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Assume that n=p, so that the number of samples matches the number of covariates, and that \mathbb {X} has rank n. Recall that the Least-Squares Estimator \hat{{\boldsymbol \beta }} has the formula
\hat{{\boldsymbol \beta }} = (\mathbb {X}^ T \mathbb {X})^{-1} \mathbb {X}^ T \mathbf Y.
If we assume that the vector {\boldsymbol \varepsilon }= (\epsilon _1, \ldots , \epsilon _ n) is uniformly distributed in the n-dimensional box [-1,+1]^{n}, then:
“The model is homoscedastic ; i.e. \varepsilon _1, \ldots , \varepsilon _ n are i.i.d."
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“In the deterministic design setting, \mathbf Y is also deterministic."
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“In the deterministic design setting, the LSE \hat{{\boldsymbol \beta }} is a uniformly distributed random variable."
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(You may use the following fact: in the 1-dimensional case, consider a \sim \text {Uniform}([0,1]) and let \lambda > 0. Intuitively enough, the distribution of b = \lambda a is uniform over the interval [0,\lambda ]. More generally, if a is uniformly distributed over a rectangular region R \subset \mathbb {R}^ n and M is an n \times n matrix of full rank, then b is uniformly distributed over the region M(R) \subset \mathbb {R}^ n, the image of R under the transformation M.)
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