Consider a Poisson regression model where $y^{\ast }$ has density $f^{\ast }\left(y^{\ast }\right)=$ $e^{-\mu }{\mu }^y/y^{\ast !}!,y_i^{\ast }=0,1,2,\dots$, and we have independence over $i$. Because of coding error we only fully observe $y^{\ast }$ when $y^{\ast }\ge 2$. When $y^{\ast }=0$ or 1 we only observe that $y^{\ast }\le 1$. Suppose this is coded as $y^{\ast }=1$. Define the observed data $y=y^{\ast }$ for $y_i^{\ast }\ge 2$ and $y=1$ for $y_i^{\ast }=0$ or 1 .

(a) Obtain the density $f(y)$ of the observed $y$.

(b) Obtain E[y]. [There is some algebra here.] Now introduce regressors with $\mathrm{E}[y^{\ast }\mid \mathbf{x}]=\exp \left({\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }\right)$ and define the indicator variable $d=1$ for $y^{\ast }\ge 2$ and $d=0$ for $y^{\ast }=0$ or 1 .

(c) Give the exact formula for this example of the objective function of an estimator that provides a consistent estimator of $\beta$ using data on $y_i,d_i$, and ${\mathbf{x}}_i$.

(d) Give the exact formula for this example of the objective function of an estimator that provides a consistent estimator of $\beta$ using data on only $d_i$ and ${\mathbf{x}}_i$.

(e) Is it possible to consistently estimate $\beta$ using data on only $d_i$ and ${\mathbf{x}}_i$ ? Explain your answer.