Do Firms Maximize? Evidence from Professional Football


 Thomas Campbell
 1 years ago
 Views:
Transcription
1 Do Frms Maxmze? Evdence from Professonal Football Davd Romer Unversty of Calforna, Berkeley and Natonal Bureau of Economc Research Ths paper examnes a sngle, narrow decson the choce on fourth down n the Natonal Football League between kckng and tryng for a frst down as a case study of the standard vew that competton n the goods, captal, and labor markets leads frms to make maxmzng choces. Playbyplay data and dynamc programmng are used to estmate the average payoffs to kckng and tryng for a frst down under dfferent crcumstances. Examnaton of actual decsons shows systematc, clearcut, and overwhelmngly statstcally sgnfcant departures from the decsons that would maxmze teams chances of wnnng. Possble reasons for the departures are consdered. I. Introducton A central assumpton of most economc models s that agents maxmze smple objectve functons: consumers maxmze expected utlty, and frms maxmze expected profts. The argument for ths assumpton s not that t leads to perfect descrptons of behavor, but that t leads to reasonably good approxmatons n most cases. The assumpton that consumers successfully maxmze smple objectve functons frequently makes predctons about how a sngle ndvdual wll behave when confronted wth a specfc, easly descrbable decson. I am ndebted to Ben Allen, Laurel Beck, Sungmun Cho, Ryan Edwards, Maro Lopez, Peter Mandel, Travs Reynolds, Evan Rose, and Raymond Son for outstandng research assstance; to Chrstna Romer for nvaluable dscussons; to Steven Levtt and Rchard Thaler for mportant encouragement; and to numerous colleagues and correspondents for helpful comments and suggestons. An earler verson of the paper was ttled It s Fourth Down and What Does the Bellman Equaton Say? A DynamcProgrammng Analyss of Football Strategy. [ Journal of Poltcal Economy, 2006, vol. 114, no. 2] 2006 by The Unversty of Chcago. All rghts reserved /2006/ $
2 do frms maxmze? 341 Thus t can often be tested n both the laboratory and the feld. The assumpton that frms maxmze profts s much more dffcult to test, however. Partcularly for large frms, the decsons are usually complcated and the data dffcult to obtan. But the a pror case for frm maxmzaton s much stronger than that for consumer maxmzaton. As Alchan (1950), Fredman (1953), Becker (1957), Fama (1980), and others explan, competton n the goods, captal, and labor markets creates strong forces drvng frms toward proft maxmzaton. A frm that fals to maxmze profts s lkely to be outcompeted by more effcent rvals or purchased by ndvduals who can obtan greater value from t by pursung dfferent strateges. And managers who fal to maxmze profts for the owners of ther frms are lkely to be fred and replaced by ones who do. Thus the case for frm maxmzaton rests much more on logcal argument than emprcal evdence. As Fredman puts t, unless the behavor of busnessmen n some way or other approxmated behavor consstent wth the maxmzaton of returns, t seems unlkely that they would reman n busness for long.... The process of natural selecton thus helps to valdate the hypothess [of return maxmzaton] (1953, 22). Ths paper takes a frst step toward testng the assumpton that frms maxmze profts by examnng a specfc strategc decson n professonal sports: the choce n football between kckng and tryng for a frst down on fourth down. Examnng strategc decsons n sports has two enormous advantages. Frst, n most cases, t s dffcult to thnk of any sgnfcant channel through whch strategc decsons are lkely to affect a team s profts other than through ther mpact on the team s probablty of wnnng. Thus the problem of maxmzng profts plausbly reduces to the much smpler problem of maxmzng the probablty of wnnng. Second, there are copous, detaled data descrbng the crcumstances teams face when they make these decsons. 1 The predctons of smple models of optmzaton appear especally lkely to hold n the case of fourthdown decsons n professonal football. There are three reasons. Frst, the market for the coaches who make these decsons s ntensvely compettve. Salares average roughly $3 mllon per year, and annual turnover exceeds 20 percent. 2 Second, wnnng s valued enormously (as shown by the very hgh salares commanded by hghqualty players). And thrd, the decsons are unusually amenable to learnng and mtaton: the decsons arse repeatedly, and 1 Thaler (2000) stresses the potental value of sports decson makng n testng the hypothess of frm optmzaton. 2 The salary fgure s based on the 23 coaches (out of 32) for whom 2004 salary nformaton could be obtaned from publcly avalable sources. The turnover data pertan to
3 342 journal of poltcal economy nformaton about others decsons s readly avalable. Thus a falure of maxmzaton n ths settng would be partcularly strkng. Ths paper shows, however, that teams choces on fourth downs depart n a way that s systematc and overwhelmngly statstcally sgnfcant from the choces that would maxmze ther chances of wnnng. One case n whch the departure s partcularly strkng and relatvely easy to see arses when a team faces fourth down and goal on ts opponent s 2yard lne early n the game. 3 In ths stuaton, attemptng a feld goal s vrtually certan to produce 3 ponts, whle tryng for a touchdown has about a threesevenths chance of producng 7 ponts. The two choces thus have essentally the same expected mmedate payoff. But f the team tres for a touchdown and fals, ts opponent typcally gans possesson of the ball on the 2yard lne; f the team scores a touchdown or a feld goal, on the other hand, the opponent returns a kckoff, whch s consderably better for t. Thus tryng for a touchdown on average leaves the opponent n consderably worse feld poston. I show later that ratonal rsk averson about ponts scored, concern about momentum, and other complcatons do not notceably affect the case for tryng for a touchdown. As a result, my estmates mply that the team should be ndfferent between the two choces f the probablty of scorng a touchdown s about 18 percent. They also mply that tryng for a touchdown rather than a feld goal would ncrease the team s chances of wnnng the game by about three percentage ponts, whch s very large for a sngle play. In fact, however, teams attempted a feld goal all nne tmes n my sample they were n ths poston. Analyzng the choce between kckng and tryng for a frst down or touchdown n other cases s more complcated: the mmedate expected payoffs may be dfferent under the two choces, and the attractveness of the dstrbutons of ball possesson and feld poston may be dffcult to compare. Fortunately, however, the problem can be analyzed usng dynamc programmng. The choce between kckng and gong for t leads to an mmedate payoff n terms of ponts (whch may be zero) and to one team havng a frst down somewhere on the feld. That frst down leads to addtonal scorng (whch agan may be zero) and to another possesson and frst down. And so on. Secton II of the paper therefore uses data from over 700 Natonal Football League (NFL) games to estmate the values of frst downs at each pont on the feld (as well as the value of kckng off). To avod the complcatons ntroduced when one team s well ahead or when the end of a half s approachng, I focus on the frst quarter. Secton III uses the results of ths analyss to examne fourthdown decsons over the entre feld. To estmate the value of kckng, I use 3 The Appendx summarzes the rules of football that are relevant to the paper.
