F^3: F tests, Functional Forms and Favorite Coefficient Models

F^3: F tests, Functional Forms and Favorite Coefficient Models Favorite coefficient model: otherteams use "nflpricedata Bdta", clear *Favorite coefficient model: otherteams reg rprice pop pop2 rpci wprcnt1 po5 otherpro cap95 newstad tempstad exp reloc Source SS df MS Number of obs = 599 -------------+---------------------------------- F(11, 587) = 3868 Model 68234497 11 620313609 Prob > F = 00000 Residual 941315547 587 1603604 R-squared = 04203 -------------+---------------------------------- Adj R-squared = 04094 Total 162366052 598 271515137 Root MSE = 12663 rprice Coef Std Err t P>t [95% Conf Interval] pop 3637686 6960611 523 0000 2270613 500476 pop2-0565243 0257712-219 0029-1071394 -0059093 rpci 001093 0000975 1121 0000 0009015 0012844 wprcnt1 1120213 3253936 344 0001 4811356 175929 po5 1329569 4351338 306 0002 4749606 2184178 otherpro -4932958 7706291-640 0000-6446484 -3419432 cap95 3665552 1185358 309 0002 1337493 5993611 newstad 1228442 288925 425 0000 6609888 1795894 tempstad 1734893 5005793 035 0729-8096552 1156634 exp -1368493 9465654-145 0149-322756 4905743 reloc 2260889 7544666 030 0765-1255694 1707872 _cons 8280417 3948694 210 0036 5251289 1603571 *F tests: *-testing one parameter and t-tests test pop ( 1) pop = 0 F( 1, 587) = 2731 Prob > F = 00000 di sqrt(2731) 52258971 test exp ( 1) exp = 0 F( 1, 587) = 209 Prob > F = 01488 di sqrt(209) 14456832

*-testing joint hypotheses (multiple parameters) test (pop=0) (pop2=0) ( 1) pop = 0 ( 2) pop2 = 0 F( 2, 587) = 2705 Prob > F = 00000 test pop pop2 ( 1) pop = 0 ( 2) pop2 = 0 F( 2, 587) = 2705 Prob > F = 00000 test wprcnt1 po5 ( 1) wprcnt1 = 0 ( 2) po5 = 0 F( 2, 587) = 2003 Prob > F = 00000 * and functional forms: reg rprice pop pop2 yr Source SS df MS Number of obs = 599 -------------+---------------------------------- F(3, 595) = 13163 Model 647712753 3 215904251 Prob > F = 00000 Residual 975947764 595 164024834 R-squared = 03989 -------------+---------------------------------- Adj R-squared = 03959 Total 162366052 598 271515137 Root MSE = 12807 rprice Coef Std Err t P>t [95% Conf Interval] pop 2846925 439818 647 0000 198314 3710709 pop2-0755177 0225328-335 0001-1197711 -0312643 yr 1473496 0963911 1529 0000 1284187 1662804 _cons -2893024 1931758-1498 0000-3272414 -2513635 gen yr2=yr^2 2

reg rprice pop pop2 yr yr2 Source SS df MS Number of obs = 599 -------------+---------------------------------- F(4, 594) = 10027 Model 654432808 4 163608202 Prob > F = 00000 Residual 969227709 594 163169648 R-squared = 04031 -------------+---------------------------------- Adj R-squared = 03990 Total 162366052 598 271515137 Root MSE = 12774 rprice Coef Std Err t P>t [95% Conf Interval] pop 2851166 438675 650 0000 1989623 3712708 pop2-0756423 022474-337 0001-1197805 -0315041 yr 1608629 7854038 205 0041 6612274 3151135 yr2-0397471 0195857-203 0043-0782126 -0012815 _cons -1626834 7873816-207 0039-3173225 -8044385 reg rprice pop pop2 iyr Source SS df MS Number of obs = 599 -------------+---------------------------------- F(20, 578) = 2090 Model 681500815 20 340750407 Prob > F = 00000 Residual 942159702 578 163003409 R-squared = 04197 -------------+---------------------------------- Adj R-squared = 03997 Total 162366052 598 271515137 Root MSE = 12767 rprice Coef Std Err t P>t [95% Conf Interval] pop 2849277 4386113 650 0000 198781 3710743 pop2-0756375 0224693-337 0001-1197689 -0315061 yr 1997 4730331 329658 143 0152-1744405 1120507 1998 7278587 3296527 221 0028 8039558 1375322 1999 1000391 3269817 306 0002 3581739 1642608 2000 1377885 3269828 421 0000 7356655 2020104 2001 1754961 3269856 537 0000 1112736 2397185 2002 1213741 3244808 374 0000 5764362 1851046 2003 1383357 3244738 426 0000 7460653 2020648 2004 1426077 3244817 439 0000 7887699 2063384 2005 1701298 3244971 524 0000 106396 2338635 2006 1877821 3245007 579 0000 1240477 2515165 2007 2207489 3245171 680 0000 1570113 2844866 2008 2473209 3245344 762 0000 1835799 311062 2009 2605907 3245522 803 0000 1968461 3243352 2010 2818382 3245692 868 0000 2180903 345586 2011 2653022 3245879 817 0000 2015507 3290538 2012 2583445 3246101 796 0000 1945886 3221004 2013 2777627 3246339 856 0000 2140021 3415233 2014 292944 3246568 902 0000 2291789 3567091 _cons 4343622 265721 1635 0000 3821726 4865519 predict yhat (option xb assumed; fitted values) scatter yhat pop 3

