0.1 HAR-RV Model
True long-memory volatility models (ARFIMA-type, with slowly decaying autocorrelation of squared/absolute returns) are accurate but awkward to estimate and forecast with. Corsi’s (2009) Heterogeneous Autoregressive model of Realized Volatility (HAR-RV) approximates long memory with a simple, linear, OLS-estimable model — the standard practitioner benchmark for volatility forecasting from realized volatility data.
Let \(RV_t\) be the daily realized variance. Define the (overlapping) weekly and monthly averages \[ RV_t^{(w)} = \frac{1}{5}\sum_{i=1}^5 RV_{t-i}, \qquad RV_t^{(m)} = \frac{1}{22}\sum_{i=1}^{22} RV_{t-i}. \] The HAR-RV model regresses (typically on \(\sqrt{RV}\) or \(\log RV\) for better distributional behavior) next-day realized volatility on the daily, weekly, and monthly components: \[ RV_{t+1} = c + \beta_d\, RV_t + \beta_w\, RV_t^{(w)} + \beta_m\, RV_t^{(m)} + \varepsilon_{t+1}. \]
Why it works. The model is motivated by the Heterogeneous Market Hypothesis: different types of traders (high-frequency, daily, institutional/long-horizon) operate and react to volatility over different horizons. Cascading these three horizons in a simple linear-in-parameters regression generates an ACF for \(RV_t\) that decays slowly, mimicking true long-memory behavior without needing a fractionally-integrated model.
Extensions. - HAR-RV-J / CHAR: adds a separate jump component (from bipower variation, see Realized Volatility) since jumps and continuous volatility have different persistence and forecasting implications. - HAR-RV with leverage: adds signed/asymmetric past returns to capture the leverage effect (as in EGARCH). - HARQ: lets the coefficient on \(RV_t\) vary with the estimated precision of \(RV_t\) (measurement-error-robust HAR).
HAR-RV sits alongside GARCH and stochastic volatility as one of the three main practical approaches to volatility forecasting, distinguished by using realized (high-frequency-based) rather than daily-return-based volatility proxies.