Testing exogeneity in the functional linear regression model

We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of the squared difference of projections of the two estimators for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and consistency under general alternatives. Moreover, we prove bootstrap consistency results for residual-based bootstraps. In simulations, we investigate the finite sample performance of the proposed testing approach and illustrate the superiority of bootstrap-based approaches. In particular, the bootstrap approaches turn out to be much more robust with respect to the choice of the regularization parameter.