Inference for the Lagged Cross‐Covariance Operator Between Functional Time Series

Abstract

When considering two or more time series of functional data objects, 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 currently 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 the 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 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 principal component analysis to achieve this, which is shown to perform well in a simulation study. We illustrate the proposed methodology with an application to densely observed intraday price data of stocks listed on the New York stock exchange‐20.40

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…It would be interesting to deduce our results also on separable Banach spaces, see, e. g., [39] who dealt with the estimation of AR operators in Banach spaces, to derive the asymptotic distribution of our estimation errors (see [36]) as well as their asymptotic lower bounds.…”

“…A comprehensive study of lag-h-cross-covariance operators C X,Y;h of stationary L 2 [0, 1]-valued processes X = (X k ) k∈Z , Y = (Y k ) k∈Z can be found in Rice & Shum [36]. They established operator estimates, discussed methods measuring their significance and deduced their limit distribution.…”

“…This article studies lagged covariance and cross-covariance operators of stationary processes in Cartesian products of abstract Hilbert spaces based on ideas in [36] and [2]. The focus is on deducing moment estimators and asymptotic upper bounds of the estimation errors for these operators.…”

“…Remarks 3.4. (a) Although estimating (lagged) cross-covariance operators is widely discussed, see e. g [36],[5],[2],[18],. Theorems 3.3-3.4 are new in many ways.…”

Functional ARCH and GARCH Models: A Yule-Walker Approach

“…Further probabilistic features and asymptotics of (lag-h-)covariance and cross-covariance operators can be found in [5] and [16], and [28] thoroughly studied the asymptotics of lag-h-cross-covariance operators.…”