The Australian Rules Football Fixed Odds and Line Betting Markets: Econometric Tests for Efficiency and Simulated Betting Systems

Transcription

1 Th Australian Ruls Football Fixd Odds and Lin Btting Markts: Economtric Tsts for Efficincy and Simulatd Btting Systms by Adi Schnytzr and Guy Winbrg a Papr to b prsntd at: Th 4 th Binnial Equin Industry Program Intrnational Acadmic Confrnc Louisvill, Kntucky (USA), Jun 2005 Abstract Th purpos of this papr is to tst th fficincy of two Australian Ruls Football btting markts mploying two approachs commonly adoptd in th litratur and to compar both th markts and th mthods mployd. Th two markts ar a lin btting markt and a fixd odds win btting markt and th two approachs ar conomtric tsting and btting simulation. W conduct our comparisons by subjcting data for th Australian Football Lagu (AFL) to highly dtaild scrutiny. W rstrict ourslvs to wak-form markt fficincy, as this provs sufficintly complicatd to show that nithr th connction btwn th two forms of fficincy tsting nor that btwn th fficincy of th two markts, is clar-cut. Nonthlss, taking advantag of th fact that many gams in th AFL ar playd on nutral grounds, w ar abl to rjct th xistnc of any significant favorit-longshot bias in ithr markt for ach of th four sasons individually and for th priod as a whol, xcpt for a rvrs bias in th lin markt in 2001 as rflctd by significant profits in btting simulations, and to dmonstrat th xistnc of a significant bias in favor of tams with an apparnt hom ground advantag for thr of th four sasons in th win btting markt simulations. Th lin markt is fr of such a bias. Th rsults of conomtric tsts for this bias ar mor ambiguous. Finally, w suggst that th diffrnc in apparnt fficincy in th two markts may b du to th assumption, on th part of bookmakrs, of a linar rlationship btwn lins and prics. Kywords: markt fficincy, btting markts, sports conomics a Dpartmnt of Economics, Bar-Ilan Univrsity, Ramat-Gan, Isral (-mail addrsss: schnyta@biu.013.nt.il, guy@winbrgdoron.co.il) Th authors wish to thank Michal Baily, Hamish Davidson, John Kyriakopoulos, Damon Rashd and Eric Sornsn for thir hlp with data collction.

2 1. Introduction Th purpos of this papr is to tst th fficincy 1 of two Australian Ruls football btting markts mploying two approachs commonly adoptd in th litratur and to compar both th markts and th mthods mployd. Th two markts ar a lin btting markt and a fixd odds win btting markt and th two approachs ar conomtric tsting and btting simulation. W conduct our comparisons by subjcting data for th Australian Football Lagu (AFL) to highly dtaild scrutiny. W rstrict ourslvs to wak-form markt fficincy, as this provs sufficintly complicatd to show that nithr th connction btwn th two forms of fficincy tsting nor that btwn th fficincy of th two markts, is clar-cut. Nonthlss, taking advantag of th fact that many gams in th AFL ar playd on nutral grounds, w ar abl to rjct th xistnc of any significant favorit-longshot bias in ithr markt for ach of th four sasons individually and for th priod as a whol, xcpt for a rvrs bias in th lin markt in 2001 as rflctd by significant profits in btting simulations, and to dmonstrat th xistnc of a significant bias in favor of tams with an apparnt hom ground advantag for thr of th four sasons in th win btting markt simulations. Th lin markt is fr of such a bias. Th rsults of conomtric tsts for this bias ar mor ambiguous. Finally, w suggst that th diffrnc in apparnt fficincy in th two markts may b du to th assumption, on th part of bookmakrs, of a linar rlationship btwn lins and prics. Th rmaindr of this papr is organizd as follows: A discussion of lin and fixd odds btting, th conomics of bookmaking and a brif survy of th litratur ar prsntd in Sction 2. Sction 3 provids a summary of th basics of Australian Ruls football rlvant to an undrstanding of this papr, sction 4 dscribs our data st and highlights th problms to b xplaind, sction 5 discusss th wak fficincy conomtric tsts and th btting systms, sction 6 dscribs th rlationship btwn th two btting markts, and sction 7 concluds th papr. 2. Th Basics and th Litratur Thr ar svral altrnativ btting mthods offrd in th diffrnt sports btting markts worldwid. In tam sports, two mthods prdominat: "Point sprad" (also known as "lin") wagring is th dominant form of wagring on basktball and Amrican football contsts. 2 An altrnativ mthod is "fixd odds" win btting, which prdominats in US basball. 3 Basst (1981) xamind why bookmakrs usd th point sprad mthod xclusivly for wagring on th National Football Lagu (NFL) whn th odds mthod is also fasibl. Givn simpl spcifications of blifs and btting bhavior, and givn that point sprad and odds ar qually thrilling, h dmonstrats that th point sprad bt can stand alon as th bt producd by a profit maximizing bookmakr. Woodland and Woodland (1991) suggstd that th markt structur of having lin (and not odds) btting on th NFL was a consqunc of a risk-avrs attitud of bttors. History has ovrtakn ths thortical analyss and th ral rason for th availability of th two diffrnt typs of btting in diffrnt sports appars to b unrlatd to considrations of ithr maximizing bhavior or attitud to risk. Currntly, both typs ar availabl for most tam sports, and on can vn bt according to ithr of thm with th sam bookmakr. Nonthlss, mpirical rsarch comparing ths two altrnativ mthods has, to th bst of our knowldg, not yt bn publishd. A typical point sprad wagr in th AFL rquirs that th bttor risk $1 for th chanc to rciv $1.9 if succssful. 4 Th lin on a gam spcifis th favord tam and th point sprad. A bttor placing a wagr on th favorit wins th bt if th favorit wins by a margin of victory gratr than th point sprad. A bttor placing a wagr on th longshot wins th bt if th longshot loss by lss than th point sprad or wins th gam outright. Th abov-mntiond $1.9-for-$1 dividnd implis that, if th markt is fficint, no btting stratgy should win mor than prcnt of th tim, or, in othr words, th $1.9-for-$1 dividnd rquirs that bttors must pick winnrs in prcnt of bts to brak vn S Osborn (2001) for a list of studis that invstigat th Efficint Markts Hypothsis in diffrnt markts. 2 S Basst (1981), Dobra, Cargill and Myr (1990) and Woodland and Woodland (1991). 3 S Woodland and Woodland (1994). 4 In contrast to th US markt, th winning dividnd pr $1 point sprad wagr in th AFL is not fixd. Th rang of this dividnd in our data of was $1.78-$2.05, whil in 66% of th gams it was $1.9, and th avrag was $1.9 as wll. 5 Th prcntag of winning bts (WP) ncssary to brak vn, prcnt, is obtaind by stting th xpctd valu of th random variabl, a gambl WP * (1 WP) * (-1), qual to zro. 6 S, for furthr discussion, Vrgin and Scriabin (1978), Gandar t al (1988), and Dana and Knttr (1994). 1