4 do frms maxmze? 343 the outcomes of actual feld goal attempts and punts. Decsons to go for t on fourth down (.e., not to kck) are suffcently rare, however, that they cannot be used to estmate the value of tryng for a frst down or touchdown. I therefore use the outcomes of thrddown plays nstead. I then compare the values of kckng and gong for t to determne whch decson s better on average as a functon of where the team s on the feld and the number of yards t needs for a frst down or touchdown. Fnally, I compare the results of ths analyss wth teams actual choces. I fnd that teams choces are far more conservatve than the ones that would maxmze ther chances of wnnng. Secton IV consders varous possble complcatons and bases and fnds that none change the basc conclusons. Secton V consders the results quanttatve mplcatons. Because the analyss concerns only a small fracton of plays, t mples that dfferent choces on those plays could have only a modest mpact on a team s chances of wnnng. But t also mples that there are crcumstances n whch teams essentally always kck even though the case for gong for t s clearcut and the benefts of gong for t are substantal. Fnally, Secton VI dscusses the results broader mplcatons. The hypothess that frms maxmze smple objectve functons could fal as a result of ether the pursut of a dfferent, more complex objectve functon or a falure of maxmzaton. I dscuss how ether of these possbltes mght arse and how one mght be able to dstngush between them. 4 II. The Values of Dfferent Stuatons A. Framework The dynamcprogrammng analyss focuses on 101 stuatons: a frst down and 10 on each yard lne from a team s 1 to ts opponent s 10, a frst and goal on each yard lne from the opponent s 9 to ts 1, a kckoff from the team s 30 (followng a feld goal or touchdown, or at the begnnng of the game), and a kckoff from ts 20 (followng a safety). Let V denote the value of stuaton. Specfcally, V s the expected longrun value, begnnng n stuaton, of the dfference between the ponts scored by the team wth the ball and ts opponent when the two teams are evenly matched, average NFL teams. 4 Two recent papers that apply economc tools to sports strategy and n dong so use sports data to test hypotheses about maxmzaton are the study of serves n tenns by Walker and Wooders (2001) and the study of penalty kcks n soccer by Chappor, Levtt, and Groseclose (2002). In contrast to ths paper, these papers fnd no evdence of large departures from optmal strateges. Carter and Machol (1971, 1978) and Carroll, Palmer, and Thorn (1998, chap. 10) are more closely related to ths paper. I dscuss how my analyss s related to these studes below.
5 344 journal of poltcal economy By descrbng the values of stuatons n terms of expected pont dfferences, I am mplctly assumng that a team that wants to maxmze ts chances of wnnng should be rskneutral over ponts scored. Although ths s clearly not a good assumpton late n a game, I show n Secton IV that t s an excellent approxmaton for the early part. For that reason, I focus on the frst quarter. Focusng on the frst quarter has a second advantage: t makes t reasonable to neglect effects nvolvng the end of a half. Because play n the second quarter begns at the pont where the frst quarter ended, the value of a gven stuaton n the frst quarter almost certanly does not vary greatly wth the tme remanng. Let g ndex games and t ndex stuatons wthn a game. Let D gt be a dummy that equals one f the tth stuaton n game g s a stuaton of type. For example, suppose that p 100 denotes a kckoff from one s ; then, snce all games begn wth a kckoff, Dg1 p 1 for all g and Dg1 p 0 for all g and for all ( 100. Let Pgt denote the net ponts scored by the team wth the ball n stuaton g,t before the next stuaton. That s, P gt s the number of ponts scored by the team wth the ball mnus the number scored by ts opponent. Fnally, let B gt be a dummy that equals one f the team wth the ball n stuaton g,t also has the ball n stuaton g, t 1 and that equals mnus one f the other team has the ball n stuaton g, t 1. The realzed value of stuaton g,t as of one stuaton later has two components. The frst s the net ponts the team wth the ball scores before the next stuaton, P gt. The second s the value of the new stuaton. If the same team has the ball n that stuaton, ths value s smply the V correspondng to the new stuaton. If the other team has the ball, ths value s mnus the V correspondng to the new stuaton (snce the value of a stuaton to the team wthout the ball s equal and opposte to the value of the stuaton to ts opponent). In terms of the notaton just ntroduced, the value of stuaton g, t 1 to the team wth the ball n stuaton g,t s Bgt Dgt 1V. The value of stuaton g,t as of that stuaton must equal the expectaton of the stuaton s realzed value one stuaton later. We can wrte the value of stuaton g,t as DV gt. Thus we have [ ] DVp gt EP gt Bgt Dgt 1V, (1) where the expectaton s condtonal on stuaton g, t. Now defne as the dfference between the realzed value of stuaton e gt
6 do frms maxmze? 345 g,t one stuaton later and the expectaton of the realzed value condtonal on beng n stuaton g,t: [ ] [ ] egt p Pgt Bgt DgtV EP gt Bgt Dgt 1V. By constructon, egt s uncorrelated wth each of the Dgt s. If e were cor related wth a D, ths would mean that when teams were n stuaton, the realzed value one stuaton later would dffer systematcally from V; but ths would contradct the defnton of V. Usng ths defnton of, we can rewrte (1) as e gt DVp gt Pgt Bgt Dgt 1V e gt, (2) or Pgt p V(D gt BgtD gt 1) e gt. (3) To thnk about estmatng the V s, defne Xgt p Dgt BgtDgt 1. Then (3) becomes Pgt p VX gt e gt. (4) Ths formulaton suggests regressng P on the X s. But e may be correlated wth the X s. Specfcally, e gt s lkely to be correlated wth the BgtDgt 1 terms of the Xgt s. Recall, however, that egt s uncorrelated wth the Dgt s. Thus the Dgt s are legtmate nstruments for the Xgt s. Further, snce they enter nto the X gt s, they are almost surely correlated wth them. We can therefore estmate (4) by nstrumental varables, usng the D s as the nstruments. 5 gt There s one fnal ssue. There are 101 V s to estmate. Even wth a large amount of data, the estmates of the V s wll be nosy. But the value of a frst down s almost certanly a smooth functon of a team s poston on the feld. Thus forcng the estmates of the V s to be smooth wll mprove the precson of the estmates whle ntroducng mnmal bas. I therefore requre the estmated V s to be a quadratc splne as a functon of the team s poston on the feld, wth knot ponts at both 9, 17, and 33yard lnes and at the 50. I do not mpose any restrctons 5 There s another way of descrbng the estmaton of the V s. Begn wth an ntal set of V s (such as V p 0 for all ). Now for each, compute the mean of the realzed values of all stuatons of type one stuaton later usng the assumed V s and the actual Pgt s. Repeat the process usng the new V s as an nput, and terate untl the process converges. One can show that ths procedure produces results that are numercally dentcal to those of the nstrumental varables approach.
