关于GJR- GARCH模型中Nelson Cao不等式约束的注记：是否存在杠杆效应？
A note on the Nelson Cao inequality constraints in the GJR-GARCH model: Is there a leverage effect?
摘要：
金融时间序列的大多数程式化事实和几个风险价值度量都是通过单变量或多变量GARCH过程建模的。由于计算的原因，高级的Gjr模型不能收敛并不罕见，并且通常的简约方法是GJR- GARCH模型。在文献和专门的计量经济学软件中，对于参数应使用哪些约束条件存在分歧，间接地介绍了不对称和杠杆之间的区别。我们表明，各种软件包所使用的方法是不一致的纳尔逊曹不等式约束。实施蒙特卡罗模拟，尽管结果是经验正确的，估计的参数是不符合理论上的纳尔逊-曹约束，以确保积极的条件方差。另一方面，在排除了杠杆假设的情况下，如果加上二阶矩和四阶矩的存在条件等典型的限制，GJR模型中的非对称项也可能为负值。
英文摘要：
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.
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