Consider the following special one-parameter case of the gamma distribution, (f(y)=left(y / lambda^{2} ight) exp (-y /

Consider the following special one-parameter case of the gamma distribution, \(f(y)=\left(y / \lambda^{2}\right) \exp (-y / \lambda), y>0, \lambda>0\). For this distribution it can be shown that \(\mathrm{E}[y]=2 \lambda\) and \(\mathrm{V}[y]=2 \lambda^{2}\). Here we introduce regressors and suppose that in the true model the parameter \(\lambda\) depends on regressors according to \(\lambda_{i}=\exp \left(\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}\right) / 2\). Thus \(\mathrm{E}\left[y_{i} \mid \mathbf{x}_{i}\right]=\exp \left(\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}\right)\) and \(\mathrm{V}\left[y_{i} \mid \mathbf{x}_{i}\right]=\left[\exp \left(\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}\right)\right]^{2} / 2\). Assume the data are independent over \(i\) and \(\mathbf{x}_{i}\) is nonstochastic and \(\boldsymbol{\beta}=\boldsymbol{\beta}_{0}\) in the dgp.

(a) Show that the log-likelihood function (scaled by \(N^{-1}\) ) for this gamma model is \(Q_{N}(\boldsymbol{\beta})=N^{-1} \sum_{i}\left\{\ln y_{i}-2 \mathbf{x}_{i}^{\prime} \boldsymbol{\beta}+2 \ln 2-2 y_{i} \exp \left(-\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}\right)\right\}\).

(b) Obtain plim \(Q_{N}(\beta)\). You can assume that assumptions for any LLN used are satisfied. [ E[In \(\left.y_{i}\right]\) depends on \(\beta_{0}\) but not \(\beta\).]

(c) Prove that \(\widehat{\boldsymbol{\beta}}\) that is the local maximum of \(Q_{N}(\boldsymbol{\beta})\) is consistent for \(\boldsymbol{\beta}_{0}\). State any assumptions made.

(d) Now state what LLN you would use to verify part (b) and what additional information, if any, is needed to apply this law. A brief answer will do. There is no need for a formal proof.