Consider the multiple linear regression model $\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\boldsymbol{\varepsilon}$. Show that the least-squares estimator can be written as

\[\hat{\beta}=\beta+\mathbf{R} \varepsilon \quad \text { where } \quad \mathbf{R}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime}\]