On the identification of structural linear functionals

This paper asks which aspects of a structural Nonparametric Instrumental Variables Regression (NPIVR) can be identified well and which ones cannot. It contributes to answering this question by characterising the identified set of linear continuous functionals of the NPIVR under norm constraints. Each element of the identified set of NPIVR can be written as the sum of a common 'identifiable component' and an idiosyncratic 'unidentifiable component'. The identified set for any continuous linear functional is shown to be a closed interval, whose midpoint is the functional applied to the 'identifiable component'. The formula for the length of the identified set extends the popular omitted variables formula of classical linear regression. Some examples illustrate the wide applicability and utility of our identification result, including bounds and a new identification condition for point-evaluation functionals. The main ideas are illustrated with an empirical application of the effect of children on labour market outcomes.