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

We often estimate a parameter of interest psi when the identifying conditions involve a finite-dimensional nuisance parameter theta is an element of R-d. 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 (psi) over cap that solve unbiased estimating equations including theta which is also estimated by solving unbiased estimating equations. This article presents four additional results for settings where (theta) over cap solves (partial) score equations and psi does not depend on theta. This includes many causal inference settings where theta describes the treatment probabilities, missing data settings where theta describes the missingness probabilities, and measurement error settings where theta describes the error distribution. These four additional results are: (1) Counter-intuitively, the asymptotic variance of (psi) over cap is typically smaller when theta is estimated. (2) If estimating theta is ignored, the sandwich estimator for the variance of (psi) over cap is conservative. (3) A consistent sandwich estimator for the variance of (psi) over cap. (4) If (psi) over cap with the true theta plugged in is efficient, the asymptotic variance of (psi) over cap does not depend on whether theta 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.