3 Th fixd odds wagr in th AFL rquirs that th bttor risk $1 for th chanc to rciv a fixd sum if succssful. 7 Th odds on a tam spcify th odds that th tam will win th gam. A bttor placing a wagr on a tam wins th bt if this tam wins. As in th lin btting markt, th bookmakr sts odds to arn around fiv prcnt of th total amount bt if his book is balancd. 8 Thrfor th bttor must pick around 52.5 prcnt of winnrs to brak vn. 9 Whil th initial lin/odds ar basd on xprt opinion of th gam's outcom, thraftr th bookmakr adjusts th lin/odds. 10 Thr is no consnsus in th litratur whthr ths adjustmnts ar mad to rflct th collctiv judgmnt of gamblrs about th outcom or bcaus th bookmakrs ar stting prics in ordr to xploit bttors' biass. Lvitt (2004), using data on prics and quantitis of bts placd, found support for th lattr hypothsis; i.., th bookmakrs do not appar to b trying to st prics to qualiz th amount of mony bt on ithr sid of a wagr. Our rsults do not prmit us to shd any maningful light on this issu. Th qustion of whthr organizd sports btting markts ar wak-form fficint according to Fama's (1970) dfinition has rcivd considrabl attntion in th litratur. Zubr t al (1985), Saur t al (1988), and Gandar, Zubr, O'Brin and Russo (1988) implmntd th wak fficincy tst for point sprads in th NFL. 11 Schnytzr and Winbrg (2004) implmntd it using data on National Basktball Association (NBA) gams in th US for 1999/ /4 priod. Whil Zubr t al (1985) found this tst to b too wak to stablish dfinit conclusions, and Gandar t al (1988) concludd that statistical tsts ar not powrful nough to dtct infficincis, 12 Saur t al (1988) and Schnytzr and Winbrg (2004) could not rjct th null hypothsis that th NFL and th NBA gambling markts ar wakly fficint. Bfor concluding that fficincy prvails, Zubr t al (1985) considrd an "xtrm" altrnativ that th lin is unrlatd to th actual point sprad. Thy could not rjct this hypothsis for 15 of th 16 wks in thir sampl, noting that th xtrm altrnativ hypothsis is as consistnt with thir sampl data as is fficincy. Thir conclusion was that an altrnativ tsting stratgy is rquird. Unlik Zubr t al (1985), both Saur t al (1988) and Schnytzr and Winbrg (2004) rjctd this xtrm altrnativ hypothsis. Saur t al (1988) qustiond th validity of tsting wak-form fficincy on a wk-by-wk basis and argud that Zubr t al's (1985) tst fails to provid sufficint vidnc to support th argumnt that spculativ infficincis xist in th btting markt for NFL gams. 13 Golc and Tamarkin (1991), on th othr hand, showd that sprads st in th NFL btting markt ar systmatically biasd prdictors of actual rsults. Thn again, in his rviw of th litratur on sports btting markts, Saur (1998) outlind th substantial vidnc that prics st in ths markts ar fficint forcasts of outcoms. Prics (in th form of btting odds or btting lins) appar to aggrgat scarc information from divrs sourcs, and ar, almost invariably, unbiasd stimators of actual gam outcoms (in trms of both point sprads and winnr's idntity). Whil Saur (1998) notd that a numbr of studis, particularly of point sprad btting markts, hav offrd sightings of profitabl trading stratgis, h also notd that ths sightings frquntly disappar upon furthr invstigation. Thalr and Zimba (1988) rviwd th arly litratur on US ractrack btting markts, noting a consistnt favorit-longshot bias. Woodland and Woodland (1994) found that th favorit-longshot bias in 7 Th rang of actual payouts in our data st is $1.03-$10, whil th avrag sum is $ Th avrag bookmakrs' commission during our data was 3.9%, whil th avrag during was 5.5%. Baily and Clark (2004) notd that th commission could b as low as 2-3%. 9 In th vnt th outcom is idntical to th lin, known as a "push" or a "no bt", th gamblr's wagr is rfundd. In th vry rar vnt whr th outcom is a ti, th fixd odds bttor wins half th amount h would hav won had his tam won. Our data contains only 5 tid gams (0.7%). Not that on can bt on a ti for most gams at odds of 65 to 1, but this is part of an xotic catgory of bts. Not that this sms a high pric givn that thr ar ovr 700 gams in our sampl! 10 Nonthlss, Lvitt (2004) notd that th adjustmnts in th lin/odds ar typically small and rlativly infrqunt; in th fiv days prcding an NFL gam, th postd pric changing an avrag of 1.4 tims pr gam, and in 85 prcnt of thos changs, th lin movd by th minimum incrmnt of on-half of a point. Yt, h notd that in hors racing, th odds st by bookmakrs chang far mor frquntly. 11 Exampls for othr paprs that tst th NFL's wak-form fficincy ar Pankoff (1968), who usd sasons to show that wak-form fficincy prvails, and Amoako-adu, Marmr and Yagil (1985), who showd that th NFL is wakly infficint, using th sasons. 12 Thy not that th statistical tsts ar too wak to rjct rationality in a markt whr irrationality appars to xist. Thy also indicat that thir rsults ar strikingly consistnt with thos of Summrs (1986), who simulatd a modl of stock prics incorporating non-rational xpctations and thn showd that standard statistical tsts ar too wak to dtct th absnc of rationality formd xpctations. 13 It should b notd that th bulk of both Zubr t al (1985) and Saur t al (1988) ar dvotd to an analysis of smistrong fficincy. 2

4 ractrack btting xists in rvrs for basball bttors, but that no btting stratgy admits profits in xcss of commissions. Dar and Holland (2004) modifid prvious rsarch to gnrat a spcification that thy argu yilds th most rliabl stimats of infficincy in th NFL. Thir rsults indicat a btting lin bias favoring bts on hom longshots that dos not appar consistntly from sason to sason, and th xpctd profits arising from this bias may b too small to b xploitd. Th vidnc with rspct to Australian Ruls football btting markts is limitd. 14 Stfani and Clark (1992) usd tam ranking and hom ground advantag to prdict winnrs. Thy rportd a hom ground advantag for ach of th AFL tams during , and a biggr advantag for tams outsid of Mlbourn and for thos that do not shar a hom ground. Thy wr abl to prdict th corrct winning tam in 68 prcnt of gams. Brailsford t al (1995) xamind th AFL and th Australian Rugby Lagu (ARL), focusing on two diffrnt kinds of btting markt. For th ARL, thy considrd lin btting whil for th AFL thy xamind an xotic btting mthod, whrby bttors ar rquird to slct th point sprad to within a 12-point rang, known as "bins". 15 Thy prdictd gam outcoms and tstd btting stratgis and rportd a favorit-longshot bias in th AFL. Thir succss rat from btting on hom tams during was 58 prcnt, yt th rturn rat was ngativ. Othr stratgis gnratd positiv rturns, and btting on th prdictd bin yildd an avrag rturn of 23 prcnt. Ths rsults imply markt infficincy, yt th authors raisd doubts as to whthr this apparnt infficincy is xploitabl. Baily and Clark (2004) not that no attmpt has bn mad to utiliz all past matchs to stablish a prdiction procss. Thy dvlop modls using all prvious match rsults, and invstigat th additional bnfits of incorporating individual playr statistics in th prdiction procss. Thy us data from and prsnt modls prdicting corrctly up to 67 prcnt of winnrs, and producing btting profits of up to 15 prcnt. Clark (2005) invstigats th hom advantag in th AFL using svral modls, and dmonstrats that individual clubs hav hom ground advantags to diffrnt dgrs, non-victorian tams having a largr advantag. His rsults lnd support to th conclusion that crowd ffcts ar th main dtrminant of hom ground advantag. H did not tst fficincy, yt his findings ar rlvant to this papr. Th bottom lin from this brif survy of th litratur is that thr is no consnsus concrning th xistnc of th wak-form fficincy and th possibility of making spculativ profits in sports btting markts. 3. Australian Ruls football Australian Ruls football is a high scoring, continuous-action gam. For th sasons, th avrag gam scor pr tam was 95 points, with a minimum of 25, a maximum of 196, and a standard rror of 28. Each tam has 18 playrs on th fild at any givn tim and 4 substituts ar availabl for unrstrictd, rpatd substitutions as dmd fit by th tam coach. Th hom and away sason compriss 176 gams playd ovr 22 wkly rounds of ight gams ach, btwn 16 tams. Following this is a final sris btwn th top ight tams. Th two surviving tams from this phas play for th prmirship titl in what is known as th "Grand Final". In total, 185 gams ar playd in an ntir sason. 16 Th AFL is a national lagu which bgan as th Victorian Football Lagu. Excpt for Glong, all th othr tams in this lagu drivd from Mlbourn and had thir own hom grounds. Th addition of nw tams from othr stats has bn accompanid by a policy of stadium consolidation in Mlbourn. Thus, it has not bn tru for som yars that ach tam has its own stadium. Of th sixtn tams that mak up th AFL, only thr hav stadiums which ar uniquly hom grounds, whr it may b said that thy hav an advantag; Brisban Lions, Sydny Swans and Glong Cats (although vn Glong do not play all of thir official hom gams at this hom ground). All th othr tams play at grounds shard with on or mor tams. Thus, daling with th hom advantag in th AFL rquirs furthr car and a complimnt to th official hom dsignation is ncssary. Whnvr w rfr to hom tams as ithr substs of th data or dummy variabls in rgrssions, w man tams with an a priori hom ground advantag. Tams which ar officially dsignatd as hom tams but hav no a priori hom ground advantag ar rfrrd to as Nutral S Brailsford, Easton, Gray and Gray (1995). 15 This "bins" btting mthod is vry diffrnt from th mthods studid in our papr. Also, th dividnds in th "bins" mthod ar calculatd, in pari-mutul fashion, aftr th outcom of th match, such that a crtain prcntag of th total amount wagrd is rturnd to succssful bttors. Brailsford t al (1995) not that th avrag commission ovr thir sampl priod was around 20% and that lin btting was not offrd by th AFL at th tim of thir papr. 16 For furthr information about th AFL, s its official wbsit: 17 In what is probably a uniqu, albit bizarr, fatur of th AFL, thr ar vn gams in which th official hom tam is playing an opponnt with a gnuin a priori hom ground advantag! A rcnt xampl is providd in Round 9 of th 3

