The Australian Rules Football Fixed Odds and Line Betting Markets: Econometric Tests for Efficiency and Simulated Betting Systems
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The Australian Rules Football Fixed Odds and Line Betting Markets:
Econometric Tests for Efficiency and Simulated Betting Systems
by
Adi Schnytzer and Guy Weinberga
Paper to be presented at:
The 4th Biennial Equine Industry Program International Academic Conference
Louisville, Kentucky (USA), June 2005
Abstract
The purpose of this paper is to test the efficiency of two Australian Rules Football betting markets
employing two approaches commonly adopted in the literature and to compare both the markets and the
methods employed. The two markets are a line betting market and a fixed odds win betting market and the
two approaches are econometric testing and betting simulation. We conduct our comparisons by subjecting
2001-2004 data for the Australian Football League (AFL) to highly detailed scrutiny. We restrict ourselves
to weak-form market efficiency, as this proves sufficiently complicated to show that neither the connection
between the two forms of efficiency testing nor that between the efficiency of the two markets, is clear-cut.
Nonetheless, taking advantage of the fact that many games in the AFL are played on neutral grounds, we are
able to reject the existence of any significant favorite-longshot bias in either market for each of the four
seasons individually and for the period as a whole, except for a reverse bias in the line market in 2001 as
reflected by significant profits in betting simulations, and to demonstrate the existence of a significant bias in
favor of teams with an apparent home ground advantage for three of the four seasons in the win betting
market simulations. The line market is free of such a bias. The results of econometric tests for this bias are
more ambiguous. Finally, we suggest that the difference in apparent efficiency in the two markets may be
due to the assumption, on the part of bookmakers, of a linear relationship between lines and prices.
Keywords: market efficiency, betting markets, sports economics
a
Department of Economics, Bar-Ilan University, Ramat-Gan, Israel (e-mail addresses: schnyta@biu.013.net.il,
guy@weinbergdoron.co.il)
The authors wish to thank Michael Bailey, Hamish Davidson, John Kyriakopoulos, Damon Rasheed and Eric Sorensen
for their help with data collection.1. Introduction
The purpose of this paper is to test the efficiency 1
of two Australian Rules football betting markets
employing two approaches commonly adopted in the literature and to compare both the markets and the
methods employed. The two markets are a line betting market and a fixed odds win betting market and the
two approaches are econometric testing and betting simulation. We conduct our comparisons by subjecting
2001-2004 data for the Australian Football League (AFL) to highly detailed scrutiny. We restrict ourselves
to weak-form market efficiency, as this proves sufficiently complicated to show that neither the connection
between the two forms of efficiency testing nor that between the efficiency of the two markets, is clear-cut.
Nonetheless, taking advantage of the fact that many games in the AFL are played on neutral grounds, we are
able to reject the existence of any significant favorite-longshot bias in either market for each of the four
seasons individually and for the period as a whole, except for a reverse bias in the line market in 2001 as
reflected by significant profits in betting simulations, and to demonstrate the existence of a significant bias in
favor of teams with an apparent home ground advantage for three of the four seasons in the win betting
market simulations. The line market is free of such a bias. The results of econometric tests for this bias are
more ambiguous. Finally, we suggest that the difference in apparent efficiency in the two markets may be
due to the assumption, on the part of bookmakers, of a linear relationship between lines and prices.
The remainder of this paper is organized as follows: A discussion of line and fixed odds betting, the
economics of bookmaking and a brief survey of the literature are presented in Section 2. Section 3 provides a
summary of the basics of Australian Rules football relevant to an understanding of this paper, section 4
describes our data set and highlights the problems to be explained, section 5 discusses the weak efficiency
econometric tests and the betting systems, section 6 describes the relationship between the two betting
markets, and section 7 concludes the paper.
2. The Basics and the Literature
There are several alternative betting methods offered in the different sports betting markets worldwide. In
team sports, two methods predominate: "Point spread" (also known as "line") wagering is the dominant form
of wagering on basketball and American football contests.2 An alternative method is "fixed odds" win
betting, which predominates in US baseball.3 Basset (1981) examined why bookmakers used the point spread
method exclusively for wagering on the National Football League (NFL) when the odds method is also
feasible. Given simple specifications of beliefs and betting behavior, and given that point spread and odds
are equally thrilling, he demonstrates that the point spread bet can stand alone as the bet produced by a profit
maximizing bookmaker. Woodland and Woodland (1991) suggested that the market structure of having line
(and not odds) betting on the NFL was a consequence of a risk-averse attitude of bettors. History has
overtaken these theoretical analyses and the real reason for the availability of the two different types of
betting in different sports appears to be unrelated to considerations of either maximizing behavior or attitude
to risk. Currently, both types are available for most team sports, and one can even bet according to either of
them with the same bookmaker. Nonetheless, empirical research comparing these two alternative methods
has, to the best of our knowledge, not yet been published.
A typical point spread wager in the AFL requires that the bettor risk $1 for the chance to receive $1.9 if
successful.4 The line on a game specifies the favored team and the point spread. A bettor placing a wager on
the favorite wins the bet if the favorite wins by a margin of victory greater than the point spread. A bettor
placing a wager on the longshot wins the bet if the longshot loses by less than the point spread or wins the
game outright. The above-mentioned $1.9-for-$1 dividend implies that, if the market is efficient, no betting
strategy should win more than 52.63 percent of the time, or, in other words, the $1.9-for-$1 dividend requires
that bettors must pick winners in 52.63 percent of bets to break even. 5 6
1
See Osborne (2001) for a list of studies that investigate the Efficient Markets Hypothesis in different markets.
2
See Basset (1981), Dobra, Cargill and Meyer (1990) and Woodland and Woodland (1991).
3
See Woodland and Woodland (1994).
4
In contrast to the US market, the winning dividend per $1 point spread wager in the AFL is not fixed. The range of
this dividend in our data of 2001-2004 was $1.78-$2.05, while in 66% of the games it was $1.9, and the average was
$1.9 as well.
5
The percentage of winning bets (WP) necessary to break even, 52.63 percent, is obtained by setting the expected value
of the random variable, a gamble WP * 0.9 + (1 – WP) * (-1), equal to zero.
6
See, for further discussion, Vergin and Scriabin (1978), Gandar et al (1988), and Dana and Knetter (1994).
