Business Finance – Economics
Explain in your own words how to do the tests and whether or not you reject the hypotheses. For all tests, report p-values.
· Cut and paste your software output for all commands that you execute. In Stata, an easy way to copy is to highlight, right-click, and choose ‘Copy as Picture.’
· . You don’t need to provide any data, but we do need to see the regression estimates and the commands that you executed.
Use the data in the file mlbsal17f. The file contains a cross section of 353 observations on salaries and performance measures for baseball players. (When importing the data into Stata, make sure to check ‘first row is variable names.’)
Suppose you want to model a player’s salary as a function of the player’s total games played, at-bats, runs scored, hits, runs batted in, home runs, doubles, and triples. The units of salary are thousands of dollars.
The log-linear model takes the form:
ln(salary) = β0 + β1games + β2ab + β3hits + β4runs + β5rbi + β6doubles + β7triples + β8hr + u.
The linear model takes the form:
salary = b0 + b1games + b2ab + b3hits + b4runs + b5rbi + b6doubles + b7triples + b8hr + v
The double-log model takes the form:
ln(salary)=B0 + B1ln(games) + B2ln(ab) + B3ln(hits) + B4ln(runs) + B5ln(rbi) + B6doubles + B7triples + B8hr + w.
(We don’t take the logs of doubles, triples, or hr because some observations are zero.)
a. For all three models, report and interpret the estimated coefficient for runs.
b. Which of the three models fits the data best?
c. Consider the 2nd player in the sample, with 918 games played, 3333 at-bats, etc. What salary does the double-log model predict for such a player?
d. Compute a test statistic for testing the hypothesis that the double-log model has no explanatory power.
H0: R2 = 0.
State the distribution of the test statistic (Remember also for every test in this exam to report the p-value.)
e. For the double-log model, test at the .10 level the hypothesis that, all else equal, doubles has no influence on ln(salary).
H0: B6 = 0.
Now do the same for hr.
H0: B8 = 0.
The test statistics have how many degrees of freedom?
f. Can you use the results from part e to make any inferences about the hypothesis that doubles and hr together have no effect on ln(salary),
H0: B6 = B8 = 0?
Explain.
Perform an F-test of the above hypothesis at the .10 level. State the degrees of freedom for the test, and show how the test statistic can be computed from sums of squared residuals. Can you reject H0?
g. In the presence of heteroscedasticity, are the coefficient estimates still BLUE? Are conventional test statistics still valid? Describe the properties of the estimators of the coefficients and the standard errors under heteroscedasticity.
h. Use a Breusch-Pagan test to test for heteroscedasticity in the double-log model. After estimating the model in Stata, you can perform the test using estat hettest followed by the list of explanatory variables. Do you find evidence of heteroscedasticity?
i. Can you show how the same Breusch-Pagan test statistic in part h can be computed from a regression involving the squared (and scaled) residuals?
j. Perform a heteroscedasticity-consistent test of the hypothesis that, in the double-log model, the coefficient on ln(ab) equals zero
H0: B2 = 0.
Does correcting for heteroscedasticity change your inference on this coefficient in this model?
k. Find the double-log model that maximizes the adjusted R2. State the rule that gives a necessary condition for the adjusted R2 to be maximized. Does the adjusted R2 have interpretation as a percentage of variation?
l. Stata reports the Standard Error of the Regression (aka Standard Error of the Estimate) under “Root MSE.” Show for the double-log model how this statistic can be computed from the sum of squared residuals.
m. Show that the bias in the coefficient on ln(games) due to omitting from the model ln(ab) equals the product of two terms: (1) the coefficient on ln(ab) and (2) the coefficient on ln(games) from a regression of ln(ab) on all the other explanatory variables.