5 4. Th data Th data usd in this papr ar drivd from publicly availabl sourcs, 18 i.., intrnt-basd sports statistical information. Thus, th gam data com from and whil th closing odds and lins ar from and Our data consist of gam prformancs, dats, grounds, odds and lins. W us data from th 2001 to 2004 sasons, for a total of 740 gams (all hom and away gams plus th finals), and 1480 tam obsrvations. Lin data ar missing for 86 gams, sinc th bookmakrs do not publish a lin in a match whr both tams hav qual (or vry clos) btting odds. 19 W dnot th tam from whos prspctiv th sprad and rsult ar dfind as th tam of rcord. Thr is no singl corrct way of choosing th tam of rcord, and thr diffrnt mthods hav bn usd. 20 First, th tam of rcord can b dfind to b th favorit. Scond, th tam of rcord can b dfind to b th hom tam. Third, th tam of rcord can b chosn randomly, avoiding any systmatic ffcts. W us th official hom tam as th tam of rcord, as do Gandar t al (1988) and most of th othr studis. Nonthlss, sinc, as notd abov, th official hom dfinition in th AFL dos not automatically imply a hom ground advantag, this is somwhat akin to a random slction. Morovr, w analyz diffrnt substs of th data sparatly in ordr to tst dirctly for various possibl biass. Som basic proprtis of our data st ar prsntd in Tabl 1, including th winning prcntag in diffrnt catgoris and diffrnt apparnt biass in th markts. 21 Thus, th rat of succssful btting in th win markt on hom tams, favorits, and hom favorits xcds th prcntag ncssary to brak vn during all yars in our sampl, whil hom longshots and nutral longshots sm ovrpricd vis-à-vis thir winning frquncis. Th rat of succssful btting in th lin markt is lss consistnt, as it xcds th prcntag ncssary to brak vn for hom and hom longshot tams in all sasons bar 2001, hom favorits vry sason, not at all for favorits, and in 2001 and 2002 for nutral longshots. Our hom tam win rat of 64 prcnt for compars with th 58 prcnt as rportd by both Stfani and Clark (1992) for and Brailsford t al (1995) for , and th 60 prcnt from Clark (2005) for But it should b notd that w rfr to ral hom tams, whras th othr paprs rport for official hom tams. This xplains both why our winning rat is highr and th prcptibl upward trnd in official hom tam winning frquncis as nw non-victorian tams hav ntrd th AFL ovr th past two dcads, thrby incrasing th proportion of tams with ral hom ground advantags. Mor puzzling in Tabl 1 ar th apparnt inconsistncis among th biass as btwn prics and lins and across sasons. Th cass of hom and nutral longshots tams will suffic to illustrat th problm. Hom tams ar, on avrag, ovr-pricd in fixd odds btting rlativ to winning frquncis in 2001, corrctly pricd in 2002 and undr-pricd in 2003 and And yt for both 2001 and 2002, th lin on avrag undrstats th point sprad! Nutral longshots ar tams which ar longshots in gams whr nithr tam has any hom ground advantag. Thus, th rsults in this catgory show th prsnc or absnc of favorit/longshot biass, bing uncontaminatd by hom ground advantag considrations. Nutral longshots ar vidntly ovrpricd vis-àvis winning frquncis in all sasons, yt in th lin markt thy ar undrpricd during 2001, 2002 and ovr th whol priod. A priori, thn, thr sms much to xplain! But, as w show in th nxt sction, fw of ths contradictory ar statistically significant ithr undr conomtric tsting or whn attmpts ar mad to xploit thm in a btting systm sason, whn th Wstrn Bulldogs, a Mlbourn-basd tam, wr th official hom tam in thir gam against th Sydny Swans, whil th gam was playd in Sydny! 18 Although it should notd that not all data from prvious sasons or gams ar availabl on-lin today. 19 Rspons to a qury from on of th authors by on of Australia s lading bookmakrs, Sportsbt Pty Ltd. 20 S Golc and Tamarkin (1991). 21 Not that th succssful win rat is not dirctly corrlatd with nt rturns from btting, as th dividnd for a succssful bt diffrs across tams (this is mor rlvant for WIN as th dividnd changs dramatically within gams, as mntiond abov). 4

