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After examining various correlations between longitudinal data and illustrating high correlation between one or more variables, I realized that I could only show that the data was correlated but could not prove causality. Because I can not show ceteris paribus through an experiment by holding a control, how could I mathematically illustrate that there is a likelihood that $y$ is caused by $x$.

I have shown that

$E[y|\mathbf{x},u] = \mathbf{\hat{\beta} x} + u$ , where $\mathbf{x}$ is a vector of explanatory variables, $\hat{\beta}$ is a vector of coefficients and $u$ is the error or unobserved variables.

but what would be the next step? If you could point me toward the name of the technique or a journal or any guidance at all would be greatly appreciated. One thing to keep in mind is that the only data I have are longitudinal.

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You could look at the following books on causality: Pearl Freedman et al Morgan and Winship

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It is'nt possible to give a short answer! This kind of thing is discussed a lot on the following blog:

so you could search there.

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