Source: Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2003) · Econometrica 71(2), 579–625 · doi:10.1111/1468-0262.00418
TL;DR
Volatility, traditionally a latent variable filtered with GARCH or stochastic-volatility models, can instead be measured directly by summing squared (and cross-) high-frequency intraday returns — "realized volatility." Grounded in the theory of quadratic variation, realized volatility is a consistent ex-post estimator of the integrated variance/covariance. Treated as observable, daily realized vols have clean time-series properties (log realized vols are approximately Gaussian and long-memory), and a simple long-memory Gaussian VAR for the log realized volatilities forecasts future volatility better than daily GARCH or RiskMetrics.
What it models
The conditional covariance matrix of asset returns. Building on continuous-time arbitrage-free price processes (semimartingales), the paper formally links the latent integrated covariance matrix to an observable empirical object — realized volatility cumulated from intraday returns — so that daily and lower-frequency volatility can be modeled with standard time-series tools instead of restrictive parametric multivariate ARCH/SV models that suffer a curse of dimensionality.
Specification
For an n-vector of intraday returns sampled at frequency Δ (the empirical work uses Δ = 1/48, i.e. 30-minute returns, 48 per day), the h-day realized volatility (covariance) matrix is
where R_{t,h} stacks the (h/Δ) intraday return vectors. As Δ → 0, V_t converges (by the theory of quadratic variation) to the integrated covariance matrix, making realized volatility an unbiased and consistent ex-post estimator. Forecasting is done with a fifth-order long-memory (fractionally-integrated) Gaussian VAR for the daily logarithmic realized volatilities (VAR-RV): (1−L)^d applied with a common d, lag order 5. Coupled with the empirically supported assumption that returns standardized by realized vols are Gaussian, this yields a lognormal–normal mixture for the return distribution and well-calibrated density/quantile (VaR) forecasts.
Estimation
Data: continuously recorded Olsen & Associates spot quotes for DM/$ and Yen/$ exchange rates, ~13 years, Dec 1, 1986 – Jun 30, 1999 (3,045 days). In-sample estimation Dec 1, 1986 – Dec 1, 1996 (2,449 days); out-of-sample Dec 2, 1996 – Jun 30, 1999 (596 days).
Long-memory parameter estimated via the GPH log-periodogram regression (Robinson 1995): common d ≈ 0.401 across the three (log) volatility/covariance series; fractional-integration estimates on individual series are 0.356, 0.424, 0.393.
VAR lag polynomials set to order 5; fractionally differenced with the estimated d.
What it captures
Long memory / persistence: log realized vols show slowly decaying autocorrelation, well captured by fractional integration (d ≈ 0.4).
Approximate log-normality: raw realized vols are right-skewed, but their logarithms are very nearly Gaussian.
Gaussian standardized returns: returns divided by realized vol are approximately normal — the basis for the lognormal-normal mixture.
Realized correlations are also persistent and tractable, giving an observable covariance matrix.
Use & extensions
Out-of-sample forecast-evaluation regressions: the VAR-RV model delivers the highest R² among competitors and dominates GARCH(1,1), RiskMetrics (IGARCH, λ=0.94), and a VAR on long-memory-filtered absolute returns (VAR-ABS), at both 1-day and 10-day horizons.
Turned volatility into a near-observable, seeding the high-frequency volatility literature: HAR model (Corsi 2009), realized-covariance estimation, jump separation, and modern risk forecasting.
Limitations
Microstructure noise (bid-ask bounce, discreteness) biases naive realized variance at very fine sampling; the paper samples at 30 minutes to balance this, and later work (realized kernels, pre-averaging, two-scale estimators) corrects it.
Requires clean high-frequency data; jumps, overnight gaps, and illiquid assets need separate treatment.
Demonstrated on two liquid FX rates; the bivariate/trivariate covariance scaling, while better than multivariate ARCH, still has practical limits.
Key references
Andersen, T., Bollerslev, T., Diebold, F. & Labys, P. (2003) — Modeling and Forecasting Realized Volatility — Econometrica
Andersen, Bollerslev, Diebold & Labys (2001) — The Distribution of Realized Exchange Rate Volatility — JASA (companion, in-sample distributional results)
Barndorff-Nielsen, O. & Shephard, N. (2002) — Econometric Analysis of Realized Volatility — JRSS B
Corsi, F. (2009) — A Simple Approximate Long-Memory Model of Realized Volatility (HAR) — Journal of Financial Econometrics
Provenance: verified/generated from the paper's full text.
