Keskustelualoitteita #65 Joensuun yliopisto, Taloustieteet. Market effiency in Finnish harness horse racing. Niko Suhonen

Transcription

1 Keskustelualoitteita #65 Joensuun yliopisto, Taloustieteet Market effiency in Finnis arness orse racing Niko Suonen ISBN ISSN no 65

2 Market Efficiency in Finnis Harness Horse Racing Niko Suonen * University of Joensuu (niko.suonen@joensuu.fi) August 29 Abstract Te paper analyzes te efficiency of te betting markets in Finnis arness orse racing. We first present te basic properties of te betting markets. We focus on te organization structures, te market efficiency, and te well-known inefficiencies. Next we review different metods to test te market efficiency. An alternative metod proposed based on actual and on teoretical net returns is introduced. Finally te betting market efficiency is tested wit different tests using database accumulated from te Finnis arness orse race tracks. Robustness of results is controlled wit bootstrap metods. Te results imply tat te markets are semi-strong efficient and caracterised by te favourite long-sot bias. However, consistent evidence for oter game market anomalies like end of te day effect or gambler s fallacy is not found. * I tank Mika Linden, Timo Kuosmanen, and Tuukka Saarimaa for teir elpful comments and suggestions. I also wis to tank Jani Saastamoinen for providing data. Financial support from Yrjö Jansson Foundation, Finnis Cultural Foundation, and Te Finnis Foundation for Gaming Researc is gratefully acknowledged. 1

3 1. Introduction Gaming is a large business, and te general attitudes for gambling ave been liberated from te curtain of sin. Nowadays, betting is more acceptable and it can be seen as entertainment. However, te political discussion about te market structures in gaming is passionate. Typically, in many European countries gaming is organized by full legal monopolies. However, te pressure to open te market for free competition widens everyday. Free competition in te betting markets may lead to ricer gambling menus and even lower prices at least in some gambles. On te oter and, gaming and its problems, suc as gambling addictions, may easier be controlled by a monopoly. However, we do not take policy approac. Instead, we examine ow gamblers beave in tese markets. Our first aim is to briefly discuss market efficiency conditions and different testing metods in betting markets. Secondly, we introduce an alternative testing metod in wic te confidence intervals for net return rates are calculated wit bootstrap metods to secure more appropriate statistical testing. Finally, we test ow efficient te Finnis arness orse race betting markets are. Te paper is organized as follows. In section 2, we take a look at some te key properties of betting markets suc as te organization structures, te market efficiency, and te well-known inefficiencies. Next, we present te metods used to test te betting market efficiency. Section 4 includes descriptions of our dataset and assumptions. In section 5, we introduce our metod. Te empirical results from te different test types are presented in section 6. Te last section concludes te paper. 2. Market Efficiency in Betting Markets 2.1 How Betting Markets Work? Tere are a lot of similarities between financial markets and gambling markets. For instance, bot of tese markets ave a large number of investors or bettors tat ave an opportunity to access te set of information, wic may elp to predict an outcome of te game. However, tere are also differences. In te gambling markets bettors do not ave an opportunity to sell teir ticket in a secondary market. Moreover, gambling markets constitute an environment tat as targeted investments and fast profit clearance. Tis is different compared to te financial markets at least some of tem. For a more precise discussion of similarities between gambling and financial markets, see e.g. Sauer (1998). 2

4 Altoug financial and gambling markets are analogous, tere exists one conceptual difference: te enjoyment of gambling. Gambling brings positive (enjoyment) consumption for te gamblers and tey are also prepared to pay for it. However, in bot markets gamblers try also to maximize (minimize) teir profits (losses) or utility. As consumption of gambling is part of gambling, te gambling markets include bot te demand and supply side. Gamblers are te consumers tat ask for te gambling service. Typically te bookies offer game opportunities based on some sport event or in lotteries. Bookies will return te gamblers an amount of money tat is dependent, at least at some level, on te probability of te outcome. Tere are two main traditions to organize te supply side: pari-mutuel and bookmaker mecanism. In pari-mutuel system (or totalizator) gamblers winning bets sare te pool of all bets. Tus, te pool sares constitute te odds for every possible outcome and te odds determine te payoffs for te gambler. However, te odds are not directly contractual after betting. For instance, if te gambler bets five minutes before closing time, it is possible tat te odd varies because of te oter gamblers betting beaviour. Terefore, te final odd and te payoff are fixed wen te betting is closed. In te pari-mutuel system, te operator takes a sare of te pool to cover te costs, taxes and profits. Tis is te so-called take-out rate and it decreases te odds and payoffs. Terefore, te establised pari-mutuel system is risk-free for te operator. Actually te bettors gamble against eac oter. Te bookmaker system operates somewat differently. In tis system, te bookmaker offers fixedodds to te bettor. Tis means tat te bettors know for sure teir possible payoffs after participating, altoug te bookmaker can offer different odds after gamblers decision. For tis reason, if te betting volumes go aead in some direction, te bookmaker tries to equalize te betting volumes by canging te odds. Terefore, even te bookmaker offers odds independently, te decisions are affected by te gamblers betting beaviour. Due to te bookmaker system, te operator as some risk and bettors gamble directly against te bookie. Similarly as in te parimutuel system, te bookmaker s take-out rate decreases te odds. In general, bot systems are used in orse race track betting. For instance, pari-mutuel system is used in te Nort America, France, Germany, Japan, Hong Kong and Finland. Respectively, te betting markets are organized by a bookmaking system at least in te Great Britain and Australia. As te odds are inversed winning probabilities adjusted by te take-out rate, te probabilities reflect market information. Odds in te pari-mutuel system equal te gamblers subjective probabilities. 3

