## Abstract

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.

Original language | English |
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Pages (from-to) | 72-92 |

Number of pages | 21 |

Journal | Stata Journal |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2024 Mar |

### Bibliographical note

Publisher Copyright:© StataCorp LLC 2024.

## Keywords

- instrument score
- IVE
- OLS
- overlap weight
- propensity score
- psr
- st0740

## ASJC Scopus subject areas

- Mathematics (miscellaneous)