Testing the stability of the functional autoregressive process

The functional autoregressive process has become a useful tool in the analysis of functional time series data. It is defined by the equation X n + 1 = Ψ X n + ε n + 1 , in which the observations X n and errors ε n are curves, and Ψ is an operator. To ensure meaningful inference and prediction based on this model, it is important to verify that the operator Ψ does not change with time. We propose a method for testing the constancy of Ψ against a change-point alternative which uses the functional principal component analysis. The test statistic is constructed to have a well-known asymptotic distribution, but the asymptotic justification of the procedure is very delicate. We develop a new truncation approach which together with Mensov’s inequality can be used in other problems of functional time series analysis. The estimation of the principal components introduces asymptotically non-negligible terms, which however cancel because of the special form of our test statistic (CUSUM type). The test is implemented using the R package fda, and its finite sample performance is examined by application to credit card transaction data.