1.9.4 Realized Volatility & HAR

Basic Prerequisites: Econometrics & Time Series (ARMA, long memory, regression) and Statistics & Inference (OLS).

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1. Intuition & Practical Objective

GARCH infers one latent ${\sigma }_t$ per day from a single daily return. But with intraday data you can measure daily variance directly: sum the squared intraday returns. That quantity, the realized volatility, is a far less noisy estimate of the day’s latent variance - it uses $n$ returns instead of one, and it turns volatility modeling into something closer to estimation of an observable than inference about a hidden state.

The payoff compounds: because $R{V}_t$ is (nearly) observable, volatility forecasting becomes a plain regression problem - no constrained likelihood, no flat surface. The HAR-RV model (Corsi 2009) is the punchline: a simple three-timescale linear cascade - daily, weekly, monthly averages of past $RV$ - that reproduces the long-memory (slowly, hyperbolically decaying) autocorrelation of volatility without fractional integration, price to pay zero extra estimation pain, and beats GARCH on forecast accuracy in many studies.

The practical objective: give the practitioner the fastest reasonable volatility forecast - a 3-regressor OLS - and the understanding of why averaging over three horizons generates long memory.

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2. Mathematical Ground Truth & Derivations

Realized variance (Tsay §3.15.1). With $n$ intraday returns $r_{t,i}$ in day $t$,

$$R{V}_t=\sum _{i=1}^n{r}_{t,i}^2\underset{n\to \infty }{\to }{\int }_t^{t+1}{\sigma }^2(s)ds(\text{quadratic variation}),$$

and $\mathbb{E}[R{V}_t\mid {\mathcal{F}}_{t-1}]={\sigma }_t^2$ when intraday returns are conditionally zero-mean - i.e. $R{V}_t$ is an (almost) unbiased estimator of daily variance. Its log, $\ln R{V}_t$, is close to Gaussian and behaves like an ARIMA(0,1,$q$) with a weakly negative MA term: near-unit-root persistence at the daily level, the signature of long memory in volatility ($\hat{d}\approx 0.4$ in fractionally-integrated terms; Bollerslev–Jubinski; Ray–Tsay).

Microstructure bias and the sampling choice. In the limit $n\to \infty$ the noise in intraday returns (bid–ask bounce, discreteness) dominates and $R{V}_t$ diverges with sampling frequency. The classic fix is to sample at a moderate frequency; the empirical optimum (Tsay §3.15.1) is 4–15 minutes. (Note: this 4–15 min rule is a Ch3 result; Ch5 of Tsay does not re-derive it - attribute it here.) Alternatives: the two-scale / multi-scale estimators and bipower variation $B{V}_t\propto \sum |r_{t,i}||r_{t,i-1}|$, which is robust to jumps.

Range estimators (Tsay §3.15.2). When only OHLC data exist, ranges recover efficiency: Parkinson ${\hat{\sigma }}^2=(H-L{)}^2/(4\ln 2)\approx 0.3607(H-L{)}^2$ (≈5× more efficient than close-to-close), Garman–Klass, Rogers–Satchell, and the Yang–Zhang combination

$${\hat{\sigma }}_{yz}^2={\hat{\sigma }}_o^2+k{\hat{\sigma }}_c^2+(1-k){\hat{\sigma }}_{rs}^2,k=\frac{0.34}{1.34+(n+1)/(n-1)}.$$

HAR-RV (Corsi 2009). Regress tomorrow’s realized variance on averages of past $RV$ over three horizons:

$$R{V}_{t+1}=c+{\beta }_{d}R{V}_t+{\beta }_w{\overline{RV}}_t^{(5)}+{\beta }_m{\overline{RV}}_t^{(22)}+{ϵ}_{t+1},$$

where ${\overline{RV}}_t^{(h)}=\frac{1}{h}{\sum }_{j=1}^{h}R{V}_{t-j+1}$ (daily / weekly / monthly averages, the last two being the interesting ones; Corsi includes $R{V}_t$ itself as the daily term). The model is linear in three positive, slowly-different horizons, and the sum ${\beta }_d+{\beta }_w+{\beta }_m$ is close to 1. Fitting in logs (log-HAR) is common because $\ln RV$ is far closer to Gaussian and stabilises the residuals.

