Bonus Problem. $(30$pt$)$

We try to build a deeper understanding of the endogeneity issue in the regression model

$y=a+bx+lon.$

Suppose that $x$ and $lon$ are random variables, and parameters a and $b$ are constants.

$($a$).$ Show that $b=\frac{\text{C}\text{o}\text{v}(x,y)-\text{C}\text{o}\text{v}(x,lon)}{\text{V}\text{a}\text{r}(x)}.$

Hint: the covariance of the random variables Cov$(x,y)$ has the following

properties

Cov$(x,y)=$Cov$(x,a+bx+lon)=$Cov$(x,a)+$Cov$(x,bx)+$Cov$(x,lon),$

Cov$(x,bx)=$bVar$(x).$

$($b$).$ If $x$ and $lon$ are independent, what is Cov$(x,lon)?$