= Solution
For the full-column-rank <design matrix>, the normal-error <maximum-likelihood estimator> is the least-squares estimator
$$
\boxed{\widehat\theta=(X^TX)^{-1}X^TY.}
$$
Indeed maximizing the Gaussian likelihood over $\theta$ minimizes $\|Y-X\theta\|^2$, whose <normal equations> are $X^TX\widehat\theta=X^TY$; rank $p$ makes $X^TX$ invertible. Substitution of the model gives
$$
\widehat\theta=\theta+(X^TX)^{-1}X^T\epsilon.
$$
Since $\mathbb E\epsilon=0$, it follows that $\boxed{\mathbb E\widehat\theta=\theta}$, proving unbiasedness.
Back to article page