5 Contrary to tis, te bookmaker s odds reflect bot is own and gamblers estimations of winning probabilities. Tis builds up te simultaneous supply and demand for a game price in ex-ante sense. If all available information used efficiently tis market price is close to te true ex-post odds, i.e. a market is expected to operate efficiently. Now, as we ave istorical information concerning te past game outcomes and odds related to particular orses, we can test in ex-post sense ow correct tese market estimations ave been. Tus we can test ow efficient or unbiased te orse race betting markets are. 2.2 Definition of Market Efficiency For now on, we discuss te pari-mutuel orse race track betting in wic te market information plays a crucial role. Tus, te odds and te bettor s subjective probabilities refer to te gambling market s estimates of eac orse s cance of winning te race and tese odds can be used to analyze te betting market efficiency. Market efficiency as several concepts and definitions. Next we proceed wit te definition by Taler & Ziemba (1988). Te following tree definitions of te market efficiency (ME) seem appropriate Weak ME: Does te subjective probability predict te objective probability? 2. Semi-strong ME: Do some systematic bets ave positive expected value? 3. Strong ME: Does every bet ave te same expected value? In finance, market efficiency analysis is based on te following conditional (rational) expectation ypotesis P E = E[ P Ω ], t+ 1 t+ 1 t were is te observed market price at next period and P E t + 1 t 1 EP [ + Ω ] is conditional expectation t (prediction) for it based on te relevant present and past market information Ω t. Te efficiency condition is also formulated in following way E EP [ t+ 1 EP [ ]] t+ 1 Ω =, i.e. no systematic prediction t errors exist. In te context of game markets we can give a similar example tat illustrates te basic relation between subjective probability and objective probability. Te market efficiency conditions 1 (ME1) and 3 (ME3) can be analyzed wit te following simple regression model 1 In fact, Taler and Ziemba (1988) used only conditions 2 and 3. However, we added te condition 1 for a logical clarification. 4

6 Objective probabilityit, = β + β1 Subjective probability it, + εit,, (2.1) were E[ ε it, ] = for it,. Te condition ME1 is almost trivial as subjective and objective probabilities are igly correlated. Tus, if β 1 >, ten we can say tat te implicit odds predict in average te true winner in te correct direction. Condition ME3 is more informative. If every bet as te same expected value or eac bet s expected value is known, ten te subjective probability sould predict te objective probability witout bias for i, i.e. β = and β 1 = 1. Equation (2.1) does not give a direct answer to condition ME2. However if it is possible to reac a positive expected value wit systematic betting ten β and β 1 >. Testing te betting market efficiency seems to be straigtforward but te main problem is to ave an appropriate estimate for te objective probability in orse race betting. For instance, te lotto game and te orse race betting differ fundamentally from eac oter 2. However, if we ave enoug istorical data from te orse race betting for a particular orse, we can estimate te objective probabilities tat give us some tools to make comparisons. In Section 3, we discuss te estimation metods in details. 2.3 Well-known Inefficiencies Te best-known and documented inefficiency in te gambling markets is te Favourite-longsot Bias: te favourites are under-bet wile te long-sots are overbet (FLB for sort) 3. As a result, if gamblers bet systematically for te favourites, tey do not lose as muc as gamblers tat prefer te longsots. In te pari-mutuel system, tis penomenon clearly reflects te gamblers beaviour. However, FLB is also manifested in te bookmaker market as well, and, terefore, it is probably affected by bot te demand and supply side elements. For te survey of FLB, see, for instance, Taler & Ziemba (1988), Sauer (1998), Coleman (24). We illustrate by a Finnis orse racetrack dataset ow FLB contributes to te gambler s expected returns. Figure 1 sows tat te rate of net return (RR) on betting orses wit ig odds (6-7) is 2 Te most lotto games use te pari-mutuel system but te objective probabilities for every winning outcome are known. Tus, in teory, it is possible to calculate te exact payoff for every outcome if we ave information about gamblers betting beaviour. 3 Te bias was first noted by Griffit (1949). 5

