Parameter Constraints in EGARCH and GJR-GARCH Models

Summary

The document asks what parameter restrictions apply when estimating EGARCH and GJR-GARCH volatility models, contrasting them with the familiar positivity and stationarity conditions for standard GARCH. It presents the two asymmetric model specifications but does not provide an answer or derive bounds for their parameters.

For EGARCH, the question includes log variance, a centered absolute standardized shock term, a signed shock term, and lagged log variance. For GJR-GARCH, it adds a negative-shock indicator interaction to the variance equation. The text therefore identifies the model components whose restrictions a practitioner would need to investigate, especially how asymmetry terms affect admissible values. It offers no empirical evidence, estimation procedure, or discussion of distributional assumptions, so it should be treated as an unanswered question rather than a guide to definitive constraints.

Key ideas

• The document asks how EGARCH and GJR-GARCH parameter constraints compare with standard GARCH restrictions.
• EGARCH models log variance and include both shock magnitude and signed-shock effects.
• GJR-GARCH adds a term that changes the variance response for negative shocks.
• The document poses the estimation question but supplies no solution or evidence.

Tags

# Constraints by estimating GARCH, EGARCH, GJR-GARCH models


# Constraints by estimating GARCH, EGARCH, GJR-GARCH models












I know that by estimating an GARCH model, given by: $$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2,$$

$\omega, \alpha, \beta >0$ and $\alpha + \beta <1$. But what are the constraints and bounds for a EGARCH and GJR-GARCH model, given by:

$$ln(\sigma_t^2) = \omega + \alpha \Big(\frac{|\epsilon|}{\sigma_t} - \sqrt{\frac{2}{\pi}}\Big) +\gamma \frac{\epsilon_{t-1}}{\sigma_{t-1}} + \beta ln(\sigma_{t-1}^2)$$ $$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \gamma \epsilon_{t-1}^2 \cdot 1_{\epsilon_{t-1}<0} + \beta \sigma_{t-1}^2$$

Thanks in advance!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.