On the Irrelevance of Impossibility Theorems: The Case of the Long-run Variance
On the Irrelevance of Impossibility Theorems: The Case of the Long-run Variance
复制标题
论不可能性定理的无关性：长期方差的案例
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Linxia Ren
中科院分区：
文献类型：
--
作者：
Pierre Perron;Linxia Ren
有人认为，估计零频率下平稳随机过程的谱密度函数（或所谓的长期方差）是一个不适定问题，因此任何估计都将具有无限的极小极大风险（例如，Pötscher 2002）。最常见的是，它是一些统计量的极限分布中存在的令人讨厌的参数，然后需要对其进行估计以获得具有关键分布的检验统计量。在这种情况下，我们认为这种不可能的结果是无关紧要的。我们表明，在存在导致长期方差估计问题不适定的不连续性的情况下，使用零频率处谱密度函数的真实值会导致测试大小为 0 或 100%，因此导致置信区间完全没有信息。另一方面，基于长期方差标准估计的检验将具有明确定义的极限分布，因此信息量也更大。
It has been argued that estimating the spectral density function of a stationary stochastic process at the zero frequency (or the so-called long-run variance) is an ill-posed problem so that any estimate will have an infinite minimax risk (e.g., Pötscher 2002). Most often it is a nuisance parameter that is present in the limit distribution of some statistic and one then needs an estimate of it to obtain test statistics that have a pivotal distribution. In this context, we argue that such an impossibility result is irrelevant. We show that, in the presence of the discontinuities that cause the ill-posedness of the estimation problem for the long-run variance, using the true value of the spectral density function at frequency zero leads to tests that have either 0 or 100% size and, hence, lead to confidence intervals that are completely uninformative. On the other hand, tests based on standard estimates of the long-run variance will have well defined limit distributions and, accordingly, be more informative.