Reader question: Standard GARCH learns volatility mainly from daily returns. If high-frequency realized volatility is available as an additional noisy measurement, how can both data streams be modeled together rather than using one only as an external regressor?

Realized GARCH introduces a measurement equation that links a realized volatility measure to the same latent conditional variance that drives daily returns. The model therefore becomes a joint system: one equation for returns, one recursion for latent variance, and one equation describing how the realized measure observes that variance with noise and leverage effects.

This article owns one precise computational job: joint modelling of daily returns and high-frequency realized volatility through a latent-variance recursion plus measurement equation. It does not own plain GARCH/EWMA, HAR-RV, GARCH-MIDAS, realized-kernel construction, or generic forecast comparison.

This is public mathematical and computational education. It is not a volatility-trading strategy, personalized financial advice, or a claim that a fitted latent variance is directly observable truth.

1. The three-equation architecture

A common log-linear Realized GARCH(1,1) specification is:

r_t = √h_t z_t,

log h_t = ω + β log h_t−1 + γ log x_t−1,

log x_t = ξ + φ log h_t + τ(z_t) + u_t.

Here:

r_t is the daily return;
h_t is latent conditional variance;
x_t is a realized volatility/variance measure built from intraday data;
z_t is the standardized return shock;
u_t is measurement noise.

2. Why the measurement equation is the defining feature

In ordinary GARCH, h_t is inferred indirectly from returns.

Realized GARCH adds a second noisy view:

x_t ≈ measurement of h_t.

The measurement equation explicitly states how closely the realized measure tracks latent variance and how return shocks affect the realized measure contemporaneously.

Hansen, Huang and Shek emphasize that this equation “completes” the model by connecting high-frequency-based realized measures to the conditional variance driving returns.

3. The leverage function

A standard quadratic leverage term is:

τ(z) = τ₁z + τ₂(z²−1).

The linear term captures asymmetry between negative and positive standardized return shocks. The quadratic centered term captures additional nonlinear dependence while preserving zero mean under standardized Gaussian z.

If τ₁ is negative, a negative return shock can increase the realized measure relative to h_t more than an equally sized positive shock.

4. Why log form is attractive

Using logs has several advantages:

h_t remains positive automatically after exponentiation;
x_t is also positive;
realized variance is strongly right-skewed in levels but often closer to Gaussian after logging;
multiplicative relationships become linear.

A level model is possible, but positivity constraints become more delicate.

5. Effective persistence

Substitute the measurement equation into the variance recursion schematically:

log h_t ≈ constant + (β+γφ)log h_t−1 + shock terms.

Thus in the common log-linear (1,1) setup, an important effective persistence quantity is:

P = β + γφ.

Under the simplest stationarity interpretation, |P| should be below one.

This is a strong diagnostic because β alone is not the persistence parameter once the realized measure feeds back into h.

6. Why realized data can improve volatility inference

Daily returns contain only one noisy observation per day.

A realized measure aggregates many intraday returns and can contain much more information about the day’s variation.

Realized GARCH lets that information enter the variance recursion directly while still acknowledging that the realized measure is noisy and may exclude overnight variation or contain microstructure bias.

7. Measurement bias is allowed

The equation:

log x_t = ξ + φ log h_t + …

does not require x_t to be an unbiased one-for-one measure of h_t.

If intraday RV misses overnight variance, or if microstructure noise distorts scale, ξ and φ can absorb systematic differences.

This is one reason the model is more realistic than simply setting x_t=h_t.

8. Joint likelihood

Because both r_t and x_t are modelled conditionally on h_t, parameters can be estimated jointly.

Under Gaussian assumptions:

z_t ~ N(0,1)

and:

u_t ~ N(0,σ_u²).

The log likelihood is the sum of:

the return-density contribution;
the realized-measure conditional-density contribution.

Heavy-tailed return innovations can be substituted when justified.

9. Parameter identification

The realized measure can improve identification because it gives direct information about the latent variance process.

But parameters still interact:

β and γφ both affect persistence;
ξ and φ affect measurement scale;
τ coefficients and return-distribution skew/tails can overlap in explaining asymmetry;
σ_u can absorb poor realized-measure quality.

Falsifier: inspect the Hessian/Jacobian, profile likelihood and parameter stability across multiple optimizer starts.

10. Initialization

The latent h₁ must be initialized.

Common choices include:

unconditional log variance implied by parameters;
sample variance;
first realized measure adjusted by the measurement relation;
backcast.

In long samples, initialization may matter little. In short or highly persistent samples it can materially affect early likelihood contributions.

11. One-step forecasting

At the end of day t, x_t is observed.

The next latent variance forecast follows directly:

log h_t+1 = ω + β log h_t + γ log x_t.

This is an important contrast with ordinary GARCH: the most recent high-frequency realized measure is a direct state update.

12. Multi-step forecasting

Future x values are not yet observed, so multi-step forecasts require the model-implied measurement relation.

The expected dynamics depend on the effective persistence β+γφ and the expected leverage/measurement shocks.

Long-horizon forecasts therefore converge toward the model’s long-run log-variance level when the persistence condition holds.

13. Inputs and outputs

Inputs can include:

daily returns;
realized variance/kernel/range measure;
intraday sampling/cleaning rule;
return innovation distribution;
measurement-error distribution;
initial variance;
parameter bounds;
training window;
optimization settings.

Outputs can include:

latent variance h_t;
return-volatility forecasts;
realized-measure forecasts;
ω,β,γ,ξ,φ,τ parameters;
measurement-noise variance;
effective persistence;
standardized return residuals;
measurement residuals;
out-of-sample loss metrics;
rolling stability diagnostics.

