Modelling football match results and the efficiency of fixed-odds betting

Transcription

1 Moelling football match results an the efficiency of fixe-os betting John Goar * University of Wales Swansea Ioannis simakopoulos University of Wales Bangor * Corresponing author ress: Department of Economics University of Wales Swansea Singleton Park Swansea S2 8PP UK Tel: +44 (0) Fax: +44 (0) [email protected] Key wors: Sports betting, efficiency, orere probit bstract n orere probit regression moel estimate using 15 years ata is use to moel English league football match results. s well as past match results ata, the significance of the match for en-ofseason league outcomes; the involvement of the teams in cup competition; the geographical istance between the two teams home towns; an the average attenances of the two teams all contribute to the moel s performance. The moel is use to test the weak-form efficiency of prices in the fixeos betting market, an betting strategies with a positive expecte return are ientifie.

2 Moelling football match results an the efficiency of fixe-os betting 1. Introuction The preictability of match results is the main concern of research on the efficiency of sports betting markets. The recent applie statistics literature has focusse primarily on moelling goal scoring (Dixon an Coles, 1997; Rue an Salvesen, 2000; Crower et al., 2002). Recently some econometricians have suggeste moelling match results irectly (rather than inirectly through scores) using iscrete choice regression moels (Forrest an Simmons, 2000a,b; Koning, 2000; Kuypers, 2000; Dobson an Goar, 2001). focus on match results rather than scores can be justifie partly on grouns of simplicity: fewer parameters are require; estimation proceures are simpler; an the resulting moels len themselves to the inclusion of a variety explanatory variables. This paper presents a regression-base moel to explain an preict match results that is more extensive an comprehensive than any previously evelope, as regars both the range of explanatory variables that are incorporate, an the extent of the ata set use to estimate the moel. Research into the efficiency of prices set by bookmakers in betting markets has provie a small but increasing contribution to the financial economics literature on market efficiency. Much of this literature focuses on racetrack betting, but betting on team sports match results has also attracte attention. Early researchers sought evience of inefficiencies in the form of systematic biases in bookmakers prices, such as home-away team or favourite-longshot biases. More recently forecasting moels have been use to establish whether historical information available in previous match results can be extrapolate to formulate profitable betting strategies. This paper uses the moel outline above to test the weak-form efficiency of the prices quote by high street bookmakers, an to ientify potentially profitable betting strategies. 1

3 The paper is structure as follows. Section 2 reviews the literature on moelling an forecasting football match results. Section 3 escribes the specification an estimation of an orere probit regression moel to explain an preict match results. Section 4 reviews the literature on betting market efficiency. Section 5 investigates the efficiency of the prices quote by high street bookmakers over four English football seasons from to , using both regression-base tests an irect economic tests of the profitability of betting strategies base on the moel s evaluation of expecte returns. Section 6 conclues. 2. Moelling football match results limite but increasing number of acaemic researchers have attempte to moel match results ata for football. Early contributions by Moroney (1956) an Reep et al. (1971) use the poisson an negative binomial istributions to moel at an aggregate level the istributions of the numbers of goals score per game. This aggregate approach preclues the generation of specific forecasts for iniviual matches base on information about the respective strengths of the two teams. By comparing final league placings with experts pre-season forecasts, however, ill (1974) emonstrates that iniviual match results o have a preictable element, an are not etermine solely by chance. Maher (1982) evelops a moel in which the home an away team scores follow inepenent poisson istributions, with means reflecting the attacking an efensive capabilities of the two teams. full set of attacking parameters an a set of efensive parameters for each team are estimate ex post, but the moel oes not preict scores or results ex ante. tenency to unerestimate the proportion of raws is attribute to interepenence between the home an away scores, an correcte using the bivariate poisson istribution to moel scores. Dixon an Coles (1997) evelop a forecasting moel capable of generating ex ante probabilities for scores an match outcomes. The home an away team scores follow inepenent poisson 2

4 istributions, but for low-scoring matches an a hoc ajustment allows for interepenence. Using a similar framework, Rue an Salvesen (2000) assume that the time-varying attacking an efensive parameters of all teams vary ranomly over time. The prior estimates of these parameters are upate as new match results information is obtaine. Markov chain Monte Carlo iterative simulation techniques are use for inference. Crower et al. (2002) evelop a proceure for upating the team strength parameters that is computationally less emaning. Researchers who have examine the impact of specific factors on match results inclue Barnett an ilitch (1993), who investigate whether artificial playing surfaces, use by several clubs uring the 1980s an early 1990s, conferre an aitional home-team avantage. Rier et al. (1994) show that player ismissals have a negative effect on the match result for the teams concerne. Clarke an Norman (1995) use a range of non-parametric techniques to ientify the effect of home avantage on match results. Dixon an Robinson (1998) investigate variations in the scoring rates of the home an away teams uring the course of a match. The scoring rates at any time epen partly upon the number of minutes elapse, but also upon which (if either) team is leaing at the time. Recently, several researchers have use iscrete choice regression moels to moel match results irectly, rather than inirectly through scores. part from its computational simplicity, a major avantage of this approach is its avoiance of the thorny problem of interepenence between the home an away team scores. 1 Forrest an Simmons (2000a,b) investigate the preictive quality of newspaper tipsters' match results forecasts, an the performance of the pools panel in proviing hypothetical results for matches that were postpone. Koning (2000) estimates a moel to escribe a set of match results ex post, as part of a broaer analysis of changes in competitive balance in Dutch 1 If scores are the focus, the ifference between a 0-0 raw an a 1-0 home win is the same as the ifference between a 1-0 an a 2-0 home win, or between a 2-0 an a 3-0 home win. If results are the focus, there is a large ifference between a 0-0 raw an a 1-0 win, but no ifference between home wins of ifferent magnitues. What is crucial is which (if either) team won; the precise numbers of goals score an concee by either team are inciental. 3

