Hentschel (1995) All in the Family: Nesting Symmetric and Asymmetric GARCH Models

Summary

Hentschel (1995) derives a parametric family of GARCH models that nests eight widely-used specifications — EGARCH, TGARCH, AGARCH, standard GARCH, GJR-GARCH, NA-GARCH, NARCH, and A-PARCH — under a single variance equation governed by four parameters: a Box-Cox power $\lambda$ for the conditional standard deviation, an exponent $\nu$ for the shock function, a shift parameter $b$, and a rotation parameter $c$. The shift and rotation independently capture two distinct types of asymmetry in the news impact curve. Estimated on 17,486 daily Center for Research in Security Prices (CRSP) excess returns (1926–1990), all standard models are rejected; the best fit is approximately $\lambda \approx 1$, $\nu \approx 1.5$ with significant shift but insignificant rotation.

Key Claims

• The unified variance equation is $(\sigma_t^\lambda - 1)/\lambda = \omega + \alpha \sigma_{t-1}^{\lambda-\nu} f^\nu(\varepsilon_t) + \beta(\sigma_{t-1}^\lambda - 1)/\lambda$, where $f(\varepsilon_t) = |\varepsilon_t - b| - c(\varepsilon_t - b)$ is the shifted and rotated absolute value function (eq. 7).
• The nesting is achieved by restricting $(\lambda, \nu, b, c)$: EGARCH $(\lambda=0, \nu=1, b=0)$; TGARCH $(\lambda=\nu=1, b=0)$; standard GARCH $(\lambda=\nu=2, b=c=0)$; GJR-GARCH $(\lambda=\nu=2, b=0)$; NA-GARCH $(\lambda=\nu=2, c=0)$; NARCH $(\lambda=\nu, b=c=0)$; A-PARCH $(\lambda=\nu, b=0)$.
• Shift ($b$): moves the news impact curve minimum rightward, dominating the asymmetric response for small shocks. Rotation ($c$): changes the slopes on either side of the minimum, dominating for large shocks. These are empirically distinct: current models conflate them by allowing only one at a time.
• All symmetric GARCH models are rejected against asymmetric alternatives with likelihood-ratio (LR) statistics exceeding 200. Among asymmetric models: EGARCH, AGARCH, and GARCH are all rejected in favor of a freely estimated power $(\lambda \approx 1, \nu \approx 1.5)$. The standard GARCH model is rejected in favor of a shift (but not rotation) of the news impact curve.
• The estimated $\lambda \approx 1$, $\nu \approx 1.5$ implies the conditional variance is driven by $|\varepsilon_t - b|^{1.5}$ — intermediate between the Taylor-Schwert absolute-value GARCH and the Bollerslev squared-residual GARCH, closer to the AGARCH than the GARCH.
• Asymmetry in U.S. equity volatility is primarily a small-shock phenomenon: the shift $b$ is significant across all maintained models; the rotation $c$ is not significant in the GARCH class. This inverts the prior assumption that leverage effects are driven by large negative returns.
• The Box-Cox power choice maps directly to a continuous-time diffusion limit for $\sigma_t$; empirical tests of $\lambda$ thus provide guidance for the choice of diffusion process in option pricing models.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH

Entities Mentioned

• Ludger Hentschel

Quotes

"Current GARCH models only permit either a shift or a rotation, but not both. In principle, these two types of asymmetry are distinct, and should not be treated as substitutes for each other."

"The data prefer models in which large shocks increase volatility by more than they would in either the AGARCH or EGARCH models, but by less than they would in a GARCH model."

My Take

The nesting framework is the canonical reference for understanding how GARCH models relate to each other. The key insight — that shift and rotation are independent dimensions of the news impact curve and map to different shock-size regimes — is underappreciated. Most empirical work tests EGARCH against GARCH without acknowledging that both restrict a jointly estimable parameter space. The empirical finding that the shift (small-shock asymmetry) dominates the rotation (large-shock asymmetry) challenges the standard "leverage effect" narrative, which focuses on crash-like large negative returns. The Box-Cox power test also provides a rare principled guide for choosing the diffusion limit for option pricing; the empirical $\lambda \approx 1$ suggests the Taylor-Schwert absolute-value family is empirically closer than the standard square-root diffusion underlying the Heston model.