Consider a dataset that examines the relationship between a company's advertising expenditure and its sales revenue. The dataset contains the natural logarithm of advertising expenditure and the corresponding sales revenue for $5$ companies.

The semi$-$logarithmic regression model $($levels$-$log$)$ is specified as follows:

Where:

Sales is the sales revenue in thousands of dollars.

Advertising is the advertising expenditure in thousands of dollars.

${}_0$ and ${}_1$ are the parameters to be estimated.

$u$ is the error term.

$\backslash$table$[[$Company$,ln($Advertising$),$Sales $($thousands $$)],[$A$,4\mathrm{.}605,250],[$B$,5\mathrm{.}298,300],[$C$,5\mathrm{.}991,400],[$D$,4\mathrm{.}7,220],[$E$,5\mathrm{.}521,350]]$

Calculate the following using the formulas covered in this chapter:

$($a$)$ The OLS estimates for ${}_0$ and ${}_1.$ Interpret the coefficient ${}_1.$

$($b$)$ The SST$,$ SSE and SSR components of the total variation.

$($c$)$ The R$-$squared. Interpret the value obtained.