Ordinary least squares and instrumental-variables estimators for any outcome and heterogeneity

Given an exogenous treatment d and covariates x , an ordinary least-squares (OLS) estimator is often applied with a noncontinuous outcome y to find the effect of d , despite the fact that the OLS linear model is invalid. Also, when d is endogenous with an instrument z , an instrumental-variables estimator (IVE) is often applied, again despite the invalid linear model. Furthermore, the treatment effect is likely to be heterogeneous, say, µ 1 ( x ), not a constant as assumed in most linear models. Given these problems, the question is then what kind of effect the OLS and IVE actually estimate. Under some restrictive conditions such as a “saturated model”, the estimated effect is known to be a weighted average, say, E { ω ( x ) µ 1 ( x )}, but in general, OLS and the IVE applied to linear models with a noncontinuous outcome or heterogeneous effect fail to yield a weighted average of heterogeneous treatment effects. Recently, however, it has been found that E { ω ( x ) µ 1 ( x )} can be estimated by OLS and the IVE without those restrictive conditions if the “propensity-score residual” d − E ( d| x ) or the “instrument-score residual” z−E ( z| x ) is used. In this article, we review this recent development and provide a command for OLS and the IVE with the propensity- and instrument-score residuals, which are applicable to any outcome and any heterogeneous effect.