5 football. Kuypers (2000) uses a variety of explanatory variables rawn from current-season match results to estimate an ex ante forecasting moel n orere probit regression moel for match results In Section 3, orere probit regression is use to moel an preict football match results. 3 The result of the match between teams i an j, enote y, epens on the unobserve variable inepenent an ientically istribute (NIID) isturbance term, ε, as follows: * y an a normal ome win: y = 1 if µ 2 < Draw: y = 0.5 if µ 1 < * y + ε * y + ε < µ 2 way win: y = 0 if * y + ε < µ 1 (1) y epens on the following systematic influences on the result of the match between teams i an j: * P i,y,s = ome team i s average win ratio (1=win, 0.5=raw, 0=loss) from matches playe 0-12 months (y=0) or months (y=1) before current match; within the current season (s=0) or previous season (s=1) or two seasons ago (s=2); in the team s current ivision (=0) or one (=±1) or two (=±2) ivisions above or below the current ivision. R i,m = Result (1=win, 0.5=raw, 0=loss) of m th most recent home match playe by home team i (m=1 M). R i,n = Result of n th most recent away match playe by home team i (n=1 N). 2 These inclue the average points per game an the cumulative points attaine by the home an away teams in the current season; the league positions an goal ifferences of the two teams; an the points an goal ifferences obtaine by the two teams from the last three matches. 3 Dobson an Goar (2001) escribe an early prototype of the present moel. 4

6 SIG = 1 if match has championship, promotion or relegation significance for home team i but not for away team j; 0 otherwise. SIG = 1 if match has significance for away team j but not for home team i; 0 otherwise. CUP i = 1 if home team i is eliminate from the F Cup; 0 otherwise. 4 DIST = natural logarithm of the geographical istance between the grouns of teams i an j. TTPOS i,k = resiual for team i from a cross-sectional regression of the log of average home attenance on final league position (efine on a scale of 92 for the PL winner to 1 for the bottom team in FLD3) k seasons before the present season, for k=1,2. LST1 = result (1 for a home win, 0.5 for a raw an 0 for an away win) of the corresponing fixture between teams i an j if that fixture took place in the preceing season. LST1 =0 if the fixture i not take place in the previous season. LST0 =1 if the fixture i not take place in the previous season, an 0 if it i. j,y,s j,n P, R, R j,m, CUP j, TTPOS j,k = as above, for away team j. Match results ata for the Premier League (PL) an three ivisions of the Football League (FLD1 to FLD3) were obtaine from various eitions of Rothmans Football Yearbook. Forecasts for the season were obtaine using a version of the orere probit moel estimate over the previous 15 seasons from to inclusive, an reporte in Table 1. Forecasts for the , an seasons were obtaine using versions of the same moel, each estimate using ata for the previous 15 seasons. These estimations are not reporte. The contribution to the moel of each set of explanatory variables is now consiere. 4 The F Cup is a suen-eath knock-out tournament involving both league an non-league teams. Up to an incluing the 1999 season, teams from FLD2 an FLD3 entere the cup in November, an teams from the PL an FLD1 in January. The final is playe at the en of the league season, usually in early- or mi-may. 5

7 Team quality inicators The win ratio variables P i,y,s (for team i, an their counterparts for team j) are the main team quality inicators. The higher the value of y the higher is the probability of a home win, so positive home * team an negative away team coefficients are expecte. It is assume that team i s unerlying quality is capture by its win ratio over the previous 12 months, 0 P i,0, = -1 i,0,1 P, an its win ratio between 12 an 24 months ago, + = 1-1 i,1,1 2 P + + = -2 P i,1,2. The moel allows the iniviual components of these sums to make ifferent contributions to the team quality measure. For example, if current-season results are a better inicator of current team quality than previous-season results in the same ivision within the 0 0 same 12-month perio, the coefficient on P i,0,0 shoul excee the coefficient on P i,0,1. If previousseason results from a higher ivision inicate higher quality than those from a lower ivision, the 1 coefficient on P + 0 i,1,1 shoul excee the coefficient on P i,1,1 ; an so on. In general, the estimates of the i,y, s coefficients on { P, P j,y,s } reporte in panel 1 of Table 1 are well efine, an conform accurately to prior expectations. i,y, s Experimentation inicate that the coefficients on { P, P j,y,s } were strongly significant for y=0,1; but not for y=2 (efine in the same way as escribe above for matches that took place months prior to the match in question). χ 2 (20) in panel 5 of Table 1 is the omitte variables version of the Lagrange Multiplier (LM) test escribe by Weiss (1997) for the joint significance of the 20 aitional coefficients on { P, P i,2, s j,2,s information matrix). The test is not significant. } for = 2,...,+2, s=2,3 (using the scoring form of the 6

