A note on the Nelson Cao inequality constraints in the GJR-GARCH model: Is there a leverage effect?

TL;DR

This paper examines the Nelson-Cao inequality constraints in GJR-GARCH models, revealing inconsistencies in software implementations and discussing implications for leverage effects and parameter coherence.

Contribution

It clarifies the correct application of Nelson-Cao constraints in GJR-GARCH models and highlights discrepancies in existing software implementations.

Findings

Software often uses inconsistent constraints with Nelson-Cao inequalities.

Monte Carlo simulations show empirical correctness but theoretical incoherence.

Negative asymmetry parameters can occur under typical constraints, challenging leverage effect assumptions.

Abstract

The majority of stylized facts of financial time series and several Value-at-Risk measures are modeled via univariate or multivariate GARCH processes. It is not rare that advanced GARCH models fail to converge for computational reasons, and a usual parsimonious approach is the GJR-GARCH model. There is a disagreement in the literature and the specialized econometric software, on which constraints should be used for the parameters, introducing indirectly the distinction between asymmetry and leverage. We show that the approach used by various software packages is not consistent with the Nelson-Cao inequality constraints. Implementing Monte Carlo simulations, despite of the results being empirically correct, the estimated parameters are not theoretically coherent with the Nelson-Cao constraints for ensuring positivity of conditional variances. On the other hand ruling out the leverage hypothesis, the asymmetry term in the GJR model can take negative values when typical constraints like the condition for the existence of the second and fourth moments, are imposed.