6 5. Masuring wak fficincy: Economtric tsts v. Btting systms 5.1 Economtric Tsts Th conomtric wak fficincy tsts ar not as asy to apply and intrprt as is gnrally implid in th litratur. Th main rason is that thy rais many subtl mthodological issus. First, th issu of a suitabl tam of rcord has alrady bn mntiond. Golc and Tamarkin (1991) us thr diffrnt dfinitions: favorits, hom tams and random slction. W dfin (1) OFFICIAL HOME, (2) HOME, (3) FAV, (4) HOMEFAV, (5) HOMELONG, and (6) NEUTRALLONG as binary variabls, qual to on if th rlvant tam (1) is dsignatd officially as th hom tam rgardlss of th ground at which th gam is playd, (2) is playing whr thy a priori hav a ral hom ground advantag, (3) is th favorit according to th bookmakrs in th fixd odds btting markt, (4) has an a priori ral hom ground advantag and is th favorit, (5) has a ral hom advantag and is th longshot in th fixd odds btting markt, and (6) is th longshot in a gam playd at a ground whr nithr tam has a ral hom ground advantag, and zro othrwis. As Golc and Tamarkin (1991) point out, if on is looking for a bias, thn choosing th tam of rcord according to HOME or FAV may intrfr with th rsults. As alrady notd, w hav a natural solution in our data st. Sinc, in contrast to most othr tam gams studid in th litratur, OFFICIAL HOME and HOME ar not idntical in th AFL, w opt for OFFICIAL HOME to dfin th tam of rcord and us HOME to tst for any biass in this dirction whnvr w do not brak th data into substs according to charactristics (2) through (6). Scond, what lvl of significanc is significant? i.., what lvl of significanc is rquird to rjct th null hypothsis of wak fficincy? Is it th usual fiv prcnt simply bcaus that is usual? Or is a markt only infficint if it prmits profitabl btting basd on past and prsnt prics or lins and hom ground information? As dmonstratd blow, thr ar diffrnt infrncs for th fiv prcnt and tn prcnt lvls of significanc. Third, ar F and χ 2 -tsts for joint tsts of hypothss on rgrssion cofficints compatibl with T and Z- tsts on th individual cofficints? If not, which ar th mor rliabl? Our tsts will prsnt incompatibilitis btwn ths two tsts. Bfor procding to list othr difficultis, it will prov usful to dfin som trms. Lt LINE dnot th point sprad in th bookmakr's btting lin and PS dnot th actual point sprad btwn th two tams (dfind in a way that is consistnt with th dfinition of LINE, i.., according to OFFICIAL HOME), and lt WIN dnot th actual winnr in th gam (dfind as a variabl qual to 1 for th winning tam, 0 for th losing tam and 0.5 for tis). Furthr, lt: PRICE = 1 (1 + odds) NPRICE = normalizd pric = PRICE pr _ gam PRICE If bttors us th availabl information fficintly, thn w would xpct th point sprad/fixd odds to b th bst unbiasd forcast of th gam's outcom. Lt i and j dnot two diffrnt tams playing in gam t. Thn, in gnral, th Efficint Markt Hypothsis rquirs that: (1a) (1b) Mdian [PS Ω t-1 ] = LINE Mdian [P(WIN) Ω t-1 ] = NPRICE whr Ω t-1 is th st of all information availabl to th bttor prior to th gam. Strn (1991) found that th distribution of th margin of victory ovr th point sprad (dfind as th numbr of points scord by th favorit minus th numbr of points scord by th longshot minus th point sprad) is not significantly diffrnt from th normal distribution. Thrfor th tru outcom of a gam can b modld as a normal random variabl with man qual to th point sprad, and quations (1a) and (1b) imply that: (2a) (2b) E t-1 [PS Ω t-1 ] = LINE E t-1 [P(WIN) Ω t-1 ] = NPRICE 5

7 Equations (1a-1b) and (2a-2b) rflct th most gnral dfinition of fficincy, and a varity of fficincy tsts hav bn prformd basd on quations (1a) and (2a), although quations (1b) and (2b) rmain untstd in th contxt of tam sports btting to th bst of our knowldg. A natural tst is basd on th information containd in th st Ω t-1. In gnral, Ω t-1 will contain th currnt lins and odds, past lins and odds, past outcoms, known gam conditions (.g., ground, hom tam), past gam statistics, othr public information (.g., injuris, rfrs), and privat information. For th tsts of wak-form markt fficincy with which w ar concrnd, Ω t-1 should snsu stricto contain only prics, but it is convntional to includ information rgarding hom ground advantag as wll. Th basic statistical tst of wak fficincy for th lin btting markt involvs stimating th following modl: (3a) PS = a 0 + a 1 LINE + ε whr a 0 is a constant and ε is an indpndntly and idntically distributd random rror. Support for th linar spcification in th spcific cas of Australian Ruls Football is providd by Baily and Clark s (2004) dmonstration that point sprads in AFL gams ar normally distributd. Thus, quation (3a) is stimatd using Ordinary Last Squars (OLS), as w tst th linar rlationship btwn point sprads and lins. Th paralll tst in th fixd odds btting markt is stimating P[WIN Ω t-1 ]. Th xpctd winning tam in a gam is th on with P>0.5, and this raiss th nxt difficulty: What is th propr functional spcification for this tst? It is wll known that OLS is not th bst choic of stimator for a probability modl. Among othr things, it maks no us of th fact that th fittd dpndant variabl rprsnts a probability and must b btwn zro and on and th probabilitis sum to on pr gam. But ths ar tchnical issus and th Linar Probability Modl (LPM) at last somtims provids a solution. 22 Th two stimation tchniqus commonly usd in rgrssions on dummy variabls ar Probit and Logit. But, as alrady notd, th win variabl is spcial bcaus it sums to on across tams pr gam. Furthr, owing to th possibility of drawn gams, it is not, strictly spaking, a dummy variabl. In th conditional logit rgrssion (CLOGIT) basd on McFaddn (1973), 23 ach gam is tratd as an indpndnt drawing from a multinomial distribution in which vry tam, i, has its associatd probability of winning, Pi. Thrfor, th probability of tam i winning tam j in round t is stimatd as: P = β xit βxit + β x jt whr β is a vctor of cofficints to b stimatd, and X it is a matrix of obsrvabl variabls of prformanc indics for tam i in gam t. Th P satisfy 0 P 1 and P + P jit = 1. Th stimats of β maximiz th stimatd liklihood of th occurrnc of th actual rsults of all gams: L = * t * 1 all _ gams _ t ( ) P i P kt whr i* is th indx of th winning tam and k is that of th losing tam, in ach gam. Thus, th statistical tst of wak fficincy for th fixd odds markt involvs stimating th following spcific modl: (3b) P = b1 ln NPRICEit b1 ln NPRICEit + b1 ln NPRICE jt ϕ whr ϕ is an indpndntly and idntically distributd random rror and w tst th null hypothsis that th cofficint of th log of th normalizd pric quals 1. It is vidnt from quation (3b), that for a unitary 22 Whn not too many obsrvations ar lost in corrcting for htroskdasticity. 23 S also Figlwski (1979) and Schnytzr and Shilony (1995). 6

