具有成对依赖的马尔可夫随机场中的无意义关联
Nonsense associations in Markov random fields with pairwise dependence
AI总结：
本文首次严谨研究马尔可夫随机场中成对依赖导致的无意义关联，发现方差膨胀、收缩或不变等多种情况，并给出忽略依赖的OLS推断在正依赖下仍有效的一般条件。
摘要：
Yule (1926) 指出了时间序列数据中“无意义相关”的问题，即当两个随机向量各自内部存在依赖时，会导致衡量两者之间依赖关系的指标出现过度离散——即方差膨胀。在此后近一个世纪里，关于无意义相关的文献大量涌现——但几乎全部局限于时间序列领域。据我们所知，本文首次对更一般形式的（正）依赖下的这一现象进行了严谨研究，具体针对格点和图上的马尔可夫随机场。我们同时考虑二元和连续随机向量，并采用三种不同的关联度量：相关性、协方差，以及将一个随机向量投影到另一个上所得的普通最小二乘（OLS）系数。在某些设置中，我们观察到与 Yule 的无意义相关一致的方差膨胀。然而，令人惊讶的是，我们还在某些设置中发现方差收缩，而在其他设置中，方差在依赖下保持不变。或许最值得注意的是，我们找到了在回归误差存在正依赖时，忽略依赖的 OLS 推断仍然有效的一般条件，这与无数教科书和课程中对 OLS 的表述相矛盾。
英文摘要：
Yule (1926) identified the issue of "nonsense correlations" in time series data, where dependence within each of two random vectors causes overdispersion -- i.e. variance inflation -- for measures of dependence between the two. During the near century since then, much has been written about nonsense correlations -- but nearly all of it confined to the time series literature. In this paper we provide the first, to our knowledge, rigorous study of this phenomenon for more general forms of (positive) dependence, specifically for Markov random fields on lattices and graphs. We consider both binary and continuous random vectors and three different measures of association: correlation, covariance, and the ordinary least squares coefficient that results from projecting one random vector onto the other. In some settings we find variance inflation consistent with Yule's nonsense correlation. However, surprisingly, we also find variance deflation in some settings, and in others the variance is unchanged under dependence. Perhaps most notably, we find general conditions under which OLS inference that ignores dependence is valid despite positive dependence in the regression errors, contradicting the presentation of OLS in countless textbooks and courses.
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