1The fixed odds wager in the AFL requires that the bettor risk $1 for the chance to receive a fixed sum if
successful.7 The odds on a team specify the odds that the team will win the game. A bettor placing a wager
on a team wins the bet if this team wins. As in the line betting market, the bookmaker sets odds to earn
around five percent of the total amount bet if his book is balanced.8 Therefore the bettor must pick around
52.5 percent of winners to break even. 9
While the initial line/odds are based on expert opinion of the game's outcome, thereafter the bookmaker
adjusts the line/odds.10 There is no consensus in the literature whether these adjustments are made to reflect
the collective judgment of gamblers about the outcome or because the bookmakers are setting prices in order
to exploit bettors' biases. Levitt (2004), using data on prices and quantities of bets placed, found support for
the latter hypothesis; i.e., the bookmakers do not appear to be trying to set prices to equalize the amount of
money bet on either side of a wager. Our results do not permit us to shed any meaningful light on this issue.
The question of whether organized sports betting markets are weak-form efficient according to Fama's
(1970) definition has received considerable attention in the literature. Zuber et al (1985), Sauer et al (1988),
and Gandar, Zuber, O'Brien and Russo (1988) implemented the weak efficiency test for point spreads in the
NFL.11 Schnytzer and Weinberg (2004) implemented it using data on National Basketball Association
(NBA) games in the US for 1999/00-2003/4 period. While Zuber et al (1985) found this test to be too weak
to establish definite conclusions, and Gandar et al (1988) concluded that statistical tests are not powerful
enough to detect inefficiencies,12 Sauer et al (1988) and Schnytzer and Weinberg (2004) could not reject the
null hypothesis that the NFL and the NBA gambling markets are weakly efficient.
Before concluding that efficiency prevails, Zuber et al (1985) considered an "extreme" alternative that the
line is unrelated to the actual point spread. They could not reject this hypothesis for 15 of the 16 weeks in
their sample, noting that the extreme alternative hypothesis is as consistent with their sample data as is
efficiency. Their conclusion was that an alternative testing strategy is required. Unlike Zuber et al (1985),
both Sauer et al (1988) and Schnytzer and Weinberg (2004) rejected this extreme alternative hypothesis.
Sauer et al (1988) questioned the validity of testing weak-form efficiency on a week-by-week basis and
argued that Zuber et al's (1985) test fails to provide sufficient evidence to support the argument that
speculative inefficiencies exist in the betting market for NFL games.13 Golec and Tamarkin (1991), on the
other hand, showed that spreads set in the NFL betting market are systematically biased predictors of actual
results. Then again, in his review of the literature on sports betting markets, Sauer (1998) outlined the
substantial evidence that prices set in these markets are efficient forecasts of outcomes. Prices (in the form of
betting odds or betting lines) appear to aggregate scarce information from diverse sources, and are, almost
invariably, unbiased estimators of actual game outcomes (in terms of both point spreads and winner's
identity). While Sauer (1998) noted that a number of studies, particularly of point spread betting markets,
have offered sightings of profitable trading strategies, he also noted that these sightings frequently disappear
upon further investigation.
Thaler and Ziemba (1988) reviewed the early literature on US racetrack betting markets, noting a
consistent favorite-longshot bias. Woodland and Woodland (1994) found that the favorite-longshot bias in
7
The range of actual payouts in our data set is $1.03-$10, while the average sum is $2.36.
8
The average bookmakers' commission during our 2001-2004 data was 3.9%, while the average during 1998-2004 was
5.5%. Bailey and Clarke (2004) noted that the commission could be as low as 2-3%.
9
In the event the outcome is identical to the line, known as a "push" or a "no bet", the gambler's wager is refunded. In
the very rare event where the outcome is a tie, the fixed odds bettor wins half the amount he would have won had his
team won. Our data contains only 5 tied games (0.7%). Note that one can bet on a tie for most games at odds of 65 to 1,
but this is part of an exotic category of bets. Note that this seems a high price given that there are over 700 games in
our sample!
10
Nonetheless, Levitt (2004) noted that the adjustments in the line/odds are typically small and relatively infrequent; in
the five days preceding an NFL game, the posted price changing an average of 1.4 times per game, and in 85 percent of
those changes, the line moved by the minimum increment of one-half of a point. Yet, he noted that in horse racing, the
odds set by bookmakers change far more frequently.
11
Examples for other papers that test the NFL's weak-form efficiency are Pankoff (1968), who used 1956-1965 seasons
to show that weak-form efficiency prevails, and Amoako-adu, Marmer and Yagil (1985), who showed that the NFL is
weakly inefficient, using the 1979-1981 seasons.
12
They note that the statistical tests are too weak to reject rationality in a market where irrationality appears to exist.
They also indicate that their results are strikingly consistent with those of Summers (1986), who simulated a model of
stock prices incorporating non-rational expectations and then showed that standard statistical tests are too weak to
detect the absence of rationality formed expectations.
13
It should be noted that the bulk of both Zuber et al (1985) and Sauer et al (1988) are devoted to an analysis of semi-
strong efficiency.
2racetrack betting exists in reverse for baseball bettors, but that no betting strategy admits profits in excess of
commissions. Dare and Holland (2004) modified previous research to generate a specification that they argue
yields the most reliable estimates of inefficiency in the NFL. Their results indicate a betting line bias
favoring bets on home longshots that does not appear consistently from season to season, and the expected
profits arising from this bias may be too small to be exploited.
The evidence with respect to Australian Rules football betting markets is limited.14 Stefani and Clarke
(1992) used team ranking and home ground advantage to predict winners. They reported a home ground
advantage for each of the AFL teams during 1980-1989, and a bigger advantage for teams outside of
Melbourne and for those that do not share a home ground. They were able to predict the correct winning
team in 68 percent of games. Brailsford et al (1995) examined the AFL and the Australian Rugby League
(ARL), focusing on two different kinds of betting market. For the ARL, they considered line betting while
for the AFL they examined an exotic betting method, whereby bettors are required to select the point spread
to within a 12-point range, known as "bins".15 They predicted game outcomes and tested betting strategies
and reported a favorite-longshot bias in the AFL. Their success rate from betting on home teams during
1987-1995 was 58 percent, yet the return rate was negative. Other strategies generated positive returns, and
betting on the predicted bin yielded an average return of 23 percent. These results imply market inefficiency,
yet the authors raised doubts as to whether this apparent inefficiency is exploitable. Bailey and Clarke (2004)
note that no attempt has been made to utilize all past matches to establish a prediction process. They develop
models using all previous match results, and investigate the additional benefits of incorporating individual
player statistics in the prediction process. They use data from 1997-2003 and present models predicting
correctly up to 67 percent of winners, and producing betting profits of up to 15 percent. Clarke (2005)
investigates the home advantage in the AFL using several models, and demonstrates that individual clubs
have home ground advantages to different degrees, non-Victorian teams having a larger advantage. His
results lend support to the conclusion that crowd effects are the main determinant of home ground advantage.
He did not test efficiency, yet his findings are relevant to this paper. The bottom line from this brief survey
of the literature is that there is no consensus concerning the existence of the weak-form efficiency and the
possibility of making speculative profits in sports betting markets.
