Within the inferential context of predicting a distribution of potential outcomes P[y(t)] under a uniform treatment assignment t ∈ T, this paper deals with partial identification of the α-quantile of the distribution of interest Qα[y(t)] under relatively weak and credible monotonicity-type assumptions on the individual response functions and the population selection process. On the theoretical side, the paper adds to the existing results on non-parametric bounds on quantiles with no prior information and under monotone treatment response (MTR) by introducing and studying the identifying properties of α-quantile monotone treatment selection (α-QMTS), α-quantile monotone instrumental variables (α-QMIV) and their combinations. The main result parallels that for the mean; MTR and α-QMTS aid identification in a complementary fashion, so that combining them greatly increases identification power. The theoretical results are illustrated through an empirical application on the Italian returns to educational qualifications. Bounds on several quantiles of ln(wage) under different qualifications and on quantile treatments effects (QTE) are estimated and compared with parametric quantile regression (α-QR) and α-IVQR estimates from the same sample. Remarkably, the α-QMTS & MTR upper bounds on the α-QTE of a college degree versus elementary education imply smaller year-by-year returns than the corresponding α-IVQR point estimates.
