Consider a latent variable modeled by \(\boldsymbol{y}_{i}^{*}=\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}+\varepsilon_{i}\), with \(\varepsilon_{i} \sim \mathcal{N}[0,1]\). Suppose we observe only \(y_{i}=1\) if \(y_{i}^{*}

(a) Find \(\operatorname{Pr}\left[y_{i}=1 \mid \mathbf{x}_{i}\right]\). [Hint: Note that this differs from the standard case both due to presence of \(U_{i}\) and because the equalities are reversed with \(y_{i}=1\) if \(y_{i}^{*}

(b) Provide details on an estimation method to consistently estimate \(\beta\).

(c) Suppose you estimate this model and find that the third regressor \(x_{3 i}\) has estimated coefficient \(\widehat{\beta}_{3}=0.2\). Provide a meaningful interpretation of \(\widehat{\beta}_{3}\).