HAR-RV Model Explained: Daily, Weekly, and Monthly Volatility Forecasts

The heterogeneous autoregressive model of realized volatility, usually shortened to HAR-RV, predicts a future realized-volatility measure from recent measures summarized over different horizons. A common daily specification uses one completed day, a five-trading-day average, and a 22-trading-day average. Those windows are inputs to the regression; a next-day HAR-RV forecast is still a next-day forecast.

The model is useful when a reader wants a transparent baseline for persistence in realized volatility. Its additive structure makes it straightforward to estimate with a linear regression and to compare with a return-based model such as GARCH. It is a forecasting specification, not a rule for when to enter a trade, how large a position to take, or whether an asset is attractive.

Corsi’s original paper calls this an additive cascade across volatility components at different time scales. It was motivated by heterogeneous market horizons, but the regression does not identify individual traders or prove that particular investor groups caused the observed persistence. Treat the daily, weekly, and monthly terms as statistical summaries of past volatility.

For a completed session \(t\), one common realized-variance measure adds squared intraday returns:

\[ RV_t = \sum_{i=1}^{M} r_{t,i}^{2} \]

Here \(r_{t,i}\) is the return over intraday interval \(i\), and \(M\) is the number of intervals included in the session. The sum has squared-return units. This guide intentionally uses realized variance on that scale in every HAR component and forecast example. Corsi’s original HAR-RV uses realized-volatility measures on the square-root scale instead; the two conventions are related but cannot be mixed in one regression. Keep the daily, weekly, and monthly components and the forecast target on one consistent scale. The realized-volatility calculation guide covers the measurement choices behind that label.

The sampling interval, price source, market session, and treatment of overnight returns all define what \(RV_t\) measures. For example, a sum built only from regular-session intraday returns does not automatically include the close-to-open move. Intraday sampling can also be affected by quote noise and other market microstructure features. Hansen and Lunde (2006) study how market microstructure noise affects realized-variance measurement. Andersen, Bollerslev, Diebold, and Labys develop the link between high-frequency observations, realized volatility, and lower-frequency return variation; their framework does not make every sampling rule interchangeable.

Use one consistent definition for the history, regressors, and forecast target. If the input changes from five-minute to one-minute returns, or from regular-session to close-to-close coverage, the resulting series can change. This is not merely a formatting detail: it can alter estimated coefficients and the forecast being evaluated.

Let \(RV_t^{(d)}=RV_t\) denote the daily measure. One standard construction averages the latest five completed daily values for the weekly component and the latest 22 for the monthly component:

\[ RV_t^{(w)} = \frac{1}{5}\sum_{j=0}^{4}RV_{t-j}, \qquad RV_t^{(m)} = \frac{1}{22}\sum_{j=0}^{21}RV_{t-j} \]

The divisor makes these components average daily realized variance over their windows. A sum would instead represent accumulated realized variance across the window. Either convention can be used in a separately defined model, but the formula, target, and units must agree.

These windows overlap. The latest daily observation is also included in the weekly and monthly averages, and the weekly observations are part of the monthly window. The regressors therefore summarize related parts of the same history; their labels do not make them independent measurements. Overlap also means the coefficients should not be read as clean causal effects of three separate horizons.

The five- and 22-observation windows conventionally refer to trading sessions, not necessarily calendar weeks and months. A market holiday, a missing session, or a futures contract with a different trading calendar requires an explicit rule. Record the observation calendar and how incomplete windows are handled before comparing estimates across assets.

A basic one-step daily HAR-RV regression is

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

The left side is the realized variance of the next completed session. Each predictor is known by the end of session \(t\). The intercept \(\beta_0\) and coefficients \(\beta_d,\beta_w,\beta_m\) are estimated from historical observations; \(\varepsilon_{t+1}\) is the regression error that remains after the predictors are used.

In an ordinary least-squares fit, the coefficients are chosen to minimize the in-sample sum of squared forecast errors for the defined target. This produces a compact linear benchmark without estimating a separate coefficient for every daily lag. Corsi (2009) develops the HAR-RV construction and reports forecasting results for the paper’s data and specifications. Those empirical results do not guarantee that the same coefficients or accuracy will carry to another asset, period, or realized-volatility measure.

The notation varies across papers. Some HAR specifications use logarithms, other realized-volatility estimators, additional lags, or different windows. State the chosen target and transformation rather than treating every model with “HAR” in its name as the same equation.

