Interpreting Negative Coefficients in GARCH Models
Summary
The document explains how to interpret negative estimated coefficients in GARCH variance equations. A negative coefficient does not automatically make conditional variance negative, so the relevant checks are whether fitted variance becomes negative and whether the estimated parameters satisfy the model’s applicable admissibility conditions. In a standard GARCH model, constraining coefficients to be positive during likelihood maximization is a sufficient condition for positivity, but it is not always necessary.
The answer cautions against mechanically deleting terms with negative estimates. Restrictions differ across model families, including EGARCH and GJR-type specifications, and must be assessed for the particular formulation. The cited discussion points to work on inequality constraints and the original standard GARCH formulation, but supplies no estimation example or comparison of alternative constraints. Its practical message is to validate the variance path and model-specific conditions rather than apply one universal sign rule.
Key ideas
- A negative estimated coefficient does not by itself imply that conditional variance becomes negative.
- Check the fitted variance path and any known parameter restrictions for the chosen model.
- Positive coefficients are a sufficient, but not always necessary, condition in standard GARCH.
- Do not automatically remove every term whose estimated coefficient is negative.
- EGARCH and other extensions have their own admissibility conditions.
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Full text
# Negative signs in GARCH equation
# Negative signs in GARCH equation
When one try to fit a GARCH on a time series it may happen that one or more coefficients in the estimation output have negative sign. In these cases:
- all the negative coefficients (and relative orders) must necessary be removed from the equation, even if they are significant?
- Is this eventually true for both ARCH and GARCH components?
- The fact that variance equation can't contain negative coefficients is even true for the other garch extensions (EGARCH, TARCH, ...)?
Thank you
## Answer by Malick (score 2, accepted)
https://quant.stackexchange.com/a/41540
A negative coefficient does not necessarily entail a negative $\sigma^{2}$. Usually we do not impose positivity constraints during estimation, then we check if $\sigma^{2}$ takes some negative values or if coefficients respect some known positivity constraints (when these constraints are known).
Regarding the standard Garch model, you can force all the coefficients to be positive during maximisation of the likelihood but it is not a necessary condition (just a sufficient condition), the original formulation of Bollerslev (86) imposes this sufficient constraint. If you employ this constraint, you should not "remove" negative coefficients as they should never appear as a plausible result during estimation.
For each Garch-type of model, sufficient conditions (i.e parameter restrictions) are different, as an example see Tsai, H., & Chan, K. S. (2008) for Garch model. Egarch, GJR... have different constraints.
Tsai, H., & Chan, K. S. (2008). A note on inequality constraints in the garch model. Econometric Theory, 24(3), 823–828. http://doi.org/10.1017/S0266466608080432
Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307–327.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.