8 Recent performance inicators The recent match results variables R an R (for team i, an their counterparts for team j) allow i,m i,n for the inclusion of each team s few most recent home an away results in the calculation of * y. lthough R an R also contribute towars the values of i,m i,n P i,y,s an are to some extent correlate with these variables, the possibility of short-term persistence in match results suggests R an R i,m i,n may have particular importance (over an above the information they convey about team quality) in helping preict the result of the current match. Generally the estimate coefficients on { i, m R, R, R, R } reporte in panel 2 of Table 1 are more erratic than those on { P i,y, s, P i,n j,n j,m terms of both their numerical magnitues an the statistical significance of iniviual coefficients. j,y,s }, in Experimentation inicate that the home team s recent home results are more useful as preictors than its recent away results; an similarly the away team s recent away results are more useful than its recent home results. Statistically significant estimate coefficients are obtaine for some (but not all) values of m 9, an for some n 4. M=9 an N=4 are the chosen lag lengths. χ 2 (10) in panel 5 of Table 1 is the omitte variables LM test statistic (Weiss, 1997) for the joint significance of the 10 aitional coefficients on { R, R, R, R } for m=10,11,12 an n=5,6. The test is not significant. 5 m i, i,n j,n j,m Other explanatory variables, cut-off parameters an iagnostic tests The ientification of matches with significance for championship, promotion an relegation issues has been an important issue in the literature on the estimation of the eman for match attenance (Jennett 1985, Peel an Thomas, 1988). It is also relevant in the present context, since match outcomes are likely to be affecte by incentives: if a match is significant for one team an insignificant for the other, the incentive ifference is likely to influence the result. The algorithm use here to assess whether or not a match is significant is crue but simple. match is significant if it is still possible 5 Other similar tests were carrie out using various permutations of aitional variables for m>9 an n>4, with the same result. 7

9 (before the match is playe) for the team in question to win the championship or be promote or relegate, assuming that all other teams currently in contention for the same outcome take one point on average from each of their remaining fixtures. 6 Estimate coefficients on SIG an SIG that are positive an negative (respectively) are consistent with incentive effects of the kin escribe above. Both coefficients reporte in Table 1 are signe accoringly an significant at the 1% level. Early elimination from the F Cup may have implications for a team s results in subsequent league matches, although the irection of the effect is ambiguous. On the one han, a team eliminate from the cup may be able to concentrate its efforts on the league, suggesting an improvement in league results. On the other han, elimination from the cup may entail loss of confience (while cup progress fosters team spirit), suggesting a eterioration in league results. In Table 1 the estimate coefficient on CUP i is negative, the coefficient on CUP j is positive, an both are significant at the 1% level. This suggests that the secon of the two effects escribe above ominates. Contrary to football folklore, elimination from the cup appears to have a harmful effect on the team s subsequent league results. Clarke an Norman (1995) emonstrate that geographical istance is a significant influence on match results. This fining is confirme by a positive an highly significant coefficient on DIST in Table 1. The greater intensity of competition in local erbies may partially offset home avantage in such matches, or the psychological or practical ifficulties of long istance travel for both teams an spectators may increase home avantage in matches between teams from opposite ens of the country. TTPOS i,k an TTPOS j,k are positive for teams that ten to attract higher-than-average attenances after controlling for league position (an negative in the opposite case). These variables allow for a big team effect on match results: regarless of the values of other controls, big teams are more likely (an small teams less likely) to win. The avantages of a large support might be psychological 6 Several alternative efinitions of significance were consiere, but the chosen algorithm prouce values of SIG an SIG with the most explanatory power in the orere probit moel. The algorithm succees in ientifying those matches, mostly playe uring the last few weeks of the season, in which there is a major ifference between the incentives facing the teams. 8

