Both the GJR and the GARCH-specifications are used quite often in the finance literature. The GARCH is defined as:
$$ \sigma^2_t = \omega + \alpha\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}$$
and the GJR-GARCH reads as follows
$$ \sigma^2_t = \omega + (\alpha+\gamma \mathbb{I}_{t-1})\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}$$
where $\mathbb{I}_{t-1}$ is the indicator function:
$\mathbb{I}_{t-1}(\varepsilon_{t-1})=\varepsilon_{t-1}$ for $\varepsilon_{t-1}>0$ and
$\mathbb{I}_{t-1}(\varepsilon_{t-1})=0$ otherwise.
According to research (Laurent et al. and Brownlees et al.) the GJR models generally perform better than the GARCH specification. Thus, including a leverage effect leads to enhanced forecasting performance.
My question is: what are reasons for the GJR specification to perform better than the GARCH model. Also, it can occur that both models perform on par. What could be reasons for this observation of equal forecasting performance?
Thank you very much.
References:
Brownlees, C. T., Engle, R. F., & Kelly, B. T. (2011). A practical guide to volatility forecasting through calm and storm. Available at SSRN 1502915.
Laurent, S., Rombouts, J. V., & Violante, F. (2012). On the forecasting accuracy of multivariate GARCH models. Journal of Applied Econometrics, 27(6), 934-955.
Best Answer
GJR-GARCH offers what vanilla GARCH has to offer, plus the leverage effect. In general, a richer model (e.g. GJR-GARCH) will fit the sample data better (at least not worse) than a simpler model (e.g. GARCH) -- when fitted using unconstrained maximization such as (unpenalized) maximum likelihood. So in sample GJR-GARCH cannot lose to a vanilla GARCH.
Out of sample two things matter:
Regarding 1., leverage effect is among the stylized facts of financial returns, so the added feature in the GJR-GARCH model seems relevant. Then it is enough to estimate the model parameters with sufficient precision for a GJR-GARCH to beat vanilla GARCH.
Regarding 2., when using daily data with hundreds or thousands of observations, apparently the estimation precision is not that big of an issue, which allows GJR-GARCH to beat vanilla GARCH.
On the other hand, if the performance of the two models is on par, either the estimation precision (small sample?) or the relevance of leverage (not that characteristic to the asset at hand?), or a combination of both is to blame.