A result from a standard linear model course is that the variance of the ordinary least squares (OLS) coefficient of a variable will never decrease when including additional covariates into the regression. The variance inflation factor (VIF) measures the increase of the variance. Another result from a standard linear model or experimental design course is that including additional covariates in a linear model of the outcome on the treatment indicator will never increase the variance of the OLS coefficient of the treatment at least asymptotically. This technique is called the analysis of covariance (ANCOVA), which is often used to improve the efficiency of treatment effect estimation. So we have two paradoxical results: adding covariates never decreases the variance in the first result but never increases the variance in the second result. In fact, these two results are derived under different assumptions. More precisely, the VIF result conditions on the treatment indicators but the ANCOVA result averages over them. Comparing the estimators with and without adjusting for additional covariates in a completely randomized experiment, I show that the former has smaller variance averaging over the treatment indicators, and the latter has smaller variance at the cost of a larger bias conditioning on the treatment indicators. Therefore, there is no real paradox.

中文翻译：

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线性模型中两个看似矛盾的结果：方差膨胀因子和协方差分析

标准线性模型过程的结果是，当在回归中包括其他协变量时，变量的普通最小二乘（OLS）系数的方差将永远不会减小。方差膨胀因子（VIF）衡量方差的增加。标准线性模型或实验设计过程的另一个结果是，在治疗指标的结果的线性模型中包括其他协变量，至少不会渐近地增加治疗的OLS系数的方差。这项技术称为协方差分析（ANCOVA），通常用于提高治疗效果估计的效率。因此，我们有两个矛盾的结果：添加协变量永远不会减少第一个结果的方差，而永远不会增加第二个结果的方差。实际上，这两个结果是在不同的假设下得出的。更准确地说，VIF结果取决于治疗指标，但ANCOVA结果是这些指标的平均值。在完全随机化的实验中，比较有和没有调整其他协变量的估计量，我发现前者在治疗指标上的平均方差较小，而后者在治疗指标上存在较大偏差条件的情况下，方差较小。因此，没有真正的悖论。我表明前者在治疗指标上的平均方差较小，而后者在治疗指标上存在较大偏差条件的情况下，方差较小。因此，没有真正的悖论。我表明前者在治疗指标上的平均方差较小，而后者在治疗指标上存在较大偏差条件的情况下，方差较小。因此，没有真正的悖论。