7 between -4% and -5%, and betting of favourite yields losses only about -15%. Teoretical loss te take out rate - is about -2%. Figure 1. Odds and te rate of net returns. Te Finnis data illustrates very well race track studies around te world from te 195s to te present day, wit very few exceptions. Ziemba (28) notices tat FLB as sligtly weakened in te course of time. However, tere is no evidence tat FLB as vanised from te racetracks (i.e., te markets ave come efficient in te long run). Note tat Kanto et al. (1992) ave previously studied Finnis orse racetrack betting. Tey used a dataset from Teivo racetrack in Tampere tat covered 4 racing days from te autumn 1986 to te summer 1987 wit a total of 395 arness races. However, tey used a double betting dataset instead of a win betting data to analyse FLB. 4 Tey found evidence for FLB. Formally, FLB is valid in te Equation (2.1) wen β < and β 1 > 1. Tus, te large probabilities are over weigted and te small probabilities are under weigted. However, altoug FLB is generally accepted, Busce & Hall (1988) found tat FLB is not valid in Hong Kong. Busce (1994) confirmed tis by dataset from Japan and Hong Kong. Tese results are important because 4 Double bets means tat te gamblers ave to coose two orses finising first and second irrespective order and it is identical quinella betting in te Nort America. 6

8 Busce & Hall (1988) actually found evidence for te opposite: te large probabilities are under weigted and te small probabilities are over weigted (i.e. β > and β 1 < 1). Wat explains bettor s systematic beaviour? Two main explanations ave emerged in te literature. Bettors ave 1) a risk-love utility function, and/or 2) a biased view of probabilities. Note also tat Sin (1991, 1992, 1993) sowed tat te presence of insiders can generate FLB in te betting markets. 5 If te gamblers (in representative sense) are risk-lovers, tey prefer to bet for orses tat ave a small probability of winning, i.e. a ig payoff. Tus, tey are prepared to pay more to generate teir utility or enjoyment compared to te risk-aversive or risk neutral gamblers, wic are not willing to participate te gamble at all (in tat case, te word bias is somewat misleading.) Weizman (1965) estimated utility function for an average expected utility maximizer. He found tat te convex utility function explains gamblers beaviour, consistent wit te assumption of a risklove attitude. After Weiztman (1965), for instance, Ali (1977), Golek & Tamarkin (1998) and earlier mentioned Kanto et al. (1992) confirmed te risk-love attitude utility function. Alternatively if bettors ave misperceptions of probabilities in te sense tat tey undervalue te ig probabilities and overvalue te small ones, favourite-longsot bias results are obtained. Tis was Griffit s (1949) explanation for te penomenon. Note tat te Prospect Teory (Kaneman & Tversky 1992, 1979) takes te biased view of probabilities as a fact. Tus, it can be argued tat FLB as noting to do wit risk attitude. Te question is about te misperception of probabilities. To make a difference between tese two approaces is difficult. Snowberg & Wolfers (27) sowed by teir win betting dataset tat te bot approaces are justified. CARA utility function wit risk-love fits te data reasonably well. Similarly, estimating a probability weigting function proposed by te Prospect Teory fits te data equally well. 6 Tus, tere is noting in te win betting data tat allows te econometrician to distinguis between risk-love and misperception. However Snowberg & Wolfers (27) used te win betting data and te exacta (two orses finising in exact order), trifecta (tree orses finising in exact order), and quinella (te gamblers ave to coose two orses finising first and second irrespective order) datasets also in teir 5 See more explanations, e.g., Sauer (1998). 6 Altoug te Prospect Teory suggest two parameter weigting function, tey used one parameter function by Prelec (1998) in estimation. 7

9 analysis. Altoug tese datasets ave similar problems as te win betting data, teir comparison is illuminating. Tus, Snowberg & Wolfers (27) estimated first bot te risk-love model and te misperception model wit te win betting data. Ten tey predicted te exacta, quinella and trifecta datasets records wit tese models and compared fitted models wit true records. As a result, tey noticed tat FLB is consistent being driven by te misperception rater tan te risk-love model. Moreover, tey argued tat te results suggest tat non-expected utility teories (e.g., te Prospect Teory) are a promising candidate for explaining racetrack bettor beaviour. However, efficiency test values can be armed also by oter type inefficiencies. End of te Day Effect (EDE) is interesting, altoug not extensively documented, regularity. In EDE gamblers beaviour canges to more aggressive in te last races of te day. In practice, tis means tat FLB is especially strong in te last races (see McGlotlin 1956, Ali 1977, and Asc et al. 1982). Tis is consistent wit te assumption of te Prospect Teory tat bettors are loss-aversive and risk-lover below te reference point. Most bettors are losing in te end of te day and te last races give te opportunity to win back teir money. However, Snowberg & Wolfers (27) does not confirm EDE in teir large dataset. Gambler s Fallacy (GM). Te bias is an incorrect belief of te probability of an independent event wen te event as recently occurred many times. Tis is well documented in experiments and in real gambling markets as well. For instance, Clotfelter & Cook (1991) noticed tat in te lotto gambling, gamblers rarely cose te number wic occurred in te previous round. Croson & Sundali (25) conducted te field experiment in casinos (roulette) and found te evidence tat supports te assumption of te gambler s fallacy. In te case of pari-mutuel orse race betting, te gambler s fallacy ypotesis can be stated as follows. Bettors underestimate te probability of te favourite orse if it as won te previous race. Metzger (1985) conducted a study in wic se validated some support of te gambler s fallacy: betting on te favourites decreases wit te lengt of teir run of success. Moreover, Terrell & Farmer (1996) used te dog race data. Tey noticed tat betting for te winning dog from te previously track will even constitute a profitable betting strategy. 8