Sheet1
games ab runs hits doubles triples hr rbi salary
1705 6705 1076 1939 320 67 231 836 6329.213
918 3333 407 863 156 38 73 342 3375
751 2807 370 840 148 18 46 355 3100
1056 3337 405 816 143 18 107 421 2900
1196 3603 437 928 19 16 124 541 1650
2032 7489 1136 2145 270 142 40 574 700
394 1293 179 303 51 13 37 141 695
432 1005 78 240 35 5 13 95 275
223 491 37 118 16 5 1 29 235
156 434 45 116 16 0 10 59 212.5
169 447 41 111 18 1 14 44 212.5
87 278 25 70 14 3 1 21 180
789 2872 312 758 124 15 85 375 3400
45 66 7 12 1 0 1 4 2000
442 1617 230 419 69 22 35 209 1250
528 1608 178 395 60 7 8 130 825
701 2088 275 610 110 11 21 254 750
478 1348 155 351 69 9 26 123 200
168 421 42 114 23 2 3 42 190
109 330 19 71 10 1 8 33 138
86 197 19 44 8 1 4 27 136
37 111 13 28 7 2 0 11 125
32 99 14 29 6 0 0 7 123
38 77 9 16 3 0 0 4 123
433 1481 207 409 79 9 58 212 3000
415 1592 234 433 62 18 21 152 1537.5
431 1573 230 422 78 17 24 136 1500
131 361 57 100 28 3 9 56 210
145 389 22 92 19 2 4 41 170
126 274 25 64 12 3 5 28 142.5
45 126 17 38 4 1 2 8 126.5
34 83 5 15 2 0 1 4 117.5
37 74 7 14 5 0 3 14 115
1046 3732 572 1040 224 37 135 596 6200
2444 9124 1343 2646 462 32 414 1562 3375
1021 4042 648 1071 124 62 18 255 2312.5
1328 5132 675 1465 251 70 48 479 2300
1279 4309 672 1092 225 17 204 672 2150
930 2935 384 817 131 29 37 265 650
411 1004 133 251 30 6 12 96 635
339 889 87 182 48 3 12 92 500
533 1555 199 422 70 8 8 136 475
180 485 45 97 23 1 8 41 195
30 108 15 24 7 1 3 10 109
931 3219 533 916 190 28 52 283 2466.667
930 2946 434 875 142 29 79 446 2450
706 2114 263 492 105 9 75 309 2416.667
788 2830 368 722 123 27 53 253 2000
980 2982 398 838 124 35 37 258 1375
904 3135 402 772 154 15 146 470 1050
500 1697 207 477 98 7 41 245 1000
698 2144 251 594 121 23 25 237 600
284 851 136 224 38 6 38 129 600
250 826 94 212 23 12 5 53 300
195 704 86 176 37 3 24 95 250
159 362 41 66 12 3 2 23 150
54 163 11 34 4 0 1 11 122.5
1390 4737 720 1308 251 82 144 688 4950
669 2444 349 627 125 26 39 239 2600
736 2134 166 574 97 5 16 243 2350
1004 3071 320 859 193 26 59 351 1066.667
1475 4888 853 1414 264 54 90 504 1000
365 1175 149 270 52 7 37 155 675
431 931 120 232 39 2 29 123 675
952 2367 215 578 114 19 26 224 360
279 840 136 218 46 7 16 110 305
38 67 7 16 1 2 0 5 120
92 229 26 58 13 0 0 11 112
12 12 1 2 0 1 0 2 109
10 7 2 4 0 0 0 4 109
2208 8087 1079 2108 347 57 22 681 3897.184
617 2317 312 644 132 12 52 280 2650
508 1487 119 379 77 8 13 151 2575
1001 2780 288 726 94 19 12 224 2050
453 1581 196 413 82 11 34 194 1025
972 2890 346 761 134 11 52 357 966.667
297 1003 131 252 38 12 20 95 508.333
343 1290 182 353 73 22 32 167 290
174 418 46 101 20 2 22 63 225
230 716 95 193 31 8 13 66 175
184 433 74 118 22 4 12 50 170
234 630 51 141 22 15 3 56 160
701 2586 402 670 131 23 111 363 3700
869 1996 381 513 53 10 6 123 2815
1240 4659 596 1274 228 33 87 633 2250
578 1769 237 464 90 15 49 207 2000
396 1370 228 369 70 8 71 240 1775
925 2770 311 726 172 11 81 413 1600
222 593 101 145 11 18 17 57 1000
