GARCH 101: An Introduction to the Use of ARCH/GARCH Models in Applied Econometrics

Summary

A practitioner-oriented introduction to ARCH/GARCH volatility models. Engle (2001) motivates GARCH through the empirical failure of constant-variance assumptions, derives the GARCH(1,1) specification and its long-run variance, demonstrates multi-step volatility forecasting by geometric mean-reversion, shows how GARCH conditional variance feeds directly into one-day value-at-risk (VaR), and surveys asymmetric extensions (exponential GARCH [EGARCH], GJR/threshold ARCH [TARCH]) and multivariate extensions (Factor-GARCH, dynamic conditional correlation [DCC], meteor showers). The empirical backbone is a Dow Jones Industrial Average (DJIA) + bond portfolio (1990–2000) with estimated persistence $\hat\alpha+\hat\beta = 0.9922$, implying that daily volatility shocks decay very slowly toward the long-run level.

Key Claims

• GARCH(1,1) specification: $\sigma^2_t = \omega + \alpha\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}$; covariance-stationary iff $\alpha+\beta < 1$; long-run variance $V_L = \omega/(1-\alpha-\beta)$.
• Multi-step forecast: $\sigma^2_{t+k|t} = V_L + (\alpha+\beta)^k(\sigma^2_t - V_L)$; forecasts mean-revert geometrically to $V_L$ at rate $(\alpha+\beta)$ per step; converges to $V_L$ as $k\to\infty$ iff $\alpha+\beta<1$.
• VaR formula: 1% one-day VaR $= -2.326\,\sigma_t \times W$ (for portfolio value $W$ under conditional normality); time-varying because $\sigma_t$ varies with GARCH.
• ARCH LM test: Regress $\varepsilon^2_t$ on its $q$ lags; test statistic $T \cdot R^2 \sim \chi^2_q$ under the null of no ARCH. Equivalent to Lagrange multiplier test.
• Maximum likelihood (MLE) / quasi-maximum likelihood (QML) estimation: Gaussian log-likelihood maximized numerically (Newton-Raphson or Berndt-Hall-Hall-Hausman, BHHH); QML is consistent and asymptotically normal under regularity conditions even if errors are non-Gaussian; robust sandwich standard errors required under non-normality.
• Asymmetric extensions: EGARCH (Nelson 1991) — models log-variance, captures leverage effect; GJR/TARCH (Glosten-Jagannathan-Runkle 1993; Rabemananjara-Zakoian 1993) — adds $\delta\varepsilon^2_{t-1}\mathbf{1}(\varepsilon_{t-1}<0)$ term so negative shocks raise variance more than positive shocks of equal magnitude.
• Factor-GARCH / DCC: Factor-GARCH reduces parameters for large $n$ by imposing $k \ll n$ common volatility factors; DCC (Engle 2002) allows time-varying correlation matrix while maintaining tractable two-step estimation.
• Meteor showers vs. heat waves (Engle-Ito-Lin 1990): Yen/dollar hourly volatility driven partly by own lagged variance (heat waves — domestic news) and partly by foreign market ARCH (meteor showers — international transmission). Test is a structured ARCH model with cross-market lags; both effects present, meteor showers dominate.
• Empirical (DJIA + bond portfolio, 1990–2000): $\hat\omega = 0.00054$, $\hat\alpha = 0.0461$, $\hat\beta = 0.9461$; $\hat\alpha+\hat\beta = 0.9922$; $V_L = 0.00054/(1-0.9922) = 0.0114$ (daily standard deviation, SD ≈ 1.07%); 1% daily VaR on $1M portfolio ranges from ≈$8,400 (low-volatility periods) to ≈$52,900 (high-volatility periods).

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Value at Risk

Entities Mentioned

• Robert Engle

Quotes

"The GARCH forecast for next period's variance is a weighted average of three components: a constant, today's squared innovation, and today's variance."

"It seems safe to say that GARCH models are now well established in the toolbox of applied economists and financial analysts."

My Take

The canonical practitioner-level introduction to GARCH. The long-run variance / geometric mean-reversion framing is cleaner and more intuitive than the original Bollerslev (1986) presentation and directly motivates the VaR application. The "meteor showers vs. heat waves" section is an elegant and teachable example of how GARCH can test an economic hypothesis rather than just fit data. Thin on asymmetric extensions (EGARCH/TARCH are mentioned but not derived) and omits the QML consistency proof — appropriate for a JEP-style article but not a reference for theoretical details.