Marcia M Schafgans and Victoria Zinde-Walsh

Published 9 May 2024

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Nonparametric kernel regression is widely used in econometrics and has been applied to models with cross-sectional, time series and panel data. As functional data analysis is gaining attention, our analysis extends to the presence of multiple functional regressors or a mixture of functional and scalar regressors. We address point-wise asymptotic properties of the estimators by relaxing restrictive smoothness assumptions on the conditioning distribution. Firstly, we establish the rate and asymptotic normality for the Nadaraya-Watson estimator for arbitrary conditioning distributions in Rq; for an absolutely continuous distribution we provide statistical guarantees for the standard rate and the asymptotic normality without requiring further smoothness. We demonstrate faster convergence associated with dimension reducing types of singularity, such as a fractal structure or a factor structure in the regressors. Second, the paper extends the limit theory of kernel functional regression to multivariate regression over a product of any number of metric spaces. Finite sample evidence confirms rate improvement due to singularity in regression over Rq. For functional regression the simulations underline the importance of accounting for multiple functional regressors.