  Granger causality, a popular method for determining causal influence between
stochastic processes, is most commonly estimated via linear autoregressive
modeling. However, this approach has a serious drawback: if the process being
modeled has a moving average component, then the autoregressive model order is
theoretically infinite, and in finite sample large empirical model orders may
be necessary, resulting in weak Granger-causal inference. This is particularly
relevant when the process has been filtered, downsampled, or observed with
(additive) noise - all of which induce a moving average component and are
commonplace in application domains as diverse as econometrics and the
neurosciences. By contrast, the class of autoregressive moving average models -
or, equivalently, linear state space models - is closed under digital
filtering, downsampling (and other forms of aggregation) as well as additive
observational noise. Here, we show how Granger causality, conditional and
unconditional, in both time and frequency domains, may be calculated simply and
directly from state space model parameters, via solution of a discrete
algebraic Riccati equation. Numerical simulations demonstrate that Granger
causality estimators thus derived have greater statistical power and smaller
bias than pure autoregressive estimators. We conclude that the state space
approach should be the default for (linear) Granger causality estimation.
