In an effort to model executive compensation for the year 1979,33 firms were selected, and data were gathered on compensation, sales, profits, and employment.
$$
\begin{array}{ccccc}
& \text { Compen- } & & & \\
\text { Firm (thousands) } & \begin{array}{c}
\text { sation, } y \\
\text { Sales, } x_{1} \\
\text { (millions) }
\end{array} & \begin{array}{c}
\text { Profits, } x_{2} \\
\text { (millions) }
\end{array} & \begin{array}{c}
\text { Employ- } \\
\text { ment, } x_{3}
\end{array} \\
\hline 1 & 450 & 4600.6 & 128.1 & 48000 \\
2 & 387 & 9255.4 & 783.9 & 55900 \\
3 & 368 & 1526.2 & 136 & 13783 \\
4 & 277 & 1683.2 & 179 & 27765 \\
5 & 676 & 2752.8 & 231.5 & 34000 \\
6 & 454 & 2205.8 & 329.5 & 26500 \\
7 & 507 & 2384.6 & 381.8 & 30800 \\
8 & 496 & 2746 & 237.9 & 41000 \\
9 & 487 & 1434 & 222.3 & 25900 \\
10 & 383 & 470.6 & 63.7 & 8600 \\
11 & 311 & 1508 & 149.5 & 21,075 \\
12 & 271 & 464.4 & 30 & 6874 \\
13 & 524 & 9329.3 & 577.3 & 39000 \\
14 & 498 & 2377.5 & 250.7 & 34300 \\
15 & 343 & 1174.3 & 82.6 & 19405 \\
16 & 354 & 409.3 & 61.5 & 3586 \\
17 & 324 & 724.7 & 90.8 & 3905 \\
18 & 225 & 578.9 & 63.3 & 4139 \\
19 & 254 & 966.8 & 42.8 & 6255 \\
20 & 208 & 591 & 48.5 & 10605 \\
21 & 518 & 4933.1 & 310.6 & 65392 \\
22 & 406 & 7613.2 & 491.6 & 89400 \\
23 & 332 & 3457.4 & 228 & 55200 \\
24 & 340 & 545.3 & 54.6 & 7800 \\
25 & 698 & 22862.8 & 3011.3 & 337119 \\
26 & 306 & 2361 & 203 & 52000 \\
27 & 613 & 2614.1 & 201 & 50500 \\
28 & 302 & 1013.2 & 121.3 & 18625 \\
29 & 540 & 4560.3 & 194.6 & 97937 \\
30 & 293 & 855.7 & 63.4 & 12300 \\
31 & 528 & 4211.6 & 352.1 & 71800 \\
32 & 456 & 5440.4 & 655.2 & 87700 \\
33 & 417 & 1229.9 & 97.5 & 14600
\end{array}
$$
Consider the model,
$$
\begin{aligned}
y_{i}=& \beta_{0}+\beta_{1} \ln x_{1 i}+\beta_{2} \ln x_{2 i} \\
&+\beta_{3} \ln x_{3 i}+\epsilon_{i}, \quad i=1,2, \ldots, 33 .
\end{aligned}
$$
(a) Fit the regression with the model above.
(b) Is a model with a subset of the variables preferable to the full model?