An LM-Type Unit Root Test for Functional Time Series

In this paper, we propose a Lagrange multiplier (LM)-type unit root test for functional time series. The key novelty lies not in introducing a new LM principle but in establishing the asymptotic validity of such a test under the functional random walk null hypothesis without relying on functional principal component analysis (FPCA) or finite-dimensional unit root subspace assumptions. We derive the limit distribution of our proposed test statistics 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. Specifically, we establish the theoretical consistency of the test under all aforementioned alternative hypotheses. Simulation studies corroborate these theoretical findings and demonstrate 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.