8 cofficint, winning probability quals pric. Not, howvr, that this tchniqu involvs dropping drawn gams from th sampl vn if this is of limitd importanc in practic. But what if th rlationship btwn P(WIN) and PRICE is linar and not logistic? Mayb it is vn somthing ntirly diffrnt? Sinc thr is no thory to guid us, w tst using th LPM as wll, and chck for consistncy with th CLOGIT rsults. Th LPM is rprsntd by: (3c) WIN = c0 + c1nprice + ς Wak-form fficincy corrsponds to th joint hypothss with rspct to quations (3a), (3b) and (3c) that a 0 =0 and a 1 =1, b 1 =1, and c 0 =0 and c 1 =1, rspctivly. Th rsults ar prsntd in Tabl 2 and dmonstrat th consistncy btwn th diffrnt markts and modls; w can not rjct th wak fficincy hypothsis for any of th modls during any of th yars at a tn prcnt lvl of significanc. Howvr, this is not th cas at a fiv prcnt lvl of significanc. Th xtrm altrnativ hypothsis, which in trms of quations (3a), (3b) and (3c), is th joint tst that a 0 =a 1 =0, b 1 =0, c 0 =c 1 =0, is always rjctd, th F and χ 2- tsts statistics bing wll abov th critical valu at any maningful lvl of significanc. But what dos th consistncy of rsults as btwn th LPM and CLOGIT spcifications man? As th following analysis will show, both functional spcifications hav important variabls missing. Onc ths ar addd, diffrncs bgin to appar. Golc and Tamarkin (1991) argud that it is advisabl to tst simultanously for numrous spcific biass by adding dummy variabls to quations such as (3a), (3b) and (3c). Th following quations purportdly idntify both favorit and hom tam biass 24 : (4a) (4b) (4c) PS = a 0 + a 1 LINE + a 2 HOME + a 3 FAV + ζ b NPRICE b HOME b FAV = 1 ln it + 2 it + 3 it b NPRICE b HOME b FAV b NPRICE jt + b HOME jt b FAV 1 ln it + 2 it + 3 it 1 ln jt P WIN + + c HOME = c0 + c1nprice 2 + c3 FAV + ς Ψ Now tsts of fficincy for quations (4a), (4b) and (4c) ar tsts of th joint null hypothss, a 1 =1,a 0 =a 2 =a 3 =0, b 1 =1,b 2 =b 3 =0 and c 1 =1,c 0 =c 2 =c 3 =0, rspctivly. Th intrcpts, a 0 and c 0, so th argumnt runs, masur any bias that xists with rspct to a visiting longshot, and a 2,b 2,c 2 and a 3,b 3,c 3 masur th hom and favorit tam biass, rspctivly. Excluding ithr HOME or FAV lads to bias in th rgrssion cofficints and rducd powr against th null. Golc and Tamarkin (1991) not furthr that quations such as (4a), (4b) and (4c) statistically isolat particular biass, although biass othr than hom and favorit tam could offst ach othr and go unmasurd, so that fficincy tsts may not b powrful with rspct to unspcifid biass. Thy tst fficincy using th thr diffrnt tams of rcord notd abov, yt thy ignor possibl intractions btwn thm. Thus, thir cofficints ar quit possibly biasd and inconsistnt, sinc hom ground advantag and favoritism ar corrlatd and thus th intractions btwn thm and with th lin cannot b xcludd without prior tsting. Accordingly, w gnraliz thir argumnt and includ all th rlvant dummy variabls and intractions in on rgrssion. Th rsults ar prsntd in Tabl 3 and thy ar bst by multicollinarity. In non of th scnarios ar w abl to rjct wak fficincy. W rjct th xtrm altrnativ hypothsis that LINE, lnnprice (and NPRICE) ar unrlatd to th actual point sprad and winnr, rspctivly. Nonthlss, th rsults in Tabl 3 do not prmit spcific infficincis to b radily idntifid sinc almost no cofficint is individually significant. Our suggstd solution to th multicollinarity problm is to divid th data into substs and tst for all possibl biass via sparat rgrssions. Thr ar tn such possibl biass: Vis-à-vis (a) hom tams, (b) favorits, (c) hom favorits, (d) hom longshots, () nutral longshots, (f) away tams, (g) longshots, (h) away favorits, (i) away longshots, and (j) nutral favorits. Sinc th last fiv ar th symmtric with rspct to th first fiv, w us th first fiv substs as th basis for conomtric tsting. W again us OLS for th lin btting markt and CLOGIT and LPM for th fixd odds markt. Rsults for lin and fixd odds markts ar rportd in Tabl 4a and Tabl 4b, rspctivly. 24 Not that Golc and Tamarkin (1991) study th lin markt only, but thir argumnt gnralizs to th fixd odds markt. 7

9 Our rsults imply that th AFL gambling markt is not consistntly wak-form fficint. W rjct th fficincy hypothsis for hom tams (in 2002 LPM, OLS and CLOGIT), hom longshots ( CLOGIT) and hom favorits (in 2002 LPM and CLOGIT, OLS, LPM and CLOGIT). On th othr hand, w rjct th xtrm altrnativ hypothss that th lin is unrlatd to th actual point sprad as th F-tst statistics for th hypothss that a 0 =a 1 =0 and c 0 =c 1 =c 2 =c 3 =0 ar wll abov th critical valu at any maningful lvl of significanc, and that th normalizd pric is unrlatd to th winning probability, with th χ 2 -tst statistics for th hypothss that b 1 =0 bing wll abov th critical valu at any maningful lvl of significanc). W xamin th nutral catgory, as it provids us with th only tst for a pur favorit-longshot bias, uncontaminatd by hom ground considrations, sinc all othr rgrssions confound this qustion. And indd, th catgory rsults imply wak-form fficincy for all modls during all yars. Accordingly, w may conclud that any biass in ths markts rlat to th prsnc of an apparnt hom ground advantag. W rturn to this issu whn rporting th rsults of our btting simulations. Our rsults also illustrat svral of th abov-mntiond problms. Thus, thr ar contradictions btwn F and χ 2 -tsts, on th on hand, and T and Z-tsts on th othr, in diffrnt scnarios. For xampl, in crtain scnarios [hom tams in 2002 in Tabl 4b(A), and nutral longshots ovr th ntir priod in Tabl 4b(B)] th F-tst implis wak fficincy, yt th significanc of othr cofficints implis a bias. In othr scnarios [most of Tabl 2, 2001, 2002, 2003 and 2004 hom longshots in Tabl 4a, 2002 favorits in Tabl 4b(A), 2001, 2002, 2003 hom longshots, 2002 favorits and 2002 nutral longshots in Tabl 4b(B)] LINE, NPRICE or lnnprice do not diffr significantly from zro, yt th F-tst fails to rjct fficincy. Th issu of th "right" lvl of significanc ariss as wll, sinc at th tn prcnt lvl thr is fficincy in all scnarios, yt at fiv prcnt thr appar to b infficincis. Th btting simulations will dmonstrat that th lvl of significanc chosn dos not maningfully prdict profitability in th fixd odds markt. 5.2 Btting systms To confirm th prsnc of th abov-notd infficincis or othrwis, fiv simpl btting systms wr constructd, both for LINE and WIN btting: (a) btting on all hom tams, (b) btting on all favorits, (c) btting on all hom longshots, (d) btting on all hom favorits, and () btting on all nutral longshots. Th simulation rsults ar prsntd in Tabl Following Tryfos t al (1984) and Gandar t al (1988), w tst th significanc of th rsults in two ways; first, a Z-tst for th null hypothsis that th succssful bt rats (s Tabl 1) ar random (th assumption bing that chanc yilds a fifty prcnt succss rat). Th scond and mor stringnt tst, is a Z-tst for th null hypothsis that a givn scnario is unprofitabl against th altrnativ that it is profitabl. If w insistnt stringntly, upon a rjction of th two null hypothss, ach at a fiv prcnt lvl of significanc, thr wr no profitabl btting systms on th lin markt, xcpt for longshots playing on nutral grounds in 2001, but thr wr significant profits btting on hom tams for th win in 2002, 2003, 2004 and th whol priod (th null hypothsis of no profits bing rjctd by both tsts at bttr than 1 in a 500). 26 Btting on hom favorits in 2002 and ovr th whol priod is also significantly profitabl. Othr profits in th tabl fail to pass our stringnt tst. Comparing th btting simulations with th conomtric tsts, w s that on all but two occasions, th conomtric tsts indicat corrctly that th lin markt is fficint (at fiv prcnt significanc), whil mistaknly not rjcting th null hypothsis of fficincy for longshots playing on nutral grounds in In all but this cas, th conomtric tsts coincid fully with th simulations at tn prcnt. Givn that linar spcification for th rgrssions of point sprads on lins is uncontrovrsial, this rlativ consnsus btwn th two approachs is hardly surprising, but th failur to pick up th rvrs favorit/longshot bias in 2001 is puzzling. Mattrs ar lss parsimonious in th comparison btwn conomtrics and simulation in th fixd odds markt. Hr th conomtric tsts, for both spcifications, corrctly 27 rjct fficincy at fiv prcnt significanc for th 2002 and hom favorits, but fail badly to find th mor significant profits to 25 W prsnt only ths fiv catgoris, as othr possibl groups, including longshots, away tams and away longshots yildd consistnt losss in both markts, whil away favorits yildd a minimal profit only in th 2001 LINE markt, and nutral favorits yildd rgular losss. 26 Th quivalnt rsults of Tryfos t al (1984) and Gandar t al (1988) wr 3 profitabl scnarios out of 70, and 0 out of 14, rspctivly. 27 In trms of significant simulatd profits, of cours. 8

