A research analyst is trying to determine whether a firm$$s price$-$earnings $($PE$)$ and price$-$sales $($PS$)$ ratios can explain the firm$$s stock performance over the past year. A PE ratio is calculated as a firm$$s share price compared to the income or profit earned by the firm per share. Generally, a high PE ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower PE ratio. The PS ratio is calculated by dividing a firm$$s share price by the firm$$s revenue per share for the trailing $12$ months. In short, investors can use the PS ratio to determine how much they are paying for a dollar of the firm$$s sales rather than a dollar of its earnings $($PE ratio$).$ In general, the lower the PS ratio, the more attractive the investment. The accompanying table shows a portion of the year$-$to$-$date returns $($Return in $\%)$ and the PE and PS ratios for $30$ firms.

Firm Return PE PS

$14\mathrm{.}414\mathrm{.}352\mathrm{.}39$

$24\mathrm{.}511\mathrm{.}000\mathrm{.}81$

$$

$3016\mathrm{.}313\mathrm{.}971\mathrm{.}96$

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a$-1.$ Estimate: Return $=\backslash$beta $0+\backslash$beta $1$PE $+\backslash$beta $2$PS $+\backslash$epsi $.($Negative values should be indicated by a minus sign. Round your answers to $2$ decimal places.$)$

a$-2.$ Are the signs on the coefficients as expected?

multiple choice $1$

Yes

No

b$.$ Interpret the slope coefficient of the PS ratio.

multiple choice $2$

As the PS ratio increases by $1$ unit, the predicted return of the firm increases by $3\mathrm{.}32\%,$ holding PE constant.

As the PS ratio increases by $1$ unit, the predicted return of the firm decreases by $3\mathrm{.}32\%,$ holding PE constant.

As the PS ratio decreases by $1$ unit, the predicted return of the firm decreases by $33\mathrm{.}17\%,$ holding PE constant.

As the PS ratio increases by $1$ unit, the predicted return of the firm decreases by $33\mathrm{.}17\%,$ holding PE constant.

c$.$ What is the predicted return for a firm with a PE ratio of $10$ and a PS ratio of $2?($Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round final answer to $2$ decimal places.$)$

d$.$ What is the standard error of the estimate? $($Round your answer to $2$ decimal places.$)$

e$.$ Interpret R$2.$

multiple choice $3$

$40\mathrm{.}08\%$ of the sample variation in y is explained by the sample regression equation.

$40\mathrm{.}08\%$ of the sample variation in x is explained by the sample regression equation.

$63\mathrm{.}31\%$ of the sample variation in x is explained by the sample regression equation.

$35\mathrm{.}65\%$ of the sample variation in y is explained by the sample regression equation. Return PE PS

$4\mathrm{.}414\mathrm{.}352\mathrm{.}39$

$-4\mathrm{.}5110\mathrm{.}81$

$5\mathrm{.}911\mathrm{.}972\mathrm{.}28$

$511\mathrm{.}311\mathrm{.}33$

$-11\mathrm{.}39\mathrm{.}81\mathrm{.}84$

$20\mathrm{.}915\mathrm{.}660\mathrm{.}79$

$64\mathrm{.}316\mathrm{.}091\mathrm{.}61$

$18\mathrm{.}98\mathrm{.}971\mathrm{.}01$

$-15\mathrm{.}211\mathrm{.}362\mathrm{.}76$

$4813\mathrm{.}711\mathrm{.}46$

$7\mathrm{.}311\mathrm{.}491\mathrm{.}18$

$20\mathrm{.}914\mathrm{.}331\mathrm{.}31$

$-18\mathrm{.}58\mathrm{.}150\mathrm{.}82$

$2\mathrm{.}99\mathrm{.}482\mathrm{.}7$

$12\mathrm{.}411\mathrm{.}171\mathrm{.}95$

$-3\mathrm{.}712\mathrm{.}632\mathrm{.}73$

$1\mathrm{.}69\mathrm{.}661\mathrm{.}98$

$15\mathrm{.}913\mathrm{.}481\mathrm{.}13$

$22\mathrm{.}814\mathrm{.}953\mathrm{.}38$

$-1\mathrm{.}28\mathrm{.}942\mathrm{.}38$

$-8\mathrm{.}110\mathrm{.}573\mathrm{.}74$

$-3\mathrm{.}37\mathrm{.}942\mathrm{.}19$

$15\mathrm{.}616\mathrm{.}44\mathrm{.}56$

$21\mathrm{.}116\mathrm{.}310\mathrm{.}89$

$6\mathrm{.}315\mathrm{.}052\mathrm{.}33$

$129\mathrm{.}160\mathrm{.}98$

$13\mathrm{.}314\mathrm{.}911\mathrm{.}39$

$15\mathrm{.}415\mathrm{.}580\mathrm{.}9$

$112\mathrm{.}560\mathrm{.}48$

$16\mathrm{.}313\mathrm{.}971\mathrm{.}96$