Functional Unit Root Test

In this paper, we propose a unit root test for functional time series. We derive a new analytical framework for nonstationary functional time series. Specifically, for the proposed test statistic, we derive its limit distribution under the null hypothesis of a random walk and its asymptotic behavior of alternative hypotheses of trend stationary, weakly dependent stationary, and autoregressive stationary models. For the trend stationary model as an alternative, a theoretical derivation of the test consistency is provided, while for other two alternatives, a combination of theoretical and experimental validation on the statistical power of the test is presented. Simulation studies are conducted to justify the theories and the desirable finite-sample performance of the proposed functional unit root test. The proposed test is also applied to real data of intraday stock price curves and the test results are plausible