Bollerslev (1986) Generalized Autoregressive Conditional Heteroskedasticity

Summary

Bollerslev generalizes Engle's (1982) ARCH process to the GARCH( $p,q$ ) process by letting the current conditional variance depend on past conditional variances as well as past squared errors. The extension is exactly analogous to going from a pure AR to an ARMA model in the conditional mean: it permits a far more flexible and parsimonious lag structure for volatility, avoiding the long, arbitrarily-restricted linear-declining lags that plagued high-order ARCH fits. The paper derives wide-sense stationarity conditions and the autocorrelation structure of the squared process, discusses maximum-likelihood estimation and testing, and illustrates the model on U.S. inflation-rate uncertainty.

Key Claims

The GARCH( $p,q$ ) model. With $\varepsilon_t \mid \psi_{t-1} \sim N(0, h_t)$ , $h_t = \alpha_0 + \sum_{i=1}^{q}\alpha_i \varepsilon_{t-i}^2 + \sum_{i=1}^{p}\beta_i h_{t-i}, \qquad \alpha_0>0,\ \alpha_i\ge 0,\ \beta_i\ge 0.$ For $p=0$ it collapses to ARCH( $q$ ); for $p=q=0$ , $\varepsilon_t$ is white noise. The lagged $h_{t-i}$ terms act as an "adaptive learning mechanism."
ARMA-for-variance analogy. Just as ARMA parsimoniously captures what a high-order AR needs many lags for, GARCH captures persistent volatility with few parameters; the process can be written as an infinite-order ARCH( $\infty$ ) distributed lag when the roots of $1-B(z)=0$ lie outside the unit circle.
Stationarity condition (Theorem 1). The GARCH( $p,q$ ) process is wide-sense stationary with $E(\varepsilon_t)=0$ , $\mathrm{var}(\varepsilon_t)=\alpha_0\,[1-A(1)-B(1)]^{-1}$ , and zero autocovariances iff $A(1)+B(1)<1$ — i.e. the sum of the ARCH and GARCH coefficients is below one.
Squared-process ACF for identification. The autocorrelations and partial autocorrelations of $\varepsilon_t^2$ play the same identification/diagnostic role for GARCH order selection that the ACF/PACF play for ARMA order selection in the mean.
Estimation and testing. Maximum likelihood for the linear-regression-with-GARCH-errors model; the asymptotic block-diagonality (independence) between mean and variance parameter estimates carries over from the ARCH regression model. As in the ARMA case (Godfrey 1978), a fully general test for GARCH presence is not feasible.
Empirical example. A simple GARCH(1,1) gives a marginally better fit and a more plausible learning mechanism for U.S. inflation-rate uncertainty than Engle-Kraft's (1983) eighth-order linear-declining ARCH.

Concepts Introduced or Extended

GARCH and BEKK-GARCH

Entities Mentioned

Quotes

"A natural generalization of the ARCH ... process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed."

"The extension of the ARCH process to the GARCH process bears much resemblance to the extension of the standard time series AR process to the general ARMA process and ... permits a more parsimonious description in many situations."

My Take

This is one of the highest-return single generalizations in time-series econometrics: adding the $\beta h_{t-1}$ term turns Engle's ARCH — which needed many tightly-restricted lags to fit persistent volatility — into a two-parameter GARCH(1,1) that has since become the default volatility model across finance and macro. The paper's own framing is the right one: GARCH is to ARCH as ARMA is to AR, and the $A(1)+B(1)<1$ stationarity condition (with the sum measuring persistence) is the single most-used diagnostic in the whole literature — its boundary case is IGARCH, and pushing it toward fractional integration gives FIGARCH. For the wiki this is the foundational source under the already-detailed GARCH concept, which had been citing "Bollerslev (1986)" inline without a page behind it; everything downstream — BEKK, DCC, EGARCH, MS-GARCH, the realized-volatility " $R^2$ paradox," the whole SV alternative — is a response to or extension of this model.