An economist is analyzing the relationship between disposable income and food expenditure for the citizens of a developing country. A survey of the citizens has been taken, and data have been collected on income and food expenditures. The following statistical model is postulated for the data:

\(Y_{i}=\beta_{1} z_{i}^{\beta_{2}} e^{\varepsilon_{i}}\), where

\(y_{i}=\) expenditure on food for the \(i\) th household;

\(Z_{i}=\) disposable income for the \(i\) th household; and

\(\varepsilon_{i}^{\prime} \mathrm{s} \sim\) iid \(N\left(0, \sigma^{2}ight)\)

(a) Transform the model into the GLM form. What parameters or functions of parameters are being estimated by the least-squares estimator applied to the transformed model?

(b) Is the least-squares estimator unbiased, the BLUE, and/or the MVUE for the parameters or functions of parameters being estimated?

(c) The actual survey consisted of 5,000 observations, and the following summary of the data is available:
\[
\begin{aligned}
\left(\mathbf{x}^{\prime} \mathbf{x}ight)^{-1} & =\left[\begin{array}{rr}
.17577542 & -.019177442 \\
-.019177442 & .0020946798 \end{array}ight] \\
\mathbf{x}^{\prime} \mathbf{y} & =\left[\begin{array}{l}
10579.646 \\
98598.324 \end{array}ight], \\
\mathbf{y}^{\prime} \mathbf{y} & =14965.67 \end{aligned}
\]
where the \(\mathbf{x}\)-matrix is a \((5,000 \times 2)\) matrix consisting of a column of 1's and a column representing the natural logarithms of the observations on income, and the \(y\)-vector refers to the corresponding natural logarithms of food expenditures. Calculate the least-squares estimate of the parameters of the transformed model. What is your estimate of the elasticity of food expenditure with respect to income?

(d) What is the probability distribution of \(\hat{\beta}_{2}\), the leastsquares estimator of \(\beta_{2}\) ? Generate the MVUE estimates of the mean and variance of this distribution. (You might find it useful to know that \(\left(y^{\prime} \mathbf{y}-\mathbf{y}^{\prime} \mathbf{x}ight.\) \(\left.\left.\left(\mathbf{x}^{\prime} \mathbf{x}ight)^{-1} \mathbf{x}^{\prime} \mathbf{y}ight) /(4,998)=25.369055ight) . \quad\) Given the assumptions of the model, and using the MVUE estimates of mean and variance, what is the (estimated) probability that the income elasticity estimated by the least-squares approach will be within \(\mp .2\) of the true income elasticity?

(e) Discuss any alterations to the specification of the relationship between food expenditure and disposable income that you feel is appropriate for this problem.