On the Construction of Bootstrap Confidence Intervals for Estimating the Correlation Between Two Time Series Not Sampled on Identical Time Points

Abstract

Two important issues characterize the design of bootstrap methods to construct confidence intervals for the correlation between two time series sampled (unevenly or evenly spaced) on different time points: (i) ordinary block bootstrap methods that produce bootstrap samples have been designed for time series that are coeval (i.e., sampled on identical time points) and must be adapted; (ii) the sample Pearson correlation coefficient cannot be readily applied, and the construction of the bootstrap confidence intervals must rely on alternative estimators that unfortunately do not have the same asymptotic properties. In this paper it is argued that existing proposals provide an unsatisfactory solution to issue (i) and ignore issue (ii). This results in procedures with poor coverage whose limitations and potential applications are not well understood. As a first step to address these issues, a modification of the bootstrap procedure underlying existing methods is proposed, and the asymptotic properties of the estimator of the correlation are investigated. It is established that the estimator converges to a weighted average of the cross-correlation function in a neighborhood of zero. This implies a change in perspective when interpreting the results of the confidence intervals based on this estimator. Specifically, it is argued that with the proposed modification of the bootstrap, the existing methods have the potential to provide a useful lower bound for the absolute correlation in the non-coeval case and, in some special cases, confidence intervals with approximately the correct coverage. The limitations and implications of the results presented are demonstrated with a simulation study. The extension of the proposed methodology to the problem of estimating the cross-correlation function is straightforward and is illustrated with a real data example. Related applications include the estimation of the autocorrelation function and the periodogram of a time series.