  Dynamical systems comprising of multiple components that can be partitioned
into distinct blocks originate in many scientific areas. A pertinent example is
the interactions between financial assets and selected macroeconomic
indicators, which has been studied at aggregate level---e.g. a stock index and
an employment index---extensively in the macroeconomics literature. A key
shortcoming of this approach is that it ignores potential influences from other
related components (e.g. Gross Domestic Product) that may exert influence on
the system's dynamics and structure and thus produces incorrect results. To
mitigate this issue, we consider a multi-block linear dynamical system with
Granger-causal ordering between blocks, wherein the blocks' temporal dynamics
are described by vector autoregressive processes and are influenced by blocks
higher in the system hierarchy. We derive the maximum likelihood estimator for
the posited model for Gaussian data in the high-dimensional setting based on
appropriate regularization schemes for the parameters of the block components.
To optimize the underlying non-convex likelihood function, we develop an
iterative algorithm with convergence guarantees. We establish theoretical
properties of the maximum likelihood estimates, leveraging the decomposability
of the regularizers and a careful analysis of the iterates. Finally, we develop
testing procedures for the null hypothesis of whether a block "Granger-causes"
another block of variables. The performance of the model and the testing
procedures are evaluated on synthetic data, and illustrated on a data set
involving log-returns of the US S&P100 component stocks and key macroeconomic
variables for the 2001--16 period.