10 (irect influence of the crow on the match result) or material (teams with a large revenue base have more resources to spen on players). Either way Table 1 suggests the big team effect is in fact a highly significant eterminant of match results. To reuce the effect of temporary variation in the attenance-performance relationship, the values of these variables for two previous seasons are inclue in the moel. Since the values over successive seasons ten to be highly correlate, TTPOS i,1 = TTPOS i,1 TTPOS i,2 (an its counterpart for team j) is use in place of TTPOS i,1. Finally, as well as their forecasts many newspaper (an other) tipsters report the results of previous meetings between the same two teams in recent seasons. Previous results may be relevant if there are influences on matches involving particular teams which persist from year to year ( jinxes or horses for courses effects); or if a efeat inspires a team to raise its efforts in an effort to exact revenge next time. LST1 is the result of the corresponing fixture if that fixture took place in the preceing season. secon ummy, LST0 =1 if the fixture i not take place an 0 if it i, is also require as a control, to istinguish between these two cases. The negative reporte coefficient on LST1 =0 in Table 1 suggests a tenency for the fortunes of the two teams to be reverse if they meet in consecutive seasons, perhaps ue to a revenge effect as ecsribe above. owever, the coefficient is only significant at the 10% level, so the effect appears to be relatively weak. 7 Panel 3 of Table 1 also reports the estimate cut-off parameters in (1), ˆµ 1 an ˆµ 2 ; Glewwe s (1997) LM test of the normality of ε in (1); Weiss s (1997) LM test (using the scoring form of the information matrix) of the null hypothesis that ε in (1) are homosceastic; 8 an the omitte variables 7 The result from two seasons before was insignificant an is not inclue in Table 1. The inclusion of the result of the first fixture in the current season among the preictors of the result of the return fixture between the same teams also faile to prouce a significant coefficient. 8 In the homosceasticity test, ε ~N(0, exp(2σz )) uner the alternative hypothesis of heterosceasticity of known form. ε are homosceastic uner 0 :σ=0. convenient choice for z is the uncertainty of match outcome measure z = ˆ (1 ˆ * * ρ ρ ) where ˆρ =1 0.5[ Φ ( µ ˆ 1 ŷ ) Φ ( µ ˆ 0 ŷ )] is the expecte result on a scale of 0 ( certain away win) to 1 ( certain home win). The alternative hypothesis allows the variance of the unsystematic component in the match result to vary irectly (σ>0) or inversely (σ<0) with uncertainty of outcome. 9

11 LM tests escribe above. The normal an homosceastic null hypotheses are both accepte at any reasonable significance level, suggesting that the orere probit moel provies a suitable representation of the match results ata. 4. The efficiency of the fixe-os betting market Section 4 reviews previous literature on the efficiency of team sports betting markets. Pankoff (1968) evelope the first regression-base test of efficiency in the National Football League (NFL) betting market, by regressing match outcomes (measure by the score ifferential) on bookmakers spreas. Intercept an slope coefficients insignificantly ifferent from zero an one respectively suggest that the sprea was an unbiase preictor of the match outcome (see below). Ganar et al. (1988), however, point out the relatively low power of regression-base efficiency tests, an propose a series of economic tests involving irect evaluation of the returns that woul have been earne by implementing technical traing rules (betting on the basis of the past performance of the teams); an behavioural rules (betting to exploit certain hypothesise behavioural patterns of the public). 9 In an NFL ata set some behavioural rules are foun to be profitable, but technical rules are not. Golec an Tamarkin (1991) test the efficiency of the spreas poste on the outcomes of NFL an college football matches. For NFL betting they fin evience of inefficiencies favouring bets on home wins an bets on unerogs. No evience of bias is foun in the college football betting spreas. Dare an MacDonal (1996) generalise the empirical methoology for regression-base tests. Several earlier tests were base on specifications that impose implicit restrictions on the general moel. Tests that search only for evience of home-away team or longshot biases, without recognising the interepenence between these team characteristics, are liable to prouce misleaing results typical behavioural rule is to back unerogs against favourites in cases where the favourite covere the sprea by a large margin the previous week. 10 In other recent US stuies, Baarinathi an Kochmann (1996) show that a strategy of betting regularly on unerogs in merican football was systematically profitable. Ganar et al. (1998) fin that the bookmakers 10

12 The prices for bets on the results of English football matches are fixe by the bookmakers several ays before the match, an are not ajuste as bets are place even if new information is receive. Bookmakers prices are in the form: a-to-b home win; c-to- raw; an e-to-f away win. If b is stake on a home win, the overall payoffs to the bettor are +a (the bookmaker pays the winnings an returns the stake) if the bet wins, an b (the bookmaker keeps the stake) if the bet loses. These quote prices can be converte to the home win, raw an away win probabilities : θ = b/(a+b); θ D = /(c+); θ = f/(e+f). The sum of these expressions invariably excees one, however, because the prices contain a margin to cover the bookmaker s costs an profits. Implicit home win, raw an away win probabilities which sum to one are bookmaker s margin is λ = θ + θ D φ = θ /( θ + θ 1. + θ D + θ ), an likewise for D φ an φ. The Pope an Peel (1989) investigate the efficiency of the prices set by four national high street bookmakers for fixe-os betting on English football. simple (though as before not very powerful) test of the weak-form efficiency hypothesis is base on regressions of match outcomes against implicit bookmaker s probabilities. Consier the linear probability moel: r r = α r + β r φ + u (2) where r =1 if the result of the match between teams i an j results is r, for r= (home win), D (raw) or (away win) an 0 otherwise. In (2) a necessary weak-form efficiency conition is {α r =0, β r =1}. Since orinary least squares (OLS) estimation of the linear probability moel prouces a heterosceastic error structure, Pope an Peel use weighte least squares (WLS) estimation, using closing prices provie a closer approximation to basketball match outcomes than their opening prices. Price movements before close of trae suggest that informe traers were active an influential. Ganar et al. (2001) are sceptical over the existence of systematic biases favouring bets on home teams in baseball an basketball. 11