After estimating the coefficients, form the next-session point forecast by substituting the latest available daily, weekly, and monthly values. The forecast omits the future regression error because its value is not known at the forecast origin:

\[ \widehat{RV}_{t+1\mid t}^{(d)} = \widehat{\beta}_0 + \widehat{\beta}_d RV_t^{(d)} + \widehat{\beta}_w RV_t^{(w)} + \widehat{\beta}_m RV_t^{(m)} \]

For a hypothetical calculation, suppose the fitted intercept is \(0.10\ \mathrm{bp}^2\), and the fitted coefficients are \(0.50\), \(0.30\), and \(0.15\). Suppose the latest daily measure is \(4\ \mathrm{bp}^2\), the five-day average is \(3\ \mathrm{bp}^2\), and the 22-day average is \(2\ \mathrm{bp}^2\). Then

\[ 0.10 + 0.50(4) + 0.30(3) + 0.15(2) = 3.30\ \mathrm{bp}^2 \]

This is an invented arithmetic example, not market data, a fitted result, or a performance claim. It is a forecast for the next session under the stated target and units. Taking its square root would produce a volatility measure in basis points for that same session; annualization would require a separate, stated convention.

To predict an entire week or month, do not simply relabel this next-day forecast. One option is to estimate a direct model whose left side is the realized variance accumulated or averaged over the intended future window. Another is to iterate one-step forecasts, which requires a clear rule for updating future daily, weekly, and monthly regressors. The horizon must be fixed before forecasts are scored. <!-- learn:illustration -->

Volatility often shows persistence across many lags. A long autoregression can represent that dependence, but it uses many parameters. HAR-RV compresses the history into a few overlapping averages and can reproduce empirical patterns that resemble long memory with a small number of coefficients.

The word “approximate” matters. Corsi’s paper explicitly describes the absence of true long-memory properties in the HAR-RV model while showing that simulations can reproduce features associated with long-memory financial returns. HAR-RV is therefore a tractable approximation to selected patterns, not a fractionally integrated process just because the data exhibit slow decay in dependence.

Likewise, daily, weekly, and monthly components are not a direct census of investors trading at those exact frequencies. The heterogeneous-horizon idea motivates the cascade, but the estimated regression does not separately identify investor types, reveal their intentions, or establish a causal mechanism. Keep the interpretation at the level of the model’s observed inputs and forecast behavior.

A large price jump may contain information about future volatility that a single total-variance input obscures. Corsi, Pirino, and Reno (2010) separate continuous and discontinuous variation with threshold bipower methods and study the resulting jump information in volatility forecasts. Their reported forecasting improvements, especially after jumps, apply to their estimators, data, and empirical design.

A jump-aware HAR specification may include separate continuous and jump-related measures or their lagged aggregates. That changes the predictors and creates new choices: how a jump is detected, how the continuous component is estimated, and whether a jump measure is available at the forecast origin. The labels “continuous” and “jump” describe estimator components under a model; they do not make the decomposition error-free.

More terms can improve a forecast in one sample and add estimation noise in another. Compare a jump extension with a simpler HAR-RV baseline on the same forecast origins, horizon, target, and information set. Do not infer that every jump predicts higher volatility or that the extension will outperform in every market.

HAR-RV coefficients are conditional on the realized-volatility series used to estimate them. Sampling frequency, trading-session boundaries, price cleaning, and missing observations can change that series. The daily, weekly, and monthly regressors inherit those choices because they are averages of the measured daily values.

Keep the forecast origin separate from the target interval. If the model is estimated at a session close, the next-session target should use the same asset, return convention, session coverage, and units as the historic target. If the measure excludes overnight returns, do not describe the forecast as a complete close-to-close variance without an adjustment. Calendar windows, overlapping multi-day horizons, and different holiday rules need their own definitions.

The same issue arises when comparing HAR-RV with another model. A daily conditional-variance forecast from GARCH and a realized-variance forecast from HAR-RV are comparable only after their target intervals and units are aligned. MIDAS regression offers a different way to relate observations at different frequencies; it does not remove the need to specify the information available at each forecast origin.

Realized variance is nonnegative, but an unconstrained linear regression does not enforce nonnegative coefficients or predictions. In some data, an ordinary least-squares fit can produce a negative out-of-sample forecast. A log specification or a constrained fit can address positivity in a different way, but each changes the model and the meaning of its coefficients. A forecast floor is also a modeling choice; set it before evaluation instead of adjusting it after seeing which model wins.

Evaluate forecasts outside the estimation sample and save the values available at each origin. Align the proxy, forecast target, horizon, session coverage, units, and common evaluation dates. Report how forecasts were estimated or updated and compare against a stated baseline. When a realized measure is used as an imperfect proxy, expected ranking results depend on its relation to the target and on the loss function; Patton (2011) explains why proxy noise does not make every comparison automatically reliable. The QLIKE and squared-error guide discusses those evaluation assumptions.

A lower forecast loss says that a model scored better under the selected target, proxy, sample, and loss. It does not show that the model causes volatility, that its daily forecast is a trade signal, or that a strategy using it will make money after execution costs and risk constraints.