379 1191 98 281 60 2 28 146 975
331 1013 109 250 51 3 16 107 925
777 1689 171 377 31 12 1 113 800
562 1353 195 336 65 10 20 127 675
423 1048 108 237 37 7 10 82 550
199 509 66 125 29 3 26 91 205
126 268 24 69 14 1 13 51 181.5
835 3122 478 924 150 30 70 368 5700
636 2091 378 626 103 22 20 158 3900
900 3109 469 859 162 20 171 548 3750
638 2277 320 637 110 12 56 258 3133.333
616 2335 342 637 153 11 80 275 3100
388 1220 166 371 76 7 27 152 1275
407 1269 110 314 67 1 32 173 1200
1310 5146 718 1338 243 85 128 574 700
554 1755 258 450 86 9 62 236 475
355 557 71 135 28 7 8 57 400
125 425 68 112 26 6 13 39 200
34 51 6 14 4 0 0 4 120
530 1845 168 461 77 4 48 219 1200
424 1365 144 347 69 6 38 176 725
340 1022 88 242 45 7 28 126 690
304 893 73 234 39 3 3 70 670
942 3407 432 909 182 16 116 472 600
882 2261 321 565 114 21 65 224 550
290 916 145 259 26 14 0 49 350
605 1755 249 432 55 17 3 109 310
2154 7918 196 2105 349 39 398 1259 225
98 236 25 58 11 1 1 20 167.5
21 21 2 5 1 0 0 1 120
694 2492 271 639 121 11 44 295 3150
645 2280 306 624 106 12 30 192 3050
544 1884 249 520 78 29 18 171 2900
641 1743 175 435 75 20 9 124 775
318 1140 166 323 60 10 33 178 655
1267 4112 509 1108 203 39 104 507 500
779 2665 283 694 122 19 72 333 500
993 3011 303 725 98 34 19 216 400
285 858 89 180 31 1 34 123 350
271 881 92 218 49 12 23 124 325
130 363 28 82 17 1 5 36 162.5
93 242 12 55 12 0 0 21 156.5
145 479 42 99 26 4 11 49 135
82 210 15 41 6 0 0 11 130
1291 4564 768 1196 217 34 285 857 3800
1767 6529 737 1694 360 31 204 905 3362.5
1678 6169 1048 1777 216 99 44 439 3333.333
715 2658 361 743 180 7 17 227 2500
852 2987 360 729 139 12 132 414 1500
932 3124 575 828 127 19 182 564 1000
477 106 135 313 56 5 8 111 970
548 1585 191 443 54 6 8 131 900
1041 3107 456 825 139 52 63 313 600
163 559 63 141 31 1 20 89 435
230 705 84 170 22 8 2 40 300
94 207 14 52 6 0 3 19 175
21 69 5 16 3 0 1 7 126
883 3003 511 839 148 13 191 515 4250
1463 5701 842 1864 278 78 59 591 4083.333
440 1667 225 471 95 6 54 233 3110
873 2866 339 716 141 28 32 292 1750
977 2912 389 739 174 10 79 348 762.5
1441 4373 398 1131 198 12 68 446 300
175 512 74 137 27 1 18 68 245
79 189 28 43 6 3 2 16 165
131 265 22 64 11 0 1 19 149
15 23 1 7 0 0 1 5 109
1028 3778 605 1139 222 35 162 636 4750
1010 3584 672 984 220 36 176 556 4416.667
1462 5669 773 1689 257 83 57 639 2312.5
682 2428 293 586 97 16 121 376 2250
983 3426 487 883 174 34 85 342 1900
836 2555 329 688 99 37 39 228 1100
315 863 66 199 30 8 5 79 735
722 2341 257 593 115 14 50 317 500
540 1425 103 332 63 6 13 19 325
373 932 107 240 37 1 31 106 300
197 577 83 137 13 4 2 34 172.5
107 347 31 75 8 4 4 26 155
1800 6942 1043 1922 369 34 273 1014 5359.973
985 3606 502 945 174 13 189 594 3800
584 2116 272 547 92 25 63 267 3025
549 1704 239 406 70 21 31 168 1855
1155 4090 543 1063 200 48 17 295 1650
1844 6744 865 1930 334 46 241 1066 1100
232 723 92 181 29 1 32 78 350
267 898 105 224 41 2 33 110 312.5
402 1122 163 257 37 8 5 102 300
