It is shown empirically that likelihood function of the GARCH is multi-modal. Hence, the maximum likelihood estimates at local and global maxima will be quantitatively different. Therefore, it is important to start an estimation method with consistent starting value that converge to global maxima. This study compares two estimation methods, BFGS and DE, on the basis of simulation and surface constructed by changing the value of GARCH [Formula: see text] model. DE is superior and consistent throughout the surface, and across distributions. PSX is used as real-world application and it has been found that the estimates obtained from DE are best and unbiased.

中文翻译：

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GARCH模型似然函数中的多模态

经验表明，GARCH 的似然函数是多模态的。因此，局部最大值和全局最大值处的最大似然估计在数量上是不同的。因此，重要的是启动一种具有一致起始值且收敛到全局最大值的估计方法。本研究在模拟的基础上比较了BFGS和DE两种估计方法，通过改变GARCH[公式：见正文]模型的值构建的表​​面。DE 在整个表面和分布范围内都非常出色且一致。PSX 被用作现实世界的应用程序，并且发现从 DE 获得的估计是最好的和无偏的。