Suppose we use an index formulation for a discrete choice model but it is felt that the latent variable is strictly positive. This is accommodated by supposing that the latent variable \(y^{*}\) has exponential density with parameter \(\gamma\), so the density \(f\left(y^{*}\right)\) is \(f\left(y^{*}\right)=\gamma^{-1} \exp \left(-y^{*} / \gamma\right)\), with \(\gamma=\exp \left(\mathbf{x}^{\prime} \boldsymbol{\beta}\right)\). We observe \(y=1\) if \(y^{*}>\mathbf{z}^{\prime} \alpha\) and \(y=0\) if \(y^{*} \leq \mathbf{z}^{\prime} \alpha\).

(a) Give the log-likelihood function for the observed data.

(b) What is the effect of a one-unit change in \(x_{j i}\) on \(\operatorname{Pr}\left[y_{i}=1\right]\) ?

(c) Suppose that \(y=1\) if \(y^{\prime \prime}>\exp \left(\mathbf{z}^{\prime} \alpha\right)\) and \(\mathbf{x}=\mathbf{z}\). Do you see any problems in identifying \(\alpha\) and/or \(\beta\) ? Explain your answer.