Estimation and inference for distribution and quantile functions in endogenous treatment effect models

Abstract

Given a standard endogenous treatment effect model, we propose nonparametric estimation and inference procedures for the distribution and quantile functions of the potential outcomes among compliers, as well as the local quantile treatment effect function. The preliminary distribution function estimator is a weighted average of indicator functions, but is not monotonically increasing in general. We therefore propose a simple monotonizing method for proper distribution function estimation, and obtain the quantile function estimator by inversion. Our monotonizing method is an alternative to Chernozhukov et al. (2010) and is arguably preferable when the outcome has unbounded support. We show that all the estimators converge weakly to Gaussian processes at the parametric rate, and propose a multiplier bootstrap for uniform inference. Our uniform results thus generalize the pointwise theory developed by Frölich and Melly (2013). Monte Carlo simulations and an application to the effect of fertility on family income distribution illustrate the use of the methods. All results extend to the subpopulation of treated compliers as well.

Keywords:

• Distribution function
• local quantile treatment effect
• monotonicity
• multiplier bootstrap
• quantile function

JEL CLASSIFICATION:

• C21
• C26

Acknowledgment

We would like to thank Blaise Melly for providing us with the data for the application. Yu-Chin Hsu gratefully acknowledges the research support from National Science Council Grant (101-2410-H-001-109-MY2) and the Career Development Award of Academia Sinica, Taiwan. All errors and omissions are our responsibility.

Notes

1 In the standard framework, one would define $Y_{d,z}$ as the potential outcome in the population that would obtain if one were to set D = d and Z = z exogenously and impose the exclusion of the instrumental assumption: $\mathrm{P}(Y_{d,1}=Y_{d,0})=1$ for $d\in \{0,1\}.$ This is equivalent to our approach where we define Y_d directly.

2 As in Donald et al. (2014b), we called q(x) as the instrument propensity score to distinguish from the conventional use of the term propensity score (the conditional probability of being treated).

3 SLE with trimming can nevertheless improve estimation results. See Donald et al. (2014b) for more details.

4 The sign of ${\widehat{\kappa }}_{1,i}$ also depends on the sign of ${\widehat{\Gamma }}_1,$ which converges in probability to ${\Gamma }_1>0.$ For simplicity we assume that ${\widehat{\Gamma }}_1$ is strictly positive. Similarly, we assume that ${\widehat{\Gamma }}_0>0.$

5 We do impose later in Assumption 3.6 that the conditional densities of the potential outcomes are bounded away from zero. However, this assumption is only needed to ensure that if one inverts the distribution function to estimate the quantile function over the entire [0,1] range, then the inverse still converges at the parametric rate. For example, if one is interested only in an “interior” range of quantiles, say, [0.2,0.8], then this assumption is again unneeded.

6 If $\mathcal{Y}=\mathbb{R},$ one can still estimate the quantile functions at the parametric rate over some compact subset of the unit interval, say, $[ϵ,1-ϵ]$ for $0<ϵ<1/2,$ provided that the density functions are strictly positive on the interval.

7 See Comments 6–8 in Donald et al. (2014b) for more details.

8 The weak convergence is in the sense of Definition 1.3.3 of van der Vaart and Wellner (1996), and ${\ell }^{\infty }(\mathcal{Y})$ denotes the set of all uniformly bounded real functions on $\mathcal{Y}.$

9 This means that for any $ϵ>0$ and $\eta >0,$ there exist $\delta >0$ small enough and N > 0 large enough such that for all n > N, $\mathrm{P}(\underset{{\nu }_d(y,y\prime )<\delta }{\sup }|\sqrt{n}({\tilde{F}}_{Y_d|\mathcal{C}}(y)-F_{Y_d|\mathcal{C}}(y))-\sqrt{n}({\tilde{F}}_{Y_d|\mathcal{C}}(y\prime )-F_{Y_d|\mathcal{C}}(y\prime ))|\le ϵ)\ge 1-\eta .$

10 The nonparametric bootstrap is potentially very time consuming given the objects of interest have to be re-estimated for each bootstrap sample. In contrast, the computational burden of the multiplier bootstrap is substantially reduced as all resampling procedures can be simulated simultaneously. However, applying multiplier bootstrap requires a consistent estimation of the functions involved in the influence function. We therefore provide such estimators in this paper.

11 The family income is reported as zero for 5.1% in 1980, 4.6% in 1990, and 4.0% in 2000 sample.

12 Note that ${\Gamma }_1^t$ is not the ATT of Z on D, but ${\Gamma }_1^t/\mathrm{P}(Z=1)$ is. Similarly, $\mathrm{E}\left\{q(X)\left[\frac{ZD1(Y\le y)}{q(X)}-\frac{(1-Z)D1(Y\le y)}{1-q(X)}\right]\right\}/\mathrm{P}(Z=1)$ is the ATT of Z on $D1(Y\le y).$ Hence, the identification result of $F_{Y_1|\mathcal{C}}^t(y)$ in (7.1(7.1) $$\begin{array}{c}{F}_{Y_1|\mathcal{C}}^t(y)=\frac{1}{{\Gamma }_1^t}\mathrm{E}\left\{q(X)\left[\frac{ZD1(Y\le y)}{q(X)}-\frac{(1-Z)D1(Y\le y)}{1-q(X)}\right]\right\},\hfill \\ F_{Y_0|\mathcal{C}}^t(y)=\frac{1}{{\Gamma }_0^t}\mathrm{E}\left\{q(X)\left[\frac{Z(D-1)1(Y\le y)}{q(X)}-\frac{(1-Z)(D-1)1(Y\le y)}{1-q(X)}\right]\right\},\hfill \\ {\Gamma }_1^t=\mathrm{E}\left\{q(X)\left[\frac{ZD}{q(X)}-\frac{(1-Z)D}{1-q(X)}\right]\right\},\hfill \\ {\Gamma }_0^t=\mathrm{E}\left\{q(X)\left[\frac{Z(D-1)}{q(X)}-\frac{(1-Z)(D-1)}{1-q(X)}\right]\right\},\hfill \end{array}$$(7.1) ) can be interpreted as of the ATT of Z on $D1(Y\le y)$ over the ATT of Z on D.