Estimation of average treatment effects for massively unbalanced binary outcomes

Abstract

The MLE of the ATE in the logit model for binary outcomes may have a significant second-order bias if the event has a low probability, which is the case we focus on in this article. We derive the second-order bias of the logit ATE estimator, and we propose a bias-corrected estimator of the ATE. We also propose a variation on the logit model with parameters that are elasticities. Finally, we propose a computational trick that avoids numerical instability in the case of estimation for rare events.

Keywords:

• Average treatment effect
• constant elasticity
• logit
• rare events
• second-order bias

Word Count::

• 8
• 324

Notes

1. See Appendix A.3 for details.

2. Our bias formula looks somewhat different from the one presented in King and Zeng (2001), which in turn is based on McCullagh and Nelder (1989). It can be shown that the two bias formulae are in fact identical. See Appendix B.

3. We assume that the treatment assignment is unconfounded given X.

4. Appendix E also notes that δ = 1 (so that $p_n\propto n^{-1}$) is not compatible with asymptotic normality and that this rate is not appropriate for logit models.

5. His Equation (2) implies that Λ(θ_n)→0, nΛ(θ_n)→∞. Note that Λ(θ_n)→0 means that $1/(1+\exp \left({\theta }_n\right))\to 1$, which in turn means that exp⁡(θ_n)→0, or ${\theta }_n\to -\mathrm{\infty }$. Also note that nΛ(θ_n)→∞ rules out the Poisson approximation, because if $\mathrm{\Lambda }\left({\theta }_n\right)\propto n^{-1}$, we cannot have nΛ(θ_n)→∞. On the other hand, nΛ(θ_n)→∞ is satisfied as long as $\mathrm{\Lambda }\left({\theta }_n\right)\propto n^{-\delta }$ with $0\le \delta <1$.

6. In Table 1, for the ${\alpha }_0=-2.5$ and n = 500 combination, the bias of $\hat{\beta }$ is 0.0168, where the true value of β is 1. So, the MLE overestimates the elasticity by 1.68%, which is reduced to 0.41% by the bias-corrected estimator.

7. When $\alpha \approx -\mathrm{\infty }$ and the support of x is bounded, we can guarantee that $\exp (\alpha +\mathit{\beta x})<1$. If the support of x is not bounded, but if we are sure that $\alpha +\mathit{\beta x}<0$ for most values of x, we may want to adopt a parameterization $P\left(x\right)=\mathrm{\Psi }(\alpha +\mathit{\beta x})$ where $\mathrm{\Psi }\left(t\right)=\frac{1}{2}\exp \left(t\right)$ if t < 0, and $\mathrm{\Psi }\left(t\right)=1-\frac{1}{2}\exp (-t)$ if t > 0.

8. It is in the sense that $\frac{t}{1+t}=t+O\left(t^2\right).$

9. p denotes the probability of y = 1 for each parameter combination.

10. As was discussed in Remark 2, the second-order bias of the ATE is zero when the propensity score is constant. In order to verify this result, we will first consider the case that the distributions of x are identical over the D = 1 and D = 0 subsamples. In Tables G1, G2, and G3 in Appendix G, we evaluate the performance of various estimators of the ATE under random assignment.

11. Our simulation results are for the case of a single covariate. Adding covariates did not change the conclusions.

12. p₁ and p₀ denote the probabilities of y=1 for D_i = 1 and D_i = 0, i.e., the treated and control sub-samples.