3. Australian Rules football
Australian Rules football is a high scoring, continuous-action game. For the 2001-2004 seasons, the
average game score per team was 95 points, with a minimum of 25, a maximum of 196, and a standard error
of 28. Each team has 18 players on the field at any given time and 4 substitutes are available for unrestricted,
repeated substitutions as deemed fit by the team coach.
The home and away season comprises 176 games played over 22 weekly rounds of eight games each,
between 16 teams. Following this is a final series between the top eight teams. The two surviving teams from
this phase play for the premiership title in what is known as the "Grand Final". In total, 185 games are played
in an entire season.16
The AFL is a national league which began as the Victorian Football League. Except for Geelong, all the
other teams in this league derived from Melbourne and had their own home grounds. The addition of new
teams from other states has been accompanied by a policy of stadium consolidation in Melbourne. Thus, it
has not been true for some years that each team has its own stadium. Of the sixteen teams that make up the
AFL, only three have stadiums which are uniquely home grounds, where it may be said that they have an
advantage; Brisbane Lions, Sydney Swans and Geelong Cats (although even Geelong do not play all of their
official home games at this home ground). All the other teams play at grounds shared with one or more
teams. Thus, dealing with the home advantage in the AFL requires further care and a compliment to the
official home designation is necessary. Whenever we refer to home teams as either subsets of the data or
dummy variables in regressions, we mean teams with an a priori home ground advantage. Teams which are
officially designated as home teams but have no a priori home ground advantage are referred to as Neutral.17
14
See Brailsford, Easton, Gray and Gray (1995).
15
This "bins" betting method is very different from the methods studied in our paper. Also, the dividends in the "bins"
method are calculated, in pari-mutuel fashion, after the outcome of the match, such that a certain percentage of the total
amount wagered is returned to successful bettors. Brailsford et al (1995) note that the average commission over their
sample period was around 20% and that line betting was not offered by the AFL at the time of their paper.
16
For further information about the AFL, see its official website: http://www.afl.com.au.
17
In what is probably a unique, albeit bizarre, feature of the AFL, there are even games in which the official home team
is playing an opponent with a genuine a priori home ground advantage! A recent example is provided in Round 9 of the
34. The data
The data used in this paper are derived from publicly available sources,18 i.e., internet-based sports
statistical information. Thus, the game data come from http://www.afl.com.au and
http://stats.rleague.com/afl/seas/season_idx.html, while the closing odds and lines are from
http://www.sportsbetting.com.au, http://www.goalsneak.com.au and http://www.centrebet.com. Our data
consist of game performances, dates, grounds, odds and lines. We use data from the 2001 to 2004 seasons,
for a total of 740 games (all home and away games plus the finals), and 1480 team observations. Line data
are missing for 86 games, since the bookmakers do not publish a line in a match where both teams have
equal (or very close) betting odds.19
We denote the team from whose perspective the spread and result are defined as the team of record. There
is no single correct way of choosing the team of record, and three different methods have been used.20 First,
the team of record can be defined to be the favorite. Second, the team of record can be defined to be the
home team. Third, the team of record can be chosen randomly, avoiding any systematic effects. We use the
official home team as the team of record, as do Gandar et al (1988) and most of the other studies.
Nonetheless, since, as noted above, the official home definition in the AFL does not automatically imply a
home ground advantage, this is somewhat akin to a random selection. Moreover, we analyze different
subsets of the data separately in order to test directly for various possible biases.
Some basic properties of our data set are presented in Table 1, including the winning percentage in
different categories and different apparent biases in the markets.21 Thus, the rate of successful betting in the
win market on home teams, favorites, and home favorites exceeds the percentage necessary to break even
during all years in our sample, while home longshots and neutral longshots seem overpriced vis-à-vis their
winning frequencies. The rate of successful betting in the line market is less consistent, as it exceeds the
percentage necessary to break even for home and home longshot teams in all seasons bar 2001, home
favorites every season, not at all for favorites, and in 2001 and 2002 for neutral longshots.
Our home team win rate of 64 percent for 2001-2004 compares with the 58 percent as reported by both
Stefani and Clarke (1992) for 1980-1989 and Brailsford et al (1995) for 1987-1995, and the 60 percent from
Clarke (2005) for 1980-1998. But it should be noted that we refer to real home teams, whereas the other
papers report for official home teams. This explains both why our winning rate is higher and the perceptible
upward trend in official home team winning frequencies as new non-Victorian teams have entered the AFL
over the past two decades, thereby increasing the proportion of teams with real home ground advantages.
More puzzling in Table 1 are the apparent inconsistencies among the biases as between prices and lines
and across seasons. The cases of home and neutral longshots teams will suffice to illustrate the problem.
Home teams are, on average, over-priced in fixed odds betting relative to winning frequencies in 2001,
correctly priced in 2002 and under-priced in 2003 and 2004. And yet for both 2001 and 2002, the line on
average understates the point spread!
Neutral longshots are teams which are longshots in games where neither team has any home ground
advantage. Thus, the results in this category show the presence or absence of favorite/longshot biases, being
uncontaminated by home ground advantage considerations. Neutral longshots are evidently overpriced vis-à-
vis winning frequencies in all seasons, yet in the line market they are underpriced during 2001, 2002 and
over the whole period. A priori, then, there seems much to explain! But, as we show in the next section, few
of these contradictory are statistically significant either under econometric testing or when attempts are made
to exploit them in a betting system.
2005 season, when the Western Bulldogs, a Melbourne-based team, were the official home team in their game against
the Sydney Swans, while the game was played in Sydney!
18
Although it should noted that not all data from previous seasons or games are available on-line today.
19
Response to a query from one of the authors by one of Australia’s leading bookmakers, Sportsbet Pty Ltd.
20
See Golec and Tamarkin (1991).
21
Note that the successful win rate is not directly correlated with net returns from betting, as the dividend for a
successful bet differs across teams (this is more relevant for WIN as the dividend changes dramatically within games, as
mentioned above).
