TEST https://doi.org/10.1007/s11749-020-00732-0 ORIGINAL PAPER Testing serial independence with functional data Zden ˇ ek Hlávka 1 · Marie Hušková 1 · Simos G. Meintanis 2,3 Received: 27 January 2020 / Accepted: 29 August 2020 © Sociedad de Estadística e Investigación Operativa 2020 Abstract We consider tests of serial independence for a sequence of functional observations. The new methods are formulated as L2-type criteria based on empirical characteristic functions and are convenient from the computational point of view. We derive asymp- totic normality of the proposed test statistics for both discretely and continuously observed functions. In a Monte Carlo study, we show that the new test is sensitive with respect to functional GARCH alternatives, investigate the choice of necessary tuning parameters, and demonstrate that critical values obtained by resampling lead to a test with good performance in any setup, whereas the asymptotic critical values may be recommended only for a sufficiently fine discretization grid. Finite-sample comparison with a distance (auto)covariance test criterion is also included, and the article concludes with application on a real data set. Keywords Functional data · Serial independence · Empirical characteristic function · Testing Mathematics Subject Classification 62R10 · 62G10 · 62H15 B Zdenˇ ek Hlávka hlavka@karlin.mff.cuni.cz Marie Hušková huskova@karlin.mff.cuni.cz Simos G. Meintanis simosmei@econ.uoa.gr 1 Department of Statistics, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic 2 Department of Economics, National and Kapodistrian University of Athens, Athens, Greece 3 Unit for Business Mathematics and Informatics, North-West University, Potchefstroom, South Africa 123