241 524 50 133 23 0 5 54 237.5
150 414 58 111 22 2 18 58 197.5
42 104 17 18 4 0 0 6 120
2310 8890 1259 2504 444 94 399 1425 4875
831 2980 418 912 167 26 84 489 3325
842 3073 444 851 190 23 103 422 3000
1624 5550 584 1481 256 26 95 614 2200
638 1940 209 452 100 9 24 187 1075
1004 3521 474 926 171 25 42 311 750
595 1746 159 410 78 6 31 185 700
1361 4411 557 1155 193 31 25 437 675
825 2361 294 610 84 20 43 264 400
337 1073 132 306 54 1 18 146 340
187 574 63 140 28 2 17 89 270
139 373 40 109 25 2 5 40 240
142 417 49 112 19 1 4 46 222
116 326 23 72 15 0 5 29 200
58 185 21 51 13 0 5 25 135
347 1287 164 335 62 7 70 247 1600
280 948 93 248 52 2 12 105 1300
427 1562 218 457 77 5 38 221 1200
277 902 118 221 35 6 39 113 975
578 1692 169 431 42 9 1 111 950
564 1441 119 329 62 3 17 136 750
359 1162 112 308 52 6 19 121 500
168 650 105 179 16 8 5 42 425
517 1523 142 387 61 5 7 104 400
221 647 71 165 30 2 27 87 350
625 1569 120 408 56 4 5 157 335
248 686 71 182 29 5 6 54 325
555 1764 199 495 84 12 26 163 300
55 197 19 51 9 2 4 19 142.5
696 2297 353 590 91 3 161 473 4200
1292 4331 652 1120 190 37 68 450 3866.667
2095 7616 1211 2088 353 62 209 930 3433.333
932 2873 416 693 121 8 137 406 3333.333
1002 3365 503 746 128 12 205 536 2066.667
1965 7179 1077 2050 356 51 162 876 1400
1106 4145 611 1120 187 38 61 390 1150
379 1448 184 388 78 8 50 214 650
1351 4838 815 1298 216 44 210 701 500
325 678 96 166 23 4 2 48 425
262 817 124 205 29 9 6 69 275
245 433 67 108 22 3 11 52 250
195 425 43 87 15 1 8 32 240
161 481 62 137 26 0 6 57 230
2729 10554 1570 3025 558 123 243 1355 2700
797 2783 314 745 119 16 32 350 2500
444 1538 227 360 71 9 72 260 1640
1425 5071 720 1353 216 39 84 491 1333.333
1656 5860 760 1454 287 29 255 856 1100
383 1134 172 331 43 13 8 148 1000
371 1138 161 298 398 12 14 125 750
363 1018 109 267 63 3 39 144 725
149 579 93 168 19 6 1 47 350
1302 4204 473 1110 183 41 70 402 250
51 164 15 38 8 0 4 25 120
47 133 17 30 3 1 2 10 118
22 9 5 1 0 0 0 1 110
946 3340 507 950 193 16 177 620 5050
799 2618 321 679 147 7 96 411 3833.333
1426 5643 808 1754 363 15 192 913 3820
1625 6213 1067 2098 422 47 85 687 2900
762 2379 269 610 83 4 117 362 2500
1277 4110 499 1000 173 52 40 354 2250
782 1916 243 448 70 10 26 174 1575
385 1062 128 263 51 6 20 104 1050
711 2144 277 605 115 18 23 204 350
520 1160 138 291 50 4 24 147 325
325 1040 146 245 35 1 55 139 255
202 616 73 144 34 6 10 50 160
187 524 53 129 22 1 12 55 152
147 581 82 149 33 6 8 60 150
1344 5201 727 1370 264 34 242 873 5500
761 2962 439 862 146 31 39 302 4833.333
921 3514 545 891 157 41 93 361 3583.333
1856 7520 1275 2281 405 86 160 790 3500
517 1433 181 361 57 9 51 168 2100
394 1278 177 344 73 2 47 182 1487.5
1203 3818 452 875 122 29 52 314 800
761 2394 278 592 119 12 63 309 500
1916 6685 744 1668 242 78 24 524 500
252 812 89 212 29 4 11 72 290
83 207 23 55 9 0 5 27 182.5
26 34 3 6 1 2 0 3 115
11 20 1 5 0 0 1 2 112.5
7 8 2 5 0 0 1 3 109
921 3094 421 800 145 24 114 434 4333.333
