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?

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    $\begingroup$ 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. $\endgroup$ 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.

When this happens, OLS is no longer unbiased or consistent. Loosely speaking, we may no longer interpet the estimates as estimates of causal effects, but rather only as correlations that may or may not be interesting.

There are many threads on this page that offer further discussions, see e.g. Understanding and interpreting consistency of OLS, Omitted variable bias: which predictors do I need to include, and why?, Why is OLS estimator of AR(1) coefficient biased?.

When the assumption is not satisfied, instrumental variables approaches may provide an alternative, see e.g. What is an instrumental variable?, how to explain instrumental variables to a layman.


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