In linear regression or Gaussian-Markov framework,

$$Y = \beta_0 + \beta_1X+\epsilon, $$

where $\epsilon\sim(0,\sigma^2)$

we usually assume $X$ is non-random. In our statistics class, professors sometimes simply mention like "here $X$ is nonrandom". I wonder what happens if $X$ is assumed to be a random variable?

Let's assume $\epsilon$ is from a normal distribution and is independent of $X$, and $X$ has mean zero.

  • $\begingroup$ If x is random wouldn't that mean that there is no relationship with y? I.e., the dependent variable is not dependent on the independent variable. Thus, the regression estimates would not be estimatable. Hypothesizing and curious about the answer. $\endgroup$ Oct 3, 2021 at 16:32
  • $\begingroup$ I marked it as a complicate of another thread that discusses this in detail, you’ll find the answer there. $\endgroup$
    – Tim
    Oct 3, 2021 at 16:34
  • $\begingroup$ Linear regression makes no assumptions on the distribution of X. All it assumes is that there exists a linear relationship between the dependent and response variables $\endgroup$ Oct 3, 2021 at 16:34
  • $\begingroup$ @Tim Thanks. So generally, techniques such as least squares, or maximum likelihood estimations are not influenced by whether we consider $X$ to be fixed or random, right? For the random $X$, we simply change a way to describe everything using conditional distribution, instead of the marginal distribution where we deem $X$ to be fixed. $\endgroup$
    – Tan
    Oct 3, 2021 at 19:13