Over the last decade, substantial interest in theoretical econometrics and microeconometrics has been directed towards nonparametric models. Much work has been devoted to the development of novel identification and estimation technieques and in particular, to the identifying power of econometric models under various types of restrictions. Notable attention has been focused on the conditional independence restriction and instrumental variable methods for both continuous and discrete data problems. This immense effort has led to tremendous outcomes in terms of theoretical findings and most importantly, new empirical practices. Nowadays, we face an apparent emphasis on minimal restrictions of nuisance parameters of the model, with a focus on specific structural features at the same time. New models permit the relaxation of implausible restrictions frequently superimposed unwillingly in empirical analysis of plain old econometric models. In this spirit, recent developments in microeconometrics have given rise to increasing interest in partially identified models. In these models, for the credibility of claims, the feature of interest is bounded to a set rather than constituting of a point in the space of parameters or functions. This in turn has its own place in economic practice. Among many appealing and commonly investigated economic circumstances, partial identification frequently arises in econometric inquiry when researchers are faced with discrete data, omnipresent in survey studies. Examples consider a very general class of the limited information discrete outcome models with endogeneity when very little is known about the genesis of the process generating endogenous variable. This thesis contributes to the aforementioned line of research and seeks to address a somewhat limited, but I believe important, range of issues in a great depth. These issues are concerned with the specification of identified sets in so-called single equation models with endogeneity. We achieve identification via instrumental variable restrictions and focus on discrete outcomes as well as discrete endogenous variables. Our focus on discrete, ordered outcome models complements the vast majority of research on econometric design under continuous variation. The latter, even though theoretically sound, often becomes practically infeasible. We believe that this study provides a level of unity to the partial identification framework as a whole and makes steps forward in understanding some aspects of single equation instrumental variable models under discrete variation.