Extremal quantile treatment effects. (English) Zbl 1454.62082

Zhang, Yichong

Ann. Stat. 46, No. 6B, 3707-3740 (2018).

Summary: 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.

MSC:

62E20	Asymptotic distribution theory in statistics
62G32	Statistics of extreme values; tail inference
62P20	Applications of statistics to economics

Keywords:

extreme quantile; intermediate quantile

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. Econometrica70 91–117. · Zbl 1104.62331 · doi:10.1111/1468-0262.00270
[2]	Abadie, A. and Imbens, G. W. (2006). Large sample properties of matching estimators for average treatment effects. Econometrica74 235–267. · Zbl 1112.62042 · doi:10.1111/j.1468-0262.2006.00655.x
[3]	Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica71 1795–1843. · Zbl 1154.62323 · doi:10.1111/1468-0262.00470
[4]	Andrews, D. W. and Cheng, X. (2012). Estimation and inference with weak, semi-strong, and strong identification. Econometrica80 2153–2211. · Zbl 1274.62160 · doi:10.3982/ECTA9456
[5]	Andrews, D. W. and Cheng, X. (2013). Maximum likelihood estimation and uniform inference with sporadic identification failure. J. Econometrics173 36–56. · Zbl 1443.62420 · doi:10.1016/j.jeconom.2012.10.003
[6]	Belloni, A., Chernozhukov, V., Fernández-Val, I. and Hansen, C. (2017). Program evaluation and causal inference with high-dimensional data. Econometrica85 233–298. · Zbl 1410.62197 · doi:10.3982/ECTA12723
[7]	Bertail, P., Haefke, C., Politis, D. N. and White, H. (2004). Subsampling the distribution of diverging statistics with applications to finance. J. Econometrics120 295–326. · Zbl 1282.62188 · doi:10.1016/S0304-4076(03)00215-X
[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. · Zbl 1072.62551 · doi:10.1080/01621459.1999.10474151
[9]	Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist.9 1196–1217. · Zbl 0449.62034 · doi:10.1214/aos/1176345637
[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. Sinica18 967–985. · Zbl 1534.62057
[11]	Billingsley, P. (1999). Convergence of Probability Measures493, 2nd ed. Wiley, Hoboken, NJ. · Zbl 0944.60003
[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. Econometrics182 87–99. · Zbl 1311.62028 · doi:10.1016/j.jeconom.2014.04.010
[15]	Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist.33 806–839. · Zbl 1068.62063 · doi:10.1214/009053604000001165
[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. · Zbl 1216.62077 · doi:10.1093/restud/rdq020
[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. Econometrica81 2205–2268. · Zbl 1326.62223 · doi:10.3982/ECTA10582
[19]	Chernozhukov, V. and Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica73 245–261. · Zbl 1152.91706 · doi:10.1111/j.1468-0262.2005.00570.x
[20]	Chernozhukov, V. and Hansen, C. (2008). Instrumental variable quantile regression: A robust inference approach. J. Econometrics142 379–398. · Zbl 1418.62154 · doi:10.1016/j.jeconom.2007.06.005
[21]	Chernozhukov, V. and Hong, H. (2004). Likelihood estimation and inference in a class of nonregular econometric models. Econometrica72 1445–1480. · Zbl 1091.62135 · doi:10.1111/j.1468-0262.2004.00540.x
[22]	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. · Zbl 1386.62073 · doi:10.1016/j.jeconom.2017.11.004
[23]	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. · Zbl 0699.62028 · doi:10.1214/aos/1176347396
[24]	Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist.2 267–277. · Zbl 0277.62034 · doi:10.1214/aos/1176342662