45. Measuring weak efficiency: Econometric tests v. Betting systems
5.1 Econometric Tests
The econometric weak efficiency tests are not as easy to apply and interpret as is generally implied in the
literature. The main reason is that they raise many subtle methodological issues. First, the issue of a suitable
team of record has already been mentioned. Golec and Tamarkin (1991) use three different definitions:
favorites, home teams and random selection. We define (1) OFFICIAL HOME, (2) HOME, (3) FAV, (4)
HOMEFAV, (5) HOMELONG, and (6) NEUTRALLONG as binary variables, equal to one if the relevant
team (1) is designated officially as the home team regardless of the ground at which the game is played, (2)
is playing where they a priori have a real home ground advantage, (3) is the favorite according to the
bookmakers in the fixed odds betting market, (4) has an a priori real home ground advantage and is the
favorite, (5) has a real home advantage and is the longshot in the fixed odds betting market, and (6) is the
longshot in a game played at a ground where neither team has a real home ground advantage, and zero
otherwise. As Golec and Tamarkin (1991) point out, if one is looking for a bias, then choosing the team of
record according to HOME or FAV may interfere with the results. As already noted, we have a natural
solution in our data set. Since, in contrast to most other team games studied in the literature, OFFICIAL
HOME and HOME are not identical in the AFL, we opt for OFFICIAL HOME to define the team of record
and use HOME to test for any biases in this direction whenever we do not break the data into subsets
according to characteristics (2) through (6).
Second, what level of significance is significant? i.e., what level of significance is required to reject the
null hypothesis of weak efficiency? Is it the usual five percent simply because that is usual? Or is a market
only inefficient if it permits profitable betting based on past and present prices or lines and home ground
information? As demonstrated below, there are different inferences for the five percent and ten percent levels
of significance.
Third, are F and χ2-tests for joint tests of hypotheses on regression coefficients compatible with T and Z-
tests on the individual coefficients? If not, which are the more reliable? Our tests will present
incompatibilities between these two tests.
Before proceeding to list other difficulties, it will prove useful to define some terms. Let LINE denote the
point spread in the bookmaker's betting line and PS denote the actual point spread between the two teams
(defined in a way that is consistent with the definition of LINE, i.e., according to OFFICIAL HOME), and let
WIN denote the actual winner in the game (defined as a variable equal to 1 for the winning team, 0 for the
losing team and 0.5 for ties). Further, let:
PRICE = 1 (1 + odds )
NPRICE = normalized price = PRICE ∑ PRICE
per _ game
If bettors use the available information efficiently, then we would expect the point spread/fixed odds to be
the best unbiased forecast of the game's outcome. Let i and j denote two different teams playing in game t.
Then, in general, the Efficient Market Hypothesis requires that:
(1a) Median [PSijt│Ωt-1] = LINEijt
(1b) Median [P(WIN)ijt│Ωt-1] = NPRICEijt
where Ωt-1 is the set of all information available to the bettor prior to the game.
Stern (1991) found that the distribution of the margin of victory over the point spread (defined as the
number of points scored by the favorite minus the number of points scored by the longshot minus the point
spread) is not significantly different from the normal distribution. Therefore the true outcome of a game can
be modeled as a normal random variable with mean equal to the point spread, and equations (1a) and (1b)
imply that:
(2a) Et-1 [PSijt│Ωt-1] = LINEijt
(2b) Et-1 [P(WIN)ijt│Ωt-1] = NPRICEijt
5Equations (1a-1b) and (2a-2b) reflect the most general definition of efficiency, and a variety of efficiency
tests have been performed based on equations (1a) and (2a), although equations (1b) and (2b) remain
untested in the context of team sports betting to the best of our knowledge. A natural test is based on the
information contained in the set Ωt-1. In general, Ωt-1 will contain the current lines and odds, past lines and
odds, past outcomes, known game conditions (e.g., ground, home team), past game statistics, other public
information (e.g., injuries, referees), and private information. For the tests of weak-form market efficiency
with which we are concerned, Ωt-1 should sensu stricto contain only prices, but it is conventional to include
information regarding home ground advantage as well.
The basic statistical test of weak efficiency for the line betting market involves estimating the following
model:
(3a) PSijt = a0 + a1LINEijt + εijt
where a0 is a constant and εijt is an independently and identically distributed random error. Support for the
linear specification in the specific case of Australian Rules Football is provided by Bailey and Clarke’s
(2004) demonstration that point spreads in AFL games are normally distributed. Thus, equation (3a) is
estimated using Ordinary Least Squares (OLS), as we test the linear relationship between point spreads and
lines.
The parallel test in the fixed odds betting market is estimating P[WINijt│Ωt-1]. The expected winning team
in a game is the one with P>0.5, and this raises the next difficulty: What is the proper functional specification
for this test? It is well known that OLS is not the best choice of estimator for a probability model. Among
other things, it makes no use of the fact that the fitted dependant variable represents a probability and must
be between zero and one and the probabilities sum to one per game. But these are technical issues and the
Linear Probability Model (LPM) at least sometimes provides a solution.22
The two estimation techniques commonly used in regressions on dummy variables are Probit and Logit.
But, as already noted, the win variable is special because it sums to one across teams per game. Further,
owing to the possibility of drawn games, it is not, strictly speaking, a dummy variable. In the conditional
logit regression (CLOGIT) based on McFadden (1973),23 each game is treated as an independent drawing
from a multinomial distribution in which every team, i, has its associated probability of winning, Pi.
Therefore, the probability of team i winning team j in round t is estimated as:
e βxit
Pijt = βx
e βxit + e jt
where β is a vector of coefficients to be estimated, and Xit is a matrix of observable variables of performance
indices for team i in game t. The Pijt satisfy 0≤ Pijt ≤1 and Pijt + Pjit = 1 . The estimates of β maximize the
estimated likelihood of the occurrence of the actual results of all games:
L= ∏ P * (1 − P )
i*t
all _ games _ t
kt
where i* is the index of the winning team and k is that of the losing team, in each game.
Thus, the statistical test of weak efficiency for the fixed odds market involves estimating the following
specific model:
e b1 ln NPRICEit
(3b) Pijt = b ln NPRICE jt
ϕ ijt
e b1 ln NPRICEit + e 1
where ϕijt is an independently and identically distributed random error and we test the null hypothesis that
the coefficient of the log of the normalized price equals 1. It is evident from equation (3b), that for a unitary
22
When not too many observations are lost in correcting for heteroskedasticity.
23
See also Figlewski (1979) and Schnytzer and Shilony (1995).
6coefficient, winning probability equals price. Note, however, that this technique involves dropping drawn
games from the sample even if this is of limited importance in practice.
But what if the relationship between P(WIN) and PRICE is linear and not logistic? Maybe it is even
something entirely different? Since there is no theory to guide us, we test using the LPM as well, and check
for consistency with the CLOGIT results. The LPM is represented by:
(3c) WIN ijt = c0 + c1 NPRICEijt + ς ijt
Weak-form efficiency corresponds to the joint hypotheses with respect to equations (3a), (3b) and (3c) that
a0=0 and a1=1, b1=1, and c0=0 and c1=1, respectively. The results are presented in Table 2 and demonstrate
the consistency between the different markets and models; we can not reject the weak efficiency hypothesis
for any of the models during any of the years at a ten percent level of significance. However, this is not the
case at a five percent level of significance. The extreme alternative hypothesis, which in terms of equations
(3a), (3b) and (3c), is the joint test that a0=a1=0, b1=0, c0=c1=0, is always rejected, the F and χ2-tests
statistics being well above the critical value at any meaningful level of significance. But what does the
consistency of results as between the LPM and CLOGIT specifications mean? As the following analysis will
show, both functional specifications have important variables missing. Once these are added, differences
begin to appear.
