How estimating nuisance parameters can reduce the variance (with consistent variance estimation)

Abstract

We often estimate a parameter of interest when the identifying conditions involve a finite‐dimensional nuisance parameter . Examples from causal inference are inverse probability weighting, marginal structural models and structural nested models, which all lead to unbiased estimating equations. This article presents a consistent sandwich estimator for the variance of estimators that solve unbiased estimating equations including which is also estimated by solving unbiased estimating equations. This article presents four additional results for settings where solves (partial) score equations and does not depend on . This includes many causal inference settings where describes the treatment probabilities, missing data settings where describes the missingness probabilities, and measurement error settings where describes the error distribution. These four additional results are: (1) Counter‐intuitively, the asymptotic variance of is typically smaller when is estimated. (2) If estimating is ignored, the sandwich estimator for the variance of is conservative. (3) A consistent sandwich estimator for the variance of . (4) If with the true plugged in is efficient, the asymptotic variance of does not depend on whether is estimated. To illustrate we use observational data to calculate confidence intervals for (1) the effect of cazavi versus colistin on bacterial infections and (2) how the effect of antiretroviral treatment depends on its initiation time in HIV‐infected patients.