0.1 Granger Causality
A series \(Y_t\) Granger-causes \(X_t\) if past values of \(Y\) improve the prediction of \(X_t\) beyond what is achievable using past values of \(X\) alone, i.e. \[ Var(X_t \mid X_{t-1}, X_{t-2},\dots) > Var(X_t \mid X_{t-1},X_{t-2},\dots,\,Y_{t-1},Y_{t-2},\dots). \] This is a statement about predictive incremental information, not structural/philosophical causation.
Testing in a VAR. In a bivariate \(VAR(p)\) for \((X_t,Y_t)\), \(Y\) fails to Granger-cause \(X\) iff all coefficients on lagged \(Y\) in the \(X\)-equation are zero; this is tested with a standard \(F\)-test (or Wald test) of that joint linear restriction.
Caveats. - Requires both series to be (or be transformed to be) stationary; testing in levels when series are \(I(1)\) and cointegrated can be misleading — the correct specification is then a VECM including the error-correction term (see Cointegration). - Sensitive to omitted variables: a third series driving both \(X\) and \(Y\) can produce spurious Granger causality. - Does not resolve simultaneity or contemporaneous causal direction, only lead-lag predictive structure.