40 60 80 100 0 5 10 15 20 pop test iyr i: operator invalid r(198); *OOPS! let's try that test a different way testparm iyr ( 1) 1997yr = 0 ( 2) 1998yr = 0 ( 3) 1999yr = 0 ( 4) 2000yr = 0 ( 5) 2001yr = 0 ( 6) 2002yr = 0 ( 7) 2003yr = 0 ( 8) 2004yr = 0 ( 9) 2005yr = 0 (10) 2006yr = 0 (11) 2007yr = 0 (12) 2008yr = 0 (13) 2009yr = 0 (14) 2010yr = 0 (15) 2011yr = 0 (16) 2012yr = 0 (17) 2013yr = 0 (18) 2014yr = 0 F( 18, 578) = 1422 Prob > F = 00000 4

*Dummies capture average residuals (what your model did not otherwise explain): reg rprice pop pop2 iyr Source SS df MS Number of obs = 599 -------------+---------------------------------- F(20, 578) = 2090 Model 681500815 20 340750407 Prob > F = 00000 Residual 942159702 578 163003409 R-squared = 04197 -------------+---------------------------------- Adj R-squared = 03997 Total 162366052 598 271515137 Root MSE = 12767 rprice Coef Std Err t P>t [95% Conf Interval] pop 2849277 4386113 650 0000 198781 3710743 pop2-0756375 0224693-337 0001-1197689 -0315061 yr 1997 4730331 329658 143 0152-1744405 1120507 1998 7278587 3296527 221 0028 8039558 1375322 1999 1000391 3269817 306 0002 3581739 1642608 2000 1377885 3269828 421 0000 7356655 2020104 2001 1754961 3269856 537 0000 1112736 2397185 2002 1213741 3244808 374 0000 5764362 1851046 2003 1383357 3244738 426 0000 7460653 2020648 skip a few 2010 2818382 3245692 868 0000 2180903 345586 2011 2653022 3245879 817 0000 2015507 3290538 2012 2583445 3246101 796 0000 1945886 3221004 2013 2777627 3246339 856 0000 2140021 3415233 2014 292944 3246568 902 0000 2291789 3567091 _cons 4343622 265721 1635 0000 3821726 4865519 gen resid = rprice - (4343622+ 2849277*pop -0756375*pop2) tabstat resid, by(yr) stat(mean) yr mean ---------+---------- 1996 201e-06 1997 4730333 1998 7278589 1999 1000391 2000 1377885 2001 1754961 2002 1213741 2003 1383357 skip a few 2010 2818382 2011 2653023 2012 2583445 2013 2777627 2014 292944 ---------+---------- Total 1804647 -------------------- 5

