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 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. Simulation experiments under a wide range of settings show that the proposed estimators perform considerably well. We apply our methodology to estimate the impact of immigration on native wages.

翻译：函数型线性回归作为研究函数型响应变量与外生解释变量之间关系的统计工具广受欢迎。然而在实际应用中，由于遗漏变量和测量误差等因素，我们很难期望所关注的解释变量完全外生。尽管这一问题具有实证相关性，但直到最近才在函数型回归文献中得到研究，且该方向的发展似乎未能充分满足实践者的需求；例如，现有讨论虽着重关注一致估计，但所提估计量的分布性质仍有待深入探索。为填补这一空白，本文提出基于函数主成分分析的一致工具变量估计量，并详细推导其渐近性质。在多种设定下的模拟实验表明，所提估计量表现优异。我们将该方法应用于估计移民对本土工资的影响。