  Multivariate functional data are becoming ubiquitous with advances in modern
technology and are substantially more complex than univariate functional data.
We propose and study a novel model for multivariate functional data where the
component processes are subject to mutual time warping. That is, the component
processes exhibit a similar shape but are subject to systematic phase variation
across their time domains. To address this previously unconsidered mode of
warping, we propose new registration methodology which is based on a
shift-warping model. Our method differs from all existing registration methods
for functional data in a fundamental way. Namely, instead of focusing on the
traditional approach to warping, where one aims to recover individual-specific
registration, we focus on shift registration across the components of a
multivariate functional data vector on a population-wide level. Our proposed
estimates for these shifts are identifiable, enjoy parametric rates of
convergence and often have intuitive physical interpretations, all in contrast
to traditional curve-specific registration approaches. We demonstrate the
implementation and interpretation of the proposed method by applying our
methodology to the Z\"urich Longitudinal Growth data and study its finite
sample properties in simulations.
