Realized GARCH Explained: Jointly Modeling Returns and Realized Volatility

ARCH and GARCH models infer changing variance from a sequence of returns. Engle introduced ARCH and Bollerslev developed its generalized form (Engle, 1982; Bollerslev, 198690063-1)). A return is a noisy signal about the amount of movement during its period: one unusually small close-to-close return does not mean the session was quiet, and one large return does not reveal the path that produced it. Intraday prices can provide a separate realized measure, such as realized variance, with more information about movement within the session.

Realized GARCH uses that measure as part of the variance dynamics. In the basic specification, yesterday's conditional variance and realized measure both help update today's conditional variance. The model also includes an equation for the realized measure itself. This extra equation is central: it describes the measure's relationship with latent variance and the return shock, rather than treating the measure as an unexplained outside input.

The original paper reports improved empirical fit over daily-return-only GARCH in its sample of Dow Jones stocks and an exchange-traded index fund. That is a sample-specific result, not a guarantee that Realized GARCH will forecast every asset or period better. For the baseline return-only model, see the GARCH guide.

Let rₜ be the return for period t, after removing a conditional mean if the model includes one. Let hₜ be the conditional variance of rₜ given the information Fₜ₋₁ available at the start of that period, and let xₜ be a positive realized measure observed during or at the end of period t. For example, xₜ could be the sum of squared intraday returns, expressed in the same squared-return units as hₜ.

The timeline matters. A forecast for hₜ uses information available before period t; after observing rₜ and xₜ, the model updates the next variance forecast hₜ₊₁. Do not use xₜ to claim you had a forecast of hₜ before period t unfolded.

Realized GARCH is a dynamic model for returns and a measured volatility proxy. It is not the same as MIDAS regression, which uses a structured lag-weight function to relate data observed at different frequencies. Both can use intraday information, but their equations, forecast targets, and timing need to be identified before comparing them.

A common Realized GARCH(1,1) model has a return equation, a variance equation, and a measurement equation:

rₜ = √hₜ zₜ log hₜ = ω + β log hₜ₋₁ + γ log xₜ₋₁ log xₜ = ξ + φ log hₜ + τ(zₜ) + uₜ

Here zₜ is the standardized return innovation, uₜ is measurement noise, and τ(zₜ) allows the observed measure to depend on the return shock. The first equation scales a unit-variance shock by the conditional standard deviation. The second updates log variance using the previous log variance and previous log realized measure. The third links the observed log measure to current latent log variance, the return shock, and a measurement innovation.

The log form is convenient because positive hₜ and xₜ can be represented on an unrestricted log scale, while exponentiating the variance forecast gives a positive value. The intercept ω is therefore an intercept for log variance; it is not itself the long-run variance. In the basic model, parameters are estimated together rather than read as isolated measures of persistence.

Substituting the measurement equation at t−1 into the variance update shows how the realized measure feeds back into log variance:

log hₜ = (ω + γξ) + (β + γφ) log hₜ₋₁ + γτ(zₜ₋₁) + γuₜ₋₁

For this (1,1) specification, persistence in log variance is summarized by β + γφ, not by β alone. The measurement relation contributes to the feedback because a higher prior variance is associated with a higher realized measure, which then enters the next update. This algebra helps interpret the fitted coefficients; it is not a stationarity proof for every Realized GARCH extension, so inference must use the restrictions for the specification actually estimated.

Hansen, Huang, and Shek introduce linear and log-linear forms of this joint model and show the three-equation structure in their paper (2012). The exact lag orders, realized measure, conditional mean, and innovation assumptions should be taken from the specification being estimated.

The measurement equation helps explain why the realized measure can move with both latent variance and the return shock. A common leverage function is

τ(zₜ) = τ₁ zₜ + τ₂ (zₜ² − 1)

The linear and centered quadratic terms can represent an asymmetric relationship between return shocks and the realized measure. The coefficients determine the shape; the formula alone does not establish that negative returns cause future volatility.

In the paper's baseline formulation, zₜ and uₜ are modeled as separate innovations, with zₜ standardized and the two innovations mutually independent. This is a tractable statistical specification, not evidence that market returns or measurement errors follow a particular distribution in every sample. Gaussian quasi-likelihood may be used for estimation even when normality is not asserted as a fact.

This equation also separates Realized GARCH from GARCH-X. GARCH-X may put an observed measure into a variance update but leave its own dynamics unspecified. Realized GARCH models the realized measure too, which supports forecasts beyond one step when future realized measures are not yet observed. The model describes a joint statistical relationship; it does not turn a realized measure into an external cause.

