  We generalize well-known results on structural identifiability of vector
autoregressive models (VAR) to the case where the innovation covariance matrix
has reduced rank. Structural singular VAR models appear, for example, as
solutions of rational expectation models where the number of shocks is usually
smaller than the number of endogenous variables, and as an essential building
block in dynamic factor models. We show that order conditions for
identifiability are misleading in the singular case and provide a rank
condition for identifiability of the noise parameters. Since the Yule-Walker
equations may have multiple solutions, we analyze the effect of restrictions on
the system parameters on over- and underidentification in detail and provide
easily verifiable conditions.
