# Simultaneity in causal diagrams

Lets assume we have simultaneity problem. Variable x causes y and y causes x. As an example i would state alcoholism: the more respondent consumes alcohol, the more 'is' alcoholic (measured for example by psychological test score). The more 'is' alcoholic, the more consumes alcohol.

Is it possible discuss such problem in the causal diagram paradigm by Judea Pearl?

My only way of thinking here is to model underlying structure using lagged values, creating structure similar to VAR models, and draw such relations:

• $$x_{t-1}$$ -> $$x_{t}$$
• $$y_{t-1}$$ -> $$y_{t}$$
• $$x_{t-1}$$ -> $$y_{t}$$
• $$y_{t-1}$$ -> $$x_{t}$$

Is this idea correct? If it is - are there alternatives? Is it possible to model such situation in cross-section way?

It may not be possible, or at least has not yet been worked out yet. See the discussion in section 4.3 in Guido Imbens's Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics working paper.

The context there is supply and demand, which is the canonical example of a simultaneous equilibrium relationship.

• Perhaps a better example of simultaneity is gravitational attraction between two masses in space at position 1 ($p_1$) and position 2 ($p_2$). Each mass simultaneously (more or less) causes change in the position of the other mass $p_1$ -> $p_2$ and $p_2$ -> $p_1$. Dec 23, 2023 at 19:33