Heterogeneous Autoregressive (HAR) Model

Definition

The Heterogeneous Autoregressive (HAR) model — specifically HAR-RV, for realized volatility — is a simple linear regression of a volatility measure on its own averages taken over several time horizons (typically daily, weekly, and monthly). Introduced by Corsi (2009), it is a short-memory model that nonetheless reproduces the slowly decaying (apparently long-memory) autocorrelation of realized volatility, and it is the workhorse specification for volatility forecasting from high-frequency data.

Key Ideas

• Multi-horizon components. From daily realized volatility $RV^{(d)}_t$, build weekly and monthly averages $RV^{(w)}_t=\frac{1}{5}\sum_{i=0}^{4}RV^{(d)}_{t-i}$ and $RV^{(m)}_t=\frac{1}{22}\sum_{i=0}^{21}RV^{(d)}_{t-i}$.
• The regression. $RV^{(d)}_{t+1} = c + \beta_d RV^{(d)}_t + \beta_w RV^{(w)}_t + \beta_m RV^{(m)}_t + \varepsilon_{t+1}$ — a constrained AR(22) fit by OLS (often on $\log RV$ or $\sqrt{RV}$ to stabilize variance).
• Volatility cascade / Heterogeneous Market Hypothesis. The specification is motivated by markets populated by agents with different horizons; longer-horizon volatility feeds shorter-horizon volatility, generating an additive cascade whose partial sums mimic long memory.
• Apparent long memory without fractional integration. A handful of components at geometrically spaced horizons approximates the hyperbolic autocorrelation decay of realized volatility, sidestepping ARFIMA estimation.

How It Works

Estimate the three coefficients by least squares on overlapping daily/weekly/monthly RV averages; forecast recursively for multi-step horizons. Common extensions:

• HAR-RV-CJ (jumps). Split realized variance into its continuous part (integrated variance, via bipower variation) and the jump part, and give each its own HAR dynamics — the continuous component is far more persistent than the jump component (Andersen–Bollerslev–Diebold "Roughing It Up").
• Leverage / asymmetry. Add signed-return or semivariance terms to capture the leverage effect.
• HARQ (Bollerslev-Patton-Quaedvlieg 2016). Realized volatility is the true integrated volatility plus measurement error, so forecasting from it is an errors-in-variables problem that attenuates the persistence coefficient. HARQ lets the daily-lag coefficient vary with the estimated error, proxied by realized quarticity $RQ_t=\tfrac{M}{3}\sum_i r_{t,i}^4$: $\beta_{1,t}=\beta_1+\beta_{1Q}RQ_{t-1}^{1/2}$ (with $\beta_{1Q}<0$), so precisely-measured days get more weight (stronger persistence, more responsive forecasts) and noisy days get shrunk. It stays a plain OLS regression and yields broad out-of-sample gains over HAR on S&P 500 and Dow stocks.

Why It Matters

• It is the default benchmark for realized-volatility forecasting: simple, robust, and hard to beat out of sample.
• It reframed the "long memory in volatility" debate — a parsimonious short-memory cascade suffices to reproduce the stylized fact, so fractional integration is not required.
• It plugs directly into risk management (VaR/ES), option pricing, and high-frequency trading applications that need one-step and multi-step volatility forecasts.

Open Questions

• Microstructure noise. The input RV must be computed from a noise-robust estimator (subsampling, realized kernels, pre-averaging) at the highest frequencies.
• Parameter instability. HAR coefficients drift across regimes; time-varying or regime-switching extensions are debated.
• Horizon/lag choice. The daily/weekly/monthly (1/5/22) convention is a rule of thumb; optimal component choice is not settled.

Related

• Realized Volatility
• Long Memory and Fractional Integration
• Bipower Variation
• Volatility Forecast Evaluation
• MIDAS Regression
• Quadratic Variation
• Corsi (2009)
• Fulvio Corsi
• Bollerslev-Patton-Quaedvlieg (2016) Exploiting the Errors
• Tim Bollerslev
• Andrew J. Patton
• Rogier Quaedvlieg