Many econometric and data-science applications require a reliable estimate of the covariance matrix, such as Markowitz portfolio selection. When the number of variables is of the same magnitude as the number of observations, this constitutes a difficult estimation problem; the sample covariance matrix certainly will not do. In this paper, we review our work in this area, going back 15+ years. We have promoted various shrinkage estimators, which can be classified into linear and nonlinear. Linear shrinkage is simpler to understand, to derive, and to implement. But nonlinear shrinkage can deliver another level of performance improvement, especially if overlaid with stylized facts such as time-varying co-volatility or factor models.

中文翻译：

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（非）线性收缩的力量：协方差矩阵估计的回顾和指南

许多计量经济学和数据科学应用都需要对协方差矩阵进行可靠的估计，例如Markowitz投资组合选择。当变量的数量与观测值的数量相同时，这将构成一个困难的估计问题；样本协方差矩阵肯定不会。在本文中，我们回顾了我们在这一领域的工作，可以追溯到15年前。我们已经推广了各种收缩率估算器，它们可以分为线性和非线性两种。线性收缩更易于理解，推导和实现。但是非线性收缩可以带来更高的性能改善水平，尤其是当被诸如时变协挥发度或因子模型之类的事实所覆盖时。