Bollerslev-Engle-Nelson (1994) ARCH Models

Summary

Comprehensive handbook chapter surveying the ARCH/GARCH literature through 1994. Organises empirical regularities of asset returns into eight stylized facts, catalogues the full family of univariate ARCH variants, develops the theory of strict stationarity vs. covariance stationarity, characterises continuous-time diffusion limits of GARCH processes, surveys multivariate extensions from vech to BEKK (Baba-Engle-Kraft-Kroner), and illustrates quasi-maximum likelihood (QML) inference on USD/DEM and a century of US stock index data.

Key Claims

Eight stylized facts: thick tails, volatility clustering, leverage effect, non-trading variance scaling, forecastable-event spikes, vol–serial-correlation linkage, cross-asset co-movements, macroeconomic effects on volatility.
Strict vs. covariance stationarity: GARCH(1,1) is strictly stationary iff $E[\ln(\alpha_1 z^2 + \beta_1)] < 0$ (Nelson 1990b), a weaker condition than $\alpha_1+\beta_1 < 1$ . Integrated GARCH (IGARCH) satisfies strict stationarity yet has $E_s(\sigma^2_t) \to \infty$ — persistence in probability and persistence in second moments are distinct concepts.
GARCH family zoo: ARCH( $q$ ), GARCH( $p,q$ ), IGARCH, nonlinear ARCH (NARCH)/Power-ARCH, augmented ARCH (AARCH)/asymmetric GARCH (AGARCH), threshold ARCH (TARCH; Zakoïan 1991), GJR-GARCH (Glosten-Jagannathan-Runkle 1993), quadratic ARCH (QARCH; Sentana 1991), quadratic threshold ARCH (QTARCH), structural ARCH (STARCH; Harvey-Ruiz-Shephard 1994), switching ARCH (SWARCH; Cai 1994; Hamilton-Susmel 1994), ARCH-in-mean (ARCH-M; Engle-Lilien-Robins 1987).
Continuous-time limits: GARCH(1,1) $\to$ mean-reverting variance diffusion $d\sigma^2 = \theta(\omega-\sigma^2)dt + \sigma^2 dW$ ; exponential GARCH (EGARCH) $\to$ log-normal diffusion for $\sigma^2$ . Different ARCH families generate different unconditional distributions (GARCH: Student- $t$ via inverted gamma mixing; EGARCH: normal–lognormal mixture).
Optimal ARCH filter (Nelson-Foster 1994): The loss-minimising filter for estimating an underlying continuous-time diffusion from discrete observations uses absolute-value residuals rather than squared residuals when the true distribution has heavy tails, providing theoretical justification for TARCH/NARCH over standard GARCH.
Temporal aggregation (Drost-Nijman 1993): Weak GARCH is closed under temporal aggregation; GARCH heteroskedasticity fades as sampling frequency decreases when $\alpha_1+\beta_1<1$ . Strict GARCH is not closed under aggregation.
Multivariate: vech-GARCH (Bollerslev-Engle-Wooldridge, BEW, 1988; no positive semi-definite (PSD) guarantee); BEKK (Engle-Kroner 1993: $H_t = C'C + A'\varepsilon_{t-1}\varepsilon'_{t-1}A + G'H_{t-1}G$ , guarantees PSD); Factor-ARCH (Engle 1987); constant conditional correlation (CCC; Bollerslev 1990); bivariate EGARCH (Braun-Nelson-Sunier 1992: time-varying conditional beta); co-persistence in variance (Bollerslev-Engle 1993).
QML inference: Bollerslev-Wooldridge (1992) sandwich standard errors (SE) $\mathcal{H}^{-1}\mathcal{G}\mathcal{H}^{-1}$ valid under non-normality; robust SEs are roughly $2\times$ non-robust on USD/DEM. Three volatility loss functions: $L_1 = \sum(\varepsilon^2-\sigma^2)^2$ , $L_2 = \sum(\varepsilon^2-\sigma^2)^2/\sigma^4$ , $L_3 = \sum[\ln\sigma^2 + \varepsilon^2/\sigma^2]$ .
Empirical 1 — USD/DEM (3,006 obs, 1981–1992): MA(1) (first-order moving average)-GARCH(1,1) with weekend dummy; $\hat\alpha_1 = 0.068$ , $\hat\beta_1 = 0.880$ ; residual kurtosis 4.892; robust SE $\approx 2\times$ non-robust.
Empirical 2 — US stocks (4 sub-samples 1885–1990): Rich EGARCH with generalised- $t$ density and extended $g(z_t,\sigma^2_t)$ allowing $\sigma^2$ -varying kurtosis and outlier-robust weighting. Two-component persistence (long-lived root half-life 119 days–4.5 years; short-lived 4–6 days); leverage $\theta_1 < 0$ always significant; standard EGARCH restrictions rejected in 3 of 4 sub-samples.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The fact that asset return volatility tends to move together across assets and markets suggests the existence of common volatility factors."

"A process may be strictly stationary without having finite second moments. The IGARCH process provides an important example."

"The optimal filter for a continuous record of observations from a diffusion process that depends only on absolute values of the residuals and not their squares."

My Take

The canonical reference for anyone entering ARCH/GARCH research. Its greatest contributions are the strict-stationarity theorem, the continuous-time limit characterisation, and the empirical distinction between strict stationarity and moment-convergence for IGARCH. The rich EGARCH applied to US stocks 1885–1990 remains one of the most thorough volatility specifications in the literature, demonstrating that standard models systematically misspecify the news impact function across all four sub-samples.