  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 duration to be useful and
are not immune to confounding effects. In this work, we address this disconnect
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 fundamental limits for reliable
identification of Granger causal influences using the proposed LASSO-based
statistic. We further characterize the false positive error probability and
test power of a simple thresholding rule for identifying Granger causal effects
and provide two methods to set the threshold in a data-driven fashion. We
present simulation studies and application to real data to compare the
performance of our proposed method to ordinary least squares and existing
LASSO-based methods in detecting Granger causal influences, which corroborate
our theoretical results.
