Suppose we have a linear model (in $\mathbb{R}^n$, say), $$y = X\beta + \epsilon $$ where $\bf{\epsilon}$ is Gaussian with mean $0$ and covariance matrix $\Sigma(\theta)$ where $\theta$ is an unknown parameter. However, suppose that the covariates $X$ are random. Assuming that $\epsilon$ and $X$ are independent, then the conditional negative log-likelihood takes the form,

$$ f(\beta, \theta) = \frac{1}{2}\log(|\Sigma(\theta)|) + \frac{1}{2}(y - X\beta)^T\Sigma(\theta)^{-1}(y- X\beta)$$

How would we argue identifiability in this case? If we want the mapping $(\beta, \theta) \mapsto f(\beta, \theta)$ to be one-to-one, then are we free to choose both $y$ and $X$ to be anything? Or do we still treat $X$ as fixed since we're using the conditional distribution $y | X$?

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    $\begingroup$ Hi: Just because $X$ is random, that doesn't mean that assumptions aren't made about $X^{\prime} X$. I just don't remember them. LOL. If you have halbert white's "asymptotic theory for econometricians", I think he has a nice discussion of this formulation in his text. It's a good text and, IMHO, doesn't get enough "hype" so I'm hyping it here a little. $\endgroup$ – mlofton Aug 20 at 4:24
  • $\begingroup$ @mlofton Thank you for the reference. I’ll try to see if my library has a copy. $\endgroup$ – Flowsnake Aug 20 at 18:00
  • $\begingroup$ I hope you can find it. Another book that may discuss it some but won't be quite as useful as White's is Hamilton's "Time Series Analysis". $\endgroup$ – mlofton Aug 21 at 18:32

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