Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / O / O4 / i / Solution

Unbiasedness as an estimator of the parameter means $DX\theta =\theta$ for every $\theta \in {\mathbb{R}}^p$, rather than just one particular value. Thus $(DX-I_p)\theta =0$ for every vector, forcing $\boxed{DX=I_p}$. Conversely that identity plainly suffices for unbiasedness.