Andersen-Bollerslev-Christoffersen-Diebold (2006) Volatility and Correlation Forecasting

Summary

A comprehensive 101-page survey of volatility and correlation forecasting, structured around three paradigms: generalized autoregressive conditional heteroscedasticity (GARCH), Stochastic Volatility (SV), and Realized Volatility (RV). The chapter covers the theory of volatility concepts, economic uses of volatility forecasts (portfolio allocation, value at risk / expected shortfall (VaR/ES), density forecasting, option pricing), univariate volatility models, and multivariate correlation-forecasting models. Published as Chapter 15 in Elliott, Granger, and Timmermann (eds.), Handbook of Economic Forecasting, Volume 1, North-Holland.

Key Claims

• GARCH zoo, volatility impulse responses. RiskMetrics (degenerate integrated GARCH (IGARCH) with fixed $\lambda$) → GARCH(1,1) → asymmetric/leverage models (GJR-GARCH, exponential GARCH [EGARCH]) → long-memory fractionally integrated GARCH (FIGARCH) → component GARCH. Volatility impulse responses via $\partial\sigma^2_{t+h|t}/\partial\varepsilon^2_t$ show stark persistence differences: RiskMetrics/IGARCH never dies; FIGARCH decays hyperbolically; GARCH decays geometrically; component GARCH shows two-speed decay.
• SV estimation: EMM vs. MCMC. Efficient Method of Moments (EMM, Gallant-Tauchen) and Markov chain Monte Carlo (MCMC) are the two principal inferential approaches for SV models. MCMC produces the posterior of the entire latent state vector as a by-product of estimation, yielding an elegant solution to the smoothing problem; for forecasting, the filter distribution $f(s_t|x_t)$ is needed rather than the smoother $f(s_t|x_T)$.
• Realized volatility. High-frequency intraday squared returns aggregate to consistent estimates of integrated variance under continuous semimartingale assumptions; microstructure noise requires subsampling or noise-correction. RV serves as a near-latent-free target for evaluating GARCH and SV forecasts, resolving the $R^2$ paradox (Andersen-Bollerslev 1998).
• VaR and Expected Shortfall. Under location-scale models, VaR and ES reduce to conditional mean plus volatility times a distributional quantile or expected shortfall factor. If $z_t$ is i.i.d., the multiplicative factor in ES is constant and depends only on tail shape. Conditional dynamics in $\sigma_{t+1|t}$ are therefore the key input to risk measurement.
• DCC and multivariate correlation. Engle's Dynamic Conditional Correlation (DCC) model decomposes $H_t = D_t R_t D_t$; $R_t$ is driven by a pseudo-correlation matrix $Q_t$ updating on standardized residuals; two-step quasi-maximum likelihood (QML) renders large-system estimation feasible. Active extensions include asymmetric correlations, regime-switching correlations, and copula-GARCH approaches.
• Portfolio allocation. Dynamic mean-variance optimization with time-varying covariances requires forecasts of the full conditional covariance matrix; DCC/BEKK enable one-period-ahead covariance forecasts that can materially improve portfolio efficiency relative to constant-covariance benchmarks.
• Option valuation. Volatility clustering in returns implies that option prices depend on the entire conditional variance path, not just unconditional variance; GARCH option pricing (Duan 1995) and SV option pricing link the return variance dynamics to the risk-neutral measure.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Stochastic Volatility
• Realized Volatility
• Conditional Correlation Models
• Long Memory and Fractional Integration
• Value at Risk
• Volatility Forecast Evaluation

Entities Mentioned

• Torben G. Andersen
• Tim Bollerslev
• Peter F. Christoffersen
• Francis X. Diebold

Quotes

"Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades."

"A key advantage of the MCMC procedure is that the distribution of the latent state vector is obtained as an inherent part of the estimation. Moreover, the inference automatically accounts for the uncertainty regarding model parameters."

My Take

This is the canonical reference-shelf survey for volatility forecasting. Its value lies in the unified three-paradigm taxonomy (GARCH / SV / RV) and in the practical emphasis: nearly every section discusses both the model's theoretical properties and its forecasting performance. The FIGARCH impulse-response comparison is one of the clearest visual demonstrations of how long-memory alters persistence. The SV/MCMC section (§4.3) is characteristically brief — the chapter is more GARCH-centric — but the pointer to the smoother/filter distinction is important for practitioners building real-time forecasting systems. The multivariate section is essentially a condensed version of Bauwens-Laurent-Rombouts (2006); for DCC in particular, the companion paper Andersen-Bollerslev-Christoffersen-Diebold (2004) is the more operational reference.