10 3. Metods of Testing Inefficiency 3.1 Basic Definitions and Notations Te following definitions are employed. Te odds, O i, are derived from te gambler s bets in eac race. Tus, O i for te orse i can be written as O i b i = n b i= 1 i 1 (1 τ ), (3.1) n were b is all bets for te orse i, i b is te sum of bets for all n orses, and i = 1 i τ is te takeout rate. Respectively, te average or representative subjective probability for orse i is 1 ρi = O i ( 1 τ ). (3.2) Te rate of net return for a one-euro bet is ER ( ) = po 1, (3.3) i i i were p i is te true winning probability of orse i. Now, if te subjective probability and te true probability are equal, ten te rate of net return is te take-out rate ( τ ), and te strong market efficiency condition 3 is establised. However, to test tis condition we need, like te bettors need, an unbiased estimate of objective probability. 3.2 Different Metods of Estimating (In)efficiency Tere are tree different tecniques to tackle te issue: 1) grouping by favourite positions, 2) grouping by te pari-mutuel odds, and 3) regressing te net return by te odds. Tese are te metods used since te 195 s. Next we briefly review tese metods. Grouping by favourite positions. Te metod was first used by Ali (1977). First, we rank orses into groups () by te favourite position in every race: te favourite orse is group one, and so on. Second, suppose tat π is an objective winning probability for a orse, wic was included in 9

11 group. Te objective probability for group 1 (te favourite) is calculated by dividing all winning cases of group 1 wit all races. Formally, it can be represent by π m Y j= 1 j =, 1,2,..., m = H (3.4) were Yj = 1, wen a orse in group wins te race j, and oterwise zero. A number of races is m. Notice tat Y j is a Bernoulli variable. Because te races are independent, Y j is Binomially distributed. Moreover, te estimators of expectation and variance of Binomial distribution ave te following unbiased properties: EY ( j ) = and Var( Y ) = π (1 π ) / m. Furtermore, te π j subjective probability in eac group can be calculated by m ρ j ρ =, = 1,..., H. (3.5) j= 1 m Te test for gambling biases is H : ρ π = for all = 1,..., H, i.e. are te subjective and objective probabilities in different groups equal. We can approximate te Binomial distribution by te Normal distribution if te observation sample is large enoug. According to te te Central Limit Teorem z = ρ π π (1 π ) / m N (,1). (3.6) Wile tis tecnique is very simple tere are also problems in using it. First, te subjective and te objective probabilities ave distributions of teir own, and tey ave also distribution specific standard errors. However, in Eq. (3.6) we ave assumed tat te subjective probability is a constant. Second, we assume tat te favourite positions in all races are Bernoulli variables. However, tere are ig variations in te race circumstances (e.g., jockey, contenders, weater etc.). Basically, te nature of probability estimation for a orse is someting else tan, for instance, estimating objective probability of eads in an experimental coin toss trial. Te violation of random experiment 1

12 assumption may bias te objective probability estimate and its standard error 7. However, te problem of variations of objective probability can be partly avoided by using odds segments. Grouping by te pari-mutuel odds. Tis is te approac first used by Griffit (1949), and followed by, for instance, Snyder (1978), and Jullien & Salanie (2). Te basic structure of te metod is close to favourite position analysis but now all information on te odds is used. We divide te betting data (orses) into nearby equal-sized groups by te odds levels (e.g ), and te objective probability is te proportion of orses wit odds group tat won teir races (similarly as in Eq. (3.4)). 8 Te benefit of tis metod is tat te variation of te odds is now controlled. Respectively, te subjective probability is calculated similarly as in Eq. (3.5) wile te rank is canged to te odds group. Now we can use estimates of te objective and te subjective probabilities to analyze te regression model (Eq. (2.1)). Tus, if we use estimates of te favourite grouping process, we ave H number of observations. Similarly, te grouping metod by te odds gives us number of observations tat are equal to te number of odds group cosen. However, Ali (1977) noticed, and Busce & Hall (1988) later discussed, tat identity of a single orse is lost in above practices, i.e. in te odds group includes several orses but only one win. Tus, te estimate of te objective probability is somewat undervalued. Moreover, te aggregation in bot metods may introduce a bias as well. Regressing te net return by te odds. Vaugan Williams & Paton (1998) responded to te critique of te grouping metods by regressing te net return by te odds. In tis analysis, te actual return for a unit stake on eac orse is calculated. In practice, if te orse loses, te return is equal to -1 euros and, in turn, if te orse wins, te return is equal to te net return R = O 1. Terefore we do not need to make any groups and avoid te problems of categorizing. However, te dependent variables, te net returns, are left-censored at -1. Consequently, te OLS estimates are biased and Censored or Tobit estimation is more appropriate alternative. Moreover, te returns for orses witin a given race are not independent. Because of tis te clustered Tobit estimation by race is used. In tis context te regression model is i i R = + O + u (3.7) ij α α1 ij ij, 7 We will discuss tis topic more precisely in te next section. 8 Te procedure of building te odds groups seems not to affect te results. For instance, Gandar et al. (21) conclude tat results were similar if tey used even-sized odds categories or even numbers of orses in eac odds category. 11