10 b had backing hom tams in ach of 2002, 2003 and 2004, and for th four sasons as a whol. Th LPM finds hom tams infficint only in 2002 and th CLOGIT rjcts fficincy only for th four sasons as a whol. In ordr to provid a possibl answr to this puzzl, w turn to a considration of th rlationship btwn bookmakrs lins and thir prics. 6. Th rlationship btwn lin and fixd odds Having dmonstratd th prsnc of a hom tam bias in th win markt alongsid almost ubiquitous fficincy in th lin markt, it is ncssary to ask what it is that accounts for this diffrnc across th two markts. On possibility is that bookmakrs assum an incorrct functional rlationship btwn lins and prics whn stting lins and odds. Th rlationship usd by bookmakrs in our sampl is prsntd in Tabl 6, and it is vidntly linar btwn point sprads and prics. 28 Nonthlss, th corrct thortical rlationship btwn lin and fixd odds is probably not linar. And if th bookmakrs ar mistakn in th linarity assumption, that could radily account for th infficincy of on markt rlativ to th scond. 7. Conclusions In this papr w hav tstd th wak-form fficincy of th lin and fixd odds btting markts for th Australian Football Lagu ovr th four sasons, 2001 through W hav shown that th null hypothsis of fficincy cannot b rjctd for th lin markt at th fiv prcnt lvl of significanc ithr via xhaustiv conomtric tsting or via btting simulation for ach sason individually (xcpt in th cas of nutral longshots in 2001 in th btting simulation) and that, in spit of som apparnt infficincis ovr th four sasons as a whol, thr ar no significant profits to b mad. On th othr hand, th fixd odds btting markt is vidntly bst by a bias that undrprics tams with an a priori hom ground advantag, as in thr of th four sasons statistically significant profits ar mad in th btting simulations. Economtric tsts provid hints of this bias but do not accord compltly with th simulation rsults and ar not consistnt as btwn th linar probability and conditional logit modls. Furthr, at th tn prcnt lvl of significanc, all conomtric tsts imply markt fficincy. W suggst that th major problm with conomtric tsting in th fixd odds markt is on of functional spcification. Without a clar undrstanding of th rlationship btwn winning probabilitis and prics and without a suitabl conomtric tchniqu capabl of handling th rlationship, conomtric tsts of fficincy will rmain of dubious valu. By taking advantag of th fact that many gams in th AFL ar playd on nutral grounds, w ar abl to rjct th xistnc of any significant pur favorit-longshot bias in ithr markt for th ach of th four sasons individually and for th priod as a whol, xcpt for a rvrs bias in th lin markt in 2001 as rflctd by significant profits obtaind by backing longshots playing on grounds whr nithr tam has a hom ground advantag. W suggst that th infficincy in th fixd odds markt rlativ to th lin markt may b du to a mistakn assumption of a linar rlationship btwn prics and lins on th part of bookmakrs, but this is a subjct rquiring furthr rsarch. Rfrncs Amoako-Adu, B., Marmr, H., and Yagil, J Th fficincy of crtain spculativ markts and gamblr bhavioral. Journal of Economics and Businss 37: Baily, M.J. and Clark, S.R Driving profit from Australian ruls football: a statistical approach. In: H. Morton (d). Procdings of th svnth Australian confrnc on mathmatics and computrs in sport. Massy Univrsity: Palmrston Nth. Basstt, G.W., Jr Points sprads vrsus odds. Journal of Political Economy 89: Brailsford, T.J., Easton, S.A., Gray, P.K., and Gray, S.F Th fficincy of Australian football btting markts. Australian Journal of Managmnt 20: Clark, S.R Hom ground advantag in th Australian football lagu. Journal of Sports Scincs (forthcoming). Dana, J.D. and Knttr, M.M Larning and fficincy in a gambling markt. Managmnt Scinc 40: Dar, W.H. and Holland, S.A Efficincy in th NFL btting markt: modifying and consolidating rsarch mthods. Applid Economics 36: This linarity implis that th bookmakrs think in trms of prics and not odds, vn though thy quot odds! 9

11 Dobra, J.L., Cargill, T.F., and Myr, R.A Efficint markts for wagrs: th cas of profssional basktball wagring. In: Goff and Tollison (d.) Sportomtrics. Txas: Collg Station. Fama, E.F Efficint capital markts: a rviw of thory and mpirical work. Journal of Financ 25: Figlwski, S Subjctiv information and markt fficincy in a btting markt. Journal of Political Economy 87: Gandar, J., Zubr, R., O'Brin, T., and Russo, B. (1988). Tsting rationality in th point sprad btting markt. Journal of Financ 43: Golc, J. and Tamarkin, M Th dgr of infficincy in th football btting markt: statistical tsts. Journal of Financial Economics 30: Lvitt, S.D Why ar gambling markts organizd diffrntly from financial markts? Economic Journal 114: McFaddn, D Conditional logit analysis of qualitativ choic bhaviour. In: Zarmbka, P. (d.) Frontirs in conomtrics. Nw York: Acadmic Prss. Osborn, E Efficint markts? Don't bt on it. Journal of Sports Economics 2: Pankoff, L.D Markt fficincy and football btting. Journal of Businss 41: Saur, R.D Th conomics of wagring markts. Journal of Economic Litratur 36: Saur, R.D., Brajr, V., Frris, S.P., and Marr, M.W Hold your bts: anothr look at th fficincy of th gambling markt for national football lagu gams. Journal of Political Economy 96: Schnytzr, A. and Shilony, Y Insid information in a btting markt. Economic Journal 105: Schnytzr, A. and Winbrg, G Is th NBA btting markt fficint? In: Papanikos, G.T. (d.) Th conomics and managmnt of mga athltic vnts: Olympic gams, profssional sports, and othr ssays. Athns: ATINER. Stfani, R. and Clark, S Prdictions and hom advantag for Australian ruls football. Journal of Applid Statistics 19: Strn, H On th probability of winning a football gam. Amrican Statistician 45: Summrs, L Dos th stock markt rationally rflct fundamntal valus? Journal of Financ 41: Thalr, R. and Zimba, W Parimutul btting markts: ractracks and lottris. Journal of Economic Prspctivs 2: Tryfos, P., Casy, S., Cook, S., Lgr, G. and Pylypiak, B Th profitability of wagring on NFL gams. Managmnt Scinc 30: Vrgin, R. and Scriabin, M Winning stratgis for wagring on national football lagu gams. Managmnt Scinc 24: Woodland, B.M. and Woodland, L.M Th ffcts of risk avrsion on wagring: point sprad vrsus odds. Journal of Political Economy 99: Woodland, B.M. and Woodland, L.M Markt fficincy and th favorit-longshot bias: Th basball btting markt. Journal of Financ 49: Zubr, R.A., Gandar, J.M., and Bowrs, B.D Bating th sprad: tsting th fficincy of th gambling markt for national football lagu gams. Journal of Political Economy 93:

12 Tabl 1 Basic proprtis of th AFL data Basic proprtis of complt AFL sasons for diffrnt substs of th data: hom tams, favorits, hom longshots hom favorits, and nutral longshots. 29 HOME FAVORITES HOME LONGSHOTS HOME FAVORITES NEUTRAL LONGSHOTS Yar No. of WIN obsrvations Succssful WIN bts rat (%) Avrag PRICE 30 No. of LINE obsrvations Succssful LINE bts rat (%) Avrag Point sprad Avrag LINE Throughout this papr w rport rsults for hom and away plus finals gams, sinc th rsults xcluding finals gams wr vry similar. 30 In this Tabl, and for th rst of th papr, pric of a bt is dfind as th rciprocal of th payout contingnt upon winning. Thus, a pric of 0.2 is (th probability) quivalnt of odds of 4 to 1 and a payout for $1 of $5. 11

