Functional instrumental variable regression with an application to estimating the impact of immigration on native wages

Functional linear regression gets its popularity as a statistical tool to study the relationship between function-valued response and exogenous explanatory variables. However, in practice, it is hard to expect that the explanatory variables of interest are perfectly exogenous, due to, for example, the presence of omitted variables and measurement error. Despite its empirical relevance, it was not until recently that this issue of endogeneity was studied in the literature on functional regression, and the development in this direction does not seem to sufficiently meet practitioners' needs; for example, this issue has been discussed with paying particular attention on consistent estimation and thus the distributional properties of the proposed estimators still remain to be further explored. To fill this gap, this paper proposes new consistent FPCA-based instrumental variable estimators and develops their asymptotic properties in detail. We also provide a novel test for examining if various characteristics of the response variable depend on the explanatory variable in our model. Simulation experiments under a wide range of settings show that the proposed estimators and test perform considerably well. We apply our methodology to estimate the impact of immigration on native wages.