13 were R ij is te actual net return to a unit bet on te i t orse in te j t race, O ij is te orse s odds, and u ij is an error term wit zero expectations. Because of te take-out rate, te net returns are expected to be negative wit all odds level according to te condition ME2. Tat is, we ave α <. In addition, te existence of FLB indicates tat te net returns are lower (iger) for iger (lower) odds levels of orses implying tat α 1 <. 4. Alternative Approac to Examine (In)efficiency 4.1 Background and Motivation Te categorisation procedures, especially favourite position metod, may produce some bias and measurement errors in te probability estimates. To motive our alternative metod we illustrate te problems found above as clearly as possible wit our Finnis dataset. In te favourite position metod te variation of odds does not matter, terefore, calculating te rate of net return by tis information is somewat ambiguous. Remember tat subjective probabilities are based on odds information but te objective probabilities are not. Figure 2a presents favourites (rank 1) odds distribution in all races. Respectively, Figure 2b sows odds distribution for te favourite wic ave actually won te race. 3,6 2, 3,2 2,8 1,6 2,4 2, 1,2 1,6 1, Odds Win Odds Figure 2a. Odds distribution of rank 1. Figure 2b. Win odds distribution of rank 1. We can notice tat bot distributions are sligtly skewed to te rigt but wit a larger extension for te win odds distribution. Tis is obvious because orses wit low odds win more probably. However, as we know tat te objective probability is te ratio of number of wins wit a favourite 12

14 position to number of races, te win odds distribution is redundant. Tus, te rate of net return calculated only by favourite odds information will lead to biased results. 9 Note tat if we use te odds grouping metod, te variation of odds can be controlled wit odds intervals. A tin (wide) interval will minimize (maximize) te variation. Consequently, wit te tin margin te bias is low, altoug not completely vanising. Alternatively, we can simply calculate all bets on te favourite and compare tis wit te actual winning returns. Formally tis may be written as Sum of Bets on Rank( ) Sum of Rank( ) Returns ER ( ) =. (4.1) Sum of Betson Rank( ) Tus, te problems mentioned above can be andled to some extension. However, te questions Wat is te true relation between te objective and te subjective probability still remains. Naturally, it would be appropriate tat tere were not any controversial arguments, for instance, on te definition of objective probability even FLB is an evident penomenon irrespective of measurement metod. Next, we sow tat an alternative metod tat will end up at te same conclusion, but from a different point of view. 4.2 Alternative Approac Te previous aggregation metods were based on estimating te probabilities. In turn, our approac focuses on te returns of bettors, and indirectly ignores te probabilities. We still keep wit te favourite position metod but instead estimating te objective probabilities, we estimate te teoretical value of return of te orse based on istorical orse racing results. To understand te procedure consider te win odds (average), WO, in every rank category. Te win odds are actually materialized and we ave a vector of win odds for eac rank. Moreover, tese different win odds categories ave teir own distributions like in Figure 2b. Consequently, te expected rate of net return for 1 euro bet for rank can be written as ER ( ) = π WO 1, (4.2) ( ) 9 Note tat calculating te rate of net return by ER ( ) = 1 τ 1, is biased as well. π ρ 13

15 wic constitutes a parallel result wit te Eq. (4.1). Note tat te objective probability does not affect win odds WO. Te probability is constant by nature because bettors cannot influence it. Contrary to tis, te win odds are directly affected by bettors. Te approac leads to questions: Are winning odds line wit teoretical assumptions of market efficiency? Te answer to tis question is two fold. Firstly, if we assume tat bettors beave as rational risk neutral gamblers wit unbiased view of probabilities, teir expected loss or rate of net return is τ. Tis sould be valid in every rank category. Tat is ER ( 1) ER ( 2)... ER ( H ) τ = = = =. (4.3) Secondly, according to tis assumption, te teoretical or objective (average) of win odds in every category can be calculated by Eq. (4.2) WO 1 τ =. (4.4) * π Next we illustrate te idea in Figure 3 1. Figure sows ow te market efficiency conditions are determined wit win odds procedure. First, actual win increases wit te teoretical value, condition ME1 is valid. Second, if te actual wins do not exceed break-even, condition ME2 is effective also. Finally, wen te wins are iger for favourites (grey area) and lower for longsot (black area) tan for te teoretical win odds, te condition ME3 ( β = and β 1 = 1) is systematically violated. Te FLB result is evident. Our testing strategy is weter te teoretical win odds are equal to actual win odds in eac rank category. Typically, we could test te equality of teoretical wins and te win odds by using information on te win odds standard errors and normal distribution assumption. In tat case, our ypotesis can be written as H : WO = WO * H WO WO * 1 :. 1 Te idea to Figure 3 is adapted from te Busce & Hall (1988). 14

