Consider the linear regression model with scalar regressor y i = x i + u i

Consider the linear regression model with scalar regressor $y_i=\beta {\mathbf{x}}_i+u_i$ with data $(y_i,x_i)$ iid over $i$ though the error may be conditionally heteroskedastic.

(a) Show that $({\hat{\beta }}_{\mathrm{O}\mathrm{L}\mathrm{S}}-\beta )={\left(N^{-1}\sum _i{x}_i^2\right)}^{-1}{N}^{-1}\sum _i{x}_i{u}_i$.

(b) Apply Kolmogorov law of large numbers (Theorem A.8) to the averages of $x_i^2$ and $x_i{u}_i$ to show that ${\hat{\beta }}_{\text{OLS }}\stackrel{p}{\to }\beta$. State any additional assumptions made on the dgp for $x_i$ and $u_i$.

(c) Apply the Lindeberg-Levy central limit theorem (Theorem A.14) to the averages of $x_i{u}_i$ to show that $N^{-1}\sum _i{x}_i{u}_i/N^{-2}\sum _{i}E\left[u_i^2{x}_i^2\right]\stackrel{p}{\to }{N}[0,1]$. State any additional assumptions made on the dgp for $x_i$ and $u_i$.

(d) Use the product limit normal rule (Theorem A.17) to show that part (c) implies $N^{-1/2}\sum _i{x}_i{u}_i\stackrel{p}{\to }{N}[0,lim{N}^{-1}\sum _{i}E\left[u_i^2{x}_i^2\right]]$. State any assumptions made on the dgp for $x_i$ and $u_i$.

(e) Combine results using (2.14) and the product limit normal rule (Theorem A.17) to obtain the limit distribution of $\beta$.

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V[yilzi] == BB- (2.14)