Consider NLS regression for the model \(y=\exp (\alpha+\beta x)+\varepsilon\), where \(\alpha, \beta\), and \(x\) are scalars and \(\varepsilon \sim \mathcal{N}[0,1]\). Note that for simplicity \(\sigma_{\varepsilon}^{2}=1\) and need not be estimated. We want to test \(H_{0}: \beta=0\) against \(H_{a}: \beta eq 0\).

(a) Give the first-order conditions for the unrestricted MLE of \(\alpha\) and \(\beta\).

(b) Give the asymptotic variance matrix for the unrestricted MLE of \(\alpha\) and \(\beta\).

(c) Give the explicit solution for the restricted MLE of \(\alpha\) and \(\beta\).

(d) Give the auxiliary regression to compute the OPG form of the LM test.

(e) Give the complete expression for the original form of the LM test. Note that it involves derivatives of the unrestricted log-likelihood evaluated at the restricted MLE of \(\alpha\) and \(\beta\). [This is more difficult than parts (a)-(d).]