*Bring on the dummies team quality effects: xi iteam iteam _Iteam_1-32 (_Iteam_1 for team==arizona omitted) reg rprice wprcnt1 po5 iyr _I* Source SS df MS Number of obs = 599 -------------+---------------------------------- F(51, 547) = 1998 Model 105644132 51 207145357 Prob > F = 00000 Residual 567219198 547 10369638 R-squared = 06507 -------------+---------------------------------- Adj R-squared = 06181 Total 162366052 598 271515137 Root MSE = 10183 rprice Coef Std Err t P>t [95% Conf Interval] wprcnt1 1102258 263116 419 0000 5854162 1619099 po5 1720806 400038 430 0000 9350076 2506605 yr 1997 4592783 2629276 175 0081-5719309 9757497 1998 7275853 2629276 277 0006 2111139 1244057 1999 1068214 2609359 409 0000 5556552 1580773 2000 1441225 2609389 552 0000 9286601 195379 2001 1827058 2609391 700 0000 1314493 2339624 2002 1251276 2590794 483 0000 7423641 1760189 2003 147252 2590943 568 0000 963578 1981461 2004 1521763 2590941 587 0000 1012821 2030704 2005 1806239 2590942 697 0000 1297298 231518 2006 1985343 2590943 766 0000 1476402 2494285 2007 2323675 2590941 897 0000 1814734 2832616 2008 2598202 2590944 1003 0000 2089261 3107144 2009 2739788 2590942 1057 0000 2230846 3248729 2010 2960092 2590944 1142 0000 245115 3469033 2011 2802938 259094 1082 0000 2293997 3311879 2012 2741714 2590941 1058 0000 2232773 3250656 2013 2944244 2590942 1136 0000 2435303 3453185 2014 3103737 2590941 1198 0000 2594796 3612678 _Iteam_2-1955611 3328334-059 0557-8493493 458227 _Iteam_3 1079848 3350316 322 0001 4217423 1737954 _Iteam_31-8364536 3334055-025 0802-7385572 5712664 _Iteam_32 2581986 3306075 781 0000 193257 3231402 _cons 3728389 3100604 1202 0000 3119334 4337444 margins, eyex(wprcnt1 po5) atmeans Conditional marginal effects Number of obs = 599 Model VCE : OLS Expression : Linear prediction, predict() ey/ex wrt : wprcnt1 po5 Delta-method ey/ex Std Err t P>t [95% Conf Interval] wprcnt1 0759302 0181303 419 0000 0403167 1115437 po5 0451939 0105095 430 0000 02455 0658378 6

*Polynomials: reg rprice pop Source SS df MS Number of obs = 599 -------------+---------------------------------- F(1, 597) = 10066 Model 234268132 1 234268132 Prob > F = 00000 Residual 138939238 597 232729043 R-squared = 01443 -------------+---------------------------------- Adj R-squared = 01429 Total 162366052 598 271515137 Root MSE = 15255 rprice Coef Std Err t P>t [95% Conf Interval] pop 1543956 1538876 1003 0000 1241729 1846183 _cons 648426 9827381 6598 0000 6291256 6677265 predict phat1 (option xb assumed; fitted values) reg rprice pop pop2 Source SS df MS Number of obs = 599 -------------+---------------------------------- F(2, 596) = 5797 Model 264417822 2 132208911 Prob > F = 00000 Residual 135924269 596 228060855 R-squared = 01629 -------------+---------------------------------- Adj R-squared = 01600 Total 162366052 598 271515137 Root MSE = 15102 rprice Coef Std Err t P>t [95% Conf Interval] pop 3341081 5172106 646 0000 2325304 4356857 pop2-0964271 0265206-364 0000-1485123 -044342 _cons 5990242 1671077 3585 0000 566205 6318433 predict phat2 (option xb assumed; fitted values) gen pop3=pop^3 7

reg rprice pop pop2 pop3 Source SS df MS Number of obs = 599 -------------+---------------------------------- F(3, 595) = 3863 Model 264669452 3 882231506 Prob > F = 00000 Residual 135899106 595 22840186 R-squared = 01630 -------------+---------------------------------- Adj R-squared = 01588 Total 162366052 598 271515137 Root MSE = 15113 rprice Coef Std Err t P>t [95% Conf Interval] pop 2751115 1851271 149 0138-8847061 6386936 pop2-0040047 2797111-001 0989-5533459 5453364 pop3-0034412 0103677-033 0740-0238029 0169205 _cons 607911 3156777 1926 0000 5459132 6699088 predict phat3 (option xb assumed; fitted values) scatter phat* pop 60 70 80 90 100 0 5 10 15 20 pop vif Variable VIF 1/VIF -------------+---------------------- pop2 128036 0000781 pop3 61200 0001634 pop 14746 0006781 -------------+---------------------- Mean VIF 67994 8

test pop2 pop3 ( 1) pop2 = 0 ( 2) pop3 = 0 F( 2, 595) = 666 Prob > F = 00014 *So drop one of the two pop variables, but not both! *Percentile dummies here, quintile dummies xtile pop5=pop, n(5) scatter pop5 pop (below) reg rprice ipop5 Source SS df MS Number of obs = 599 -------------+---------------------------------- F(4, 594) = 2911 Model 266117491 4 665293728 Prob > F = 00000 Residual 135754303 594 228542597 R-squared = 01639 -------------+---------------------------------- Adj R-squared = 01583 Total 162366052 598 271515137 Root MSE = 15118 rprice Coef Std Err t P>t [95% Conf Interval] pop5 2 3754477 1951677 192 0055-0785498 7587504 3 2152868 1951677 110 0270-1680159 5985895 4 1105344 1951677 566 0000 722041 1488646 5 1810559 1955773 926 0000 1426452 2194666 _cons 6547059 1380044 4744 0000 6276023 6818095 predict phat5 (option xb assumed; fitted values) scatter phat2 phat5 pop 5 quantiles of pop 1 2 3 4 5 0 5 10 15 20 pop 60 70 80 90 0 5 10 15 20 pop 9

