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?
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.
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