  A growth curve model (GCM) aims to characterize how an outcome variable
evolves, develops and grows as a function of time, along with other predictors.
It provides a particularly useful framework to model growth trend in
longitudinal data. However, the estimation and inference of GCM with a large
number of response variables faces numerous challenges, and remains
underdeveloped. In this article, we study the high-dimensional
multivariate-response linear GCM, and develop the corresponding estimation and
inference procedures. Our proposal is far from a straightforward extension, and
involves several innovative components. Specifically, we introduce a Kronecker
product structure, which allows us to effectively decompose a very large
covariance matrix, and to pool the correlated samples to improve the estimation
accuracy. We devise a highly non-trivial multi-step estimation approach to
estimate the individual covariance components separately and effectively. We
also develop rigorous statistical inference procedures to test both the global
effects and the individual effects, and establish the size and power
properties, as well as the proper false discovery control. We demonstrate the
effectiveness of the new method through both intensive simulations, and the
analysis of a longitudinal neuroimaging data for Alzheimer's disease.
