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

Testing exogeneity in the functional linear regression model