Estimating quantile treatments without strict overlap

Abstract

We consider the problem of estimating quantile treatment effects without assuming strict overlap , i.e., we do not assume that the propensity score is bounded away from zero. More specifically, we consider an inverse probability weighting (IPW) approach for estimating quantiles in the potential outcomes framework and pay special attention to scenarios where the propensity scores can tend to zero as a regularly varying function. Our approach effectively considers a heavy-tailed objective function for estimating the quantile process. We introduce a truncated IPW estimator that is shown to outperform the standard quantile IPW estimator when strict overlap does not hold. We show that the limiting distribution of the estimated quantile process follows an infinitely divisible law and converges at the rate $n^{1-1/γ}$, where $γ>1$ is the tail index of the propensity scores when they tend to zero. We propose a practical, data-driven procedure for selecting the truncation parameter, grounded in our asymptotic theory. The performance of our estimators is illustrated in numerical experiments and in a dataset that exhibits the presence of extreme propensity scores.

Figures (8)

• Figure 1: The left plot shows the performance of the estimator $\hat{q}_1(\tau,b_n)$ an exponentially increasing grid from $b_{n,1}=\min_{1\leq i \leq n}e(X_i,\hat{\beta})$ to $b_{n,L}=\max_{1\leq i \leq n}e(X_i,\hat{\beta})$ and grid size $L=100$. The plot reports the mean of $\hat{q}_1(\tau,b_n)$ over 100 replications plus/minus 2 bootstrap standard deviations. The right plot shows the results over the same 100 replications for the additive bias correction estimator \ref{['eq:debiased_q']} based on polynomials of order $k=2$.
• Figure 2: The left plot shows $\widehat{\textrm{MSE}(b_{n,\ell})}$ against $[L]$, where the MSE was estimated with 200 bootstrap samples of size $1000$ for one realization of model $(a)$ with $n=2000$. The corresponding subsampling distribution of the debiased estimators $\{\hat{q}_{1,j}^{\mathsf{bc}}(\tau,b_{n,\ell})\}_{j=1}^{200}$ is illustrated on the right plot. More precisely, the plot reports the bootstrap means $\bar{\hat{q}}_{1}^{\mathsf{bc}}(\tau,b_{n,\ell})$ plus/minus two bootstrap standard deviations.
• Figure 3: Estimators of the $0.9$-quantile of $Y(1)$ based on model $(a)$ i.e. Gaussian responses. The sample size increases from left to right, while the tails of the inverse propensity scores become heavier from top to bottom. The boxplots show 100 realizations of the oracle estimator that sees all the $Y_i(1)$'s (light gray), the standard IPW estimator (yellow), the truncated IPW estimator that achieves the best empirical MSE performance (light blue), the debiased truncated IPW estimator that achieves the best empirical MSE performance (dark blue) and the debiased truncated IPW estimator with our proposed bootstrap data-driven choice of the truncation parameter (dark orange).
• Figure 4: Estimators of the $0.9$-quantile of $Y(1)$ based on model $(b)$ i.e. Fréchet responses. The description give for Figure \ref{['fig:Q1_Gaussian']} also applies here.
• Figure 5: Distribution of estimated propensity score for NSW-PSID data.

Theorems & Definitions (26)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Example 4.1
• Example 4.2
• Example 4.3
• Theorem 4.4
• Remark 4.5
• Theorem 5.1