Testing Martingale Difference Hypothesis for Functional Time Series

Testing the predictability of time series is crucial for determining whether forecasting models should be developed. Existing methodologies for this problem generally fall into three categories based on the null hypothesis: white noise, independently identically distributed (i.i.d.) and martingale difference hypothesis (MDH). While numerous tests within these categories have been established for multivariate time series, research on functional time series (FTS) has predominantly concentrated on testing the white noise and i.i.d. hypotheses. This paper proposes a novel MDH test for FTS by introducing functional auto-martingale difference divergence (FAMDD), a new metric capable of effectively detecting nonlinear dependence within dependent functional data. Compared with the existing approaches, our test can detect a wider range of functional sequences with nonlinear dependence. The asymptotic behaviors of the test statistics are established under some suitable conditions. Since the limiting distribution is non-pivotal, a wild bootstrap procedure is introduced to obtain the critical values for conducting inference. Monte Carlo simulations and two real data applications are analyzed to illustrate the effectiveness of our methods.