13 Tabl 2 Wak fficincy stimats for th AFL Wak fficincy stimats for AFL sasons. Each gam in modl (2) is rprsntd by two obsrvations, on for ach tam, and ach gam in modls (1) and (3) by on obsrvation th official hom tam. T/Z valus in parnthss. Joint hypothss ar tstd at th 5% significanc lvl. (1) PS = a 0 + a 1 LINE + ε P = WIN b1 ln NPRICEit (2) b NPRICE b ln NPRICE 1 ln it 1 jt + (3) = c0 + c1nprice + ς ϕ (1) PS markt OLS CONS (-0.73) 7.37* (2.58) (0.42) 4.64 (1.49) (1.89) LINE 1.083* (8.31) 0.792* (5.85) 1.147* (8.77) 1.058* (7.85) 1.022* (15.3) F (CONS=0, LINE=1) 0.35** 3.59# 1.03** 1.65** 2.33** Adj. R No. of obs (2) WIN markt CLOGIT lnnprice 1.111* (5.14) 1.212* (5.26) 1.315* (5.73) 1.166* (5.6) 1.199* (10.88) χ 2 (lnnprice=1) 0.26*** 0.85*** 1.88*** 0.63*** 3.25*** Log liklihood No. of obs (3) WIN markt LPM CONS (-1.07) 0.1 (0.89) (-0.18) (0.68) 0131 (0.23) NPRICE 1.187* (5.48) 1.014* (4.89) 1.132* (5.63) 0.984* (5.06) 1.067* (10.51) F (CONS=0, 0.72** 4.81# 1.15** 1.77** 3.98# NPRICE=1) Adj. R No. of obs * Significant at 5% lvl. # Significant at 10% lvl. ** CONS significantly quals zro and LINE or NPRICE significantly quals on at 5% lvl (F 2,171 =3.066, F 2,652 =3.028). *** lnnprice significantly quals on at 5% lvl (χ 2 (1) =3.841). 12

14 Tabl 3 Wak fficincy stimats for th AFL dummis and all intractions Wak fficincy stimats for AFL sasons. Each gam in modl (2) is rprsntd by two obsrvations, on for ach tam, and ach gam in modls (1) and (3) by on obsrvation th official hom tam (which is our tam of rcord). T/Z valus in parnthss. Th fficincy tsts ar at th 5% significanc lvl. 7 (1) PS = a0 + anb' + ς n n= 1 P = 7 bnc' it n= 1 (2) 7 7 bnc' it bnc' jt n= 1 n= 1 + Ψ 7 (3) WIN = c0 + cnd' + ς n n= 1 whr B', C' and D' ar matrixs of obsrvabl variabls, including M, FAV, HOME, W, X, Y and Z. W = HOME *M, X =HOME *FAV, Y = HOME *M *FAV, Z =M *FAV. M quals (1) LINE, (2) lnnprice or (3) NPRICE, rspctivly (1) PS markt OLS CONS (-0.94) (0.02) (0.13) (0.17) (-0.2) LINE (0.81) (0.36) 1.76 (1.89) (1.94) 0.984* (2.8) FAV (-0.37) (0.37) (-0.46) (0.46) (-0.16) HOME (0.44) (-0.32) (0.08) (0.57) 3 (0.32) W (0.27) (-0.14) (-0.64) (-0.04) (-0.47) X (0.51) (0.3) (0.65) (-1.05) 3.74 (0.3) Y (-0.15) (0.03) (-0.2) (0.63) 0.18 (0.3) Z (0.62) (0.5) (-0.05) (-0.64) (0.24) F (CONS=FAV= = 1.36** 1.16** 0.85** 0.78** 1.2** Z=0, LINE=1) Adj. R No. of obs (2) WIN markt - CLOGIT lnnprice (-0.21) (0.56) (0.82) (0.47) (0.7) FAV (0.63) (-0.47) -1.6 (-0.3) (-0.1) (0.02) HOME (-1.43) (0.75) (-0.57) (1.27) (-0.08) W (-1.25) (0.59) (-0.83) (0.95) (-0.38) X 2.913* (2.05) 1.26 (0.78) (0.81) (-0.28) (1.92) Y 3.88 (1.65) (1.13) (0.78) (0.83) 2.785* (2.32) Z (0.62) (-0.6) (-0.27) (-0.19) (-0.07) χ 2 (FAV= =Z=0, 4.75*** 10.7*** 4.3*** 4.9*** 12.07*** lnnprice=1) Log liklihood No. of obs

15 Tabl 3 (continud) (3) WIN markt LPM CONS (-0.5) 0.848* (2.36) (0.1) (1.73) (1.6) NPRICE (1.12) (-1.16) (0.47) (-0.61) (0.26) FAV 0.24 (0.36) 0.02 (0.03) (-0.46) (-0.94) (-0.62) HOME (1.15) (-1.39) (0.28) (-1.28) (-0.22) W (-1.01) (1.38) (0) (1.71) (0.74) X (-1.23) (-0.03) (-0.21) 0.34 (0.39) (-0.64) Y (1.31) (-0.4) (0.07) -1.7 (-1.04) (0.12) Z (-0.28) (0.8) 0.86 (0.55) (1.45) (1.36) F (CONS=FAV= = 1.06** 2.32# 0.59** 1.05** 1.96** Z=0, NPRICE=1) Adj. R No. of obs * Significant at 5% lvl. # Significant at 10% lvl. ** CONS, FAV,,Z significantly quals zro and LINE or NPRICE significantly quals on at 5% lvl (F 8,165 =2.02, F 8,720 =1.97). *** FAV,,Z significantly quals zro and lnnprice significantly quals on at 5% lvl (χ 2 (7) =14.067). 14

16 Tabl 4a Wak fficincy stimats for th AFL lin markt Wak fficincy stimats for AFL sasons. Each gam is rprsntd by on obsrvation and i rprsnts (1) hom tams, (2) favorits, (3) hom favorits, (4) hom longshots, and (5) nutral longshots, in turn. T-valus ar in parnthss. PS = a 0 + a 1 LINE + ε (1) Hom tams CONS 1.76 (0.41) (1.94) (1.35) (1.62) 5.149* (2.67) LINE 1.157* (6.98) 0.797* (4.73) 0.985* (6.77) 0.991* (5.54) 0.989* (12.03) F (CONS=0, 0.92** 2** 1.08** 1.67** 4.3 LINE=1) Adj. R No. of obs (2) Favorits CONS (-0.59) (0.77) (0.44) (-0.83) (-0.23) LINE 1.18* (4.69) 0.737* (2.83) 1.067* (4.1) 1.306* (5.26) 1.092* (8.621) F (CONS=0, 0.26** 0.54** 1.04** 0.86** 0.58** LINE=1) Adj. R No. of obs (3) Hom favorits CONS (0.06) (1.09) (1.34) (-0.4) (0.83) LINE 1.212* (3.5) 0.744* (1.95) 0.714* (2.09) 1.37* (4.05) 1.059* (6.12) F (CONS=0, 1.03** 0.78** 1.32** 1.31** 3.08 LINE=1) Adj. R No. of obs (4) Hom longshots CONS (-0.36) (-0.32) (0.31) (1.07) (0.41) LINE (1.06) (0.45) (1.67) (1.55) 0.818* (2.45) F (CONS=0, 0.06** 1.83** 0.15** 1.11** 1.54** LINE=1) Adj. R No. of obs (5) Nutral longshots CONS (0.79) (-0.02) (0.48) (-0.56) (0.21) LINE 0.963* (2.06) 0.851# (1.81) 1.734* (3.03) 0.925* (2.2) 1.047* (4.44) F (CONS=0, 1.46** 0.18** 1.57** 0.32** 0.02** LINE=1) Adj. R No. of obs * Significant at 5% lvl. # Significant at 10% lvl. ** CONS significantly quals zro and LINE significantly quals on at 5% lvl (F 2,30 =3.32, F 2,106 =3.09, F 2,650 =3.03). 15

