(Linear and Nonlinear Models) Determine which of the following models can be transformed into linear models and which cannot, and identify the transformation in the cases where it is possible.

(a) \(Y_{t}=\left(\prod_{j=1}^{k} X_{j t}^{\beta_{j}}ight) V_{t}\), where \(V_{t} \sim \operatorname{Gamma}

(a, b)\)

(b) \(\frac{Y_{t}^{\lambda}-1}{\lambda}=\beta_{1}+\beta_{2}\left(\frac{X_{t}^{\delta}-1}{\delta}ight)+\varepsilon_{t}\), where \(\varepsilon_{\mathrm{t}} \sim N\left(0, \sigma^{2}ight)\)

(c) \(Y_{t}=\beta_{0}+\beta_{1} X_{t}^{\beta_{2}}+\varepsilon_{t}\) where \(\varepsilon_{t} \sim N\left(0, \sigma^{2}ight)\)

(d) \(Y_{t}=\beta_{1}\left(\beta_{2} L_{t}^{-\beta_{3}}+\left(1-\beta_{2}ight) K_{t}^{-\beta_{3}}ight)^{-\beta_{4} / \beta_{1}} \exp \left(\varepsilon_{t}ight)\), where \(\varepsilon_{\mathrm{t}} \sim f(e)\)

(e) \(Y_{t}=\left(1+\exp \left(\mathbf{X}_{t}^{\prime} \boldsymbol{\beta}+\varepsilon_{t}ight)ight)^{-1}\), where \(\varepsilon_{t} \sim \operatorname{Logistic}(0, s)\)