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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?

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2 Answers 2

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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.

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  • $\begingroup$ Thank you for this article. This is awesome reference! $\endgroup$
    – cure
    Commented Jun 16, 2020 at 9:30
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As far as I know, there is no way to model such situations in a cross-sectional way. At least the way around this issue in epidemiology is to impose a time-ordering, like with lagged variables as you mention.

A simple example is antiretroviral therapy (ART) and CD4. Before ART for recommended at diagnosis, ART therapy was dependent on CD4 T-cell count. Therefore, previous CD4 was related to current ART treatment status, which was related to future CD4 counts. By creating a lagged version of CD4 (like previous month), you can establish a clear time-order of CD4 and ART.

Even in the supply & demand example described in the linked paper, I am not sure how supply and demand could occur exactly at the same time (but maybe that's a lack of understanding of economics on my part).

Without imposing a time-order, I have trouble even thinking how you could establish a causal relationship between two variables. If they occur simultaneously, why would you think that don't have a mutual common cause that is the actual source? Furthermore, how would two things that occur at the exact same moment actually cause each other? That seems to break the temporal sequence we think is necessary for causes. I think there are relatively few things that occur exactly simultaneously. While your data may not be fine enough of a time-scale to differentiate, that's more a data issue than a causal diagram issue.

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  • $\begingroup$ 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 $p_2$ -> $p_1$. $\endgroup$
    – RobertF
    Commented Dec 23, 2023 at 19:33
  • $\begingroup$ I still think a corresponding causal diagram would be divided into sections of time. The 'effect' of gravitational attraction needs time to occur on the relative positions of the objects (a single measurement of position is not enough). It is an interesting example though, as the unique part is the units affecting each other. The diagram here would best be one for interference, where the units can affect each other (i.e., units are not independent) arxiv.org/abs/1403.1239 $\endgroup$
    – pzivich
    Commented Dec 24, 2023 at 16:08

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