GARCH vs HAR-RV for Volatility Forecasting

Worked Solution

How to Think About It: This is really a question about information efficiency: how much signal about future volatility can you extract, and what data do you need to extract it? GARCH uses only yesterday's return and yesterday's variance estimate -- it is a recursive filter on daily squared returns. HAR-RV uses realized variance computed from hundreds or thousands of intraday observations, aggregated at daily, weekly, and monthly horizons. The information gap is enormous. If you have clean tick data for a liquid asset, HAR-RV will almost always produce better forecasts. But "clean tick data" is doing a lot of heavy lifting in that sentence -- microstructure noise, missing data, and illiquidity can destroy the advantage.

Key Insight: The fundamental trade-off is information richness vs. data requirements. GARCH works with a single daily return; HAR-RV needs reliable high-frequency data and careful construction of realized variance. When the high-frequency data is good, HAR-RV wins on forecasting accuracy. When it is noisy or unavailable, GARCH is the robust fallback.

The Method:

*GARCH(1,1) Model:*

$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$

where $\epsilon_t = r_t - \mu$ is the return innovation. The parameter $\alpha$ captures the shock impact (how much yesterday's surprise moves today's variance), $\beta$ captures persistence (how much of yesterday's variance carries forward), and $\omega$ sets the unconditional variance floor. The unconditional variance is $\bar{\sigma}^2 = \omega / (1 - \alpha - \beta)$. The sum $\alpha + \beta$ controls persistence -- when it is close to 1, volatility shocks decay slowly.

• Estimation: Maximum likelihood, assuming a return distribution (usually Gaussian or Student-$t$).
• Information set: Only daily closing returns.
• Volatility proxy: Squared daily return $\epsilon_t^2$, which is an unbiased but extremely noisy estimator of daily variance.

*HAR-RV Model:*

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

where $RV_t^{(d)} = \sum_{i=1}^{M} r_{t,i}^2$ is the daily realized variance from $M$ intraday returns, $RV_t^{(w)} = \frac{1}{5} \sum_{j=0}^{4} RV_{t-j}^{(d)}$ is the weekly average, and $RV_t^{(m)} = \frac{1}{22} \sum_{j=0}^{21} RV_{t-j}^{(d)}$ is the monthly average.

• The three components capture different agent horizons: $\beta_d$ reflects day-traders, $\beta_w$ reflects medium-term portfolio managers, $\beta_m$ reflects long-horizon institutional investors.
• Estimation: Simple OLS regression -- no iterative optimization needed.
• Information set: High-frequency intraday returns (typically 5-minute or 1-minute).
• Volatility proxy: Realized variance, which converges to integrated variance as the sampling frequency increases (subject to microstructure noise corrections).

Practical Considerations:

*When to prefer GARCH:* - Only daily data available (emerging markets, OTC instruments, illiquid names). - You need a parametric conditional density for VaR or option pricing (GARCH gives you $\sigma_t^2$ in a distributional framework; HAR-RV gives you a point forecast of realized variance). - Short history -- GARCH needs fewer observations because it is a parsimonious recursive model. - You want to simulate volatility paths (GARCH has a natural generative structure; HAR-RV is a reduced-form regression).

*When to prefer HAR-RV:* - High-frequency data is available and reasonably clean (liquid equities, major FX pairs, index futures). - You care about forecast accuracy -- empirically, HAR-RV typically beats GARCH by 10-30% in out-of-sample $R^2$ for liquid assets. - You want to capture long memory in volatility without heavy parameterization (GARCH needs FIGARCH or component models to do this; HAR-RV gets it for free from the multi-horizon structure). - You want a model that is trivial to estimate and extend (add jumps, leverage effects, or additional regressors by just adding terms to the regression).

*Failure modes:* - GARCH: The squared daily return is such a noisy proxy that the model can be slow to react to regime changes. It can also produce implausibly high persistence ($\alpha + \beta > 0.999$) when the sample includes structural breaks. Check by comparing the implied unconditional volatility to historical levels. - HAR-RV: Garbage in, garbage out. Microstructure noise at high frequencies biases realized variance upward. Stale prices in illiquid names make $RV$ unreliable. The standard fix is to use 5-minute returns (Bandi-Russell optimal sampling) or a noise-robust estimator like the realized kernel. Check by comparing $RV$ to a range-based estimator (Parkinson or Garman-Klass) -- if they diverge wildly, your tick data has problems.

Answer: GARCH(1,1) is the workhorse when you only have daily data or need a parametric volatility model for simulation and risk management. HAR-RV dominates when you have reliable high-frequency data, because realized variance is a far more efficient estimator of true volatility than squared daily returns, and the multi-horizon structure naturally captures the long memory that GARCH struggles with. On a liquid equity or FX desk with good tick data, HAR-RV is the default. On a credit desk or for illiquid instruments, GARCH is often all you have.