Golec and Tamarkin (1991) argued that it is advisable to test simultaneously for numerous specific biases
by adding dummy variables to equations such as (3a), (3b) and (3c). The following equations purportedly
identify both favorite and home team biases24:
(4a) PSijt = a0 + a1LINEijt + a2HOMEijt + a3FAVijt + ζijt
eb1 ln NPRICEit + b2 HOMEit + b3 FAVit
(4b) Pijt = b ln NPRICE jt + b2 HOME jt + b3 FAV jt
Ψijt
eb1 ln NPRICEit + b2 HOMEit + b3 FAVit + e 1
(4c) WINijt = c0 + c1 NPRICEijt + c2 HOMEijt + c3 FAVijt + ς ijt
Now tests of efficiency for equations (4a), (4b) and (4c) are tests of the joint null hypotheses,
a1=1,a0=a2=a3=0, b1=1,b2=b3=0 and c1=1,c0=c2=c3=0, respectively. The intercepts, a0 and c0, so the argument
runs, measure any bias that exists with respect to a visiting longshot, and a2,b2,c2 and a3,b3,c3 measure the
home and favorite team biases, respectively. Excluding either HOME or FAV leads to bias in the regression
coefficients and reduced power against the null. Golec and Tamarkin (1991) note further that equations such
as (4a), (4b) and (4c) statistically isolate particular biases, although biases other than home and favorite team
could offset each other and go unmeasured, so that efficiency tests may not be powerful with respect to
unspecified biases. They test efficiency using the three different teams of record noted above, yet they ignore
possible interactions between them. Thus, their coefficients are quite possibly biased and inconsistent, since
home ground advantage and favoritism are correlated and thus the interactions between them and with the
line cannot be excluded without prior testing. Accordingly, we generalize their argument and include all the
relevant dummy variables and interactions in one regression. The results are presented in Table 3 and they
are beset by multicollinearity. In none of the scenarios are we able to reject weak efficiency. We reject the
extreme alternative hypothesis that LINE, lnNPRICE (and NPRICE) are unrelated to the actual point spread
and winner, respectively. Nonetheless, the results in Table 3 do not permit specific inefficiencies to be
readily identified since almost no coefficient is individually significant.
Our suggested solution to the multicollinearity problem is to divide the data into subsets and test for all
possible biases via separate regressions. There are ten such possible biases: Vis-à-vis (a) home teams, (b)
favorites, (c) home favorites, (d) home longshots, (e) neutral longshots, (f) away teams, (g) longshots, (h)
away favorites, (i) away longshots, and (j) neutral favorites. Since the last five are the symmetric with
respect to the first five, we use the first five subsets as the basis for econometric testing. We again use OLS
for the line betting market and CLOGIT and LPM for the fixed odds market. Results for line and fixed odds
markets are reported in Table 4a and Table 4b, respectively.
24
Note that Golec and Tamarkin (1991) study the line market only, but their argument generalizes to the fixed odds
market.
7Our results imply that the AFL gambling market is not consistently weak-form efficient. We reject the
efficiency hypothesis for home teams (in 2002 – LPM, 2001-2004 – OLS and CLOGIT), home longshots
(2001-2004 - CLOGIT) and home favorites (in 2002 – LPM and CLOGIT, 2001-2004 – OLS, LPM and
CLOGIT). On the other hand, we reject the extreme alternative hypotheses that the line is unrelated to the
actual point spread as the F-test statistics for the hypotheses that a0=a1=0 and c0=c1=c2=c3=0 are well above
the critical value at any meaningful level of significance, and that the normalized price is unrelated to the
winning probability, with the χ2-test statistics for the hypotheses that b1=0 being well above the critical value
at any meaningful level of significance).
We examine the neutral category, as it provides us with the only test for a pure favorite-longshot bias,
uncontaminated by home ground considerations, since all other regressions confound this question. And
indeed, the category results imply weak-form efficiency for all models during all years. Accordingly, we
may conclude that any biases in these markets relate to the presence of an apparent home ground advantage.
We return to this issue when reporting the results of our betting simulations.
Our results also illustrate several of the above-mentioned problems. Thus, there are contradictions between
F and χ2-tests, on the one hand, and T and Z-tests on the other, in different scenarios. For example, in certain
scenarios [home teams in 2002 in Table 4b(A), and neutral longshots over the entire period in Table 4b(B)]
the F-test implies weak efficiency, yet the significance of other coefficients implies a bias. In other scenarios
[most of Table 2, 2001, 2002, 2003 and 2004 home longshots in Table 4a, 2002 favorites in Table 4b(A),
2001, 2002, 2003 home longshots, 2002 favorites and 2002 neutral longshots in Table 4b(B)] LINE, NPRICE
or lnNPRICE do not differ significantly from zero, yet the F-test fails to reject efficiency.
The issue of the "right" level of significance arises as well, since at the ten percent level there is efficiency
in all scenarios, yet at five percent there appear to be inefficiencies. The betting simulations will demonstrate
that the level of significance chosen does not meaningfully predict profitability in the fixed odds market.
5.2 Betting systems
To confirm the presence of the above-noted inefficiencies or otherwise, five simple betting systems were
constructed, both for LINE and WIN betting: (a) betting on all home teams, (b) betting on all favorites, (c)
betting on all home longshots, (d) betting on all home favorites, and (e) betting on all neutral longshots. The
simulation results are presented in Table 5.25
Following Tryfos et al (1984) and Gandar et al (1988), we test the significance of the results in two ways;
first, a Z-test for the null hypothesis that the successful bet rates (see Table 1) are random (the assumption
being that chance yields a fifty percent success rate). The second and more stringent test, is a Z-test for the
null hypothesis that a given scenario is unprofitable against the alternative that it is profitable. If we insistent
stringently, upon a rejection of the two null hypotheses, each at a five percent level of significance, there
were no profitable betting systems on the line market, except for longshots playing on neutral grounds in
2001, but there were significant profits betting on home teams for the win in 2002, 2003, 2004 and the whole
period (the null hypothesis of no profits being rejected by both tests at better than 1 in a 500).26 Betting on
home favorites in 2002 and over the whole period is also significantly profitable. Other profits in the table
fail to pass our stringent test.