7 346 journal of poltcal economy Fg. 1. The estmated value of stuatons (sold lne) and twostandarderror bands (dotted lnes). The estmated value of a kckoff s 0.62 (standard error 0.04); the estmated value of a free kck s 1.21 (standard error 0.51). on the two estmated V s for kckoffs. Ths reduces the effectve number of parameters to be estmated from 101 to B. Data and Results Playbyplay accounts of vrtually all regularseason NFL games for the 1998, 1999, and 2000 seasons were downloaded from the NFL Web ste, nfl.com. 7 Snce I focus on strategy n the frst quarter, I use data only from frst quarters to estmate the V s. These data yeld 11,112 frstquarter stuatons. By far the most common are a kckoff from one s 30yard lne (1,851 cases) and a frst and 10 on one s 20 (557 cases). Because 98.4 percent of extrapont attempts were successful n ths perod, all touchdowns are counted as ponts. Fgure 1 reports the results of the nstrumental varables estmaton. It plots the estmated V for a frst and 10 (or frst and goal) as a functon of the team s poston on the feld, together wth the twostandarderror bands. The estmated value of a frst and 10 on one s 1yard lne s 1.6 ponts. The V s rse farly steeply from the 1, reachng zero at about the 15. That s, the estmates mply that a team should be ndfferent between 6 Carter and Machol (1971) also use a recursve approach to estmate pont values of frst downs at dfferent postons on the feld, usng a consderably smaller sample from There are two man dfferences from my approach. Frst, they arbtrarly assgn a value of zero to kckoffs and free kcks. Second, they dvde the feld nto 10yard ntervals and estmate the average value for each nterval. 7 Data for two games n 1999 and two games n 2000 were mssng from the Web ste.
8 do frms maxmze? 347 a frst and 10 on ts 15 and havng ts opponent n the same stuaton. The V s ncrease approxmately lnearly after the 15, rsng a pont roughly every 18 yards. The value of a frst and 10 equals the value of recevng a kckoff from the ponts around the 27yard lne. That s, recevng a kckoff s on average as valuable as a frst and 10 on one s 27. Fnally, the V s begn to ncrease more rapdly around the opponent s 10. The estmated value of a frst and goal on the 1 s 5.55 ponts; ths s about the same as the value of an 80 percent chance of a touchdown and a 20 percent chance of a feld goal. The V s are estmated relatvely precsely: except n the vcnty of the goal lnes, ther standard errors are less than 0.1. III. Kckng versus Gong for It Ths secton uses the results of Secton II to analyze the choce between kckng and gong for t on fourth down. The analyss proceeds n four steps. The frst two estmate the values of kckng and gong for t n dfferent crcumstances. The thrd compares the two choces to determne whch s on average better as a functon of the team s poston on the feld and ts dstance from a frst down. The fnal step examnes teams actual decsons. A. Kckng If one neglects the ssue of smoothng the estmates, analyzng the value of kcks s straghtforward. To estmate the value of a kck from a partcular yard lne, one smply averages the realzed values of the kcks from that yard lne as of the subsequent stuaton (where stuaton s defned as before). Ths realzed value has two components, the net ponts scored before the next stuaton and the next stuaton s value. In contrast to the prevous secton, there s no need for nstrumental varables estmaton. I constran the estmated values of kcks to be smooth n the same way as before, wth one modfcaton. Teams choces between puntng and attemptng a feld goal change rapdly around ther opponents 35 yard lne. Snce one would expect the level but not the slope of the value of kckng as a functon of the yard lne to be contnuous where teams swtch from punts to feld goal attempts, I do not mpose the slope restrcton at the opponent s 33. And ndeed, the estmates reveal a substantal knk at ths knot pont. The data consst of all kcks n the frst quarters of games. Snce what we need to know s the value of decdng to kck, I nclude not just
9 348 journal of poltcal economy actual punts and feld goal attempts, but blocked and muffed kcks and kcks nullfed by penaltes. There are 2,560 observatons. 8 The results are reported n fgure 2. Fgure 2a shows the estmated value of kckng as a functon of the team s poston on the feld. Fgure 2b plots the dfference between the estmated value of a kck and of the other team havng a frst down on the spot. From the team s 10yard lne to mdfeld, ths dfference s farly steady at around 2.1 ponts, whch corresponds to a punt of about 38 yards. It dps down n the dead zone around the opponent s 35yard lne, where a feld goal s unlkely to succeed and a punt s lkely to produce lttle yardage. It reaches a low of 1.5 (a punt of only 25 yards) at the 33 and then rses to 2.2 at the 21. As the team gets closer to the goal lne, the probablty of a successful feld goal rses lttle, but the value of leavng the opponent wth the ball rses consderably. The dfference between the values of kckng and of the opponent recevng the ball therefore falls, reachng 0.7 at the 1. The estmates are relatvely precse: the standard error of the dfference n values s typcally about B. Gong for It The analyss of the value of tryng for a frst down or touchdown parallels the analyss of kckng. There are two dfferences. Frst, because teams rarely go for t on fourth down, I use thrddown plays nstead. That s, I fnd what thrddown plays realzed values as of the next stuaton would have been f the plays had taken place on fourth down. Second, the value of gong for t depends not only on the team s poston on the feld, but also on the number of yards to go for a frst down or touchdown. If there were no need to smooth the estmates, 8 There are several mnor ssues nvolvng the data. Frst, fourthdown plays that are blown dead before the snap and for whch the playbyplay account does not say whether the kckng squad was sent n are excluded. Snce such plays are also excluded from the analyss of the decson to go for t, ths excluson should generate lttle bas. Second, t s not clear whether fake kcks should be ncluded; t depends on whether one wants to estmate the value of decdng to kck or the value of lnng up to kck. There are only fve fake kcks n the sample, however, and the results are vrtually unaffected by whether they are ncluded. The results n the text nclude fakes. Fnally, snce teams occasonally obtan frst downs on kckng plays (prmarly through penaltes), the value of a kck s affected by the number of yards the team has to go for a frst down. But there are only sx kckng plays n the sample on whch the team had 5 yards to go or less and moved the ball 5 yards or less and obtaned a frst down. Thus to mprove the precson of the estmates, I do not let the estmated value of kcks vary wth the number of yards needed for a frst down. 9 The standard errors account for the fact that the V s used to estmate the values of kcks are themselves estmated. Ths calculaton s performed under the assumpton that the dfferences between the realzed and expected values of kcks are uncorrelated wth the errors n estmatng the V s. Although ths assumpton wll not be strctly correct, t s almost certanly an excellent approxmaton.
