## The Annals of Statistics

### Extremal quantile treatment effects

Yichong Zhang

#### Abstract

This paper establishes an asymptotic theory and inference method for quantile treatment effect estimators when the quantile index is close to or equal to zero. Such quantile treatment effects are of interest in many applications, such as the effect of maternal smoking on an infant’s adverse birth outcomes. When the quantile index is close to zero, the sparsity of data jeopardizes conventional asymptotic theory and bootstrap inference. When the quantile index is zero, there are no existing inference methods directly applicable in the treatment effect context. This paper addresses both of these issues by proposing new inference methods that are shown to be asymptotically valid as well as having adequate finite sample properties.

#### Article information

Source
Ann. Statist., Volume 46, Number 6B (2018), 3707-3740.

Dates
Revised: November 2017
First available in Project Euclid: 11 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1536631288

Digital Object Identifier
doi:10.1214/17-AOS1673

Mathematical Reviews number (MathSciNet)
MR3852666

Zentralblatt MATH identifier
06965702

#### Citation

Zhang, Yichong. Extremal quantile treatment effects. Ann. Statist. 46 (2018), no. 6B, 3707--3740. doi:10.1214/17-AOS1673. https://projecteuclid.org/euclid.aos/1536631288

#### References

• [1] Abadie, A., Angrist, J. and Imbens, G. (2002). Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica 70 91–117.
• [2] Abadie, A. and Imbens, G. W. (2006). Large sample properties of matching estimators for average treatment effects. Econometrica 74 235–267.
• [3] Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71 1795–1843.
• [4] Andrews, D. W. and Cheng, X. (2012). Estimation and inference with weak, semi-strong, and strong identification. Econometrica 80 2153–2211.
• [5] Andrews, D. W. and Cheng, X. (2013). Maximum likelihood estimation and uniform inference with sporadic identification failure. J. Econometrics 173 36–56.
• [6] Belloni, A., Chernozhukov, V., Fernández-Val, I. and Hansen, C. (2017). Program evaluation and causal inference with high-dimensional data. Econometrica 85 233–298.
• [7] Bertail, P., Haefke, C., Politis, D. N. and White, H. (2004). Subsampling the distribution of diverging statistics with applications to finance. J. Econometrics 120 295–326.
• [8] Bertail, P., Politis, D. N. and Romano, J. P. (1999). On subsampling estimators with unknown rate of convergence. J. Amer. Statist. Assoc. 94 569–579.
• [9] Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196–1217.
• [10] Bickel, P. J. and Sakov, A. (2008). On the choice of $m$ in the $m$ out of $n$ bootstrap and confidence bounds for extrema. Statist. Sinica 18 967–985.
• [11] Billingsley, P. (1999). Convergence of Probability Measures 493, 2nd ed. Wiley, Hoboken, NJ.
• [12] Bitler, M., Gelbach, J. and Hoynes, H. (2006). What mean impacts miss: Distributional effects of welfare reform experiments. Am. Econ. Rev. 96 988–1012.
• [13] Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models. Handb. Econom. 6 5549–5632.
• [14] Chen, X., Ponomareva, M. and Tamer, E. (2014). Likelihood inference in some finite mixture models. J. Econometrics 182 87–99.
• [15] Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist. 33 806–839.
• [16] Chernozhukov, V. and Fernández-Val, I. (2011). Inference for extremal conditional quantile models, with an application to market and birthweight risks. Rev. Econ. Stud. 78 559–589. Supplementary data available online.
• [17] Chernozhukov, V., Fernández-Val, I. and Kaji, T. (2016). Extremal quantile regression: An overview. ArXiv Preprint ArXiv:1612.06850.
• [18] Chernozhukov, V., Fernández-Val, I. and Melly, B. (2013). Inference on counterfactual distributions. Econometrica 81 2205–2268.
• [19] Chernozhukov, V. and Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica 73 245–261.
• [20] Chernozhukov, V. and Hansen, C. (2008). Instrumental variable quantile regression: A robust inference approach. J. Econometrics 142 379–398.
• [21] Chernozhukov, V. and Hong, H. (2004). Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72 1445–1480.
