  Granger causality is among the widely used data-driven approaches for causal
analysis of time series data with applications in various areas including
economics, molecular biology, and neuroscience. Two of the main challenges of
this methodology are: 1) over-fitting as a result of limited data duration, and
2) correlated process noise as a confounding factor, both leading to errors in
identifying the causal influences. Sparse estimation via the LASSO has
successfully addressed these challenges for parameter estimation. However, the
classical statistical tests for Granger causality resort to asymptotic analysis
of ordinary least squares, which require long data durations to be useful and
are not immune to confounding effects. In this work, we close this gap by
introducing a LASSO-based statistic and studying its non-asymptotic properties
under the assumption that the true models admit sparse autoregressive
representations. We establish that the sufficient conditions of LASSO also
suffice for robust identification of Granger causal influences. We also
characterize the false positive error probability of a simple thresholding rule
for identifying Granger causal effects. We present simulation studies and
application to real data to compare the performance of the ordinary least
squares and LASSO in detecting Granger causal influences, which corroborate our
theoretical results.
