Consider the regression model (y_{i}=beta_{1}+beta_{2} x_{i}+e_{i}) where the pairs (left(y_{i}, x_{i} ight), i=1,2, ldots, N), are random

Consider the regression model \(y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}\) where the pairs \(\left(y_{i}, x_{i}\right), i=1,2, \ldots, N\), are random independent draws from a population, \(x_{i} \sim N(0,1)\), and \(E\left(e_{i} \mid x_{i}\right)=c\left(x_{i}^{2}-1\right)\) where \(c\) is a constant.

a. Show that \(E\left(e_{i}\right)=0\).

b. Using the result \(\operatorname{cov}\left(e_{i}, x_{i}\right)=E_{x}\left[\left(x_{i}-\mu_{x}\right) E\left(e_{i} \mid x_{i}\right)\right]\), show that \(\operatorname{cov}\left(e_{i}, x_{i}\right)=0\).

c. Will the least squares estimator for \(\beta_{2}\) be (i) unbiased? (ii) consistent?