(Nonlinear Least Squares Estimator) Consider the nonlinear regression model

\(\mathbf{Y}=\mathbf{g}(\mathbf{x}, \boldsymbol{\beta})+\boldsymbol{\varepsilon}=\beta_{1} \mathbf{1}_{n}+\beta_{2} \mathbf{x}+\beta_{3} \mathbf{x}^{\beta_{4}}+\boldsymbol{\varepsilon}\), for \(\boldsymbol{\beta} \in \Omega\)

where \(\mathbf{Y}\) is \(n \times 1, \mathbf{1}_{n}\) is an \(n \times 1\) vector of 1 's, the \(n \times 1\) vector \(\mathbf{x}\) is not a vector of identical constants, \(E(\boldsymbol{\varepsilon} \mid \mathbf{x})=\mathbf{0}\) and \(\operatorname{Cov}(\boldsymbol{\varepsilon} \mid \mathbf{x})=\sigma^{2} \mathbf{I}\).

(a) Is a parameter vector with \(\beta_{4}=0\) identifiable in this model?

(b) Is a parameter vector with \(\beta_{4}=1\) identifiable in this model?

(c) Define an admissible parameter space, \(\boldsymbol{\Omega}\), for this model

(d) Given a random sample of data \((\mathbf{Y}, \mathbf{x})\), describe how you would go about estimating this model based on the least squares principle.

(e) Identify an asymptotic distribution for the least squares estimator of \(\boldsymbol{\beta}\).

(f) Suppose that \(\beta_{4}\) was "in a close neighborhood", but not exactly equal to one of the values 0 or 1 . Could there be any issues relating to "multicollinearity" affecting the least squares estimator of the parameter vector? Explain.