  Granger causality has been employed to investigate causality relations
between components of stationary multiple time series. Here, we generalize this
concept by developing statistical inference for local Granger causality for
multivariate locally stationary processes. Thus, our proposed local Granger
causality approach captures time-evolving causality relationships in
nonstationary processes. The proposed local Granger causality is well
represented in the frequency domain and estimated based on the parametric
time-varying spectral density matrix using the local Whittle likelihood. Under
regularity conditions, we demonstrate that the estimators converge weakly to a
Gaussian process. Additionally, the test statistic for the local Granger
causality is shown to be asymptotically distributed as a quadratic form of a
multivariate normal distribution. The finite sample performance is confirmed
with several simulation studies for multivariate time-varying VAR models. For
practical demonstration, the proposed local Granger causality method uncovered
new functional connectivity relationships between channels in brain signals.
Moreover, the method was able to identify structural changes of Granger
causality in financial data.
