  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. 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.
