Write a Jupyter notebook in code python for the following simulation exercise. Consider the linear model

yi $=\backslash$alpha $+\backslash$beta xi $+$ ei where xi $$ fx$(\backslash$theta x$),$ ei $$ fe$(\backslash$theta e$)$ and fx$,$ fe are PDFs of some random

variable. xi and ei are independent. For this question, assume that fx and fe are both

standard normal distribution but we will consider different distributions below. assume that xi and ei are correlated with a correlation

coefficient of $\backslash$rho xe $=0\mathrm{.}5.$

$($a$)$ What are the theoretical properties of the OLS estimators b$\backslash$alpha and b$\backslash$beta $?$ What is their

theoretical distribution given a sample size N$?$

$($b$)$ Draw M i$.$i$.$d$.$ random samples of length N for xi and ei$.$ Start with M $=5000$ and

N $=50$ but you should experiment with different values.

For each m $=1,...,$M$,$ compute the OLS estimators b$\backslash$alpha m and b$\backslash$beta m and their standard

errors. Compute the means and standard deviations of the distribution of the M

estimators b$\backslash$alpha m and b$\backslash$beta m$.$

$($c$)$ Plot the histograms of the M b$\backslash$alpha m and b$\backslash$beta m$.$

$($d$)$ Compare the distribution of the simulated b$\backslash$alpha m and b$\backslash$beta m to their theoretical distributions.