Identification-Robust Testing in Endogenous Functional Linear Regression with Weak or Irrelevant Auxiliary Variables

Abstract

We develop dimension-reduction-free tests for the slope function in functional linear regression when the functional regressor may be endogenous or measured with error. The tests are based on a functional moment condition induced by an auxiliary functional variable and do not require estimation of the slope function, providing a functional analogue of Anderson--Rubin-type moment testing. They remain asymptotically valid under weak or even failed relevance of the auxiliary variable: such relevance affects power, not size. The trade-off is that power is confined to alternatives detectable through the moment operator, a subspace we characterize explicitly. We establish the asymptotic null distribution, consistency against detectable alternatives, and local power under drifting alternatives. We also derive the locally optimal test within a class of weighted test statistics. Feasible critical values for implementation of the tests are obtained from data. Simulations show reliable size control and competitive power, including under weak relevance. We illustrate the method using a functional regression analysis of residential electricity demand and temperature distributions in South Korea.