For each of the following estimated equations of the form $y=$hat$({)}_0+$hat$({)}_{1}x,$ interpret the value of hat$({)}_1.$

$($a$)y=2+0\mathrm{.}4x,$ where $y$ is the rate of return of an asset in percentage points $($e$.$g$.3),$ and $x$ is the rate of return of the market, also in percentage points. Determine the type of relationship as well as the interpretation of hat$({)}_1.$

$($b$)y=10\mathrm{.}5-0\mathrm{.}2x,$ where $y$ is a firm's market value $($in million $ $),$ and $x$ is the number of competitors or firms in the market. Determine the type of relationship as well as the interpretation of hat$({)}_1.$

$($c$)log(y)=-1\mathrm{.}4+1\mathrm{.}8log(x),$ where $y$ is the firm's stock price $($in $ $),$ and $x$ is the market return. Determine the type of relationship as well as the interpretation of hat$({)}_1.$ Keep in mind that both variables are in logs$.$

$($d$)y=530\mathrm{.}4-378\mathrm{.}4log(x),$ where $y$ is the firm's stock price $($in $ $),$ and $x$ is the oil price per barrel $($in $ $).$ Determine the type of relationship as well as the interpretation of hat$({)}_1.$ Keep in mind that one of the variables is in logs$.$ If this was a realistic example, what kind of firms and in which industry would experience this type of relationship with the oil price?