How Does Correlation Relate to Causation: Seven Insights from an Analytic Relationship Between the Two

“Correlation does not imply causation,” the adage warns. But how does correlation relate to causation? For linear stochastic dynamical systems driven by white noise, a widely used class of models across the natural and social sciences, the question has an exact answer. At steady state, correlation (more precisely covariance) and causation are linked by the Lyapunov equation. From this equation we derive closed-form expressions for pairwise correlation coefficients as explicit functions of the causal parameters and noise variances. We analyze these expressions for nine network topologies and unpack seven insights regarding the relationship between correlation and causation. For instance, we illustrate through some examples that causation is transitive, while correlation is not; and that the mapping from causation to correlation is generally many-to-one. The formulae also reveal that the covariance between any two variables is a linear combination of the noise variances, with weights that are highly nonlinear functions of the causal parameters. This is one of the reasons that the relationship between correlation and causation resists simple intuition. We show that the formulae from the fully connected three-node network serve as the “master formulae” for 3 ×3 networks, because of the structural symmetry of its topology, and can be reduced to the formulae for every sparser three-node network as causal connections are set to zero. Finally, we note that the Lyapunov equation is a unifying mathematical object for causal modeling frameworks such as Granger causality and dynamic causal modeling, with additive white noise assumption.