  Scientists often want to learn about cause and effect from hierarchical data,
collected from subunits nested inside units. Consider students in schools,
cells in patients, or cities in states. In such settings, unit-level variables
(e.g. each school's budget) may affect subunit-level variables (e.g. the test
scores of each student in each school) and vice versa. To address causal
questions with hierarchical data, we propose hierarchical causal models, which
extend structural causal models and causal graphical models by adding inner
plates. We develop a general graphical identification technique for
hierarchical causal models that extends do-calculus. We find many situations in
which hierarchical data can enable causal identification even when it would be
impossible with non-hierarchical data, that is, if we had only unit-level
summaries of subunit-level variables (e.g. the school's average test score,
rather than each student's score). We develop estimation techniques for
hierarchical causal models, using methods including hierarchical Bayesian
models. We illustrate our results in simulation and via a reanalysis of the
classic "eight schools" study.
