Nonlinear GARCH Model
Nonlinear Generalized Autoregressive Conditional Heteroscedasticity Model · Also known as: NL-GARCH, asymmetric GARCH, GJR-GARCH, nonlinear volatility model
The Nonlinear GARCH model extends the standard GARCH framework to capture asymmetric and nonlinear responses of conditional volatility to past shocks. It allows negative returns (bad news) to amplify volatility more than positive returns of equal magnitude, a phenomenon known as the leverage effect, which is empirically pervasive in financial markets.
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When to use it
Use a Nonlinear GARCH model when you have a financial return series that shows ARCH effects AND evidence of leverage or sign asymmetry — the sign bias test or a clear visual difference between the impact of positive and negative shocks. It is the preferred volatility specification for equity indices, individual stocks, energy prices, and exchange rates where downside risk amplification is economically important. Prefer standard GARCH when the sign bias test shows no asymmetry or when the sample is short (fewer than 500 observations), as the extra parameters reduce precision. Avoid for non-financial series where volatility clustering is absent.
Strengths & limitations
- Captures the leverage effect — negative return shocks amplify volatility more than positive ones — which standard GARCH misses.
- Parsimonious: only one additional parameter (gamma) beyond standard GARCH(1,1) in the GJR specification.
- Improves density forecasts and Value-at-Risk estimates in equity and commodity markets.
- Nested within standard GARCH: a likelihood ratio test directly checks whether the nonlinear extension is warranted.
- EGARCH variant models log-volatility, guaranteeing positive conditional variance without explicit positivity constraints.
- Requires longer samples (ideally 500+ observations) for stable estimation of the extra asymmetry parameter.
- Maximum likelihood estimation is sensitive to starting values and can converge to local optima in volatile regimes.
- The GJR-GARCH positivity conditions (omega > 0, alpha >= 0, alpha + gamma >= 0, beta >= 0) must be checked post-estimation.
- Like all GARCH models, assumes a fixed parametric distribution for innovations; fat-tailed or skewed distributions require explicit specification.
Frequently asked
What is the difference between GJR-GARCH and EGARCH?
Both capture asymmetry, but differently. GJR-GARCH adds an indicator term to the variance level equation and requires positivity constraints. EGARCH models the log of conditional variance, so positivity is automatic, and uses the signed standardised shock to capture both size and sign effects. EGARCH is more flexible but harder to interpret; GJR-GARCH is simpler to estimate and nest-test against standard GARCH.
How do I test whether I need a nonlinear GARCH instead of standard GARCH?
Apply the Engle-Ng (1993) sign bias test to the squared standardised residuals from a standard GARCH fit. A significant sign bias or negative size bias indicates leverage effects. You can also perform a likelihood ratio test comparing GJR-GARCH versus GARCH(1,1) — rejection of the null (gamma = 0) confirms the asymmetric extension is needed.
Does a significant leverage effect mean the model is capturing a structural feature?
Usually yes in equity markets — the negative correlation between return and volatility innovations is well-documented. However, in samples spanning financial crises, the asymmetry parameter may absorb level shifts in volatility rather than genuine sign asymmetry. It is worth complementing the nonlinear GARCH with a structural break GARCH analysis to disentangle the two effects.
What distribution should I use for the innovations?
Financial returns are fat-tailed; using a standard normal distribution underestimates tail risk. The student-t distribution with estimated degrees of freedom is a common improvement. The skewed-t (Hansen 1994) adds an asymmetry parameter for the innovation distribution itself, distinct from the conditional-variance asymmetry captured by the GARCH structure.
Can Nonlinear GARCH be extended to multivariate settings?
Yes. The most common approach is to estimate univariate GJR-GARCH or EGARCH models for each series individually, extract the conditional standard deviations, and then feed them into a DCC-GARCH framework to model time-varying correlations. Full multivariate Nonlinear GARCH systems exist but are rarely used beyond two or three series due to the curse of dimensionality.
Sources
- Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1779-1801. DOI: 10.1111/j.1540-6261.1993.tb05128.x ↗
- Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347-370. DOI: 10.2307/2938260 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Generalized Autoregressive Conditional Heteroscedasticity Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-garch-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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