# Identifiability of parameters in a linear model when covariates are random

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 $$y | X$$?

• 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. – mlofton Aug 20 at 4:24
• @mlofton Thank you for the reference. I’ll try to see if my library has a copy. – Flowsnake Aug 20 at 18:00
• 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". – mlofton Aug 21 at 18:32