17 Tabl 4b Wak fficincy stimats for th AFL fixd odds markt Wak fficincy stimats for AFL sasons in two modls: A. CLOGIT and B. LPM. Y rprsnts (1) hom tams, (2) favorits, (3) hom favorits, (4) hom longshots, and (5) nutral longshots. Z valus in parnthss. b1 ln NPRICEit + b2yit P = ϕ A. b NPRICE b Y b NPRICE jt b Y jt 1 ln it + 2 it 1 ln B. WIN = c 0 + c 1 NPRICE + ε A. C L O G I T (1) Hom tams lnnprice 1.144* (4.96) 1.082* (4.58) 1.242* (5.25) 1.085* (5.02) 1.13* (9.89) HOME (-0.43) 0.457* (2.19) (1.13) (1.29) 0.22* (2.09) χ 2 (lnnprice=1, HOME=0) 0.44** 5.42** 3.03** 2.22** 7.39 Log liklihood No. of obs (2) Favorits lnnprice 1.089* (3.01) (1.62) 1.485* (3.87) 1.332* (3.65) 1.137* (6.15) FAV (0.07) 0.543# (1.86) (-0.56) (-0.56) (0.41) χ ** 4.25** 2.12** 0.92** 3.45** (lnnprice=1, FAV=0) Log liklihood No. of obs (3) Hom favorits lnnprice 1.098* (3.98) 0.767* (2.83) 1.216* (4.27) 1.092* (4.23) 1.041* (7.68) HOMEFAV (0.07) 0.873* (2.6) (0.56) (0.47) 0.304# (1.89) χ ** ** 0.85** 6.66 (lnnprice=1, HOMEFAV=0) Log liklihood No. of obs (4) Hom longshots lnnprice 1.103* (4.87) 1.3* (5.18) 1.457* (5.7) 1.297* (5.68) 1.283* (10.74) HOMELONG (-0.11) (1) (1.55) 0.671# (1.85) 0.397* (2.15) χ ** 1.8** 4.02** 3.94** 7.58 (lnnprice=1, HOMELONG=0) Log liklihood No. of obs (5) Nutral longshots lnnprice 1.119* (4.69) 1.181* (4.69) 1.31* (5.28) 1.108* (4.79) 1.179* (9.75) NEUTRALLONG (0.08) (-0.3) (-0.05) (-0.53) (-0.39) χ ** 0.94** 1.89** 0.91** 3.41** (lnnprice=1, NEUTRALLONG=0) Log liklihood No. of obs

18 Tabl 4b (continud) B. L P M (1) Hom tams CONS (-0.97) (0.12) (0.24) (0.43) (0.08) NPRICE 1.224* (4.54) 1.154* (4.5) 1.038* (4.45) 0.999* (3.97) 1.074* (8.56) F (CONS=0, 0.54## ## 1.1## 2.77## NPRICE=1) Adj. R No. of obs (2) Favorits CONS (-0.17) (1.62) (-0.72) (-0.65) (-0.01) NPRICE 1.087* (3) (1.46) 1.287* (3.57) 1.25* (3.41) 1.045* (5.83) F (CONS=0, 0.13## 2.84## 0.43## 0.25## 1.33## NPRICE=1) Adj. R No. of obs (3) Hom favorits CONS (-1.1) (0.72) (-1.04) (-1.07) (-1.31) NPRICE 1.622* (2.92) 0.846# (1.68) 1.581* (3.29) 1.632* (3.03) 1.432* (5.5) F (CONS=0, 0.63## ## 0.92## 4.25 NPRICE=1) Adj. R No. of obs (4) Hom longshots CONS (0.92) (0.75) (0.49) (-0.47) (0.82) NPRICE (0.14) (0.57) 0.79 (0.95) 1.838# (1.86) 0.793# (1.84) F (CONS=0, 0.56## 0.41## 0.48## 0.28## 1.18## NPRICE=1) Adj. R No. of obs (5) Nutral longshots CONS (0.06) (1.64) (-0.36) (0.86) 0.612* (1.94) NPRICE (1.53) (-0.28) (1.52) 0.32 (0.56) 0.612# (1.94) F (CONS=0, 0.02## 1.89## 0.12## 1.13## 1.54## NPRICE=1) Adj. R No. of obs * Significant at 5% lvl. # Significant at 10% lvl. ** lnnprice significantly quals on and HOME / FAV / HOMEFAV / HOMELONG / NEUTRALLONG significantly quals zro at 5% lvl (χ 2 (2) =5.991). *** lnnprice significantly quals on at 5% lvl (χ2 (1) =3.841). ## CONS significantly quals zro and NPRICE significantly quals on at 5% lvl (F 2,368 =3.05, F 2,1478 =3). 17

19 HOME FAV HOME LONG HOME FAV NEUT LONG Tabl 5 Wak fficincy btting simulation Btting simulation rsults for AFL gams playd during , for hom tams, favorits, hom longshots, hom favorits, and nutral longshots. Assum $1 is bt on ach gam that mts th abov-mntiond critrions. Th fficincy is at 5% significanc lvl. 31 Yar No. of WIN gams No. of WIN bts WIN wak fficincy CLOGIT WIN wak fficincy LPM WIN rturn rat S1* S2* No. of LINE gams No. of LINE bts LINE wak fficincy LINE rturn rat S1* S2* Ys No Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys No No Ys Ys Ys Ys Ys Ys Ys No Ys No Ys Ys Ys Ys Ys Ys Ys Ys Ys No Ys Ys Ys Ys Ys No No No Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys Ys * S1 and S2 ar, rspctivly, th significanc lvls for th null hypothss of randomnss and non-profitability. Th rlvant tst statistics ar givn by Z 1 and Z 2, rspctivly: 0. 5 Z 1 = [ W 0.5B] *[ B( p)(1 p) ] [ D( W / B) ( L / B) ], Z 2 = 2 2 [ 1/ B{ ( D ( W / B) + ( L / B) ) ( D( W / B) ( L / B) ) }] 0. 5 whr W, L and B ar, rspctivly, th numbr of wins, losss and total bts for a givn scnario, and p=0.5. D is th man dividnd pr scnario Evrything is fficint at 10% significanc lvl. 32 This diffrs from Tryfos, Casy, Cook, Lgr and Pylypiak (1984) and Gandar t al (1988), as thy tstd th NFL lin btting markt whr th winning odds ar fixd at 10 to

20 Tabl 6 Th rlationship btwn point sprads and prics for th AFL Rlationship btwn point sprad and V, which rprsnts (1) PRICE and (2) NPRICE for AFL sasons. Each gam is rprsntd by two obsrvations, on for ach tam. T valus in parnthss. PS = a 0 + a 1 V + Ξ (1) PRICE CONS * (-95.6) * ( ) * (-136.1) * ( ) * (-196.7) PRICE * (103.25) * (150.27) * (147.98) * (150.54) * (212.61) Adj. R No. of obs (2) NPRICE CONS * (-96.35) * ( ) * ( ) * ( ) * ( ) NPRICE * (103.96) * (151.6) * (149.38) * (161.83) 109.7* (249.95) Adj. R No. of obs * Significant at all convntional lvls. 19