Instrumental variable estimation in functional linear models

In an increasing number of empirical studies, the dimensionality measured e.g. as the size of the parameter space of interest, can be very large. Two instances of large dimensional models are the linear regression with a large number of covariates and the estimation of a regression function with many instrumental variables. An appropriate setting to analyze high dimensional problems is provided by a functional linear model, in which the covariates belong to Hilbert spaces. This paper considers the case where covariates are endogenous and assumes the existence of instrumental variables (that are functional as well). The paper shows that estimating the regression function is a linear ill-posed inverse problem, with a known but data-dependent operator. The first contribution is to analyze the rate of convergence of the penalized least squares estimator. Based on the result, we discuss the notion of “instrument strength” in the high dimensional setting. We also discuss a generalized version of the estimator, when the problem is premultiplied by an instrument-dependent operator. This extends the technology of Generalized Method of Moments to high dimensional, functional data. A central limit theorem is also established on the inner product of the estimator. The studied estimators are easy and fast to implement, and the finite-sample performance is discussed through simulations and an application to the impact of age-specific fertility rate curves on yearly economic growth in the United Kingdom.