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When considering two or more time series of functions or curves, for instancethose derived from densely observed intraday stock price data of severalcompanies, the empirical cross-covariance operator is of fundamental importancedue to its role in functional lagged regression and exploratory data analysis.Despite its relevance, statistical procedures for measuring the significance ofsuch estimators are undeveloped. We present methodology based on a functionalcentral limit theorem for conducting statistical inference for thecross-covariance operator estimated between two stationary, weakly dependent,functional time series. Specifically, we consider testing the null hypothesisthat two series possess a specified cross-covariance structure at a given lag.Since this test assumes that the series are jointly stationary, we also developa change-point detection procedure to validate this assumption, which is ofindependent interest. The most imposing technical hurdle in implementing theproposed tests involves estimating the spectrum of a high dimensional spectraldensity operator at frequency zero. We propose a simple dimension reductionprocedure based on functional PCA to achieve this, which is shown to performwell in a small simulation study. We illustrate the proposed methodology withan application to densely observed intraday price data of stocks listed on theNYSE.

Inference for the cross-covariance operator of stationary functional time series