Linear Probability Model Revisited: Why It Works and How It Should Be Specified

Lee, Myoung‐jae; Lee, Goeun; Choi, Jin-young

doi:10.1177/00491241231176850

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…One might think that (1.4) and (1.5) are "artifacts" due to the restrictive condition E(D j |X) = L(D j |X), but that is not the case. For binary D and any Y , Lee et al (2023) showed that the estimand of the D-slope in the OLS of Y on (D, X) is a weighted average of E(Y 1 − Y 0 |X) plus a bias, and the OLS D-slope is inconsistent because its estimand is not zero even when Angrist (1998) and Angrist and Pischke (2009), however, then the bias is zero, and the OLS estimand becomes a weighted average of E(Y 1 − Y 0 |X). Doing analogously, we imposed the condition…”

Section: Introductionmentioning

confidence: 99%

Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable

Lee

Choi

Lee

2023

Journal of Causal Inference

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When a binary treatment D D is possibly endogenous, a binary instrument δ \delta is often used to identify the “effect on compliers.” If covariates X X affect both D D and an outcome Y Y , X X should be controlled to identify the “ X X -conditional complier effect.” However, its nonparametric estimation leads to the well-known dimension problem. To avoid this problem while capturing the effect heterogeneity, we identify the complier effect heterogeneous with respect to only the one-dimensional “instrument score” E ( δ ∣ X ) E\left(\delta | X) for non-randomized δ \delta . This effect heterogeneity is minimal, in the sense that any other “balancing score” is finer than the instrument score. We establish two critical “reduced-form models” that are linear in D D or δ \delta , even though no parametric assumption is imposed. The models hold for any form of Y Y (continuous, binary, count, …). The desired effect is then estimated using either single index model estimators or an instrumental variable estimator after applying a power approximation to the effect. Simulation and empirical studies are performed to illustrate the proposed approaches.

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Section: Introductionmentioning

confidence: 99%

Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable

Lee

Choi

Lee

2023

Journal of Causal Inference

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

Lee,

Han

2024

The Stata Journal: Promoting communications on statistics and S

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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.

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Linear Probability Model Revisited: Why It Works and How It Should Be Specified

Cited by 2 publications

References 18 publications

Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable

Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable

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

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