General Comments

(Also drawing from Prof.Pohlmeier's lecture notes.)
What econometricians are doing seems to be just a problem of optimization, optimizing prediction by choosing among different predictors, and based on different criteria of optimum. Analogies can be made with microeconomic optimization problems. I wonder whether there can be an analogous concept of equilibrium in econometrics, e.g.equilibrium of prediction
The optimization part can actually explain for modeler's particular interests in linear functions/regressions, whether a mathematical modeler or statistical one.

Moments

To be conditional means having one dimension of variability fixed, like conditional probability, conditional expectation.

Distribution Theory

normal distribution can describe distribution of the mean when the variance is (assumed as) known.
chi-square can describe distribution of the variances whether mean is known or not.
Student's t can describe distribution of the mean whether variance is known or not.