10 Fg. 2. a, The estmated value of kcks. b, The estmated value of the dfference between the values of kcks and of turnng the ball over. The dotted lnes show the twostandarderror bands.
11 350 journal of poltcal economy one could use averages to estmate the value of gong for t for a specfc poston and number of yards to go. That s, one could consder all cases n whch the correspondng crcumstance occurred on thrd down, fnd what the plays realzed values would have been f they had been fourthdown plays, and average the values. In fact, however, there are over a thousand dfferent cases n the sample. Smoothng the estmates s therefore essental. To smooth the estmates, I focus on the dfference between the values of gong for t and of turnng the ball over on the spot rather than estmatng the value of gong for t drectly. In general, ths dfference depends on three factors. The frst s the dfference between the values of havng a frst down on the spot and of the other team havng a frst down there. Snce the V s are essentally symmetrc around the 50yard lne, ths factor s essentally ndependent of the team s poston on the feld. The second factor s the probablty that the team succeeds when t goes for t. As long as the team s not close to ts opponent s goal lne, there s no reason for ths probablty to vary greatly wth the team s poston. The thrd (and least mportant) factor s the average addtonal beneft from the yards the team gans when t goes for t. Agan, as long as the team s not close to the opponent s goal lne, there s no reason for ths factor to vary substantally wth ts poston. Close to the opponent s goal lne, however, the team has less room to work wth, and so ts chances of success and average number of yards ganed are lkely to be lower. On the other hand, because the value of a touchdown s much larger than the value of a frst down on the 1, the addtonal beneft from ganng yards may be hgher. Thus near the goal lne, we cannot be confdent that the dfference between the values of gong for t and of turnng the ball over does not vary substantally wth the team s poston. The dfference between the values of gong for t and of turnng the ball over on the spot s Gy ( V ), or Gy V, where Gy denotes the value of gong for t on yard lne wth y yards to go and denotes the yard lne opposte yard lne. From the team s goal lne to the opponent s 17, I assume that ths dfference s ndependent of and quadratc n y: 2 Gy V p a 0 a1y a2y. (5) From the opponent s 17 to ts goal lne, I let the dfference depend quadratcally on both and y: G V p b by b b y b y b b y b y y b8y. (6) At the 17, where the two functons meet, I constran both ther level
12 do frms maxmze? 351 Fg. 3. The estmated dfference between the values of gong for t and of the other team havng the ball on the spot at a generc yard lne outsde the opponent s 17 (sold lne) and at the opponent s 5 (dashed lne). The dotted lnes show the twostandarderror bands. and ther dervatve wth respect to to be equal for all y. Ths creates sx restrctons. The data consst of all thrddown plays n the frst quarter; there are 4,733 observatons. 10 Fgure 3 summarzes the results. The sold lne shows the estmates of Gy V as a functon of y for a generc poston on the feld not nsde the opponent s 17, and the dashed lne shows the estmates at the opponent s 5. Outsde the opponent s 17, the estmate of Gy V for a team facng fourth and 1 s On thrdand 1 plays from the goal lne to the opponent s 17, teams are successful 64 percent of the tme, and they gan an average of 3.8 yards; ths corresponds to an expected value of 2.66 ponts. 11 Thus the estmate of 2.64 s reasonable. The estmated dfference falls roughly lnearly wth the number of yards to go. It s 2.05 wth 5 yards to go (equvalent to a 45 percent chance of success and an average gan of 6.3 yards), 1.49 wth 10 yards to go (a 30 percent chance of success and an average gan of 6.6 yards), and 1.08 wth 15 yards to go (an 18 percent chance of 10 To parallel the analyss of kckng, plays that are blown dead before the snap for whch t would not have been possble to determne whether the kckng team had been sent n are excluded (see n. 8). And to prevent outlers that are not relevant to decsons about gong for t from affectng the results, plays on whch the team had more than 20 yards to go are excluded. 11 The translatons of average outcomes nto pont values n ths paragraph are done for a team at mdfeld. Snce the V s are not exactly symmetrc around the 50 or exactly lnear, choosng a dfferent poston would change the calculatons slghtly.
13 352 journal of poltcal economy success and an average gan of 7.7 yards). These estmates are smlar to what one would obtan smply by lookng at the average results of the correspondng types of plays. At the opponent s 5, the estmate of Gy V wth 1 yard to go s 2.94 (equvalent to a 38 percent chance of a frst down wth an average gan of 2 yards plus a 25 percent chance of a touchdown), whch s slghtly hgher than the estmate elsewhere on the feld. The estmate falls more rapdly wth the number of yards to go than elsewhere on the feld, however. Wth 5 yards to go, t s 1.42 (equvalent to a 26 percent chance of a touchdown). The estmate for 5 yards to go s qute smlar to what one would obtan by lookng at averages; the estmate for 1 yard to go s somewhat hgher, however. The dotted lnes show the twostandarderror bands. For the range n whch Gy V s constraned to be ndependent of, the standard errors are small: for 15 yards to go or less, they are less than 0.1. Insde the 17, where fewer observatons are beng used, they are larger, but stll typcally less than 0.2. C. Recommended Choces Fgure 4 combnes the analyses of kckng and gong for t by showng the number of yards to go where the estmated average payoffs to the two choces are equal as a functon of the team s poston. On the team s own half of the feld, gong for t s better on average f there s less than about 4 yards to go. After mdfeld, the gan from kckng falls, and so the crtcal value rses. It s 6.5 yards at the opponent s 45 and peaks at 9.8 on the opponent s 33. As the team gets nto feld goal range, the crtcal value falls rapdly; ts lowest pont s 4.0 yards on the 21. Thereafter, the value of kckng changes lttle whle the value of gong for t rses. As a result, the crtcal value rses agan. The analyss mples that once a team reaches ts opponent s 5, t s always better off on average gong for t. The two dotted lnes n the fgure show the twostandarderror bands for the crtcal values. 12 The crtcal values are estmated farly precsely. Although these fndngs contradct the conventonal wsdom, they are qute ntutve. As descrbed n Secton I, one case for whch the ntuton s clear s fourth and goal on the 2. The expected payoffs n terms of mmedate ponts to the two choces are very smlar, but tryng for a touchdown on average leaves the other team n consderably worse feld poston. Another farly ntutve case s fourth and 3 or 4 on the 50. If the team goes for a frst down, t has about a chance of success; 12 For example, the lower dotted lne shows the pont where the dfference between the estmated values of gong for t and kckng s twce ts standard error.