*Use areg to run fixed effect but only for one variable: reg rprice ipop5 rpci wprcnt1 po5 otherpro cap95 newstad tempstad exp reloc Source SS df MS Number of obs = 599 -------------+---------------------------------- F(13, 585) = 2830 Model 626811825 13 482162942 Prob > F = 00000 Residual 996848692 585 170401486 R-squared = 03860 -------------+---------------------------------- Adj R-squared = 03724 Total 162366052 598 271515137 Root MSE = 13054 rprice Coef Std Err t P>t [95% Conf Interval] pop5 2 2541772 1758804 145 0149-9125669 5996111 3 8769561 1896003 046 0644-2846847 4600759 4 5966707 2235847 267 0008 1575442 1035797 5 1048341 2947759 356 0000 4693934 1627289 rpci 0009766 0000988 989 0000 0007826 0011706 wprcnt1 107572 3363099 320 0001 415198 1736242 po5 1375161 4491841 306 0002 4929515 2257371 otherpro -1212273 5141579-236 0019-2222093 -2024526 cap95 4354861 1231306 354 0000 1936543 6773179 newstad 1194611 2981379 401 0000 6090599 1780162 tempstad -8271498 5148519-016 0872-1093898 9284683 exp -1268215 9783231-130 0195-3189668 6532384 reloc 1876172 7807957 024 0810-1345887 1721121 _cons 1518201 4236342 358 0000 6861714 235023 areg rprice rpci wprcnt1 po5 otherpro cap95 newstad tempstad exp reloc, absorb(pop5) Linear regression, absorbing indicators Number of obs = 599 F( 9, 585) = 2352 Prob > F = 00000 R-squared = 03860 Adj R-squared = 03724 Root MSE = 130538 rprice Coef Std Err t P>t [95% Conf Interval] rpci 0009766 0000988 989 0000 0007826 0011706 wprcnt1 107572 3363099 320 0001 415198 1736242 po5 1375161 4491841 306 0002 4929515 2257371 otherpro -1212273 5141579-236 0019-2222093 -2024526 cap95 4354861 1231306 354 0000 1936543 6773179 newstad 1194611 2981379 401 0000 6090599 1780162 tempstad -8271498 5148519-016 0872-1093898 9284683 exp -1268215 9783231-130 0195-3189668 6532384 reloc 1876172 7807957 024 0810-1345887 1721121 _cons 1914491 4503953 425 0000 1029902 279908 pop5 F(4, 585) = 4581 0001 (5 categories) 10

*Why hold back after all, it's a favorite coefficient model! xtile rpci5=rpci, n(5) reg rprice ipop5 irpci5 wprcnt1 po5 otherpro cap95 newstad tempstad exp reloc Source SS df MS Number of obs = 599 -------------+---------------------------------- F(16, 582) = 2456 Model 654410648 16 409006655 Prob > F = 00000 Residual 969249868 582 166537778 R-squared = 04030 -------------+---------------------------------- Adj R-squared = 03866 Total 162366052 598 271515137 Root MSE = 12905 rprice Coef Std Err t P>t [95% Conf Interval] pop5 2 2893204 1782792 162 0105-6082849 6394693 3 1033402 1916853 054 0590-273139 4798194 4 6223562 2259975 275 0006 1784861 1066226 5 9793596 2976267 329 0001 3948062 1563913 rpci5 2 4189854 1739856 241 0016 7726917 7607016 3 5881469 1792729 328 0001 2360463 9402475 4 1120414 1902813 589 0000 7466918 1494135 5 2303001 221697 1039 0000 1867577 2738424 wprcnt1 1002677 3358905 299 0003 3429723 1662383 po5 1228393 4479319 274 0006 3486326 2108153 otherpro -1385896 5150659-269 0007-239751 -3742817 cap95 3924648 1233576 318 0002 1501846 6347451 newstad 1219293 295795 412 0000 6383374 1800249 tempstad 07546 5132032 001 0988-100041 1015502 exp -1428308 9684361-147 0141-3330363 4737477 reloc 1453795 7751601 019 0851-1377072 1667831 _cons 5300211 2122683 2497 0000 4883306 5717116 xtile wprcnt5=wprcnt1, n(5) reg rprice ipop5 irpci5 iwprcnt5 po5 otherpro cap95 newstad tempstad exp reloc Source SS df MS Number of obs = 599 -------------+---------------------------------- F(19, 579) = 2100 Model 662419204 19 348641686 Prob > F = 00000 Residual 961241313 579 166017498 R-squared = 04080 -------------+---------------------------------- Adj R-squared = 03886 Total 162366052 598 271515137 Root MSE = 12885 rprice Coef Std Err t P>t [95% Conf Interval] pop5 2 2946497 1782272 165 0099-5540087 6447003 3 1260088 1923869 065 0513-2518526 5038701 4 6509895 2264744 287 0004 2061781 1095801 5 1017253 2984338 341 0001 4311083 1603398 rpci5 2 43131 1739807 248 0013 8959984 7730202 3 6032934 1797499 336 0001 2502521 9563347 4 1087504 1911532 569 0000 7120657 1462942 11

