  We consider statistical inference for impulse responses in sparse, structural
high-dimensional vector autoregressive (SVAR) systems. We introduce consistent
estimators of impulse responses in the high-dimensional setting and suggest
valid inference procedures for the same parameters. Statistical inference in
our setting is much more involved since standard procedures, like the
delta-method, do not apply. By using local projection equations, we first
construct a de-sparsified version of regularized estimators of the moving
average parameters associated with the VAR system. We then obtain estimators of
the structural impulse responses by combining the aforementioned de-sparsified
estimators with a non-regularized estimator of the contemporaneous impact
matrix, also taking into account the high-dimensionality of the system. We show
that the distribution of the derived estimators of structural impulse responses
has a Gaussian limit. We also present a valid bootstrap procedure to estimate
this distribution. Applications of the inference procedure in the construction
of confidence intervals for impulse responses as well as in tests for forecast
error variance decomposition are presented. Our procedure is illustrated by
means of simulations.
