  Standard regression adjustment gives inconsistent estimates of causal effects
when there are time-varying treatment effects and time-varying covariates.
Loosely speaking, the issue is that some covariates are post-treatment
variables because they may be affected by prior treatment status, and
regressing out post-treatment variables causes bias. More precisely, the bias
is due to certain non-confounding latent variables that create colliders in the
causal graph. These latent variables, which we call phantoms, do not harm the
identifiability of the causal effect, but they render naive regression
estimates inconsistent. Motivated by this, we ask: how can we modify regression
methods so that they hold up even in the presence of phantoms? We develop an
estimator for this setting based on regression modeling (linear, log-linear,
probit and Cox regression), proving that it is consistent for a reasonable
causal estimand. In particular, the estimator is a regression model fit with a
simple adjustment for collinearity, making it easy to understand and implement
with standard regression software. The proposed estimators are instances of the
parametric g-formula, extending the regression-with-residuals approach to
several canonical nonlinear models.
