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Suppose there are two time series, $x_t$ and $y_t$, that capture daily counts of some sort. $x_t$ is believed to have causal impact on $y_t$. Suppose further that a simple regression is fit to the data, disregarding the time aspect:

$$y_t = \alpha + \beta x_t + \epsilon.$$

There are at least the following two features that make this case difficult to reason about: the treatment being non-binary and the time aspect.

What assumptions are needed in order to legitimately give $\beta$ a causal interpretation?

Ivan
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  • This seems similar to this question https://stats.stackexchange.com/questions/493211/ In my opinion these type of considerations are a bit circular. It is interpreted to be causal when it is assumed to be causal. The $\beta$ has a causal interpretation when the role of $x_t$ is considered solely causal and potential correlations/dependencies are not influenced by other indirect effects (effects that are not like $x$ causing $y$) or can be considered negligible. – Sextus Empiricus Nov 13 '20 at 00:02
  • Thank you! I missed that post. It is indeed a difficult topic. In the majority of cases mentioned, the observations are presumably assumed to be independent, like patients and so. Here they are not due to the time structure. Does it not add complexity? – Ivan Nov 15 '20 at 19:09

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