a bad meeting this afternoon). Combining all three — a fast signal, a medium signal, and a slow signal — captures both the quick reaction and the slow-building undertow without needing a complicated model of exactly how memory decays. HAR-RV does exactly this for volatility: today's realized vol, this week's average, and this month's average, combined in one regression.

The mechanics, one symbol at a time

Let $RV_t$ be realized volatility (or variance) measured on day $t$. Define the weekly average $RV_t^{(w)} = \frac{1}{5}\sum_{i=0}^{4} RV_{t-i}$ and the monthly average $RV_t^{(m)} = \frac{1}{22}\sum_{i=0}^{21} RV_{t-i}$ (5 and 22 being typical trading-day counts for a week and month). The HAR-RV model forecasts tomorrow's realized volatility as:

$$RV_{t+1} = \beta_0 + \beta_d\, RV_t + \beta_w\, RV_t^{(w)} + \beta_m\, RV_t^{(m)} + \epsilon_{t+1}.$$

In plain English: tomorrow's volatility is a weighted combination of today's volatility, the past week's average volatility, and the past month's average volatility, with the three coefficients $\beta_d, \beta_w, \beta_m$ estimated by ordinary linear regression — no options pricing, no complicated fractional-differencing math, just three predictors capturing three different memory horizons. Empirically, all three tend to come out positive and statistically significant, meaning volatility genuinely depends on information from all three timescales simultaneously, not just the most recent day.

Worked example 1: a one-step forecast by hand

Suppose fitted coefficients are $\beta_0 = 0.02$, $\beta_d = 0.35$, $\beta_w = 0.40$, $\beta_m = 0.20$ (in daily realized-vol percentage units), and today's readings are $RV_t = 1.8\%$, $RV_t^{(w)} = 1.4\%$, $RV_t^{(m)} = 1.1\%$ — meaning volatility has spiked today relative to its recent averages. The forecast:

$$RV_{t+1} = 0.02 + 0.35(1.8) + 0.40(1.4) + 0.20(1.1) = 0.02 + 0.63 + 0.56 + 0.22 = 1.43\%.$$

Notice the forecast (1.43%) sits between today's elevated reading (1.8%) and the calmer monthly average (1.1%) — the model correctly expects some reversion from today's spike, but not all the way back to the monthly baseline, because the weekly and daily terms still carry real predictive weight.

Worked example 2: a calm day forecast

Now suppose today is unusually calm relative to recent history: $RV_t = 0.8\%$, $RV_t^{(w)} = 1.3\%$, $RV_t^{(m)} = 1.5\%$ (a lull after a more volatile month). Using the same coefficients:

$$RV_{t+1} = 0.02 + 0.35(0.8) + 0.40(1.3) + 0.20(1.5) = 0.02 + 0.28 + 0.52 + 0.30 = 1.12\%.$$

Here the forecast (1.12%) sits above today's calm reading (0.8%), pulled up by the still-elevated weekly and monthly averages — the model expects some reversion back up toward the recent higher-volatility regime rather than assuming today's calm persists. This asymmetric pull toward multiple horizons at once is exactly the long-memory behavior HAR-RV is built to approximate.

What this means in practice

HAR-RV is one of the most widely used volatility forecasting models in practice precisely because it's simple to estimate (ordinary least squares), robust, and consistently competitive with far more complex models in forecast accuracy comparisons (see Forecast Evaluation and Accuracy Metrics). It's a standard baseline any new volatility model needs to beat, and it's often used directly in production for options market-making and volatility-targeting strategies.