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

Myoung-jae Lee and Chirok Han

Stata Journal, 2024, vol. 24, issue 1, 72-92

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, μ1(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)μ1(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)μ1(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.

Keywords: psr; OLS; IVE; propensity score; instrument score; overlap weight (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1177/1536867X241233645

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