Consider the nonlinear panel data model (y_{i t}=alpha_{i}+exp left(mathbf{x}_{i t}^{prime} beta ight)+u_{i t}), where (beta) are parameters

Consider the nonlinear panel data model \(y_{i t}=\alpha_{i}+\exp \left(\mathbf{x}_{i t}^{\prime} \beta\right)+u_{i t}\), where \(\beta\) are parameters to be estimated, \(\alpha_{i}, i=1, \ldots, N\), are individual specific effects, \(u_{i t}\) are iid \(\left[0, \sigma_{\varepsilon}^{2}\right]\) errors, and the panel is short.

(a) Suppose that all \(\alpha_{i}=0\). Can \(\beta\) be consistently estimated? If yes, provide the formula or objective function for a consistent estimator. If no, give a brief explanation of why \(\beta\) cannot be consistently estimated.

(b) Suppose the individual-specific effects \(\alpha_{i}\) are random and are iid \(\left[0, \sigma_{\alpha}^{2}\right]\) distributed independently of the regressors. Can \(\beta\) be consistently estimated? If yes, provide the formula or objective function for a consistent estimator. If no, give a brief explanation of why \(\beta\) cannot be consistently estimated.

(c) Suppose the individual specific effects \(\alpha_{i}\) are random but are correlated with the regressors. Can \(\beta\) be consistently estimated? If yes, provide the formula or objective function for a consistent estimator. If no, give a brief explanation of why \(\beta\) cannot be consistently estimated.