Suppose the dgp is y i = 0 x i + u i , u i

Suppose the dgp is $y_i={\beta }_0{x}_i+u_i,u_i=x_i{\epsilon }_i,x_i\sim {N}[0,1]$, and ${\epsilon }_i\sim {N}[0,1]$. Assume that data are independent over $i$ and that $x_i$ is independent of ${\epsilon }_i$. Note that the first four central moments of ${N}[0,{\sigma }^2]$ are $0,{\sigma }^2,0$, and $3{\sigma }^4$.

(a) Show that the error term $u_i$ is conditionally heteroskedastic.

(b) Obtain plim $N^{-1}{\mathbf{X}}^{\mathrm{\prime }}\mathbf{X}$. [Obtain $E\left[x_i^2\right]$ and apply a law of large numbers.]

(c) Obtain ${\sigma }_0^2=\mathrm{V}\left[u_i\right]$, where the expectation is with respect to all stochastic variables in the model.

(d) Obtain plim $N^{-1}{\mathbf{X}}^{\mathrm{\prime }}{\mathrm{\Omega }}_0\mathbf{X}=lim{N}^{-1}E\left[{\mathrm{X}}^{\mathrm{\prime }}{\mathrm{\Omega }}_0\mathrm{X}\right]$, where ${\mathrm{\Omega }}_0=\mathrm{Diag}[\mathrm{V}[u_i\mid X_i]$.

(e) Using answers to the preceding parts give the default OLS result (4.22) for the variance matrix in the limit distribution of $\sqrt{N}({\hat{\beta }}_{\text{OLS }}-{\beta }_0)$, ignoring potential heteroskedasticity. Your ultimate answer should be numerical.

(f) Now give the variance in the limit distribution of $\sqrt{N}({\hat{\beta }}_{\text{OLS }}-{\beta }_0)$, taking account of any heteroskedasticity. Your ultimate answer should be numerical.
(g) Do any differences between answers to parts (e) and (f) accord with your prior beliefs?