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

Myoung-jae Lee and Chirok Han

Gospodarka Narodowa-The Polish Journal of Economics, 2025, vol. 2024, issue 1

Abstract: Given an exogenous treatment d and covariates x, an ordinary leastsquares (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: Teaching/Communication/Extension/Profession (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:ags:polgne:361264

DOI: 10.22004/ag.econ.361264

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