Comparing the betting simulations with the econometric tests, we see that on all but two occasions, the
econometric tests indicate correctly that the line market is efficient (at five percent significance), while
mistakenly not rejecting the null hypothesis of efficiency for longshots playing on neutral grounds in 2001.
In all but this case, the econometric tests coincide fully with the simulations at ten percent. Given that linear
specification for the regressions of point spreads on lines is uncontroversial, this relative consensus between
the two approaches is hardly surprising, but the failure to pick up the reverse favorite/longshot bias in 2001 is
puzzling.
Matters are less parsimonious in the comparison between econometrics and simulation in the fixed odds
market. Here the econometric tests, for both specifications, correctly27 reject efficiency at five percent
significance for the 2002 and 2001-2004 home favorites, but fail badly to find the more significant profits to
25
We present only these five categories, as other possible groups, including longshots, away teams and away longshots
yielded consistent losses in both markets, while away favorites yielded a minimal profit only in the 2001 LINE market,
and neutral favorites yielded regular losses.
26
The equivalent results of Tryfos et al (1984) and Gandar et al (1988) were 3 profitable scenarios out of 70, and 0 out
of 14, respectively.
27
In terms of significant simulated profits, of course.
8be had backing home teams in each of 2002, 2003 and 2004, and for the four seasons as a whole. The LPM
finds home teams inefficient only in 2002 and the CLOGIT rejects efficiency only for the four seasons as a
whole. In order to provide a possible answer to this puzzle, we turn to a consideration of the relationship
between bookmakers’ lines and their prices.
6. The relationship between line and fixed odds
Having demonstrated the presence of a home team bias in the win market alongside almost ubiquitous
efficiency in the line market, it is necessary to ask what it is that accounts for this difference across the two
markets. One possibility is that bookmakers assume an incorrect functional relationship between lines and
prices when setting lines and odds. The relationship used by bookmakers in our sample is presented in Table
6, and it is evidently linear between point spreads and prices.28 Nonetheless, the correct theoretical
relationship between line and fixed odds is probably not linear. And if the bookmakers are mistaken in the
linearity assumption, that could readily account for the inefficiency of one market relative to the second.
7. Conclusions
In this paper we have tested the weak-form efficiency of the line and fixed odds betting markets for the
Australian Football League over the four seasons, 2001 through 2004. We have shown that the null
hypothesis of efficiency cannot be rejected for the line market at the five percent level of significance either
via exhaustive econometric testing or via betting simulation for each season individually (except in the case
of neutral longshots in 2001 in the betting simulation) and that, in spite of some apparent inefficiencies over
the four seasons as a whole, there are no significant profits to be made. On the other hand, the fixed odds
betting market is evidently beset by a bias that underprices teams with an a priori home ground advantage, as
in three of the four seasons statistically significant profits are made in the betting simulations. Econometric
tests provide hints of this bias but do not accord completely with the simulation results and are not consistent
as between the linear probability and conditional logit models. Further, at the ten percent level of
significance, all econometric tests imply market efficiency. We suggest that the major problem with
econometric testing in the fixed odds market is one of functional specification. Without a clear understanding
of the relationship between winning probabilities and prices and without a suitable econometric technique
capable of handling the relationship, econometric tests of efficiency will remain of dubious value.
By taking advantage of the fact that many games in the AFL are played on neutral grounds, we are able to
reject the existence of any significant pure favorite-longshot bias in either market for the each of the four
seasons individually and for the period as a whole, except for a reverse bias in the line market in 2001 as
reflected by significant profits obtained by backing longshots playing on grounds where neither team has a
home ground advantage.
We suggest that the inefficiency in the fixed odds market relative to the line market may be due to a
mistaken assumption of a linear relationship between prices and lines on the part of bookmakers, but this is a
subject requiring further research.
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10Table 1 Basic properties of the AFL data
Basic properties of complete 2001-2004 AFL seasons for different subsets of the data: home teams, favorites, home
longshots home favorites, and neutral longshots.29
Year No. of Successful Average No. of Successful Average Average
WIN WIN bets PRICE30 LINE LINE bets Point LINE
observations rate (%) observations rate (%) spread
HOME 2001-2004 1480 64 0.61 1308 55 15.65 10.63
2001 370 58 0.63 320 52 15.78 12.12
2002 370 67 0.67 342 58 13.77 8.43
2003 370 64 0.59 306 54 15.20 10.78
2004 370 65 0.60 340 57 17.97 11.38
FAVORITES 2001-2004 1480 69 0.69 1308 50 20.84 19.70
2001 370 68 0.72 320 48 21.43 21.23
2002 370 72 0.71 342 48 17.67 18.26
2003 370 69 0.67 306 51 23.31 19.50
2004 370 69 0.68 340 51 21.22 19.88
HOME 2001-2004 1480 42 0.37 1308 58 -10.8 -16.4
LONGSHOTS 2001 370 34 0.39 320 47 -18.9 -17.5
2002 370 41 0.39 342 63 -7.6 -16.1
2003 370 42 0.35 306 58 -14.4 -17.6
2004 370 49 0.37 340 64 -3.15 -14.2
HOME 2001-2004 1480 75 0.72 1308 54 27.19 22.28
FAVORITES 2001 370 71 0.74 320 54 30.36 24.59
2002 370 81 0.73 342 56 24.23 19.99
2003 370 75 0.69 306 53 27.25 22.33
2004 370 74 0.70 340 54 27.15 22.35
NEUTRAL 2001-2004 1480 32 0.37 1308 52 -17.77 -17.97
LONGSHOTS 2001 370 34 0.38 320 63 -9.98 -18.62
2002 370 32 0.39 342 57 -14.75 -17.11
2003 370 33 0.36 306 46 -22.85 -16.13
2004 370 28 0.33 340 44 -23.47 -19.63
29
Throughout this paper we report results for home and away plus finals games, since the results excluding finals
games were very similar.
30
In this Table, and for the rest of the paper, price of a bet is defined as the reciprocal of the payout contingent upon
winning. Thus, a price of 0.2 is (the probability) equivalent of odds of 4 to 1 and a payout for $1 of $5.
11Table 2 Weak efficiency estimates for the AFL
Weak efficiency estimates for 2001-2004 AFL seasons. Each game in model (2) is represented by two observations, one
for each team, and each game in models (1) and (3) by one observation – the official home team. T/Z values in
parentheses. Joint hypotheses are tested at the 5% significance level.