14 Fg. 4. The number of yards to go where the estmated values of kckng and gong for t are equal (sold lne) and twostandarderror bands (dotted lnes), and the greatest number of yards to go such that when teams have that many yards to go or less, they go for t at least as often as they kck (dashed lne).
15 354 journal of poltcal economy thus both the team and ts opponent have about a 50 percent chance of a frst and 10. But the team wll gan an average of about 6 yards on the fourthdown play; thus on average t s better off than ts opponent f t goes for t. If the team punts, ts opponent on average wll end up wth a frst and 10 around ts 14. Both standard vews about football and the analyss n Secton II suggest that the team and ts opponent are about equally well off n ths stuaton. Thus, on average the team s better off than ts opponent f t goes for a frst down, but not f t punts. Gong for the frst down s therefore preferable on average. The very hgh crtcal values n the dead zone also have an ntutve explanaton. The chances of makng a frst down declne only moderately as the number of yards to go ncreases. For example, away from the opponent s end zone, the chance of makng a frst down or touchdown on thrd down s 64 percent wth 1 yard to go, 44 percent wth 5 yards to go, and 34 percent wth 10 yards to go. As a result, the large decrease n the gan from kckng n the dead zone causes a large ncrease n the crtcal value. D. Actual Choces Teams actual choces are dramatcally more conservatve than those recommended by the dynamcprogrammng analyss. On the 1,604 fourth downs n the sample for whch the analyss mples that teams are on average better off kckng, they went for t only nne tmes. But on the 1,068 fourth downs for whch the analyss mples that teams are on average better off gong for t, they kcked 959 tmes. 13 The dashed lne n fgure 4 summarzes teams choces. It shows, for each pont on the feld, the largest number of yards to go wth the property that when teams have that many yards to go or less, they go for t at least as often as they kck. Over most of the feld, teams usually kck even wth only 1 yard to go. Teams are slghtly more aggressve n the dead zone, but are stll far less aggressve than the dynamcprogrammng analyss suggests. On the lne summarzng teams choces, the null hypothess that the average values of kckng and gong for t are equal s typcally rejected wth a tstatstc between three and seven These fgures exclude the 28 cases for whch we cannot observe the team s ntent because of a penalty before the snap. 14 Carter and Machol (1978) and Carroll et al. (1998, chap. 10) also examne fourthdown decsons. Carter and Machol consder only decsons nsde the opponent s 35yard lne. They use estmates from ther earler work (descrbed n n. 6 above) to assgn values to dfferent stuatons. To estmate the payoff from gong for t, they pool thrddown and fourthdown plays. They assume that all successful plays produce exactly the yards needed for a frst down, that all unsuccessful plays produce no yards, and that the probablty of success does not depend on the team s poston on the feld. They then compare the estmated payoffs to gong for t wth the payoffs to feld goal attempts and punts. They
16 do frms maxmze? 355 IV. Complcatons A. Ratonal Rsk Averson I have assumed that a wnmaxmzng team should be rskneutral concernng ponts scored. Ths s clearly not exactly correct. The analyss may therefore overstate the value of a touchdown relatve to a feld goal, and thus overstate the benefts of gong for t on fourth down. Three consderatons suggest that ths effect s not mportant. Frst, as I show n Secton V, teams are conservatve even n stuatons n whch wnmaxmzng behavor would be rsklovng over ponts scored. Second, t s essentally rrelevant to decsons n the mddle of the feld. Near mdfeld, a team should maxmze the probablty that t s the frst to get close to the opponent s goal lne, snce that s necessary for ether a feld goal or a touchdown. But teams are conservatve over the entre feld. Thrd, drect evdence about the mpact of ponts on the probablty of wnnng suggests that rsk neutralty s an excellent approxmaton for the early part of the game. Because teams adjust ther play late n the game on the bass of the score, one cannot just look at the dstrbuton of actual wnnng margns. Instead, I try to approxmate what the dstrbuton of wnnng margns would be n the absence of lategame adjustments and use ths to estmate the value of a feld goal or touchdown early n the game. I begn by dvdng the games nto decles accordng to the pont spread. I then fnd the score for the favorte and the underdog at the end of the frst half; the dea here s that these scores are relatvely unaffected by adjustments n response to the score. I then construct synthetc fnal scores by combnng the frsthalf scores of each par of games wthn a decle. Ths yelds a total of 74(73)/2 or 73(72)/2 synthetc games for each decle, for a total of 26,718 observatons. I use the results to estmate the mpact of an addtonal feld goal or touchdown n the frst quarter. For example, the estmated effect of a feld goal on the probablty of wnnng s the sum of the probablty that a team would tral by 1 or 2 ponts at the end of the game plus half the probablty that the score would be ted or the team would tral by 3 ponts. conclude that teams should be consderably more aggressve than they are. Carroll et al. consder decsons over the entre feld. They do not spell out ther method for estmatng the values of dfferent stuatons (though t appears related to Carter and Machol s), and t yelds mplausble results. Smlarly to Carter and Machol, they pool thrddown and fourthdown plays and assume that successful plays produce one more yard than needed for a frst down, that unsuccessful plays yeld no gan, and that the chances of success do not vary wth feld poston. They agan conclude that teams should be consderably more aggressve. Ther specfc fndngs about when gong for t s preferable on average are qute dfferent from mne, however. Fnally, nether Carter and Machol nor Carroll et al. nvestgate the statstcal sgnfcance of ther results.
