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Say we were interested in SAT scores for high school students as our dependent variable in a multiple regression. Now, assume we are God and can include literally all relevant covariates in the model (race, gender, socioeconomic status, parent education, etc ad infinitum), such that the residual for every estimate is 0. Also assume we have appropriate data to fit this model, no computational or time constraints, the relationship is linear, multicollinearity not a problem, etc etc. We're all familiar with the old adage "correlation doesn't imply causation," but in this hypothetical (and impossible) scenario, would the resulting coefficients imply causation? For example, if it turned out that the coefficient for parental education was statistically significant and positive, could we conclude that a higher level of parental education causes SAT scores of the child to increase?

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    $\begingroup$ With equal validity you could claim that increases in childrens' SAT scores cause higher levels of parental education. $\endgroup$ – whuber May 29 at 16:06
  • $\begingroup$ @whuber so could you say higher levels of parental education cause higher SAT scores or the reverse, but there's no other possibility? $\endgroup$ – kjakeb May 29 at 17:05
  • $\begingroup$ @whuber, +1, but I guess not in this example since there is the time flow and parental education is predetermined w.r.t. SAT scores. Or am I wrong? $\endgroup$ – Richard Hardy May 29 at 17:21
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    $\begingroup$ @Richard Right: the regression knows nothing about time flow. Thus, the logic of the assertion "these regression results imply causality" is incorrect; at a minimum, it must be "these regression results PLUS information about temporal relationships ARE PART OF a demonstration of causation." I feel obliged to write "are part of" in recognition that additional criteria may need to be invoked, such as the Bradford Hill criteria. This also implies I am denying that "all relevant covariates" has any meaning for real systems. $\endgroup$ – whuber May 29 at 17:46
  • $\begingroup$ @whuber, right, this is roughly what I though. I could not tell immediately whether you had excluded the information about temporal relationships from your consideration, hence the question. Thanks for your detailed response! $\endgroup$ – Richard Hardy May 29 at 18:16
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The key question in your problem statement is defining what the "relevant variables" are. To do this properly, you first need a formal definition of a causal model, and to be precise about what is the causal effect of interest. Given a causal model, we can tell which variables are the "relevant" variables so the causal effect can be identified via covariate adjustment (if possible). Mathematically, this is a solved problem.

Please check these answers:

Is a regression causal if there are no omitted variables?

Confounder - definition

Omitted variable bias: which predictors do I need to include, and why?

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