Modeling Volatility and GARCH Family › Limitations and Impact

GARCH models successfully capture several stylized facts:

• Volatility clustering: The autoregressive structure ensures that large shocks lead to high variance
• Persistence: High $\alpha + \beta$ creates the slow decay observed in practice
• Fat tails: Even with normal errors, the mixture of variances produces unconditionally fat-tailed returns
• Leverage effects: Asymmetric variants like GJR-GARCH capture the negative correlation between returns and volatility

GARCH has important limitations, though:

• No jumps: GARCH assumes volatility evolves continuously. Major events like central bank announcements or geopolitical shocks can cause discrete volatility jumps that GARCH cannot capture.
• Univariate: GARCH doesn't naturally extend to capturing correlations across assets. Multivariate extensions (DCC-GARCH, BEKK) exist but come at the cost of substantial complexity.
• Returns-only information: GARCH models volatility based solely on past returns, ignoring other relevant information like option prices, macroeconomic indicators, or sentiment measures.
• Extreme tails: While GARCH captures unconditional fat tails through time-varying variance, it may not fully account for the extreme tails observed in practice, particularly during crisis periods.

Several practical issues arise when applying GARCH models. Model selection between GARCH, GJR-GARCH, EGARCH, and other variants requires judgment. Information criteria like AIC and BIC help, but out-of-sample forecast evaluation is the gold standard. The degrees-of-freedom parameter for Student-t errors and the lag orders (p, q) require careful selection.

Parameter stability is another concern. GARCH parameters estimated over one period may not remain stable, particularly across different volatility regimes. Rolling or expanding window estimation can reveal parameter drift. Very high persistence (near-integrated GARCH with α + β ≈ 1) can cause numerical issues and unrealistic long-run variance estimates.

For intraday applications, standard daily GARCH may be insufficient. High-frequency volatility modeling uses realized variance measures and represents an active research area.

Despite its limitations, GARCH transformed how practitioners and academics think about volatility. Before Engle's work, constant volatility was the default assumption. GARCH provided a tractable, estimable framework for time-varying volatility that:

• Improved risk measurement and capital allocation
• Enhanced derivative pricing and hedging
• Enabled volatility forecasting for trading strategies
• Established volatility itself as an object of study and trading

Robert Engle's 2003 Nobel Prize recognized these contributions. Today, GARCH and its variants remain the workhorses of volatility modeling, forming the foundation upon which more sophisticated approaches build.