= Nonidentifiability prevents unbiased coefficient estimation
If a <design matrix> $X$ has a nonzero null vector $z$, the <normal linear model> has identical observation distributions at $\beta$ and $\beta+z$. Any estimator has the same expectation at those two parameters, and therefore cannot be an <unbiased estimator> of both coefficient vectors. Full column rank is required for unbiased estimation of the entire unconstrained coefficient vector; identifiable linear combinations can still be estimated without it.
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