Suppose a fitted log-linear model has ω = −0.7824, β = 0.75, and γ = 0.15. At the end of period t, assume the current conditional variance is hₜ = 0.0004 and the observed realized measure is xₜ = 0.0009. All values are hypothetical, positive, and expressed on the same daily decimal-return variance scale.

Use the variance equation to update the forecast for period (t+1):

log hₜ₊₁ = −0.7824 + 0.75 log(0.0004) + 0.15 log(0.0009) ≈ −7.7024 hₜ₊₁ = exp(−7.7024) ≈ 0.0004517

The implied conditional standard deviation is √0.0004517 ≈ 0.02125, or about 2.13% for the stated return period. The realized measure affects the next variance forecast through its fitted coefficient; this calculation does not update any coefficients or use the measurement equation to claim that xₜ equals hₜ.

The example demonstrates the arithmetic only. It has no parameter uncertainty, estimation error, or out-of-sample observations, so it says nothing about statistical significance or whether this model beats a benchmark <!-- learn:illustration -->

Estimation uses aligned return and realized-measure observations. A likelihood or quasi-likelihood combines the return equation with the measurement equation, so a change in the assumed innovation distributions or dependence structure can change the fitted parameters. Researchers should state the estimation method, innovation assumptions, conditional-mean treatment, realized measure, and data-cleaning rules rather than presenting the fitted recursion without its measurement model.

After each period closes, the model can use the new return and realized measure to update its state and produce the next variance forecast. For a multi-step forecast, future (x) values are unknown. A complete Realized GARCH specification supplies dynamics for the realized measure, so the forecast must account for its future distribution or expectation instead of inserting future realized values from the test sample.

An evaluation needs a target that matches the forecast: for example, next-period conditional return variance or a realized measure over the same interval. Keep the target, proxy, units, horizon, and evaluation dates aligned. An in-sample likelihood compares fit under a model; it does not replace a chronological forecast test.

Save each forecast at the time it would have been available, refit only with information available at that origin, and compare against a preselected baseline on a common sample. Choose the loss before reviewing the results and report the paired loss difference with an uncertainty estimate that accounts for dependence when needed. The QLIKE and squared-error guide explains why proxy properties and loss choice affect the comparison.

The model is only as useful as the measure and time interval it receives. A realized measure made from intraday prices may omit overnight movement even when rₜ is a close-to-close return. The original paper explicitly allows a realized-measure window shorter than the return interval; its measurement equation need not force xₜ to be an unbiased estimate of hₜ. That flexibility does not make mismatched windows harmless: the interpretation of φ, the forecast target, and any comparison still depend on the intervals.

The paper gives a concrete example: a measure built from about 6.5 trading hours can be paired with a conditional variance for a 24-hour close-to-close return. In that setup, the authors expect φ < 1 because the intraday measure captures only part of the full return interval. A fitted φ is therefore tied to the measure, session, and return definition; it is not a universal calibration constant that can be transferred unchanged to another asset or sampling design.

High-frequency data can also carry bid–ask bounce, asynchronous trading, price discreteness, stale quotes, and recording errors. The two-scale realized-volatility guide describes one estimator designed to address some high-frequency microstructure noise. Hansen and Lunde document how market microstructure noise can materially affect realized-variance measurement (2006). A different proxy can change the fit, forecasts, and leverage estimates.

Because the log-linear model uses log xₜ, the chosen measure must be strictly positive. If data can be zero or nonpositive, a transformation, filter, or floor changes the model and should be justified and disclosed. Check whether the realized measure includes jumps, which session it covers, how missing bars are handled, and whether its units match the return variance. More intraday observations do not automatically remove measurement error.

Ordinary GARCH uses lagged returns or squared innovations and lagged variance. Realized GARCH adds a realized measure to the variance dynamics and models that measure through an equation tied to latent variance and return shocks. GARCH-X usually treats its added measure as an outside input. MIDAS regression uses a parameterized lag-weight curve across frequencies; GARCH-MIDAS is a related hybrid whose equations should be read separately. These labels are not interchangeable.

The model forecasts a defined conditional variance or realized measure under its assumptions. It does not forecast return direction, prove that an intraday measure causes tomorrow's variance, or establish a profitable strategy. A strong in-sample likelihood does not guarantee better future forecasts. Trading conclusions require a separately specified signal and an after-cost evaluation that includes spread, fees, market impact, financing, and risk limits.

For a reproducible report, specify the asset and sample, return definition and conditional mean, realized-measure construction and session window, model equations and lag orders, innovation assumptions, estimation window, forecast origin and horizon, benchmark, scoring loss, and uncertainty method. Report sensitivity to plausible realized-measure choices and disclose any data exclusions or positivity adjustments