16 Figure 3. Te market efficiency conditions wit win odds. Now, a confidence interval based on WO as described above is * WO z [ se( WO)] < WO < WO + z [ se( WO)], (4.5) 1 α/2 1 α/2 WO z1 α /2 were se( ) is te standard error and is te value from te N(,1) distribution wit excess of probability 1 α /2. Te usual values for α are.1,.5 and.1. is rejected if WO exceeds te upper limit or is less tan te lower limit. H However, as we saw in Figure 2b te win odds are not normally distributed. Tis rises te question of te appropriate confidence intervals of equality tests. Tus, te conventional statistical interference can be misleading. To secure te robustness of te results, we estimate te confidence intervals for eac rank category by Bootstrap, and particularly wit te percentile metod. First, we compute 1, bootstrapped samples from te win odds data by randomly sampling wit 15

17 replacement. For eac bootstrapped sample, we compute mean,, and sort tem from te smallest to te largest value. Consequently, te 99 % confidence interval can ten be estimated simply by selecting te bootstrapped statistics at te low.5-t and ig 99.5-t percentiles. Furtermore, in a two-tailed test, ˆ b WO * H : WO = WO ˆ blow *, *, is rejected if WO WO or WO WO. ˆ big According to te bootstrapped confidence intervals, we can calculate confidence intervals to te actual rate of net returns wic can be written as ˆ blow, low WO 1 E( R π = ) and (4.6) ˆ b, ig ig π WO 1 = E( R ). (4.7) Finally, if te take-out rate τ is between te above confidence intervals, te condition ME3 and Eq. 4.3 are establised. 5. Data Descriptions and Necessary Assumptions Our dataset is provided by Suomen Hippos (te Finnis Trotting and Breeding Association) WebSite wit computer programme. In Finland, Suomen Hippos as a monopoly, by law, to organize pari-mutuel betting in orse racing (arness), and te total yearly turnover is over 14 million euros. Tere are 44 racetracks in Finland and te main track of te country, Vermo, is situated in Helsinki, and it osts over 6 race events every year. Te gambling menu covers te traditional Win, Sow and Quinella betting types tat can be bet on every race. Trifecta, as well as Superfecta, can be bet on particular races. Tere is possible off-track betting. Moreover, betting on almost every race in Finland via te internet is possible. Te tote s take-out rate on average is 2.97 % (calculated from te dataset). Te dataset includes win betting information and consists of orse race runs in Finland from to On te wole, te data includes 27,595 arness races, and te total number of included orses is 345,38. Tus, our data contains odds for te orses in every race and information on te exact finising order of orses. We dropped out te races in wic numbers of running orses was under ten. Moreover, if te take-out rate was an obvious outlier (i.e. too ig or 16

18 low), we dropped also te race out of te data. Te data does not contain any information about te caracteristics of an individual bettor or average amounts of te bet per race per person. Terefore we assume an agent or a bettor tat typifies all te bettors, te so-called representative bettor Results 6.1 FLB wit Different Tests We replicated different test types for FLB by using Finnis dataset. First, te results of grouping by favourite positions test (Eq. 4.6) are sown in Table 1. Te results confirm te FLB. Favourites are under bet wile long-sots are over bet. Te cross point is in te 4 t and 5 t ranked orse, in wic te difference between te objective and te subjective probability is not statistical significant. Tese results are line wit te previous studies. Table 1. Objective and subjective probabilities by favourite position. Rank No. races No. wins Average Odds Subjective Probability Objective Standard Probability Deviation z-value ( ρ π ) *** * * *** *** *** *** *** *** *** *** *** *** * * Indicates tat te difference is significant at te 1 % level, ** at te 5 %, *** at te 1 %. Note tat rank positions are in logical order, i.e., number of winnings decrease monotonically wit te rank positions. Tis is, indeed, a necessary assumption for te metod but at te same time it supports validity of te ME1 condition. Moreover, since we ave a data on te exact finising order for every orse, we can ceck te link between rank and finising position (e.g. did te orse ranked 4 t really finised te line in 4 t place). In most cases te rank positions predicted finising positions quite accurately. Te finising positions were sligtly distorted at te top places. Tis is 11 Naturally, racetrack bettors ave eterogeneous preferences and information. 17

