Let’s say you run the most basic regression Y = X beta + epsilon. The X is chosen out of the set all possible regressors Z (say you run income ~ age + sex, wher

Let’s say you run the most basic regression Y = X beta + epsilon. The X is chosen out of the set all possible regressors Z (say you run income ~ age + sex, where you also could have used education, location, whatever). Is that not equivalent to a prior that the coefficient on variables in Z but not in X is zero?