Consider a latent variable modeled by y = x + , with N [ 0 , 1 ] .

Consider a latent variable modeled by ${\mathit{y}}^{\ast }={\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }+\epsilon$, with $\epsilon \sim \mathcal{N}[0,1]$. Suppose we observe only $y=2$ if $y^{\ast }<\alpha ,y=1$ if $\alpha \le y^{\ast }<U$, and $y=0$ if $y^{\ast }\ge U$, where the upper limit $U$ is a known constant for each individual (i.e., data) and may differ over individuals, but $\alpha$ is unknown.

(a) Obtain the conditional probabilities that $y=0,y=1$, and $y=2$.

(b) Provide details on a method to consistently estimate $\beta$ and $\alpha$.