How HAR-RV Algorithms Forecast Realized Volatility Across Time Scales: Daily–Weekly–Monthly Components, Long-Memory Approximation, Measurement Error, Jumps and Forecast Failure

Reader question: Realized volatility is persistent across days, weeks and months. Can a simple regression capture that slow decay without fitting a complicated fractional-memory model?

The Heterogeneous Autoregressive model of Realized Volatility (HAR-RV) answers with a deliberately simple cascade. It forecasts future realized volatility from daily, weekly and monthly averages of past realized volatility. The model is not truly long-memory in the strict stochastic-process sense, but its overlapping time-scale components can approximate the slow autocorrelation decay observed in volatility.

This article owns one precise computational job: multi-horizon forecasting of realized volatility using the HAR-RV cascade and its validation/measurement-error diagnostics. It does not own ex-post high-frequency volatility measurement, GARCH/EWMA conditional-variance recursions, GARCH-MIDAS macro mixing, jump detection or generic forecast-comparison tests.

This is public mathematical and computational education. It is not a volatility-trading strategy, personalised financial advice or a claim that one historical autoregression can forecast every future volatility regime.

1. Start with a realized-volatility measurement

Suppose intraday returns for day t are r_t,i.

A simple realized variance is:

RV_t = Σ_i r_t,i².

That measurement can be replaced by a noise-robust realized kernel, pre-averaged measure or another justified estimator.

HAR-RV begins after the realized-volatility measurement exists. It forecasts the time series of those daily measures.

2. The three canonical time scales

Define:

RV_t^(d) = RV_t,

RV_t^(w) = (1/5)Σ_j=0⁴ RV_t−j,

RV_t^(m) = (1/22)Σ_j=0²¹ RV_t−j.

The exact weekly/monthly day counts can be adapted to the data calendar, but the conventional daily/weekly/monthly interpretation is central.

3. The basic HAR-RV regression

A one-day-ahead model is:

RV_t+1 = β₀ + β_dRV_t^(d) + β_wRV_t^(w) + β_mRV_t^(m) + ε_t+1.

The regression is linear and can be estimated by ordinary least squares under the baseline specification.

The surprising strength of HAR-RV is that these three simple averages can generate a slowly decaying effective impulse response resembling long-memory volatility behavior.

4. Why the regressors overlap

The daily observation RV_t appears inside the weekly average, and both appear inside the monthly average.

This creates substantial regressor correlation.

That overlap is intentional: the variables represent nested horizons, not independent factors.

But it means individual coefficient interpretation can be unstable even when the combined forecast is stable.

Diagnostic: inspect condition numbers, coefficient covariance and rolling estimates. Do not treat β_d, β_w, β_m as separately structural if the design matrix is poorly conditioned.

5. The heterogeneous-market intuition

Corsi’s original motivation is that market participants operate at different horizons.

• short-horizon participants react to recent volatility;
• weekly-horizon participants react to a smoother component;
• longer-horizon participants respond to even slower volatility information.

The cascade of these horizons produces persistent aggregate volatility even though the fitted regression itself is finite-order.

The economic story is useful intuition, but the forecast should still be evaluated statistically rather than accepted because the narrative sounds plausible.

6. HAR-RV is not a true fractional-memory process

A genuine long-memory process can have hyperbolically decaying autocorrelations and fractional integration.

The three-component HAR model has finite-dimensional regression structure.

Its achievement is approximation:

a short-memory AR-type specification can mimic important long-memory features over practical horizons.

This distinction matters when extrapolating beyond the horizons represented by the regressors.

7. Direct multi-step forecasts

For an h-day target, one approach is to define future average realized volatility:

RV_t+1:t+h = (1/h)Σ_j=1^hRV_t+j

and estimate a direct regression:

RV_t+1:t+h = β_0,h + β_d,hRV_t^(d) + β_w,hRV_t^(w) + β_m,hRV_t^(m) + ε_t,h.

This avoids recursively feeding one-day predictions into future regressors.

But multi-day target averages overlap through time, creating serial correlation in regression residuals.

8. HAC inference for overlapping forecast horizons

If h-day target windows overlap, ε_t,h and ε_t+1,h share future realized-volatility observations.

Ordinary OLS coefficient estimates may still be useful, but naive independent-error standard errors can be misleading.

Heteroskedasticity-and-autocorrelation-consistent covariance estimators can be appropriate for inference.

The forecast itself and the coefficient significance test are separate objects; robust standard errors do not change the point forecast unless the estimation procedure changes.

9. Log-HAR

Realized variance is nonnegative and strongly right-skewed.

A common transformation is:

y_t = log(RV_t).

Then fit:

y_t+1 = β₀ + β_dy_t^(d) + β_wy_t^(w) + β_my_t^(m) + ε.

This stabilises variance and ensures that exponentiating the fitted value gives a positive number.

But:

E[RV|X] ≠ exp(E[log RV|X])

in general.

A log model therefore requires a retransformation decision when the desired target is the conditional mean of RV rather than the conditional median.

10. Retransformation bias

If log residuals were conditionally normal with variance s²:

E[RV|X] ≈ exp(ŷ + ½s²).

