lets say we have a perfect linear regression, i.e. we have included all relevant variables (to prevent OVB: omitted variable bias), and also such that there is no other problems like mutli-collinearity.

in this case, can we infer causality between a predictor variable x and the outcome variable y?

i understand that in normal cases, we cannot assume causality due to the possibility of there being a confounding varaible. however, if we solve OVB, we naturally solve for the lack of confounding variable, since it will be included.

therefore in this case, is it okay to infer causality?

  • $\begingroup$ i ask this because i also understand that when we choose y and x, we try to choose it in a way that makes logical, casual sense e.g. wage as y, education as x and not the other way around $\endgroup$ – Ian Petrus Tan Oct 9 '20 at 12:31
  • $\begingroup$ ALSO since, from an economic POV, coefficient of a predictor variable captures the effect of xi on y, holding all else i.e. other variables constant $\endgroup$ – Ian Petrus Tan Oct 9 '20 at 12:45
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    $\begingroup$ Having a baby is a beautiful explanatory variable for gender. Would you conclude that one's gender is caused by giving birth?? $\endgroup$ – whuber Oct 9 '20 at 15:01

Let's say I run a model of Depression (Y) as a function of Drug use (X), and I have somehow magically controlled for all other variable that might be correlated with both X and Y, and the beta for drug use is positive. Does that mean drug use causes depression? Maybe, but it might also be that depression causes drug use. In fact both of these causal relationships might exist simultaneously, but even this magical model can't distinguish between them. This is sometimes referred to as reciprocal causation.

In general a regression model (linear or otherwise) run on observational data at a single point in time can't distinguish between X->Y or Y->X. Even though we TRY to choose our X and Y variable so that X is causally prior to Y, that's often not possible, as the depression example shows - and really many of the most interesting questions in psychology, political science, sociology, and economics involve relationships where reciprocal causation is one of the major threats to validity.

A true randomized control design solves this problem because it ensures that NOTHING (not even Y itself) is correlated with assignment to the treatment X, but you can't replicate this in a standard regression model because you can't have Y on both sides of the equation at the same time. So the best alternative is to bring in "time:" if having certain X value at time 1 is associated with having a certain Y value at time 2, or that a change in X is associated with a subsequent change in Y, you can be sure that this relationship is not due to Y causing X.

This explains the popularity of research designs that incorporate time, such as difference-in-difference models, time series analysis, etc.

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    $\begingroup$ I would suggest that you correct your point about RCTs: treatment assignment is not correlated to potential outcomes but the treatment (x) is correlated with OBSERVED outcome (y). Otherwise, you will never OBSERVE treatment effects (i.e., mean differences for y by levels of x). $\endgroup$ – Student Oct 9 '20 at 16:18

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