17 356 journal of poltcal economy Ths exercse suggests that 7 ponts are n fact slghtly more than seventhrds as valuable as 3. An addtonal 3 ponts are estmated to rase the probablty of wnnng by 6.8 percentage ponts; an addtonal 7 ponts are estmated to rase the probablty by 16.2 percentage ponts, or 2.40 tmes as much. The source of ths result s that the dstrbuton of synthetc margns s consderably hgher at 4 and 7 ponts than at 1 or 2. To put t dfferently, to some extent what s mportant about a touchdown s not that ts usual value s 7 ponts, but that ts usual value s between two and three tmes the value of a feld goal. B. Thrd Down versus Fourth Down There are two ways to nvestgate the approprateness of usng thrddown plays to gauge what would happen f teams went for t on fourth down. The frst s to consder how teams ncentves are lkely to affect outcomes on fourth downs relatve to thrd downs. Relatve payoffs to dfferent outcomes are dfferent on the two downs. In partcular, the beneft from a long gan relatve to just makng a frst down s smaller on fourth down. As a result, both the offense and defense wll behave dfferently: the offense wll be wllng to lower ts chances of makng a long gan n order to ncrease ts chances of just makng a frst down, and the defense wll be wllng to do the reverse. Ths suggests that the drecton of the bas from usng thrddown plays should depend on whch team has more nfluence on the dstrbuton of outcomes. Snce t seems unlkely that the defense has substantally more nfluence than the offense on the dstrbuton of outcomes, t follows that the use of thrd downs s unlkely to lead to substantal overestmates of the value of gong for t. More mportant, the relatve payoffs to dfferent outcomes do not dffer greatly between thrd and fourth downs. For example, consder a team that s on ts 30 and needs 2 yards for a frst down. On thrd down (under the realstc assumpton that the team wll punt f t fals to make a frst down), the beneft of ganng 15 yards rather than none s 1.4 tmes as large as the beneft of ganng 2 yards rather than none. On fourth down, the beneft of ganng 15 yards rather than none s 1.2 tmes as large as the beneft of ganng 2 yards rather than none. Thus one would not expect ether sde to behave very dfferently on the two downs. And when a team has goal to go, the payoff on ether thrd down or fourth down depends almost entrely on whether the team scores a touchdown. Thus one would expect both sdes behavor to be essentally the same on the two downs. These consderatons suggest that any bas from the use of thrddown plays s lkely to be small. The second approach s to drectly compare the realzed values of plays where teams went for t on fourth downs (.e., the mmedate ponts
18 do frms maxmze? 357 scored plus the value of the resultng feld poston) wth what one would expect on the bass of the analyss of thrd downs. Ths comparson s potentally problematc, however, for two reasons. Frst, as descrbed above, teams went for t only 118 tmes n the sample. Second, tmes when teams choose to go for t may be unusual: the teams may know that they are partcularly lkely to succeed, or they may be desperate. To ncrease the sample wthout brngng n fourthdown attempts that are lkely to be especally unusual, I nclude the entre game except for the last two mnutes of each half (and overtmes). Ths ncreases the sample to 1,338 plays. And as a partal remedy for the second problem, I experment wth controllng for the amount the team wth the ball s tralng by and the amount t s favored by. The results suggest that fourth downs are vrtually ndstngushable from thrd downs. The mean of the dfference between the realzed value of the fourthdown attempts and what s predcted by the analyss of thrd downs s (wth a standard error of 0.7), whch s essentally zero. When controls for the pror pont spread and the current pont dfferental are ncluded, the coeffcent falls to and remans hghly nsgnfcant. The pont estmate corresponds to the probablty of success beng one percentage pont lower on fourth downs than on thrd downs, whch would have almost no mpact on the analyss. C. Addtonal Informaton In makng fourthdown decsons, a team has more nformaton than the averages used n the dynamcprogrammng analyss. Thus t would not be optmal for t to follow the recommendatons of the dynamcprogrammng analyss mechancally. Addtonal nformaton cannot, however, account for the large systematc departures from the recommendatons of the dynamcprogrammng analyss. Over wde ranges, teams almost always kck n crcumstances n whch the analyss mples that they would be better off on average gong for t. For example, on the 512 fourth downs n the sample n the offense s half of the feld for whch the dynamcprogrammng analyss suggests gong for t, teams went for t only seven tmes. Smlarly, on the 175 fourth downs wth 5 or more yards to go for whch the analyss suggests gong for t, teams went for t only 13 tmes. Addtonal nformaton can account for ths behavor only f teams know on a large majorty of fourth downs that the expected payoff to gong for t relatve to kckng s consderably less than average, and know on the remander that the expected payoff s dramatcally larger than average. Ths possblty s not at all plausble. Further, t predcts that when teams choose to go for t, the results wll be far better than
19 358 journal of poltcal economy one would expect on the bass of averages. As descrbed above, ths predcton s contradcted by the data. D. Momentum Falng on fourth down could be costly to a team s chances of wnnng not just through ts effect on possesson and feld poston, but also through ts effect on energy and emotons. Thus t mght be more costly for the other team to have the ball as a result of a faled fourthdown attempt than for t to have the ball at the same place n the course of a normal drve or because of a punt. The analyss mght therefore overstate the average payoff to gong for t. There are two reasons to be skeptcal of ths possblty. Frst, the same reasonng suggests that there could be a motvatonal beneft to succeedng on fourth down, and thus that the analyss could understate the benefts of a successful fourthdown attempt. Second, studes of momentum n other sports have found at most small momentum effects (e.g., Glovch, Vallone, and Tversky 1985; Albrght 1993; Klaassen and Magnus 2001). More mportant, t s possble to obtan drect evdence about whether outcomes dffer systematcally from normal after plays whose outcomes are ether very bad or very good. To obtan a reasonable sample sze, for very bad plays I consder all cases n whch from one stuaton to the next (where a stuaton s defned as before), possesson changed and the ball advanced less than 10 yards. For very good plays, I consder all cases n whch the offense scored a touchdown. These crtera yeld 636 very bad plays and 628 very good plays. I then examne what happens from the stuaton mmedately followng the extreme play to the next stuaton, from that stuaton to the next, and from that stuaton to the subsequent one. In each case, I ask whether the realzed values of these stuatons one stuaton later dffer systematcally from the V s for those stuatons. That s, I look at the means of the relevant e gt s (always com puted from the perspectve of the team that had the ball before the very bad or very good play). The results provde no evdence of momentum effects. All the pont estmates are small and hghly nsgnfcant; the largest tstatstc (n absolute value) s less than 1.3. Moreover, the largest pont estmate (agan n absolute value) goes the wrong drecton from the pont of vew of the momentum hypothess: from the stuaton mmedately followng a very bad play to the next, the team that lost possesson does somewhat better than average The workng paper verson of the paper (Romer 2005) consders two addtonal complcatons. The frst s the possblty of sample selecton bas n the estmaton of the V s
