  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.
