Let observations on the quantity supplied of a certain commodity be generated by \(Y_{i}=x_{i} \beta+V_{i}\), where \(\left|x_{i}ight| \in[a\), b] \(\forall i\) are scalar observations on fixed prices, \(\beta\) is an unknown slope coefficient, and the \(V_{i}^{\prime}\) 's are iid random variables having a mean of zero, a variance of \(\sigma^{2} \in(0, \infty)\), and \(P\left(\left|V_{i}ight| \leq might)=1 \forall i

(a, b\), and \(m\) are finite positive constants). Two functions of the \(x_{i}{ }^{\prime}\) s and \(Y_{i}\) 's are being considered for generating an estimate of the unknown value of \(\beta\) :

\(\hat{\beta}=\left(\mathbf{x}^{\prime} \mathbf{x}ight)^{-1} \mathbf{x}^{\prime} \mathbf{Y}\) and \(\hat{\beta}_{\mathrm{r}}=\left(\mathbf{x}^{\prime} \mathbf{x}+\mathrm{k}ight)^{-1} \mathbf{x}^{\prime} \mathbf{Y}\), where \(\mathrm{k}>0, \mathbf{x}\) is an \((n \times 1)\) vector of observations on the \(x_{i}{ }^{\prime}\) s and \(\mathbf{Y}\) is an \((n \times 1)\) vector of the corresponding \(Y_{i}^{\prime}\) s.

(a) Define the means and variances of the two estimators of \(\beta\).

(b) Is it true that \(\lim _{n ightarrow \infty} \mathrm{E}(\hat{\beta})=\beta\) and/or \(\lim _{\mathrm{n} ightarrow \infty}\) \(\mathrm{E}\left(\hat{\beta}_{\mathrm{r}}ight)=\beta\) ?

(c) Define the expected squared distances of the two estimators from \(\beta\).

(d) Which, if either, of the estimators converges in mean square to \(\beta\) ?

(e) Which, if either, of the estimators converges in probability to \(\beta\) ?

(f) Define asymptotic distributions for each of the estimators.

(g) Under what circumstances would you prefer one estimator to the other for generating an estimate of the unknown value of \(\beta\) ?