  When considering two or more time series of functions or curves, for instance
those derived from densely observed intraday stock price data of several
companies, the empirical cross-covariance operator is of fundamental importance
due to its role in functional lagged regression and exploratory data analysis.
Despite its relevance, statistical procedures for measuring the significance of
such estimators are undeveloped. We present methodology based on a functional
central limit theorem for conducting statistical inference for the
cross-covariance operator estimated between two stationary, weakly dependent,
functional time series. Specifically, we consider testing the null hypothesis
that two series possess a specified cross-covariance structure at a given lag.
Since this test assumes that the series are jointly stationary, we also develop
a change-point detection procedure to validate this assumption, which is of
independent interest. The most imposing technical hurdle in implementing the
proposed tests involves estimating the spectrum of a high dimensional spectral
density operator at frequency zero. We propose a simple dimension reduction
procedure based on functional PCA to achieve this, which is shown to perform
well in a small simulation study. We illustrate the proposed methodology with
an application to densely observed intraday price data of stocks listed on the
NYSE.