1643 6031 745 1523 287 28 231 872 3000
753 2740 426 818 96 41 13 242 2475
1590 5698 79 1538 275 28 197 818 1750
482 1069 125 249 36 3 13 91 600
758 1798 250 442 67 17 10 147 600
248 690 63 164 38 1 14 72 525
370 1192 176 308 52 7 12 85 450
195 632 91 148 22 1 3 45 265
139 441 59 114 16 2 10 46 220
119 306 30 62 13 0 3 25 160
23 79 8 14 1 0 2 6 115
47 151 14 39 5 0 1 12 112.5
7 14 1 2 1 0 0 2 110
63 159 10 27 5 1 2 8 109
1485 5713 778 1613 291 32 252 938 3650
1704 6465 1138 1923 315 96 108 656 3500
583 2059 258 563 61 38 4 176 2266.667
1095 3841 432 1021 143 42 10 338 2000
1705 6797 891 1915 273 46 53 541 1975
816 2421 317 597 117 14 110 366 1750
480 1736 230 470 83 3 44 254 1700
459 1171 145 265 55 3 36 145 1016.667
534 1908 286 476 70 14 112 327 910
378 1323 251 426 88 7 63 255 900
254 630 68 160 40 6 10 75 632.5
722 2794 440 785 160 27 93 387 500
308 710 104 175 20 6 0 43 285
995 3780 521 1079 206 13 123 584 4400
1341 4892 660 1307 251 29 196 744 3666.667
1140 3386 452 844 183 35 69 335 3666.667
2562 9789 1514 3005 634 134 298 1520 3100
779 2816 292 717 111 23 8 249 2400
439 1566 190 449 81 10 33 210 1525
484 1456 160 365 92 9 42 195 1175
55 58 4 11 4 0 0 6 1090
290 475 58 124 15 5 11 60 460
960 2424 271 596 78 23 8 179 400
347 1330 170 331 59 17 17 139 378.5
1536 5745 637 1546 276 31 147 786 250
172 457 40 109 18 0 3 50 225
22 38 6 10 2 0 3 4 115
1382 5645 820 1812 304 51 142 785 5200
1543 5526 809 1580 290 17 258 950 3100
585 1712 271 514 82 21 46 230 3050
2707 10047 1551 2866 493 83 432 1710 2700
788 2363 264 697 145 6 47 323 2400
702 2177 258 580 124 11 31 247 600
306 1165 182 337 43 12 3 106 500
1044 3290 380 774 165 14 121 434 400
252 633 89 165 26 4 8 59 275
200 641 72 175 27 5 19 102 253.6
1184 3000 387 756 152 26 92 406 250
87 178 21 51 14 1 4 22 172
1060 4144 588 1160 230 44 156 673 4200
916 3123 504 772 128 5 220 608 4000
1859 6879 1472 2000 329 54 199 725 3250
733 2484 278 675 114 9 61 330 2800
1375 4550 646 1191 253 16 172 624 2600
367 798 133 185 40 2 7 69 627.5
728 2417 341 657 101 21 17 222 625
889 3289 482 955 163 25 38 336 600
269 753 83 208 24 5 3 69 400
24 53 8 14 3 0 3 9 120
578 2165 311 652 132 12 87 344 4000
521 1805 295 561 122 8 45 204 3016.667
490 1599 212 393 72 9 81 261 2772.5
1495 5227 624 1367 247 21 162 709 2187.5
711 2079 231 479 85 10 59 255 1866.667
502 1551 155 388 46 13 4 100 1112.5
776 1938 271 509 79 9 39 189 900
732 1803 277 464 50 25 13 140 850
434 971 84 272 58 5 15 135 475
196 640 68 156 25 2 20 80 287.5
291 614 51 141 26 5 10 71 200
1092 3216 480 889 138 19 10 240 150
972 3655 614 974 171 8 235 734 4800
886 3270 463 968 194 23 95 421 4550
1402 5416 807 1630 249 40 86 679 2937.5
753 2152 231 547 67 17 16 199 1900
750 1741 168 469 78 9 23 179 650
346 1279 172 331 68 4 75 230 525
1064 3258 561 813 170 47 84 319 500
443 1070 133 257 37 9 3 80 450
439 1098 150 260 41 8 6 83 312
211 700 63 183 32 1 11 64 275
249 828 112 176 36 2 41 110 250
667 2087 217 510 92 5 13 168 200
12 24 1 4 0 0 0 2 109
