For the gamma regression model of Exercise 5.2, $\mathrm{E}[\mathit{y}\mid \mathbf{x}]=\exp \left({\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }\right)$ and $\mathrm{V}[\mathit{y}\mid \mathbf{x}]=$ ${(\exp \left({\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }\right))}^2/2$

(a) Show that these conditions imply that $\mathrm{E}\left[\mathbf{x}\{{(y-{\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta })}^2-{(\exp \left({\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }\right))}^2/2\}\right]=\mathbf{0}$.

(b) Use the moment condition in part

(a) to form a method of moments estimator ${\hat{\beta }}_{\text{MM }}$.

(c) Give the asymptotic distribution of ${\hat{\beta }}_{\text{MM }}$ using result (6.13) .

(d) Suppose we use the moment condition $\mathrm{E}\left[\mathbf{x}(y-\exp \left({\mathbf{x}}^{\mathrm{\prime }}\mathit{\beta }\right))\right]$ in addition to that in part (a). Give the objective function for a GMM estimator of $\mathit{\beta }$.

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NOMM-00) N [0, G'So(G)']. (6.13)