Estimating nuisance parameters often reduces the variance (with consistent variance estimation)

In many applications, to estimate a parameter or quantity of interest psi, a finite-dimensional nuisance parameter theta is estimated first. For example, many estimators in causal inference depend on the propensity score: the probability of (possibly time-dependent) treatment given the past. theta is often estimated in a first step, which can affect the variance of the estimator for psi. theta is often estimated by maximum (partial) likelihood. Inverse Probability Weighting, Marginal Structural Models and Structural Nested Models are well-known causal inference examples, where one often posits a (pooled) logistic regression model for the treatment (initiation) and/or censoring probabilities, and estimates these with standard software, so by maximum partial likelihood. Inverse Probability Weighting, Marginal Structural Models and Structural Nested Models have something else in common: they can all be shown to be based on unbiased estimating equations. This paper has four main results for estimators psi-hat based on unbiased estimating equations including theta. First, it shows that the true limiting variance of psi-hat is smaller or remains the same when theta is estimated by solving (partial) score equations, compared to if theta were known and plugged in. Second, it shows that if estimating theta using (partial) score equations is ignored, the resulting sandwich estimator for the variance of psi-hat is conservative. Third, it provides a variance correction. Fourth, it shows that if the estimator psi-hat with the true theta plugged in is efficient, the true limiting variance of psi-hat does not depend on whether or not theta is estimated, and the sandwich estimator for the variance of psi-hat ignoring estimation of theta is consistent. These findings hold in semiparametric and parametric settings where the parameters of interest psi are estimated based on unbiased estimating equations.