• [22] Chernozhukov, V. V. (2000). Conditional extremes and near-extremes: Concepts, asymptotic theory, and economic applications. Ph.D. thesis, Stanford Univ., Stanford, CA.
• [23] D’Haultfoeuille, X., Maurel, A. and Zhang, Y. (2018). Extremal quantile regressions for selection models and the black-white wage gap. J. Econometrics. 203 129–142.
• [24] Dekkers, A. L. M. and de Haan, L. (1989). On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17 1795–1832.
• [25] Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist. 2 267–277.
• [26] Falk, M. (1991). A note on the inverse bootstrap process for large quantiles. Stochastic Process. Appl. 38 359–363.
• [27] Feigin, P. D. and Resnick, S. I. (1994). Limit distributions for linear programming time series estimators. Stochastic Process. Appl. 51 135–165.
• [28] Firpo, S. (2007). Efficient semiparametric estimation of quantile treatment effects. Econometrica 75 259–276.
• [29] Firpo, S. and Rothe, C. (2014). Semiparametric estimation and inference using doubly robust moment conditions. Technical Report, IZA Discussion Paper.
• [30] Frölich, M. and Melly, B. (2013). Unconditional quantile treatment effects under endogeneity. J. Bus. Econom. Statist. 31 346–357.
• [31] Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66 315–331.
• [32] Hall, P. and Van Keilegom, I. (2009). Nonparametric “regression” when errors are positioned at end-points. Bernoulli 15 614–633.
• [33] Hansen, B. E. (2014). Nonparametric sieve regression: Least squares, averaging least squares, and cross-validation. In The Oxford Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics 215–248. Oxford Univ. Press, Oxford.
• [34] Hirano, K., Imbens, G. W. and Ridder, G. (2003). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71 1161–1189.
• [35] Hirano, K. and Porter, J. R. (2003). Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica 71 1307–1338.
• [36] Imai, K. and Ratkovic, M. (2014). Covariate balancing propensity score. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 243–263.
• [37] Khan, S. and Nekipelov, D. (2013). On uniform inference in nonlinear models with endogeneity. Economic Research Initiatives at Duke (ERID) Working Paper 153.
• [38] Knight, K. (2001). Limiting distributions of linear programming estimators. Extremes 4 87–103.
• [39] Lee, S. and Seo, M. H. (2008). Semiparametric estimation of a binary response model with a change-point due to a covariate threshold. J. Econometrics 144 492–499.
• [40] Lehmann, E. L. (1974). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco, CA.
• [41] Newey, W. K. and McFadden, D. (1994). Large sample estimation and hypothesis testing. Handb. Econom. 4 2111–2245.
• [42] Pitman, E. J. G. (1949). Lecture Notes on Nonparametric Statistical Inference. Columbia Univ.
• [43] Portnoy, S. and Jurečková, J. (1999). On extreme regression quantiles. Extremes 2 227–243.
• [44] Robins, J. M. and Rotnitzky, A. (1995). Semiparametric efficiency in multivariate regression models with missing data. J. Amer. Statist. Assoc. 90 122–129.
• [45] Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika 70 41–55.
• [46] Rubin, D. B. (1976). Inference and missing data. Biometrika 63 581–592.
• [47] Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist. 6 34–58.
• [48] Smith, R. L. (1994). Nonregular regression. Biometrika 81 173–183.
• [49] Staiger, D. and Stock, J. H. (1997). Instrumental variables regression with weak instruments. Econometrica 65 557–586.
• [50] Stock, J. H. and Yogo, M. (2005). Testing for weak instruments in linear IV regression. In Identification and Inference for Econometric Models 80–108. Cambridge Univ. Press, Cambridge.
• [51] van der Vaart, A. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
• [52] Zarepour, M. and Knight, K. (1999). Bootstrapping point processes with some applications. Stochastic Process. Appl. 84 81–90.
• [53] Zhang, Y. (2018). Supplement to “Extremal quantile treatment effects.” DOI:10.1214/17-AOS1673SUPP.

#### Supplemental materials

• Supplement to “Extremal quantile treatment effects”. This supplement contains all the proofs, two empirical applications, and simulation results.