Consider a latent variable modeled by y i = x i + i

Consider a latent variable modeled by $y_i^{\ast }={\mathbf{x}}_i^{\mathrm{\prime }}\mathit{\beta }+{\epsilon }_i$ with ${\epsilon }_i\sim \mathcal{N}[0,{\sigma }^2]$. Suppose $y_i^{\ast }$ is censored from above so that we observe $y_i=y_i^{\ast }$ if $y_i^{\ast }<U_i$ and $y_i=U_i$ if $y_i^{\ast }\ge U_i$, where the upper limit $U_i$ is a known constant for each individual (i.e., data) and may differ over individuals.

(a) Give the log-likelihood function for this model. [Hint: Note that this differs from the standard case both owing to presence of $U_i$ and because the equalities are reversed with $y_i=y_i^{\ast }$ if $y_i^{\ast }<U_i$.]

(b) Obtain the expression for the truncated mean $\mathrm{E}[y_i\mid {\mathbf{x}}_i,y_i<U_i]$. [Hint: For $z\sim$ $\mathcal{N}[0,1]$, we have $\mathrm{E}[z\mid z>c]=\varphi (c)/[1-\mathrm{\Phi }(c)]$. Also, $\mathrm{E}[z\mid z<c]=-\mathrm{E}[-z\mid -$ $z>-c]$ and $-z\sim \mathcal{N}[0,1]$.

(c) Hence give Heckman's two-step estimator for this model.

(d) Obtain the expression for the censored mean $\mathrm{E}[y_i\mid {\mathbf{x}}_i]$. [An essential part is the answer in part (b).]