13 rˆ (1 rˆ ) as weights, where rˆ are the fitte values of the epenent variable obtaine from (preliminary) estimation of the moel using OLS. 11 There is some evience of epartures from {α r =0, β r =1}. Recently Cain et al. (2000) report evience of longshot bias in the fixe-os betting market for match results an scores in English football. Comparisons of estimate fair prices with actual prices for specific scores suggest some evience of biases. 5. Efficiency of prices in the fixe-os betting market: empirical results In Section 5, tests are presente for weak-form efficiency in the prices quote by high street bookmakers for fixe-os betting on match results uring four English league football seasons, from to inclusive. If the orere probit moel prouces information about the match outcome probabilities that is not alreay reflecte in the bookmakers prices, then the latter fail to satisfy stanar weak-form efficiency criteria: that all historical information relevant to the assessment of the match outcome probabilities shoul be reflecte in the quote price. The fixe-os betting ata set comprises the prices quote by five bookmakers (enote B1 to B5) for all league matches playe uring the four seasons. 12 From a total of 8,144 league matches playe, the moel is capable of generating preictions for 7,782. ll matches involving teams that entere the league within the previous two calenar years are iscare, because the moel requires match results for a full two-year perio prior to the match in question. In , one FLD3 fixture was rescheule with insufficient notice for the bookmakers to quote prices, an is also iscare. The final sample comprises 7,781 matches. Estimate ex ante probabilities for the 1,945 matches available for the season are obtaine by substituting the full set of covariate values for the home an away teams for each match into the orere probit moel estimate using ata for the 15 seasons to 11 lternatively, the moel can be estimate as a logit regression, in which case ifferent numerical estimates of the coefficients ρ 1 an ρ 2 are expecte. 12 The ata on the bookmakers prices were obtaine from -tables.co.uk 12

14 (inclusive) as reporte in Table 1, generating a fitte value for the match between home team i an away team j, enote ŷ *. The estimate home win, raw an away win probabilities are: p =1 Φ( ˆµ 2 ŷ * ), D p =Φ( 2 ˆµ ŷ * ) Φ( ˆµ 1 ŷ * ) an p =Φ( 1 ˆµ ŷ * ), where Φ is the stanar normal istribution function. Probabilities for matches in the other three seasons are obtaine in the same way, in each case using the orere probit moel estimate over the previous 15 seasons. Table 2 summarises features of the the implicit bookmaker probabilities, the moel s estimate match result probabilities, an the match outcomes. In Table 2 the implicit bookmaker probabilities are obtaine by applying the proceure escribe in Section 4 to the arithmetic mean of the prices quote by the five bookmakers for each outcome. 13 When the bookmaker an the moel s probabilities are isaggregate by month an by ivision, there is little systematic variation in the mean probabilities for each outcome, but the cross-sectional stanar eviations of both the bookmaker an the moel s probabilities increase systematically uring the course of the season. It is possible to make a more specific assessment of the probabilities for any iniviual match (base mainly on the current season s results) towars the en of the season than at the start (when results from previous seasons provie the only guiance). The cross-sectional stanar eviations are highest in the PL an lowest in FLD3. This apparently reflects a lesser egree of competitive balance within the PL than is the case within each ivision of the Football League. Overall the moel appears effective in replicating the main features of the bookmaker probabilities. s an overall inicator of forecasting accuracy, Rue an Salvesen (2000) suggest the pseuolikelihoo measure, calculate as the geometric mean of the probabilities for the observe 13 Several important changes affecting the UK fixe-os betting market took place uring the observation perio. The UK government abolishe betting uty previously levie at the rate of 10 pence per 1 bet, for bets place via the Internet in 2000, an for bets place in the high street in t the start of the perio, bookmakers require bettors to place combination bets on at least five matches (for bets on home wins) or three matches (for raws or away wins). This restriction was progressively relaxe, an iniviual bets on any match are currently accepte. pparently in response to these changes, the bookmakers have increase their margins by between 1% an 2% (see Table 4 below). 13

15 results. This can be calculate using both the bookmaker an the moel s probabilities. In all cases the ifference in forecasting performance appears very small. Regression-base weak-form efficiency tests Panel 1 of Table 3 reports WLS estimates of α r an β r in (2) for r=,d,. Estimations are carrie out over all four seasons an for each season iniviually. In the estimations over all seasons, the acceptance or rejection of 0 :α r =0; 0 :β r =1; an 0 :{α r, β r }={0,1} is borerline in most cases. In the estimations for the iniviual seasons, 0 :β r =1 is rejecte at the 5% level for r=d in one of the four seasons; but elsewhere 0 :α r =0, 0 :β r =1, an 0 :{α r, β r }={0,1} are always accepte. Overall there seems to be little or no evience of systematic eparture from these necessary weak-form efficiency conitions. The availability of the evaluate probabilities obtaine from the moel permits an extension of these conventional regression-base weak-form efficiency tests. If the moel prouces no aitional relevant information (beyon what is alreay containe in the bookmaker probabilities) a term in r ( p φ r ) shoul be insignificant when ae to the regressions escribe above. The aitional weak-form efficiency conitions are γ r =0 an {α r, β r, γ r }={0,1,0} for r=,d, in the linear probability moel: r r r = α r + β r φ + γ r ( p φ r ) + u (3) Panel 2 of Table 3 reports the WLS estimation results of (3). In the estimations over all seasons, 0 :γ r =0 an 0 :{α r, β r, γ r }={0,1,0} are rejecte at the 5% or 1% levels for r=,d an. In the estimations for iniviual seasons the results are similar, although slightly less consistent ue to the smaller number of observations use in each regression. Overall this appears to constitute reasonably strong evience that the moel oes contain aitional information that is not impoune into the bookmaker os, an that the latter are therefore weak-form inefficient. 14