14. Evidence polarity

Evidence for confidence includes:

realized-measure residuals are centered and reasonably uncorrelated;
standardized return residuals show reduced remaining ARCH behavior;
effective persistence is stable and below the stationarity boundary;
out-of-sample forecasts improve on plain GARCH;
results survive alternative high-quality realized measures;
measurement noise behaves plausibly;
parameters are stable across nearby samples;
likelihood optimization converges repeatedly to the same solution.

Evidence against confidence includes:

measurement residual autocorrelation;
φ unstable or economically implausible;
β+γφ near/above one;
σ_u so large that the realized measure adds little information;
forecast gains disappearing out of sample;
strong sensitivity to the realized-measure construction;
optimizer dependence or weakly identified parameters.

15. Counterexample: noisy RV harms the model

If x_t is built from ultra-high-frequency returns contaminated by bid–ask bounce, it may be a poor variance measurement.

Feeding it directly into h_t+1 can inject noise into forecasts.

Falsifier: compare with a noise-robust input such as a realized kernel. If forecast performance improves materially, the original measurement layer was weak.

16. Counterexample: realized measure excludes overnight risk

Intraday RV may measure only exchange-open hours, while r_t is close-to-close.

The measurement equation must reconcile those different horizons.

Falsifier: add overnight variation or change the realized-measure definition. A large shift in ξ/φ reveals the horizon mismatch.

17. Counterexample: effective persistence is hidden

An analyst reports β=0.7 and concludes that volatility mean reverts quickly.

But if γφ=0.29, effective persistence is about 0.99.

Falsifier: always report β+γφ, not β alone, for the common log-linear specification.

18. Counterexample: measurement equation absorbs structural break

A change in market microstructure alters x_t without changing underlying economic volatility.

The model may respond by shifting ξ,φ or σ_u, giving the appearance of volatility-dynamics change.

Falsifier: test parameter stability around data-feed, tick-size or exchange-rule changes.

19. Counterexample: Gaussian innovations understate return tails

Daily returns can have heavier tails than Gaussian.

If z_t is misspecified, extreme returns can distort both h and the measurement leverage terms.

Falsifier: inspect PIT/QQ/tail diagnostics and compare Student-t or other appropriate innovations.

20. Counterexample: strong fit, weak forecast

Adding x_t nearly always improves in-sample descriptive fit when it carries information.

But forecast usefulness must be tested out of sample.

Falsifier: compare QLIKE/MSE against a simpler GARCH and HAR-RV benchmark on an untouched period.

21. Realized GARCH versus GARCH/EWMA

GARCH/EWMA algorithms own return-shock-driven conditional variance recursions.

Realized GARCH adds a realized-measure observation equation and lets that high-frequency measurement feed the latent variance state.

22. Realized GARCH versus HAR-RV

HAR-RV algorithms forecast realized volatility directly from daily/weekly/monthly realized-volatility averages.

Realized GARCH instead maintains a latent conditional variance that jointly explains returns and the realized measure.

23. Realized GARCH versus realized kernels

Realized-kernel algorithms own the ex-post measurement problem.

A realized kernel can be an input x_t to Realized GARCH, which then owns the dynamic latent-state and forecasting problem.

24. Realized GARCH versus GARCH–MIDAS

GARCH–MIDAS algorithms mix fast return dynamics with slow lower-frequency macro/financial drivers.

Realized GARCH uses same-day high-frequency volatility measurements in a measurement equation rather than a MIDAS lag polynomial.

25. Alternatives

HEAVY models: another framework that uses realized measures to drive conditional variance/covariance.

HAR-RV: direct realized-volatility forecasting.

Stochastic-volatility models with realized measures: latent state-space alternatives.

GARCH-X: treats realized measures as exogenous regressors without the same joint measurement architecture.

Realized stochastic-volatility Bayesian models: richer latent dynamics at greater computational cost.

27. What would falsify confidence?

Confidence should be withdrawn if the measurement equation leaves strong structure in residuals; if effective persistence is unstable/nonstationary; if alternative reasonable realized measures reverse the result; if the high-frequency input adds no out-of-sample forecast value; or if parameter estimates are weakly identified and optimizer-dependent.

28. Verification and update triggers

Preserve the return series, realized-measure construction, overnight treatment, innovation distribution, starting variance, parameter bounds, optimizer settings, residual diagnostics and out-of-sample evaluation design.

Revalidate when:

the realized measure changes;
market microstructure changes;
trading hours/overnight treatment changes;
volatility regime breaks;
measurement residual variance rises;
forecast horizon changes;
return-distribution specification changes.

29. Primary and high-quality references

Peter R. Hansen, Zhuo Huang and Howard H. Shek, Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility, Journal of Applied Econometrics, 2012.
Working-paper version: Realized GARCH, SSRN.
An accessible manuscript copy is hosted by Duke Economics: Hansen–Huang–Shek manuscript.
For measurement-bias discussion in realized-GARCH applications: Quantile Forecasts of Financial Returns Using Realized GARCH Models.

Educational boundary: Realized GARCH uses high-frequency realized measures to improve inference about latent daily variance. It does not turn a noisy realized measure into ground truth, and forecast value must be demonstrated out of sample.

Small Group Tutorials

How Realized-GARCH Algorithms Jointly Model Returns and High-Frequency Volatility: Latent Variance, Measurement Equations, Leverage, Persistence, Measurement Error and Forecast Failure