(1) PSijt = a0 + a1LINEijt + εijt
e b1 ln NPRICEit
(2) Pijt = b ln NPRICE
ϕ ijt
e b1 ln NPRICEit + e 1 jt
(3) WIN ijt = c0 + c1 NPRICEijt + ς ijt
2001 2002 2003 2004 2001-2004
(1) PS market – OLS
CONS -2.36 7.37* 1.216 4.64 2.859
(-0.73) (2.58) (0.42) (1.49) (1.89)
LINE 1.083* 0.792* 1.147* 1.058* 1.022*
(8.31) (5.85) (8.77) (7.85) (15.3)
F (CONS=0, LINE=1) 0.35** 3.59# 1.03** 1.65** 2.33**
Adj. R2 0.295 0.163 0.333 0.264 0.262
No. of obs 160 171 153 170 654
(2) WIN market – CLOGIT
lnNPRICE 1.111* 1.212* 1.315* 1.166* 1.199*
(5.14) (5.26) (5.73) (5.6) (10.88)
χ2 (lnNPRICE=1) 0.26*** 0.85*** 1.88*** 0.63*** 3.25***
Log likelihood -111.6 -109.2 -103.8 -107.5 -432.3
No. of obs 370 366 364 370 1470
(3) WIN market – LPM
CONS -0.137 0.1 -0.021 0.075 0131
(-1.07) (0.89) (-0.18) (0.68) (0.23)
NPRICE 1.187* 1.014* 1.132* 0.984* 1.067*
(5.48) (4.89) (5.63) (5.06) (10.51)
F (CONS=0, 0.72** 4.81# 1.15** 1.77** 3.98#
NPRICE=1)
Adj. R2 0.154 0.119 0.168 0.127 0.144
No. of obs 160 171 153 170 654
* Significant at 5% level. # Significant at 10% level.
** CONS significantly equals zero and LINE or NPRICE significantly equals one at 5% level (F2,171=3.066,
F2,652=3.028). *** lnNPRICE significantly equals one at 5% level (χ2(1)=3.841).
12Table 3 Weak efficiency estimates for the AFL – dummies and all interactions
Weak efficiency estimates for 2001-2004 AFL seasons. Each game in model (2) is represented by two observations, one
for each team, and each game in models (1) and (3) by one observation – the official home team (which is our team of
record). T/Z values in parentheses. The efficiency tests are at the 5% significance level.
7
(1) PSijt = a0 + ∑ a B'
n =1
n n ijt + ς ijt
7
∑ bn C 'it
e n =1
(2) Pijt = 7 7 Ψijt
∑ bn C 'it ∑ bn C ' jt
e n=1 + e n=1
7
(3) WINijt = c0 + ∑ cn D'n ijt + ς ijt
n =1
where B', C' and D' are matrixes of observable variables, including M, FAV, HOME, W, X, Y and Z. Wijt=
HOMEijt*Mijt , Xijt=HOMEijt*FAVijt , Yijt= HOMEijt*Mijt*FAVijt , Zijt=Mijt*FAVijt . Mijt equals (1) LINEijt , (2)
lnNPRICEijt or (3) NPRICEijt , respectively.
2001 2002 2003 2004 2001-2004
(1) PS market – OLS
CONS -14.305 0.235 2.192 2.554 -1.508
(-0.94) (0.02) (0.13) (0.17) (-0.2)
LINE 0.522 0.311 1.76 1.221 0.984*
(0.81) (0.36) (1.89) (1.94) (2.8)
FAV -7.957 7.431 -9.48 9.153 -1.619
(-0.37) (0.37) (-0.46) (0.46) (-0.16)
HOME 9.053 -5.578 1.643 10.802 3
(0.44) (-0.32) (0.08) (0.57) (0.32)
W 0.255 -0.143 -0.724 -0.036 -0.219
(0.27) (-0.14) (-0.64) (-0.04) (-0.47)
X 13.763 7.282 16.949 -25.98 3.74
(0.51) (0.3) (0.65) (-1.05) (0.3)
Y -0.182 0.039 -0.27 0.735 0.18
(-0.15) (0.03) (-0.2) (0.63) (0.3)
Z 0.618 0.536 -0.053 -0.549 0.113
(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. R2 0.313 0.144 0.327 0.251 0.262
No. of obs 160 171 153 170 654
(2) WIN market - CLOGIT
lnNPRICE -0.495 1.571 2.291 1.009 0.842
(-0.21) (0.56) (0.82) (0.47) (0.7)
FAV 2.998 -2.652 -1.6 -0.454 0.052
(0.63) (-0.47) (-0.3) (-0.1) (0.02)
HOME -1.85 1.089 -0.858 2.079 -0.057
(-1.43) (0.75) (-0.57) (1.27) (-0.08)
W -1.62 0.815 -1.228 1.492 -0.254
(-1.25) (0.59) (-0.83) (0.95) (-0.38)
X 2.913* 1.26 1.245 -0.471 1.432
(2.05) (0.78) (0.81) (-0.28) (1.92)
Y 3.88 2.959 1.879 2.089 2.785*
(1.65) (1.13) (0.78) (0.83) (2.32)
Z 4.579 -5.255 -2.221 -1.359 -0.257
(0.62) (-0.6) (-0.27) (-0.19) (-0.07)
χ2 (FAV= …=Z=0, 4.75*** 10.7*** 4.3*** 4.9*** 12.07***
lnNPRICE=1)
Log likelihood -109.3 -103.8 -102.2 -105.2 -427.3
No. of obs 370 366 364 370 1470
13Table 3 (continued)
2001 2002 2003 2004 2001-2004
(3) WIN market – LPM
CONS -0.204 0.848* 0.047 0.468 0.259
(-0.5) (2.36) (0.1) (1.73) (1.6)
NPRICE 1.155 -1.093 0.593 -0.507 0.116
(1.12) (-1.16) (0.47) (-0.61) (0.26)
FAV 0.24 0.02 -0.348 -0.664 -0.189
(0.36) (0.03) (-0.46) (-0.94) (-0.62)
HOME 0.566 -0.606 0.155 -0.522 -0.047
(1.15) (-1.39) (0.28) (-1.28) (-0.22)
W -1.247 1.572 0.002 1.932 0.407
(-1.01) (1.38) (0) (1.71) (0.74)
X -1.003 -0.237 -0.179 0.34 -0.246
(-1.23) (-0.03) (-0.21) (0.39) (-0.64)
Y 2.082 -0.611 0.125 -1.7 0.092
(1.31) (-0.4) (0.07) (-1.04) (0.12)
Z -0.368 0.977 0.86 1.905 0.816
(-0.28) (0.8) (0.55) (1.45) (1.36)
F (CONS=FAV=…= 1.06** 2.32# 0.59** 1.05** 1.96**
Z=0, NPRICE=1)
Adj. R2 0.115 0.125 0.115 0.112 0.135
No. of obs 184 185 185 183 738
* Significant at 5% level. # Significant at 10% level.