5 2315981 2233223 1037 0000 1877361 2754602 wprcnt5 2 2426796 1643006 148 0140-8001824 5653774 3 5425823 1628938 333 0001 2226476 8625171 4 5223581 1849841 282 0005 1590365 8856797 5 5567654 205616 271 0007 1529213 9606094 po5 1268031 4523751 280 0005 3795348 2156527 otherpro -1465221 5160105-284 0005-2478701 -4517404 cap95 3818626 1234124 309 0002 1394721 6242532 newstad 121339 2956871 410 0000 6326402 179414 tempstad 459044 5129468 009 0929-9615587 1053368 exp -1367471 9686333-141 0159-3269935 5349919 reloc 6656844 7769071 009 0932-1459331 1592468 _cons 5444388 1966695 2768 0000 5058116 5830661 reg rprice ipop5 irpci5 iwprcnt5 ipo5 otherpro cap95 newstad tempstad exp reloc Source SS df MS Number of obs = 599 -------------+---------------------------------- F(23, 575) = 1814 Model 682689672 23 296821596 Prob > F = 00000 Residual 940970845 575 163647103 R-squared = 04205 -------------+---------------------------------- Adj R-squared = 03973 Total 162366052 598 271515137 Root MSE = 12792 rprice Coef Std Err t P>t [95% Conf Interval] pop5 2 3605417 1780221 203 0043 1088882 7101947 3 1562707 1915575 082 0415-2199671 5325085 4 7100479 226185 314 0002 2657984 1154297 5 1028136 2971505 346 0001 4445036 1611769 rpci5 2 3687486 1742724 212 0035 2646052 7110366 3 5585828 1794864 311 0002 2060539 9111116 4 1036043 1907622 543 0000 661367 1410718 5 2317894 2231326 1039 0000 1879639 2756148 wprcnt5 2 2258604 1632596 138 0167-9479748 5465183 3 5026521 1622108 310 0002 1840542 82125 4 4917277 184122 267 0008 1300941 8533614 5 559769 2049994 273 0007 1571301 962408 po5 1 600322 1683303 357 0000 2697047 9309393 2 6959725 1720596 404 0000 3580307 1033914 3 5926108 1926655 308 0002 214197 9710247 4 7524931 2157452 349 0001 3287484 1176238 5 5372481 3261736 165 0100-1033888 1177885 otherpro -1614547 5157146-313 0002-2627461 -6016329 cap95 3703776 1229291 301 0003 1289327 6118225 newstad 1243918 2937373 423 0000 6669888 1820847 tempstad -1213641 5137001-024 0813-1130321 8875934 exp -1128534 9644683-117 0242-3022844 7657765 reloc -8824733 7727943-011 0909-1606091 1429597 _cons 5228625 2060151 2538 0000 4823991 5633259 12

* Run a few tests, will ya? testparm ipop5 ( 1) 2pop5 = 0 ( 2) 3pop5 = 0 ( 3) 4pop5 = 0 ( 4) 5pop5 = 0 F( 4, 575) = 443 Prob > F = 00015 testparm ipop5 irpci5 ( 1) 2pop5 = 0 ( 2) 3pop5 = 0 ( 7) 4rpci5 = 0 ( 8) 5rpci5 = 0 F( 8, 575) = 2074 Prob > F = 00000 testparm ipop5 irpci5 iwprcnt5 ( 1) 2pop5 = 0 ( 2) 3pop5 = 0 ( 8) 5rpci5 = 0 ( 9) 2wprcnt5 = 0 (10) 3wprcnt5 = 0 (11) 4wprcnt5 = 0 (12) 5wprcnt5 = 0 F( 12, 575) = 1540 Prob > F = 00000 13