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

Key words and phrases: Asymptotic theory, bootstrap inference, endogeneity, Hausman test, instrumental variables, inverse problem.