Suppose we fit the model $\mathbf{y}=\mathbf{X}_{1} \boldsymbol{\beta}_{2}+\boldsymbol{\varepsilon}$ when the true model is actually given by $\mathbf{y}=\mathbf{X}_{1} \boldsymbol{\beta}_{2}+\mathbf{X}_{2} \boldsymbol{\beta}_{2}+\boldsymbol{\varepsilon}$. For both models, assume $E(\boldsymbol{\varepsilon})=\mathbf{0}$ and $\operatorname{Var}(\boldsymbol{\varepsilon})=\sigma^{2} \mathbf{I}$. Find the expected value and variance of the ordinary least-squares estimate, $\hat{\boldsymbol{\beta}}_{1}$. Under what conditions is this estimate unbiased?