Two paradoxical results in linear models: the variance inflation factor and the analysis of covariance
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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 if we add 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 requires random treatment indicators. In a completely randomized experiment, the estimator without adjusting for additional covariates has smaller conditional variance at the cost of a larger conditional bias, compared to the estimator adjusting for additional covariates. Thus, there is no real paradox.
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