19 due to te fact tat in te arness races disqualifications and interrupts are very common. However, tere does not seem to be any systemic violations tat prevent te use of rank positions metod. As te grouping by odds positions metod leads to te identical results, we do not present te exact results in ere. However, we made regression on objective probabilities implied by Eq.(2.1). We used te data from te grouping by odds position. Results from te regression are sown in Table 2 Table 2. OLS estimates of te probabilities. Dependent Variable: Objective Probability wit te restrictions β = and β = 1 1 Coefficient Std. Error Lower 99 % Confidence Interval Upper 99 % Confidence Interaval Constant -.122** Subjective Probability ** No. Obs. 29 R2.99 ** Significant at te 1 % level. Te analysis gives clear and unambiguous answers to market efficient conditions ME1 and ME3. Bettors subjective probability valuations predict actual outcomes. However, te strong efficiency is violated. Subjective expectations do not equal te objective outcomes, and te markets are systemically caracterised by FLB. Next we Tobit regress net returns on odds (i.e. Eq. 4.7). Our main findings are presented in Table 3. Table 3. Clustered Tobit regression estimates for net returns. Dependent Variable: Net Return Coefficient Std. Error Slope Constant -21.3** Odds ** No. Obs Clusters Standard Error 2.93 Log-Likeliood Note: Standard errors are Huber/Wite robust estimates. ** Significant at te 1 % level. Now, constant is negative ( α < ). Tus, at te first place it indicates tat tere is no opportunity of systematic winning strategy and te take-out rate makes betting unprofitable for te gambler. Moreover, te second estimate is also negative ( α 1 < ). Terefore, wen te odds increase te net 18

20 returns decrease. FLB is evident by tis metod as well. Note tat Tobit estimates require adjusting before tey can be interpreted as marginal effects. Te correct marginal effect (slope) is estimated following Greene (23). Slope is It suggests tat an increase in te odds of one unit decreases te expected net return of one-euro bet by six cents. Te Tobit regression metod gives answer also to te condition ME2. Tere are not systematic bets tat ave a positive expected value. For te comparison, Gandar et al. (21) examined New-Zealand torougbred pari-mutuel orse races (117,775 observations), and teir results were similar to ours. Tus tere seems not to be any cultural differences in results. In summary, Finnis orse race betting markets are efficient in semi-strong level: odds reflect te outcomes of te races and tere is not opportunity to make any profits by simply systematically betting witout extended information. However, te assumption of te strong efficiency condition EM3 is rejected and te market is caracterised by FLB. 6.2 Market Efficiency wit Alternative Metod Next, we present our results of testing FLB tat is based on te alternative metod. Table 4 includes basic information about estimates and bootstrapped confidence intervals. Rank Win Odds Table 4. Actual win odds, teoretical win odds, and confidence intervals. Win Teoretical Odds Bootstrapped Lower 99 % Conf. Interval Bootstrapped Upper 99 % Conf. Interval Actual RR Bootstrapped RR Lower 99% Conf. Interval Bootstrapped RR Upper 99% Conf. Interval Sum of Wins * * * * * * * * * * * * * * Note: Table 1 sows oter information about ranking procedure (e.g., number of races and wins). Te confidence intervals for te rate of net return are calculated by te bootstrapped win odds intervals as mentioned in Section

21 Te teoretical rate of net return or take-out rate is -.29, and it fits only te bootstrapped confidence interval of te groups two and tree. Clearly, FLB is true regardless te metod but te alternative metod is more precise compared to te favourite position metod wit binomial distribution assumptions. Figure 4 illustrates te difference between te tests wit te confidence intervals of te rate of net returns. Te intervals of te rate of net returns calculated by te Binomial distribution assumptions are wider tan te bootstrapped ones. However, we adere to our metod and test oter efficiencies as well. 12 Figure 4. Binomial and bootstrapped confidence intervals of te rate of net returns. End of te Day Effect. Next we test for te end of day effect. We did te same calculations as above but only information of day s last races was used. Table 5 sows te results. Te confidence intervals for te rate of net return are calculated by te bootstrapped win odds intervals in te all races. 12 We also compared te bootstrapped confidence intervals and te Normal distribution confidence intervals based on win odds information. Surprisingly, te results were very similar. Te exact results are sown in Appendix 1. 2

