Consider the following model:

$\backslash [$

$\backslash$mathbf$\{$y$\}=\backslash$mathbf$\{$X$\}\backslash$boldsymbol$\{\backslash$beta$\}+\backslash$boldsymbol$\{\backslash$varepsilon$\}$

$\backslash ]$

known as the Classical Linear Regression Model $($CLRM$),$ where $\backslash ($ y $\backslash )$ is the dependent variable, X is the set of independent variables, $\backslash (\backslash$boldsymbol$\{\backslash$beta$\}\backslash )$ is the vector of parameters to be estimated and $\backslash (\backslash$varepsilon $\backslash )$ is the error term.

a$)$ List and discuss the assumptions you need for the Ordinary Least Squares $($OLS$)$ to be a Best Linear Unbiased Estimator $($BLUE$).$ Derive the OLS estimator.

$[25$ marks$]$

b$)$ Discuss the properties of linearity, unbiasedness and efficiency, discussing what assumption you need for each of these properties to hold.

$[25$ marks$]$

c$)$ In what sense decisions based upon the OLS framework can be regarded as rational?

$[25$ marks$]$

d$)$ What are the warnings of using the OLS approach to hypothesis testing? Discuss.

$[25$ marks$]$