20 do frms maxmze? 359 V. Quanttatve Implcatons An obvous queston s whether the potental gans from dfferent choces are mportant. There are n fact two dstnct questons. The frst s whether there are cases of clearcut departures from wn maxmzaton. If there were not, then small changes n the analyss mght reverse the conclusons. The answer s that there are clearcut departures. One example s the case of fourth and goal on the 2 dscussed above. The estmates mply that tryng for a touchdown and falng s only slghtly worse than kckng a feld goal. As a result, they mply that gong for a touchdown s preferable on average as long as the probablty of success s at least 18 percent. The actual probablty of success, n contrast, s about 45 percent. Thus there are no plausble changes n the analyss that could reverse the concluson that tryng for a touchdown s preferable on average. Moreover, the average beneft of tryng for a touchdown s substantal. The estmated value of gong for t s about 3.7 ponts, whereas the estmated value of kckng s about 2.4 ponts. Snce each addtonal pont rases the probablty of wnnng by about 2.3 percentage ponts, tryng for a touchdown on average ncreases the chances of wnnng by about three percentage ponts. Yet teams attempted a feld goal every tme n the sample they were n ths poston. Two other examples are fourth and goal on the 1 and fourth and 1 between the opponent s 35 and 40. For the frst, the estmates mply that the crtcal and actual probabltes of success are 16 percent and 62 percent, and that tryng for a touchdown on average ncreases the chances of wnnng by about fve percentage ponts. For the second, the crtcal and actual probabltes are 39 percent and 64 percent, and gong for a frst down rases the probablty of wnnng by about 2.5 percentage ponts. In these cases, teams do not always kck, but they do about half the tme. These decsons are consstent wth wn maxmzaton only f teams have substantal addtonal nformaton that allows them to dentfy tmes when ther fourthdown attempts are especally lkely to succeed. As descrbed n the prevous secton, there s no evdence of such large addtonal nformaton. The second queston s whether the analyss mples that teams could ncrease ther overall chances of wnnng substantally. Snce the analyss consders only a small fracton of plays and only a sngle decson on those plays, one would not expect t to show large potental ncreases stemmng from the fact that teams are not assgned to stuatons randomly. The second s general equlbrum effects: dfferent decsons on fourth downs could affect other choces. I conclude that the effects of sample selecton bas are small and of ambguous sgn, and that general equlbrum effects are small and most lkely strengthen the case for beng more aggressve on fourth downs.
An Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationTrade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity
Trade Adjustment Productvty n Large Crses Gta Gopnath Department of Economcs Harvard Unversty NBER Brent Neman Booth School of Busness Unversty of Chcago NBER Onlne Appendx May 2013 Appendx A: Dervaton
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationAn Empirical Study of Search Engine Advertising Effectiveness
An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan RmmKaufman, RmmKaufman
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationRESEARCH DISCUSSION PAPER
Reserve Bank of Australa RESEARCH DISCUSSION PAPER Competton Between Payment Systems George Gardner and Andrew Stone RDP 200902 COMPETITION BETWEEN PAYMENT SYSTEMS George Gardner and Andrew Stone Research
More informationKiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120
Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationHARVARD John M. Olin Center for Law, Economics, and Business
HARVARD John M. Oln Center for Law, Economcs, and Busness ISSN 10456333 ASYMMETRIC INFORMATION AND LEARNING IN THE AUTOMOBILE INSURANCE MARKET Alma Cohen Dscusson Paper No. 371 6/2002 Harvard Law School
More informationHOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*
HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt
More informationWhen Talk is Free : The Effect of Tariff Structure on Usage under Two and ThreePart Tariffs
0 When Talk s Free : The Effect of Tarff Structure on Usage under Two and ThreePart Tarffs Eva Ascarza Ana Lambrecht Naufel Vlcassm July 2012 (Forthcomng at Journal of Marketng Research) Eva Ascarza
More informationStudy on Model of Risks Assessment of Standard Operation in Rural Power Network
Study on Model of Rsks Assessment of Standard Operaton n Rural Power Network Qngj L 1, Tao Yang 2 1 Qngj L, College of Informaton and Electrcal Engneerng, Shenyang Agrculture Unversty, Shenyang 110866,
More informationPRIVATE SCHOOL CHOICE: THE EFFECTS OF RELIGIOUS AFFILIATION AND PARTICIPATION
PRIVATE SCHOOL CHOICE: THE EFFECTS OF RELIIOUS AFFILIATION AND PARTICIPATION Danny CohenZada Department of Economcs, Benuron Unversty, BeerSheva 84105, Israel Wllam Sander Department of Economcs, DePaul
More informationEstimating the Effect of the Red Card in Soccer
Estmatng the Effect of the Red Card n Soccer When to Commt an Offense n Exchange for Preventng a Goal Opportunty Jan Vecer, Frantsek Koprva, Tomoyuk Ichba, Columba Unversty, Department of Statstcs, New
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationUnderstanding the Impact of Marketing Actions in Traditional Channels on the Internet: Evidence from a Large Scale Field Experiment
A research and educaton ntatve at the MT Sloan School of Management Understandng the mpact of Marketng Actons n Tradtonal Channels on the nternet: Evdence from a Large Scale Feld Experment Paper 216 Erc
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationSTAMP DUTY ON SHARES AND ITS EFFECT ON SHARE PRICES
STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond Mke Hawkns Alexander Klemm THE INSTITUTE FOR FISCAL STUIES WP04/11 STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond (IFS and Unversty
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationStatistical Methods to Develop Rating Models
Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and
More informationGender differences in revealed risk taking: evidence from mutual fund investors
Economcs Letters 76 (2002) 151 158 www.elsever.com/ locate/ econbase Gender dfferences n revealed rsk takng: evdence from mutual fund nvestors a b c, * Peggy D. Dwyer, James H. Glkeson, John A. Lst a Unversty
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationInequity Aversion and Individual Behavior in Public Good Games: An Experimental Investigation