** CONS, FAV,…,Z significantly equals zero and LINE or NPRICE significantly equals one at 5% level (F8,165=2.02,
F8,720=1.97). *** FAV,…,Z significantly equals zero and lnNPRICE significantly equals one at 5% level (χ2(7)=14.067).
14Table 4a Weak efficiency estimates for the AFL line market
Weak efficiency estimates for 2001-2004 AFL seasons. Each game is represented by one observation and i represents
(1) home teams, (2) favorites, (3) home favorites, (4) home longshots, and (5) neutral longshots, in turn. T-values are in
parentheses.
PSijt = a0 + a1LINEijt + εijt
2001 2002 2003 2004 2001-2004
(1) Home teams
CONS 1.76 7.045 4.585 6.698 5.149*
(0.41) (1.94) (1.35) (1.62) (2.67)
LINE 1.157* 0.797* 0.985* 0.991* 0.989*
(6.98) (4.73) (6.77) (5.54) (12.03)
F (CONS=0, 0.92** 2** 1.08** 1.67** 4.3
LINE=1)
Adj. R2 0.308 0.153 0.297 0.21 0.243
No. of obs 108 120 107 113 448
(2) Favorites
CONS -3.624 4.211 2.515 -4.74 -0.666
(-0.59) (0.77) (0.44) (-0.83) (-0.23)
LINE 1.18* 0.737* 1.067* 1.306* 1.092*
(4.69) (2.83) (4.1) (5.26) (8.621)
F (CONS=0, 0.26** 0.54** 1.04** 0.86** 0.58**
LINE=1)
Adj. R2 0.117 0.04 0.094 0.137 0.101
No. of obs 160 170 153 169 652
(3) Home favorites
CONS 0.554 9.369 11.304 -3.472 3.614
(0.06) (1.09) (1.34) (-0.4) (0.83)
LINE 1.212* 0.744* 0.714* 1.37* 1.059*
(3.5) (1.95) (2.09) (4.05) (6.12)
F (CONS=0, 1.03** 0.78** 1.32** 1.31** 3.08
LINE=1)
Adj. R2 0.131 0.034 0.043 0.165 0.105
No. of obs 76 81 76 79 312
(4) Home longshots
CONS -5.252 -3.525 3.835 14.942 2.629
(-0.36) (-0.32) (0.31) (1.07) (0.41)
LINE 0.777 0.253 1.037 1.254 0.818*
(1.06) (0.45) (1.67) (1.55) (2.45)
F (CONS=0, 0.06** 1.83** 0.15** 1.11** 1.54**
LINE=1)
Adj. R2 0.004 -0.022 0.056 0.042 0.036
No. of obs 32 38 31 33 134
(5) Neutral longshots
CONS 7.939 -0.186 5.122 -5.364 1.043
(0.79) (-0.02) (0.48) (-0.56) (0.21)
LINE 0.963* 0.851# 1.734* 0.925* 1.047*
(2.06) (1.81) (3.03) (2.2) (4.44)
F (CONS=0, 1.46** 0.18** 1.57** 0.32** 0.02**
LINE=1)
Adj. R2 0.06 0.043 0.154 0.064 0.084
No. of obs 52 51 46 57 206
* Significant at 5% level. # Significant at 10% level. ** CONS significantly equals zero and LINE significantly equals
one at 5% level (F2,30=3.32, F2,106=3.09, F2,650=3.03).
15Table 4b Weak efficiency estimates for the AFL fixed odds market
Weak efficiency estimates for 2001-2004 AFL seasons in two models: A. CLOGIT and B. LPM. Y represents (1) home
teams, (2) favorites, (3) home favorites, (4) home longshots, and (5) neutral longshots. Z values in parentheses.
eb1 ln NPRICEit + b2Yit
A. Pijt = b ln NPRICE jt + b2Y jt
ϕijt
eb1 ln NPRICEit + b2Yit + e 1
B. WINijt = c0 + c1NPRICEijt + εijt
2001 2002 2003 2004 2001-2004
A. C L O G I T
(1) Home teams
lnNPRICE 1.144* 1.082* 1.242* 1.085* 1.13*
(4.96) (4.58) (5.25) (5.02) (9.89)
HOME -0.09 0.457* 0.243 0.272 0.22*
(-0.43) (2.19) (1.13) (1.29) (2.09)
χ2 0.44** 5.42** 3.03** 2.22** 7.39
(lnNPRICE=1,
HOME=0)
Log likelihood -111.5 -106.8 -103.1 -106.7 -430.1
No. of obs 370 366 364 370 1470
(2) Favorites
lnNPRICE 1.089* 0.616 1.485* 1.332* 1.137*
(3.01) (1.62) (3.87) (3.65) (6.15)
FAV 0.021 0.543# -0.158 -0.162 0.058
(0.07) (1.86) (-0.56) (-0.56) (0.41)
χ2 0.27** 4.25** 2.12** 0.92** 3.45**
(lnNPRICE=1,
FAV=0)
Log likelihood -111.6 -107.4 -103.6 -107.4 -432.2
No. of obs 370 366 364 370 1470
(3) Home favorites
lnNPRICE 1.098* 0.767* 1.216* 1.092* 1.041*
(3.98) (2.83) (4.27) (4.23) (7.68)
HOMEFAV 0.024 0.873* 0.184 0.146 0.304#
(0.07) (2.6) (0.56) (0.47) (1.89)
χ2 0.27** 7.22 2.19** 0.85** 6.66
(lnNPRICE=1,
HOMEFAV=0)
Log likelihood -111.6 -105.6 -103.6 -107.4 -430.5
No. of obs 370 366 364 370 1470
(4) Home longshots
lnNPRICE 1.103* 1.3* 1.457* 1.297* 1.283*
(4.87) (5.18) (5.7) (5.68) (10.74)
HOMELONG -0.041 0.362 0.599 0.671# 0.397*
(-0.11) (1) (1.55) (1.85) (2.15)
χ2 0.27** 1.8** 4.02** 3.94** 7.58
(lnNPRICE=1,
HOMELONG=0)
Log likelihood -111.6 -108.7 -102.6 -105.8 -430
No. of obs 370 366 364 370 1470
(5) Neutral longshots
lnNPRICE 1.119* 1.181* 1.31* 1.108* 1.179*
(4.69) (4.69) (5.28) (4.79) (9.75)
NEUTRALLONG 0.025 -0.096 -0.017 -0.175 -0.062
(0.08) (-0.3) (-0.05) (-0.53) (-0.39)
χ2 0.27** 0.94** 1.89** 0.91** 3.41**
(lnNPRICE=1,
NEUTRALLONG=0)
Log likelihood -111.6 -109.1 -103.8 -107.4 -432.2
No. of obs 370 366 364 370 1470
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