[25]	Falk, M. (1991). A note on the inverse bootstrap process for large quantiles. Stochastic Process. Appl.38 359–363. · Zbl 0727.62051 · doi:10.1016/0304-4149(91)90100-Q
[26]	Feigin, P. D. and Resnick, S. I. (1994). Limit distributions for linear programming time series estimators. Stochastic Process. Appl.51 135–165. · Zbl 0819.62070 · doi:10.1016/0304-4149(94)90022-1
[27]	Firpo, S. (2007). Efficient semiparametric estimation of quantile treatment effects. Econometrica75 259–276. · Zbl 1201.62043 · doi:10.1111/j.1468-0262.2007.00738.x
[28]	Frölich, M. and Melly, B. (2013). Unconditional quantile treatment effects under endogeneity. J. Bus. Econom. Statist.31 346–357.
[29]	Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica66 315–331. · Zbl 1055.62572 · doi:10.2307/2998560
[30]	Hall, P. and Van Keilegom, I. (2009). Nonparametric “regression” when errors are positioned at end-points. Bernoulli15 614–633. · Zbl 1200.62036 · doi:10.3150/08-BEJ173
[31]	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.
[32]	Hirano, K., Imbens, G. W. and Ridder, G. (2003). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica71 1161–1189. · Zbl 1152.62328 · doi:10.1111/1468-0262.00442
[33]	Hirano, K. and Porter, J. R. (2003). Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica71 1307–1338. · Zbl 1154.62414 · doi:10.1111/1468-0262.00451
[34]	Imai, K. and Ratkovic, M. (2014). Covariate balancing propensity score. J. R. Stat. Soc. Ser. B. Stat. Methodol.76 243–263. · Zbl 1411.62025
[35]	Knight, K. (2001). Limiting distributions of linear programming estimators. Extremes4 87–103. · Zbl 1008.62065 · doi:10.1023/A:1013991808181
[36]	Lee, S. and Seo, M. H. (2008). Semiparametric estimation of a binary response model with a change-point due to a covariate threshold. J. Econometrics144 492–499. · Zbl 1418.62504 · doi:10.1016/j.jeconom.2008.02.003
[37]	Lehmann, E. L. (1974). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco, CA. · Zbl 0354.62038
[38]	Newey, W. K. and McFadden, D. (1994). Large sample estimation and hypothesis testing. Handb. Econom.4 2111–2245.
[39]	Pitman, E. J. G. (1949). Lecture Notes on Nonparametric Statistical Inference. Columbia Univ.
[40]	Portnoy, S. and Jurečková, J. (1999). On extreme regression quantiles. Extremes2 227–243. · Zbl 0959.62047 · doi:10.1023/A:1009931219041
[41]	Robins, J. M. and Rotnitzky, A. (1995). Semiparametric efficiency in multivariate regression models with missing data. J. Amer. Statist. Assoc.90 122–129. · Zbl 0818.62043 · doi:10.1080/01621459.1995.10476494
[42]	Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika70 41–55. · Zbl 0522.62091 · doi:10.1093/biomet/70.1.41
[43]	Rubin, D. B. (1976). Inference and missing data. Biometrika63 581–592. · Zbl 0344.62034 · doi:10.1093/biomet/63.3.581
[44]	Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist.6 34–58. · Zbl 0383.62021 · doi:10.1214/aos/1176344064
[45]	Smith, R. L. (1994). Nonregular regression. Biometrika81 173–183. · Zbl 0803.62056 · doi:10.1093/biomet/81.1.173
[46]	Staiger, D. and Stock, J. H. (1997). Instrumental variables regression with weak instruments. Econometrica65 557–586. · Zbl 0871.62101 · doi:10.2307/2171753
[47]	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. · Zbl 1121.62066 · doi:10.1017/CBO9780511614491
[48]	van der Vaart, A. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[49]	Zarepour, M. and Knight, K. (1999). Bootstrapping point processes with some applications. Stochastic Process. Appl.84 81–90. · Zbl 0997.60053 · doi:10.1016/S0304-4149(99)00052-6
[50]	Zhang, Y. (2018). Supplement to “Extremal quantile treatment effects.