16 Economic weak-form efficiency tests It is also possible to test for weak-form efficiency irectly, by calculating ex post the returns that coul have been generate by following various betting strategies. For simplicity, the economic weakform efficiency tests reporte below are base on an assumption that iniviual bets coul have been place on all matches throughout the sample perio. Table 4 analyses the returns available from all available bets, while Table 5 analyses the returns from bets place only on the match outcome (home win, raw or away win) with the highest expecte return accoring to the probabilities generate from the orere probit moel. Panel 1 of Table 4 shows the average returns attainable by placing 1 bets on each possible outcome of every match, for each of the five bookmakers. The negative returns of between about 10% an 12% reflect the size of the bookmakers margins, but provie a benchmark against which to assess the performance of selective betting strategies. B3 s margin appears to be slightly but consistently higher than those of the other four bookmakers. Panel 2 of Table 4 shows the average returns attainable by placing 1 bets on each possible outcome of every match with the bookmaker offering the most favourable price for that outcome. comparison between panels 2 an 1 provies an inication of the potential for profitable exploitation of anomalies between the prices offere by ifferent bookmakers. verage losses are reuce by about 5% in , an by about 4% in This suggests some tenency for convergence between the prices offere by the five bookmakers over the course of the sample perio. Panel 3 of Table 4 shows how the returns from 23,343 available bets (three bets on each of 7,781 matches with the bookmaker offering the most favourable price for each outcome) vary if the bets are ranke in escening orer of expecte return accoring to the orere probit moel within each season, an groupe into nine bans (from the top 5% to the bottom 30%). Despite the presence of some ranom variation in the average returns between ajacent groups, the actual returns are 15

17 correlate with the rankings by expecte return accoring to the moel. 14 strategy of selecting only those bets appearing in the top 15% of the expecte returns istribution woul have prouce a (relatively generous) positive return of at least 4% in all four seasons. (perverse) strategy of selecting only those bets appearing in the bottom 30% of the expecte returns istribution woul have prouce a loss of between 14% an 17% in all four seasons, significantly worse than woul be expecte from ranom betting after allowing for the effect of the bookmakers margins. In Table 5 it is assume that bets are place only on the match outcome (home win, raw or away win) with the highest available return accoring to the probabilities obtaine from the moel. Panel 1 of Table 5 shows the average returns attainable if this strategy is use with each of the five bookmakers iniviually, an panel 2 shows the average returns if the same strategy is use with the most favourable price available for each outcome. comparison of panels 1 an 2 of Table 5 with their counterparts in Table 4 suggests that this strategy improves the expecte return by aroun 7% or 8%. ll of the returns shown in panel 2 of Table 5 are non-negative. Panels 3, 4 an 5 of Table 5 show how the returns from the 7,781 available bets vary if the bets are isaggregate by calenar month (panel 3); by ivision (panel 4); an accoring to whether either team ha playe its previous league match within five ays of the current match (panel 5). For both an , accoring to panel 3 the moel performs baly uring the first part of the season, but well uring the secon part (from January onwars). For an this pattern is reverse, with positive returns reporte for most months up to December, but preominantly negative returns from January onwars. It is not clear whether this variation is purely ranom, or whether it might be ue to ajustments in the bookmakers own price-setting rules. In particular, it seems plausible that the large positive returns the analysis suggests were available towars the en of the an seasons might have prompte some reappraisal on the part of the bookmakers, leaing to an improvement from their perspective in an For the four seasons, the correlation coefficients are 0.060, 0.055, an 0.046, respectively. ll are significantly ifferent from zero at the 1% level. 16

18 Panel 4 of Table 5 suggests there is little or no systematic variation in average returns by ivision. Panel 5 tests the importance of one potential source of inefficiency, which arises from the fact that the bookmakers prices are usually compile an publishe at least five ays before the match in question is playe. This means that if either team s previous match took place within this five-ay perio, information about the result will not be impoune into the bookmakers prices for the match in question. In contrast, the moel takes account of the results of all previous matches, whenever they were playe. The relative performance of moel (in comparison with the bookmakers) shoul therefore be higher for matches in which either team ha playe its previous match within the last five ays than elsewhere. Panel 5 provies some evience that this was the case, although the ifference between the average returns for the two cases seems to have narrowe (or isappeare altogether) over time. For the first three seasons the average return ifferential was aroun 6%, 1% an 5%, respectively. For the season the pattern was reverse, with a ifferential of 3%. s before, it is unclear whether the apparent tren is systematic, ranom or partly both. owever, a tenency for inefficiencies to iminish over time woul be consistent with several other of the finings reporte in Section Conclusion This paper has reporte the estimation of an orere probit regression moel, which has been use to preict English league football results. This paper is the first to quantify the preictive quality not only of past match results ata, but also of a wie range of other explanatory variables. The significance of each match for championship, promotion or relegation issues; the involvement of the teams in cup competition; the geographical istance between the teams home towns; an a big team effect are all foun to contribute to the moel s performance. The orere probit moel is consierably easier to implement than several of the team scores forecasting moels that have been evelope recently in the applie statistics literature, but appears capable of achieving comparable forecasting performance. 17

