Modeling Volatility and GARCH Family › Summary

This chapter developed the essential tools for modeling time-varying volatility in financial markets:

Core concepts:

• Volatility clusters because large shocks tend to be followed by large shocks, and small shocks by small shocks
• ARCH models capture this by making conditional variance depend on past squared returns
• GARCH adds lagged variance terms, achieving parsimony. GARCH(1,1) captures the same dynamics as high-order ARCH with just three parameters

Key relationships:

• The persistence $\alpha + \beta$ determines how quickly volatility reverts to its long-run mean
• Typical financial data shows persistence of 0.95 to 0.99, meaning volatility shocks decay slowly
• Asymmetric models like GJR-GARCH capture the leverage effect where negative returns increase volatility more than positive returns

Practical applications include:

• Parameter estimation via maximum likelihood, typically with Student-t errors to capture fat tails
• Model diagnostics through Ljung-Box tests on squared standardized residuals
• Volatility forecasting for risk management (dynamic VaR) and trading (volatility timing)

Limitations to remember:

• GARCH is univariate and ignores cross-asset dependencies
• Continuous volatility evolution misses discrete jumps
• Parameters may be unstable across different market regimes

The next chapter on regression analysis will extend our toolkit for understanding relationships in financial data, including how GARCH volatility forecasts relate to market-implied volatility and other explanatory variables.