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An unbiased estimator $\hat{\theta }$ satisfies ${\mathbb{E}}_{\theta }\hat{\theta }=\theta$ for every parameter value. Under the standard full-column-rank assumption on the design matrix, the Gaussian log-likelihood for the normal linear model, up to terms independent of $\beta$, is $-\Vert Y-X\beta {\Vert }^2/(2{\sigma }^2)$. Its maximum is the ordinary least squares minimum. Differentiating gives $X^{T}X\hat{\beta }=X^{T}Y$, and full column rank makes $X^{T}X$ positive definite. ThusSince $\mathbb{E}Y=X\beta$, its expectation is $(X^{T}X{)}^{-1}{X}^{T}X\beta =\beta$, proving unbiasedness. Estimating an unknown ${\sigma }^2$ as well does not change the maximizing coefficient vector.

The printed hypotheses give $p<n$ but do not explicitly require full column rank. Without that condition the maximum-likelihood estimators are not unique: they are $X^{+}Y+z$ with $z\in \ker X$, as in rank-deficient ordinary least squares. If $0\ne z\in \ker X$, the observation distributions at $\beta$ and $\beta +z$ are identical. Every estimator then has the same expectation at both parameters, so no estimator can be unbiased for both entire coefficient vectors. This proves nonidentifiability prevents unbiased coefficient estimation. The stated estimator and unbiasedness claim require the usual full-rank condition.