19 The moel has been use to test the weak-form efficiency of the prices quote by high street bookmakers for fixe-os betting on match results uring four recent football seasons. Regressionbase tests inicate that the moel contains information about match outcomes that is not impoune into the bookmakers prices. The latter are therefore weak-form inefficient. strategy of selecting bets ranke in the top 15% by expecte return accoring to the moel s probabilities woul have generate a positive return of at least 4% in each of the four seasons. strategy of betting on the match outcome for which the moel s ex ante expecte return is the highest woul have generate positive or zero gross returns in each of the four seasons. There is some evience, however, that inefficiencies in the bookmakers prices have iminishe over time. 18

20 References Baarinathi R. an Kochman L Football betting an the efficient market hypothesis, merican Economist 40: Barnett V. an ilitch S The effect of an artificial pitch surface on home team perfromance in football (soccer). Journal of the Royal Statistical Society Series 156: Cain M., Law D. an Peel D The favourite-longshot bias an market efficiency in UK football betting. Scottish Journal of Political Economy 47: Clarke S.R. an Norman J.M ome groun avantage of iniviual clubs in English soccer. The Statistician 44: Crower M., Dixon M., Lefor. an Robinson M Dynamic moelling an preiction of English Football League matches for betting. The Statistician 51: Dare W.. an MacDonal S generalize moel for testing the home an favorite team avantage in point sprea markets, Journal of Financial Economics 40: Dixon M.J. an Coles S.C Moelling association football scores an inefficiencies in the football betting market. pplie Statistics 46: Dixon M.J. an Robinson M.E birth process moel for association football matches. The Statistician 47: Dobson S. an Goar J The economics of football. Cambrige: Cambrige University Press. 19

21 Forrest D. an Simmons R. 2000a. Forecasting sport: the behaviour an performance of football tipsters. International Journal of Forecasting 16: Forrest D. an Simmons R. 2000b. Making up the results: the work of the football pools panel, The Statistician 49: Ganar J.M., Dare W.., Brown C.R. an Zuber R Informe traers an price variations in the betting market for professional basketball games, Journal of Finance 53: Ganer J.M., Zuber R.. an Lamb R.P The home fiel avantage revisite: a search for the bias in other sports betting markets. Journal of Economics an Business 53: Ganar J.M., Zuber R.., O Brien T. an Russo B Testing rationality in the point sprea betting market, Journal of Finance 43: Glewwe P test of the normality assumption in the orere probit moel. Econometric Reviews 16: Golec J. an Tamarkin M The egree of inefficiency in the football betting market: statistical tests. Journal of Financial Economics 30: ill I.D ssociation football an statistical inference, pplie Statistics 23: Jennett N ttenances, uncertainty of outcome an policy in Scottish League Football, Scottish Journal of Political Economy 31: Koning R Balance in competition in Dutch soccer. The Statistician 49:

22 Kuypers T Information an efficiency: an empiurical stuy of a fixe os betting market. pplie Economics 32: Maher M.J Moelling association football scores. Statistica Neerlanica 36: Moroney M.J Facts from figures, 3 r en. Penguin: Lonon. Pankoff L.D Market efficiency an football betting, Journal of Business 41: Peel D. an Thomas D Outcome uncertainty an the eman for football, Scottish Journal of Political Economy 35: Pope P.F. an Peel D Information, prices an efficiency in a fixe-os betting market. Economica 56: Reep C., Pollar R. an Benjamin B Skill an chance in ball games. Journal of the Royal Statistical Society Series 131: Rier G., Cramer J.S. an opstaken P Estimating the effect of a re car in soccer. Journal of the merican Statistical ssociation 89: Rue. an Salvesen O Preiction an retrospective analysis of soccer matches in a league. The Statistician 49: Weiss Specification tests in orere logit an probit moels. Econometric Reviews 16:

23 Table 2 Bookmaker s implicit probabilities, an forecast probabilities: escriptive statistics φ Bookmaker Moel ctual PsL PsL (%) D(%) (%) φ D φ p p D p Prem Div Div Div ug-oct Nov-Dec Jan-Feb Mar-May ll Notes: Data for italics. φ an r r p (r=, D, ) are cross sectional means, with stanar eviations in PsL is Rue an Salvesen s (2000) pseuolikelihoo measure of forecasting accuracy: the geometric mean of the bookmaker s or moel s probabilities for the actual results. (%), D(%) an (%) are the actual proportions of home wins, raws an away wins. 22