22 Table 5. Winning odds, te net rate of returns and confidence intervals for te day s last race. Rank No. Wins Win Odds RR Last Race Bootstrapped RR Lower 99% Confidence Level Bootstrapped RR Upper 99% Confidence Level (H) (L) (L) (H) (H) (H) (H) (L) Notes: 1) H (L) means tat te rate of net returns for te last race is iger (lower) tan on average. 2) Te total number of te last races was Te rate of net return is iger for te favourite and te longsots (e.g., 6 t and 7 t rank) compared to te rate of net return of all te races. Tus, te returns of te last races are dispersed and tere is not clear tendency. Tis migt be due to small number of observations and eterogeneity of data. For tis reason, let us consider te last two races. Table 6 presents our findings. Table 6. Winning average odds, te rate of net returns and confidence intervals for te day s last two races RR Last Race Bootstrapped RR Lower 99% Confidence Level Bootstrapped RR Upper 99% Confidence Level Rank No. Wins Win Odds (L) (L) (L) (H) (H) (H) (H) (H) (H) Notes: 1) See Table 5. 2) Te total number of te two last races was 629. Again, te results indicate tat rates of net return are lower for favourites tan on average. Te rates of net return are dispersed more equally now. Terefore, according to our data, we conclude tat tere is no meaningful end of te day effect in te Finnis orse race betting markets. 21

23 Gambler s Fallacy. We tested for te influence of te favourite winner of te previous races in betting beaviour. Table 7 sows te main results. Table 7. Te rate of net returns for te favourite wen te previous won by te favourite. Win Horse No. Races No. Wins Win Odds RR Bootstrapped RR Lower 99% Confidence Level Bootstrapped RR Upper 99% Confidence Level Previous Favourite (H) Two Previous Favourite (H) Tree Previous Favourite (H) Notes: See Table 5 and 6. Te results are interesting. Te favourite winner of previous races increases te rate of net returns for te favourite bet. Tis gives some support to te gambler s fallacy assumption. On te oter and, te favourite winner of two and tree previous races decreases te returns for te favourite bet compared to te favourite winner of first race. Tus, tis rejects te gambler s fallacy assumption tat te rate of net returns systematically increases wen favourites win te races one after anoter. Moreover, we also tested ow te returns are influenced by wins of similar orses in te previous race. Similar means ere te rank was te same in different races (e.g. te 3 t positioned orse won te two races straigt). Te rate of net return was -,26, tus tere is no tendency for gambler s fallacy. On general level te results imply tat te Finnis orse betting markets are semi-strong efficient and FLB assumption cannot rejected. However, te bias is not strong enoug for profitable bettings. Moreover, it is not possible to make profits by using some oter teoretical and/or empirical regularity, for instance, end of te day effect or gambler s fallacy strategies. In fact, we claim tat Prospect Teory assumptions like te loss aversive or risk-love beaviour in below of te reference point do not seem to be evident in our data. 7. Conclusions and Discussion We tested efficiency of gambling markets in Finland by a large dataset from arness orse races. In order to test efficiency market ypoteses we used several metods tat ave been presented in literature. We also sortly discussed te weak and strong points of different metods. An new alternative metod was presented also. It was based on te actual winning odds and te teoretical assumptions of te winnings. In tis approac we calculated confidence intervals to te rate of net returns by Bootstrap metod for te more appropriate statistical testing. Te results imply tat 22

24 markets are semi-strong efficient and caracterised by FLB. However, te meaningful evidence for oter anomalies as end of te day effect or gambler s fallacy was not found. Tis is a minor drawback for predictions from te Prospect Teory. Te weakness of our testing approac is tat we do not ave information concerning te individual betting beaviour (e.g., amount of bets). Te observed biases are not grounded wit individual betting beaviour. It can be assumed tat bettors may distribute teir total sum of bets on different orses. In tat case, te long-sots can be over-bets because of te minimum bet constrain (wic is used at least in Finnis orse racetrack) 13. Altoug tis creates some bias in betting volumes te betting beaviour could still be biased and driven by misperception of probabilities. However tis is question of future researc based on te individual betting data. We suggest tat our results concerning FLB in te orse racetrack cover also oter type of gambles in Finland as well. In Finland, for instance, RAY (Finland s Slot Macine Association) as a full legal monopoly to organize te casino table games in te game alls and in te clubs (restaurants). Terefore, te organization can determine game prices (e.g., te take-out rate) witout competition and conduct price discrimination. For instance, European roulette is typically a game wit te same take-out rate regardless of betting strategy. Usually it is -,27. However, in Finland te take-out rate of betting favourites (e.g., black or red) is also -,27, but te take-out rate of betting longsot (e.g., single number) is -,162. Te difference is remarkable. It is not know wy RAY offers tese prices. On explanation is tat tat te customers, for watever reasons, are prepared to pay more for longsots tan favourites. Tus, from te bookmaker s point of view, te information on te gambler s beaviour or on te market inefficiencies can be used, for instance, in planning te new gamble menus or setting up te rules of gambles. On te oter and, it would be appropriate to inform te gamblers about te observed game facts. 13 We tank Timo Kuosmanen for tis insigt. 23

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27 Appendix 1. Te confidence intervals of te rate of net returns based on different approaces. Rank Actual Rate of Net Return Normal Dist. RR Lower 99% Confidence Level Normal Dist. RR Upper 99% Confidence Level Binomial RR Lower 99% Confidence Level Binomial RR Upper 99% Confidence Level Bootstrapped RR Lower 99% Confidence Level Bootstrapped RR Upper 99% Confidence Level