This question presumes access to software that allows NLS and ML estimation. Consider the gamma regression model of Exercise 5-2. An appropriate gamma variate can be generated using \(y=-\lambda \ln r_{1}-\lambda \ln r_{2}\), where \(\lambda=\exp \left(\mathbf{x}^{\prime} \boldsymbol{\beta}\right) / 2\) and \(r_{1}\) and \(r_{2}\) are random draws from Uniform \([0,1]\). Let \(\mathbf{x}^{\prime} \boldsymbol{\beta}=\beta_{1}+\beta_{2} x\). Generate a sample of size 1,000 when \(\beta_{1}=-1.0\) and \(\beta_{2}=1\) and \(\mathbf{x} \sim \mathcal{N}[0,1]\).

(a) Obtain estimates of \(\beta_{1}\) and \(\beta_{2}\) from NLS regression of \(y\) on \(\exp \left(\beta_{1}+\beta_{2} x\right)\).

(b) Should sandwich standard errors be used here?

(c) Obtain ML estimates of \(\beta_{1}\) and \(\beta_{2}\) from NLS regression of \(y\) on \(\exp \left(\beta_{1}+\beta_{2} x\right)\).

(d) Should sandwich standard errors be used here?

Exercise 5-2

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.