24 Table 3 Weak-form efficiency: regression-base tests ll Observations r 1. TESTS BSED ON r = α r + β r φ + u for r =,D, ome wins Constant ** (0.021) (0.046) (0.040) (0.043) (0.042) φ *** (0.046) (0.101) (0.085) (0.092) (0.091) F ** Draws Constant ** (0.068) (0.151) (0.124) (0.137) (0.136) D φ ** ** (0.258) (0.567) (0.472) (0.523) (0.511) F ** 2.51 * way wins Constant (0.013) (0.028) (0.024) (0.026) (0.024) φ (0.046) (0.101) (0.088) (0.094) (0.089) F ** r r 2. TESTS BSED ON r = α r + β r φ + γ r ( p φ r ) + u for r =,D, ome wins Constant ** (0.021) (0.046) * (0.039) * (0.042) (0.042) φ ** * (0.046) (0.101) (0.084) (0.091) (0.091) p -φ *** *** *** ** *** (0.078) (0.163) (0.153) (0.141) (0.170) F *** 5.49 *** 3.24 ** 3.35 ** 3.37 ** Draws Constant (0.072) (0.166) (0.131) (0.143) (0.143) D φ (0.272) (0.621) (0.498) (0.548) (0.542) D D p -φ ** ** (0.227) (0.432) (0.454) (0.447) (0.515) F *** 2.48 * ** 0.01 way wins Constant (0.013) (0.028) (0.023) (0.026) * (0.024) φ (0.046) (0.101) (0.088) (0.096) (0.088) p -φ *** *** *** * ** (0.084) (0.180) (0.165) (0.153) (0.183) F *** 3.72 ** 4.24 *** 2.18 * 3.16 ** Notes: Stanar errors of estimate coefficients are shown in parentheses. t-tests on iniviual coefficients are for 0 :α r =0; 0 :β r =1; an 0 :γ r =0. F 1 is an F-test for 0 :{α r, β r }={0,1} an F 2 is an F-test for 0 :{α r, β r, γ r }={0,1,0}. *** = significant at 1% level (2-tail test); ** = significant at 5% level; * = significant at 10% level. 23

25 Table 4 Economic tests for weak form efficiency: comparison of all possible bets verage returns from each bookmaker B B B B B verage returns from selecting best available price for each bet verage returns from selecting best available price for each bet, an ranking all bets in escening orer of expecte return (accoring to the moel) Top 5% % % % % % % % % Note: Data are average returns per 1 bet. 24

26 Table 5 Economic tests for weak form efficiency: comparison of bets place only on match outcome with highest expecte return (accoring to the moel) verage returns from each bookmaker B B B B B verage returns from selecting best available price for each bet verage returns from selecting best available price for each bet, by calenar month ug Sept Oct Nov Dec Jan Feb Mar pr/may verage returns from selecting best available price for each bet, by ivision Prem Div One Div Two Div Three verage returns from selecting best available price for each bet, matches split accoring to whether either team ha playe its last match within five ays of current match Yes No Note: Data are average returns per 1 bet. 25

27 Table 1 Orere probit estimation results 1. WIN RTIOS OVER PREVIOUS 24 MONTS ( P i,y, s, P j,y, s ) ome team (i) way team (j) 0-12 months (y=0) months (y=1) 0-12 months (y=0) months (y=1) Matches playe: Current season (s=0) Last season (s=1) Last season (s=1) Two seasons ago (s=2) Current season (s=0) Last season (s=1) Last season (s=1) Two seasons ago (s=2) Two ivisions higher (=2) (0.554) (0.565) One ivision higher (=1) *** (0.250) *** (0.211) *** (0.203) *** (0.248) ** (0.210) ** (0.202) Current ivision (=0) *** (0.151) *** (0.137) *** (0.130) *** (0.130) *** (0.148) *** (0.136) *** (0.130) ** (0.128) One ivision lower (=-1) *** (0.122) *** (0.117) *** (0.108) *** (0.123) *** (0.118) (0.108) Two ivisions lower (=-2) (0.196) ** (0.201) 2. MOST RECENT MTC RESULTS ( R i, m, R i, n, R j, n, R j, m ) Number of matches ago (m,n) ome team (i) ome matches () *** way matches () ** ** way team (j) ome matches () ** ** way matches () ** ** ** *** 3. OTER EXPLNTORY VRIBLES, CUT-OFF PRMETERS ND DIGNOSTIC TESTS SIG SIG CUP CUP DIST TTPOS i,1 TTPOS i,2 TTPOS j,1 TTPOS j, *** (0.031) * (0.032) *** (0.024) *** (0.025) *** *** (0.036) *** (0.022) *** (0.036) *** (0.022) LST0 LST1 µ 1 µ 2 Normality etero. Omitte lagge win ratios & lagge results ** (0.019) * (0.021) (0.090) (0.090) χ 2 (2)=2.17 χ 2 (1)=2.58 W.R., mths (y=2): χ 2 (24)=9.11 Results, m=10-12(,); n=5-6(,) matches ago: χ 2 (10)=12.80 Notes: Estimation perio is seasons to (inclusive). Number of observations = 29,594. Stanar errors of estimate coefficients are shown in parentheses. *** = significant at 1% level (one-tail test); ** = significant at 5% level; * = significant at 10% level. 26