Bollerslev-Patton-Quaedvlieg (2016) Exploiting the Errors: A Simple Approach for Improved Volatility Forecasting

Summary

This paper introduces the HARQ family of realized-volatility forecasting models, which improve on the standard HAR model by letting the autoregressive parameters vary explicitly with the amount of measurement error in the realized-volatility (RV) regressor. Realized volatility equals the true latent integrated volatility (IV) plus a serially uncorrelated measurement error, so forecasting IV from observed RV is a classical errors-in-variables problem that attenuates the persistence parameter. Because high-frequency asymptotic theory says the size of that error is governed by integrated quarticity — consistently estimable by realized quarticity (RQ) — the models can adapt: when today's RV is precisely measured (low RQ), the model weights it more, producing stronger persistence and more responsive forecasts; when RV is noisy, it shrinks the weight. Applied to the S&P 500 and Dow constituents, HARQ delivers substantial out-of-sample forecast gains over HAR and other standard models.

Key Claims

• RV is IV plus measurement error → attenuation. $RV_t = IV_t + \eta_t$ with $\eta_t$ approximately serially uncorrelated; regressing future RV on current RV therefore suffers errors-in-variables attenuation bias, making the observed process look less persistent than the latent IV, by an amount that grows with the error variance.
• Realized quarticity measures the error. The asymptotic variance of the RV measurement error is proportional to integrated quarticity, consistently estimated by $RQ_t=\tfrac{M}{3}\sum_{i=1}^{M} r_{t,i}^4$ (with $M$ intraday returns). $RQ_t$ is a per-period, observable proxy for how noisy today's RV is.
• The HARQ model. Let the coefficient on the daily lag vary with the estimated error: $\beta_{1,t}=\beta_1+\beta_{1Q}\,RQ_{t}^{1/2}$ (with $RQ^{1/2}$ demeaned so $\beta_1$ is the average coefficient), giving $RV_{t}= \beta_0+(\beta_1+\beta_{1Q}RQ_{t-1}^{1/2})RV_{t-1}+\beta_2 RV_{t-1}^{w}+\beta_3 RV_{t-1}^{m}+u_t$. Typically $\beta_{1Q}<0$: high measurement error pulls the daily coefficient down.
• Trivial to estimate. The time-varying-coefficient specification is linear in parameters, so it is estimated by ordinary OLS — no latent-state filtering — preserving the HAR model's ease of use.
• Broad, robust gains. Across the S&P 500 index and individual DJIA stocks (1997–2013), HARQ improves out-of-sample accuracy versus HAR and a challenging benchmark set; the gains are not confined to high-noise days — adapting the persistence to measurement precision helps generally.
• A family, not one model. The same "let parameters depend on RQ" idea extends to the AR case (ARQ) and to richer specifications (varying the weekly/monthly coefficients, jump/semivariance variants).

Concepts Introduced or Extended

• Heterogeneous Autoregressive (HAR) Model
• Realized Volatility

Entities Mentioned

• Tim Bollerslev
• Andrew J. Patton
• Rogier Quaedvlieg

Quotes

"By allowing the parameters of the models to vary explicitly with the (estimated) degree of measurement error, the models exhibit stronger persistence, and in turn generate more responsive forecasts, when the measurement error is relatively low."

My Take

The move here is elegant because it turns a nuisance — the estimation error in realized volatility — into a usable signal. Everyone knew RV was a noisy proxy for IV; the insight is that high-frequency theory tells you how noisy on each specific day (via realized quarticity), so you can down-weight the days you should distrust rather than applying one fixed, attenuated coefficient. That it stays a plain OLS regression is what made HARQ catch on as a default upgrade to Corsi's HAR. It complements the wiki's realized-volatility and Andersen-Bollerslev material by taking the measurement-error problem those pages raise and building the correction directly into the forecasting equation. The main caveats are practical: realized quarticity is itself noisily estimated (a fourth moment), so the correction can be jumpy, and the linear $RQ^{1/2}$ form is a convenient approximation rather than the exact optimal weight.