Dscusson Paper No. 07034 Inequty Averson and Indvdual Behavor n Publc Good Games: An Expermental Investgaton Astrd Dannenberg, Thomas Rechmann, Bodo Sturm, and Carsten Vogt Dscusson Paper No. 07034 Inequty
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationMultiplePeriod Attribution: Residuals and Compounding
MultplePerod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationCourse outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy
Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationIntrayear Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error
Intrayear Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor
More informationWorking Paper Loss aversion and rentseeking: An experimental study
econstor www.econstor.eu Der OpenAccessPublkatonsserver der ZBW LebnzInformatonszentrum Wrtschaft The Open Access Publcaton Server of the ZBW Lebnz Informaton Centre for Economcs Kong, Xaojng Workng
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationSP Betting as a SelfEnforcing Implicit Cartel
SP Bettng as a SelfEnforcng Implct Cartel by Ad Schnytzer and Avcha Snr Department of Economcs BarIlan Unversty Ramat Gan Israel 52800 emal: schnyta@mal.bu.ac.l snrav@mal.bu.ac.l Abstract A large share
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationVasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio
Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of
More informationPrice Impact Asymmetry of Block Trades: An Institutional Trading Explanation
Prce Impact Asymmetry of Block Trades: An Insttutonal Tradng Explanaton Gdeon Saar 1 Frst Draft: Aprl 1997 Current verson: October 1999 1 Stern School of Busness, New York Unversty, 44 West Fourth Street,
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET *
ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET * Amy Fnkelsten Harvard Unversty and NBER James Poterba MIT and NBER * We are grateful to Jeffrey Brown, PerreAndre
More informationA Multistage Model of Loans and the Role of Relationships
A Multstage Model of Loans and the Role of Relatonshps Sugato Chakravarty, Purdue Unversty, and Tansel Ylmazer, Purdue Unversty Abstract The goal of ths paper s to further our understandng of how relatonshps
More informationThe impact of hard discount control mechanism on the discount volatility of UK closedend funds
Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closedend funds Abstract The mpact
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationEvaluating credit risk models: A critique and a new proposal
Evaluatng credt rsk models: A crtque and a new proposal Hergen Frerchs* Gunter Löffler Unversty of Frankfurt (Man) February 14, 2001 Abstract Evaluatng the qualty of credt portfolo rsk models s an mportant
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More informationHeterogeneous Paths Through College: Detailed Patterns and Relationships with Graduation and Earnings
Heterogeneous Paths Through College: Detaled Patterns and Relatonshps wth Graduaton and Earnngs Rodney J. Andrews The Unversty of Texas at Dallas and the Texas Schools Project Jng L The Unversty of Tulsa
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationMarginal Benefit Incidence Analysis Using a Single Crosssection of Data. Mohamed Ihsan Ajwad and Quentin Wodon 1. World Bank.
Margnal Beneft Incdence Analyss Usng a Sngle Crosssecton of Data Mohamed Ihsan Ajwad and uentn Wodon World Bank August 200 Abstract In a recent paper, Lanjouw and Ravallon proposed an attractve and smple
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More informationSketching Sampled Data Streams
Sketchng Sampled Data Streams Florn Rusu, Aln Dobra CISE Department Unversty of Florda Ganesvlle, FL, USA frusu@cse.ufl.edu adobra@cse.ufl.edu Abstract Samplng s used as a unversal method to reduce the
More informationStudent Performance in Online Quizzes as a Function of Time in Undergraduate Financial Management Courses
Student Performance n Onlne Quzzes as a Functon of Tme n Undergraduate Fnancal Management Courses Olver Schnusenberg The Unversty of North Florda ABSTRACT An nterestng research queston n lght of recent
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6  The Time Value of Money. The Time Value of Money
Ch. 6  The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21 Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
More informationTo manage leave, meeting institutional requirements and treating individual staff members fairly and consistently.
Corporate Polces & Procedures Human Resources  Document CPP216 Leave Management Frst Produced: Current Verson: Past Revsons: Revew Cycle: Apples From: 09/09/09 26/10/12 09/09/09 3 years Immedately Authorsaton:
More informationAnalysis of Covariance
Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content
More informationAdverse selection in the annuity market when payoffs vary over the time of retirement
Adverse selecton n the annuty market when payoffs vary over the tme of retrement by JOANN K. BRUNNER AND SUSANNE PEC * July 004 Revsed Verson of Workng Paper 0030, Department of Economcs, Unversty of nz.
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationChapter 8 Groupbased Lending and Adverse Selection: A Study on Risk Behavior and Group Formation 1
Chapter 8 Groupbased Lendng and Adverse Selecton: A Study on Rsk Behavor and Group Formaton 1 8.1 Introducton Ths chapter deals wth group formaton and the adverse selecton problem. In several theoretcal
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationStart me up: The Effectiveness of a SelfEmployment Programme for Needy Unemployed People in Germany*
Start me up: The Effectveness of a SelfEmployment Programme for Needy Unemployed People n Germany* Joachm Wolff Anton Nvorozhkn Date: 22/10/2008 Abstract In recent years actvaton of meanstested unemployment
More informationDepreciation of Business R&D Capital
Deprecaton of Busness R&D Captal U.S. Bureau of Economc Analyss Abstract R&D deprecaton rates are crtcal to calculatng the rates of return to R&D nvestments and captal servce costs, whch are mportant for
More informationUniversity of Kent Department of Economics Discussion Papers
Unversty of Kent Department of Economcs Dscusson Papers EXPERT ANALYSIS AND INSIDER INFORMATION IN HORSERACE BETTING: REGULATING INFORMED MARKET BEHAVIOR John Person and Mchael A. Smth November 2008 KDPE
More informationChapter 15: Debt and Taxes
Chapter 15: Debt and Taxes1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationChapter 11 Practice Problems Answers
Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make
More informationThis study examines whether the framing mode (narrow versus broad) influences the stock investment decisions
MANAGEMENT SCIENCE Vol. 54, No. 6, June 2008, pp. 1052 1064 ssn 00251909 essn 15265501 08 5406 1052 nforms do 10.1287/mnsc.1070.0845 2008 INFORMS How Do Decson Frames Influence the Stock Investment Choces
More informationProceedings of the Annual Meeting of the American Statistical Association, August 59, 2001
Proceedngs of the Annual Meetng of the Amercan Statstcal Assocaton, August 59, 2001 LISTASSISTED SAMPLING: THE EFFECT OF TELEPHONE SYSTEM CHANGES ON DESIGN 1 Clyde Tucker, Bureau of Labor Statstcs James
More information