# Random vs Fixed variables in Linear Regression Model

Reading "Econometrics" by Fumio Hayashi, from Princenton University Press, ISBN 0-691-01018-5, in page 13 by "Fixed Regressors" subtitle, it is stated:

"We have presented the classical linear regression model, treating the regressors as random. This is in contrast to the treatment in most textbooks, where X is assumed to be "fixed" or deterministic. If X is fixed, then there is no need to distinguish between conditional distribution of the error term [...] and the unconditional distribution"

I have problems understanding this paragraph. I think it says that if you consider X fixed, then it is "constant" and the conditional distribution of the error term and the unconditional distribution is the same because conditional distribution over a "constant" is equal to unconditional distribution?

On the other hand, I do not know why this is relevant, every time a run an lm with R I know X, it is always fixed! I am missing something here... may you help me?

• I once read that to assume that X is fixed is equivalent to assuming that X is independent of the error term. From independence it follows that conditional and unconditional are the same. I do not have a math proof, but I guess that to assume that X is constant is equivalent to saying that with probability one X is what it is, irrespective of what the error is, so the distribution of X - although degenerate - does not change with the value of the error. Mar 12 '19 at 18:39

The comment by Jesper indicates the formal point. You are also right that the assumption of whether $$X$$ is fixed or not has no bearing on the numerical estimates, as we regress $$y$$ on $$X$$ with the same command lm in either case.
What the assumption is crucially important for, though, is the appropriate interpretation of the results. In particular, when $$X$$ is random, this opens up the possibility that $$E(u|X)\neq0$$ or even $$E(X'u)\neq0$$, i.e., that regressors and errors are not mean independent or not uncorrelated.