Real residuals need not be homoskedastic normal, so a nonparametric smearing factor or simulation can be safer.

Falsifier: compare forecasts in log space and RV space. If the retransformation choice materially changes forecast ranking, it is part of the model specification.

11. Realized-volatility measurement error

RV is an estimator of latent integrated variance, not an error-free observation.

Microstructure noise, finite intraday sampling and estimator choice create measurement error.

If the lagged regressor is measured with noise, OLS persistence coefficients can suffer attenuation-type bias.

Bollerslev, Patton and Quaedvlieg developed the HARQ idea to account for heteroskedastic measurement error using realized quarticity.

12. HARQ intuition

Realized quarticity estimates the variability of the realized-variance measurement error.

When measurement uncertainty is high, the model should trust the latest noisy RV observation less strongly.

HARQ introduces an interaction between lagged realized variance and a quarticity-based uncertainty term.

The precise specification varies by implementation, but the principle is:

higher measurement error → less effective persistence assigned to the noisy recent RV.

13. The input measure matters

If ordinary high-frequency RV is strongly contaminated by microstructure noise, HAR-RV is forecasting a noisy target.

A realized kernel can provide a cleaner ex-post measure in noisy data.

See realized-kernel algorithms.

Improving the input measurement and improving the forecast dynamics are two different jobs.

14. Jumps: HAR-J

Realized variance can be decomposed approximately into:

• continuous variation;
• jump variation.

Realized bipower variation is one tool for estimating the continuous component under suitable assumptions.

A HAR-J extension can include jump measures separately, asking whether jumps and continuous volatility have different predictive persistence.

See realized-bipower-variation algorithms.

15. Semivariance extensions

Realized semivariance separates intraday squared returns by sign:

RS⁻ = Σ r_i²1{r_i<0},

RS⁺ = Σ r_i²1{r_i≥0}.

HAR-type models can let downside and upside realized variation carry different predictive coefficients.

This can capture leverage/asymmetry without changing the basic multi-horizon architecture.

16. Inputs and outputs

Inputs can include:

• daily realized-volatility/variance series;
• measurement method and intraday sampling rule;
• daily/weekly/monthly window definitions;
• forecast horizon;
• level or log transform;
• jump/semivariance/quarticity extensions;
• training window;
• rolling or expanding estimation rule;
• missing-day/holiday treatment.

Outputs can include:

• one-day or h-day volatility forecast;
• β coefficients;
• forecast intervals;
• residual diagnostics;
• rolling coefficient stability;
• out-of-sample MSE/MAE/QLIKE;
• Diebold–Mariano comparisons;
• measurement-error sensitivity;
• regime-break flags.

17. QLIKE and noisy volatility proxies

Future integrated variance is latent, so forecast evaluation uses an imperfect proxy such as realized variance.

Some loss functions can rank competing variance forecasts incorrectly when the proxy is noisy.

Patton’s work derives loss-function conditions under which rankings are robust to conditionally unbiased measurement noise; QLIKE is a widely used example in volatility forecasting.

A typical QLIKE loss can be written:

L = y/ŷ − log(y/ŷ) − 1,

for positive variance proxy y and positive forecast ŷ.

No single loss function should be the sole validation lens.

18. Forecast evaluation must be out of sample

HAR-RV is easy to fit and extend. That makes data mining easy too.

A proper workflow separates:

• model-development period;
• hyperparameter/specification selection;
• out-of-sample forecast period.

If dozens of HAR variants are tried and only the best one is reported, ordinary forecast p-values can overstate evidence.

See White Reality Check / Hansen SPA algorithms for the broader data-snooping problem.

19. Diebold–Mariano comparison

To compare HAR-RV with another forecast, define loss differential:

d_t = L(e_1,t) − L(e_2,t).

The Diebold–Mariano framework tests whether the expected loss differential is zero, accounting for serial dependence where required.

See Diebold–Mariano algorithms.

20. Evidence polarity

Evidence for confidence includes:

• out-of-sample performance beats simple persistence benchmarks;
• daily/weekly/monthly components are stable enough to improve forecasts jointly;
• forecast errors are not strongly autocorrelated;
• coefficient signs and magnitudes are stable across adjacent windows;
• results survive alternative realized-volatility measures;
• QLIKE and squared-error conclusions are broadly consistent;
• forecast improvement persists across volatility regimes;
• extensions such as HARQ earn their extra complexity out of sample.

Evidence against confidence includes:

• negative level forecasts;
• strong residual autocorrelation;
• rolling coefficients changing sign repeatedly;
• performance disappearing when the input RV measure changes;
• one recent RV observation dominating because of measurement error;
• large crisis-period forecast bias;
• extensions improving in sample but not out of sample.

21. Counterexample: level HAR predicts negative variance

A linear regression has no built-in positivity constraint.

During a low-volatility period, a negative intercept and coefficient combination can produce:

RV̂_t+1 < 0.

Negative variance is not meaningful.

Falsifier: count negative forecasts. Consider log-HAR, constrained regression or another positive specification if the issue is material.