Why three timescales create long memory. A sum of AR(1) components with geometrically-spaced decay rates has an autocorrelation that decays hyperbolically over a wide range of lags - an excellent approximation to true long memory. Three horizons (≈1, 5, 22 days) span roughly two decades of decay, so the HAR’s implied ACF mimics a fractional process while remaining a finite, estimable OLS. That is the “simple approximate long-memory model” of the title.

Extensions. HARQ (adds a realized-quarticity term for time-varying smoothness), HAR-J (adds the jump component $RV-BV$), and HAR with a leverage term (negative yesterday’s return raises today’s $RV$) - see 06.

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3. Computational Implementation - realized variance and a fitted HAR

Standard library only. (a) Confirms $RV$ is an unbiased daily-variance estimator by simulation; (b) builds a long-memory $\log RV$ series (fast + slow AR components) and fits log-HAR by OLS via the normal equations, solved with Gaussian elimination.

Read it: the mean $RV$ ($1.4396\text{e-}04$) nails the true daily variance ($1.4400\text{e-}04$) - unbiasedness confirmed numerically. The log-HAR fit reaches $R^2=0.68$, and all three horizon slopes are positive with a sum near 1 ($0.9420$): the daily term captures the immediate shock, the monthly term carries the slow long-memory weight, and the near-unit sum is exactly what makes multi-day forecasts decay only slowly. (On real S&P 500 data Corsi reports $R^2\approx 0.4$–$0.5$ for level-HAR - our synthetic series is smoother than reality, so treat the exact split as illustrative, not a calibration.)

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4. Failure Modes & First-Principles Breakdowns

1. The sampling-frequency trap. Too fine (1-min) $\Rightarrow$ microstructure noise inflates $RV$ without bound; too coarse (daily) $\Rightarrow$ you are back to one observation. The 4–15-minute band is a bias-variance compromise, not a law - it depends on the asset’s tick size and liquidity.
2. Jumps masquerade as volatility. A single overnight gap or macro release inflates $RV$ for that day; if you then model $RV$ as purely persistent you over-forecast for weeks. Separating continuous ($BV$) from jump ($RV-BV$) variation is the fix (HAR-J).
3. Overnight returns are dropped. $RV$ built from intraday bars misses the close-to-open move, which is a large share of total variance for equities. Include the overnight squared return where material (small for indices/FX).
4. Long memory is approximate, not exact. HAR is a finite AR approximation to a fractional process; at very long horizons (weeks–months) it understates persistence. For term-structure work, a true ARFIMA or GARCH with the appropriate persistence may be preferable.
5. Regime and volatility-of-volatility. $RV$ is itself heteroskedastic (vol-of-vol). In calm regimes $RV$ is precise; in crises it is noisy and extreme - exactly when you rely on it. HARQ’s extra term exists to model that time-varying noise.
6. Data quality is everything. Non-synchronised trades, stale quotes, and corporate-action errors contaminate $RV$ more than they contaminate daily returns - the reason $RV$ pipelines need their own hygiene (see Algorithmic & HFT / microstructure).

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5. References

• Corsi, Fulvio (2009): A Simple Approximate Long-Memory Model of Realized Volatility, J. Financial Econometrics 7(2), 174–196
• Andersen, Bollerslev, Diebold & Labys (2003): Modeling and Forecasting Realized Volatility, Econometrica 71(2)
• Barndorff-Nielsen & Shephard (2002): Econometric Analysis of Realized Volatility and its Use in Estimating Stochastic Volatility Models, JRSS-B 64(2)
• Parkinson, Michael (1980) / Garman & Klass (1980) / Yang & Zhang (2000)
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)

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6. Connected Graph Bridges

• Base: Econometrics & Time Series (long memory, ARFIMA) · Statistics & Inference (OLS)
• Prior: 03 · Asymmetric Models · Hub: Index
• Continue: 05 · Failure Modes · 06 · Advanced Extensions
• Applied: Algorithmic & HFT (microstructure noise) · Risk Parity (RV-based vol targeting)