Should instrumental variables be used as matching variables?

• Including instrumental variables in regression analysis increases inconsistency.
• Using instrumental variables in propensity score estimation maximizes inconsistency.
• Choosing between control variables and instrumental variables is challenging.

I show that for a linear model and estimating a coefficient on an endogenous explanatory variable, adding covariates that satisfy instrumental variables assumptions increases the amount of inconsistency. A special case is an endogenous binary treatment and estimating a constant treatment effect when matching on covariates that satisfy instrumental variables, rather than ignoribility, assumptions. I also establish a general result that implies that regression adjustment using the propensity score based on instrumental variables actually maximizes the inconsistency among regression-type estimators.