arXiv:1411.2056v1 [stat.ME] 7 Nov 2014 $^{∗}$Department of Economics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599. Email: juhkim@email.unc.edu. I am greatly indebted to my advisor Bernard Salani´ e for his guidance and support throughout this project. This paper has benefited from discussions with Andrew Chesher, Toru Kitagawa, Dennis Kristensen, and Ismael Mourifi´ e. I thank participants of the econometrics colloquiums at Columbia. All errors are mine. 1 I study partial identification of distributional parameters in triangular systems. The model consists of a nonparametric outcome equation and a selection equation. This allows for general unobserved heterogeneity in potential outcomes and selection on unobservables. The distribu- tional parameters considered in this paper are the marginal distributions of potential outcomes, their joint distribution, and the distribution of treatment effects. I explore different types of plausible restrictions to tighten existing bounds on these parameters. The restrictions include stochastic dominance, quadrant dependence between unobservables, and monotonicity between potential outcomes. My identification applies to the whole population without a full support condition on instrumental variables and does not rely on rank similarity. I also provide numer- ical examples to illustrate identifying power of the restrictions. Keywords: Partial Identification, Triangular Systems, Stochastic Monotonicity, Monotone Treatment Response, Quadrant Dependence JEL Classifications: C14, C21, C61, C81. Abstract July 3, 2018 Ju Hyun Kim ∗ Partial Identification of Distributional Parameters in Triangular Systems