Testing the goodness-of-fit of a functional autoregressive model

TL;DR

This paper introduces a new goodness-of-fit test for functional autoregressive models in Hilbert spaces, using empirical processes and functional CLT, with proven asymptotic properties and demonstrated finite-sample performance.

Contribution

It develops a novel GoF test for functional AR models based on empirical processes and establishes its asymptotic distribution and consistency.

Findings

The test converges to a time-changed Wiener process in Hilbert space.

The test is consistent under simple null hypotheses.

Finite-sample simulations show good performance under various alternatives.

Abstract

The proposed Goodness-of-Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space H. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space H, with subordinator given by the marginal probability of the involved strictly stationary Autoregressive Hilbertian process (ARH(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The finite-sample performance of the testing procedure, under different families of alternatives, and random projection schemes, is illustrated in the Appendix.