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Abstract:
In this work we discuss some asymptotic properties of the Gaussian kernel (GK) estimator for the correlation between two time series sampled on different time points and their relevance for the construction and interpretation of bootstrap based confidence intervals.In particular, we show that the GK estimator is asymptotically biased and converges to a weighted average of the cross-correlation function in a neighbourhood of zero. As a result, any bootstrap procedure for the construction of confidence intervals that combines the GK estimator with a standard method based on percentiles of its bootstrapped distribution(such the bias-corrected and accelerated method) asymptotically will have zero coverage even after applying recalibration. This does not imply, however, that bootstrap confidence intervals are useless. In fact, we show through an extensive simulation study, that a suitable block-bootstrap procedure can provide a useful lower bound for the absolute correlation and, in some special cases, confidence intervals with approximately the correct coverage. The ideas explored in this work apply as well to the more general problem of estimating the cross-correlation function of a bivariate time series whose components are sampled on different time points using a kernel-based estimator.