22. Counterexample: measurement error attenuates persistence

Suppose the latest daily RV estimate is extremely noisy because microstructure noise was high.

OLS treats it as an exact regressor and can underestimate the true persistence relationship.

Falsifier: compare HAR with HARQ or with a cleaner realized measure. If persistence rises systematically when measurement quality improves, the baseline was error-contaminated.

23. Counterexample: monthly component becomes stale after regime change

A sudden volatility shock can make the 22-day average much lower than the new regime.

The monthly component pulls forecasts downward just when volatility has structurally shifted higher.

Falsifier: inspect forecast errors after detected breaks and compare rolling-window or change-point-adaptive estimates.

See PELT change-point algorithms.

24. Counterexample: one market needs a different cascade

The conventional 1/5/22 horizons are economically intuitive but not mathematically sacred.

A continuously traded market or a market with different institutional horizons may have different effective components.

Falsifier: evaluate alternative lag structures out of sample under multiple-testing controls. Do not optimize dozens of horizons and then report the winner as if it were pre-specified.

25. Counterexample: overlapping regressors destabilise coefficients

Daily, weekly and monthly RV are strongly correlated.

Individual coefficient t-statistics can change markedly when the sample window moves by a few days.

Falsifier: inspect forecast stability and joint contribution rather than relying on one coefficient’s significance.

26. Counterexample: log forecast is misinterpreted as mean variance

A log-HAR predicts E[log RV|X].

Exponentiating that number gives a conditional geometric-type center, not automatically E[RV|X].

Falsifier: benchmark retransformed forecasts against realized means and use a documented smearing correction if the mean is the intended target.

27. Counterexample: jump shock is treated as persistent continuous variance

A one-off price jump can make RV enormous.

Baseline HAR spreads that observation into daily, weekly and monthly averages, potentially forecasting persistence that does not exist.

Falsifier: separate jump and continuous components with a justified estimator and compare HAR-J/HAR-C style forecasts.

28. HAR-RV versus GARCH/EWMA

GARCH/EWMA algorithms infer conditional variance primarily from lower-frequency return shocks.

HAR-RV directly models a high-frequency-based realized-volatility measure and separates persistence by daily/weekly/monthly horizons.

29. HAR-RV versus GARCH–MIDAS

GARCH–MIDAS algorithms split volatility into fast return dynamics and a slow component driven by lower-frequency variables.

HAR-RV’s slow components come from averages of realized volatility itself, not macroeconomic MIDAS regressors.

30. HAR-RV versus realized-kernel measurement

Realized kernels own the measurement question: what ex-post variance occurred despite microstructure noise?

HAR-RV owns the forecasting question: how does measured volatility propagate across horizons into future realized volatility?

31. Alternatives

ARFIMA/FIGARCH: explicit fractional/long-memory structure.

Realized GARCH: joint latent-return variance and realized-measure framework.

HARQ: measurement-error-aware HAR extension.

HAR-J / semivariance HAR: separate jumps or downside/upside variation.

Machine-learning forecasts: richer nonlinear features at higher overfitting/governance cost.

Regime-switching HAR: conditional dynamics across volatility states.

32. Weak links

• poor realized-volatility input measure;
• unhandled holidays/missing days;
• negative forecasts in levels;
• log retransformation bias;
• overlapping-horizon residual dependence;
• multicollinearity among horizon averages;
• structural breaks;
• jump contamination;
• measurement-error attenuation;
• data-snooping across many HAR variants.

33. What would falsify confidence?

Confidence should be withdrawn if HAR-RV fails to beat simple persistence out of sample; if forecast conclusions reverse with a reasonable realized-volatility estimator; if residual autocorrelation remains strong; if coefficients are unstable across regimes; if extensions do not survive hold-out evaluation; or if forecast accuracy collapses specifically when volatility is most important.

34. Verification and update triggers

Preserve the realized-volatility construction, intraday sampling rule, daily calendar, horizon windows, transformation, estimation sample, forecast horizon, coefficient history and evaluation loss functions.

Revalidate when:

• market microstructure changes;
• realized-volatility estimator changes;
• volatility regime breaks;
• forecast horizon changes;
• measurement error rises;
• new jump/semivariance features are added;
• rolling forecast performance deteriorates.

35. Primary and high-quality references

• Fulvio Corsi, A Simple Approximate Long-Memory Model of Realized Volatility, Journal of Financial Econometrics, 2009.
• Andrew J. Patton, Volatility Forecast Comparison Using Imperfect Volatility Proxies, Journal of Econometrics, 2011.
• Realized Volatility Forecasting: Robustness to Measurement Errors, International Journal of Forecasting, 2021.
• Modeling Long Memory Volatility Using Realized Measures of Volatility: A Realized HAR GARCH Model.
• Modelling and Forecasting Noisy Realized Volatility, Computational Statistics & Data Analysis.

Educational boundary: HAR-RV is a parsimonious forecasting architecture, not a law of volatility. Its usefulness must be demonstrated by clean realized-volatility inputs, out-of-sample forecast evidence and stability across regimes.