Copyright © 2007 The University of Buckingham Press

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The Journal of Prediction Markets (2007) 1, 43–59

COMPARING THE EFFECTIVENESS OF ONE-

AND TWO-STEP CONDITIONAL LOGIT MODELS
FOR PREDICTING OUTCOMES IN
A SPECULATIVE MARKET
M. Sung† and J.E.V. Johnson‡
Centre for Risk Research, School of Management, University of Southampton, Highfield,
Southampton, SO17 1BJ, United Kingdom

This paper compares two approaches to predicting outcomes in a speculative market, the horserace
betting market. In particular, the nature of one- and two-step conditional logit procedures involving a
process for exploding the choice set are outlined, their strengths and weaknesses are compared and
their relative effectiveness is evaluated by predicting winning probabilities for horse races at a UK
racetrack. The models incorporate variables which are widely recognised as having predictive power
and which should therefore be effectively discounted in market odds. Despite this handicap, both
approaches produce probability estimates which can be used to earn positive returns, but the two-step
approach yields substantially higher profits.

I. INTRODUCTION
Establishing the extent to which a financial market incorporates
information provides important clues to the manner in which it operates and
it is widely recognised that horserace betting markets, which share many
features in common with wider financial markets, can provide a valuable
window on speculative market behaviour (e.g. Snyder, 1978; Hong and Chiu,
1988; Law and Peel, 2002). Sauer (1998, p 2021), for example, observes:

“wagering markets are especially simple financial markets, in

which the pricing problem is reduced. As a result, wagering
markets can provide a clear view of pricing issues which are
complicated elsewhere.”

Two of the distinctive features of horserace betting markets which make it

possible to understand behaviour more clearly than in other financial markets
is the generation of an unequivocal outcome (a winner) within a definitive
period and the availability of a large number of essentially similar markets
(races) for analysis. These features enable market efficiency to be tested by
measuring the appropriateness of the asset’s price (market odds) against race
outcome. As a result, an extensive literature has developed addressing weak-,
semi-strong- and strong-form efficiency in horserace betting markets.
However, there have been relatively few studies exploring the extent to
which horserace bettors discount, in market odds, a combination of

†Tel: þ 44(0) 23 8059 9248 Fax: þ 44(0) 23 8059 3844 Email: ms9@soton.ac.uk
‡Tel: þ 44(0) 23 8059 2546 Fax: þ 44(0) 23 8059 3844 Email: jej@soton.ac.uk

q The University of Buckingham Press (2007)

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

fundamental variables associated with horses, their jockeys and trainers and
the race conditions. The majority of these studies employ a one-step
modelling process which involves regressing measures of past performance
(e.g. past finish position) on information derived from fundamental variables
alongside market generated probabilities (e.g. Bolton and Chapman, 1986;
Chapman, 1994; Gu, Huang and Benter, 2003). However, Benter (1994)
advocated the use of a two-step procedure, which involves developing a
model based solely on fundamental variables to predict winning probabilities.
These probabilities are then used as inputs to a second stage model which also
incorporates market generated probabilities. He argues that such a process
produces more accurate predictions. Benter’s highly successful betting
operation in Hong Kong provides anecdotal evidence to support this view and
two stage models developed by Edelman (2003) and Sung, Johnson and Bruce
(2005) have produced encouraging results. However, to date no comparison of
the accuracy of winning probabilities from one- and two-step modelling
procedures has been undertaken. This is clearly an important omission, since a
modelling technique which captures the full information content of
fundamental and market-generated variables is more likely to demonstrate
the true degree of market efficiency. This paper aims to fill this important gap
by evaluating the effectiveness of these two modelling approaches in
predicting winning probabilities at a racetrack in the UK and their ability to
reveal market inefficiency.
To achieve this, the paper is structured as follows: The one- and two-step
modelling procedures on which this study focuses are outlined in section II,
along with their relative strengths and weaknesses. The data and explanatory
variables used to develop parallel one- and two-step models are described and
justified in section III, together with the procedures used to assess the relative
predictive power of these two approaches. In section IV, the results of model
estimation and out-of-sample testing are reported and discussed. Some
implications and conclusions are developed in section V.

II. ONE- AND TWO-STEP MODELLING

(a) One- and Two-step Conditional Logit Modelling Procedures
The modelling procedure which forms the basis of the one- and two-step
procedures which are compared here is the most widely used in assessing the
degree of semi-strong-form efficiency in racetrack betting markets, namely,
conditional logit. The aim of a conditional logit model is to predict a vector of
winning probabilities peij ¼ ð pe1j ; pe2j ; . . . ; penj j Þ for race j, where peij is the
estimated model probability of horse i winning race j and nj is the number of
horses in the race j. These probabilities are estimated on the basis of a vector
of m variables: x ij ¼ ½xij ð1Þ; . . . ; xij ðmÞ
, capturing information associated
with each horse. Since horse races are competitive, an efficient probability
estimate of horse i’s chance of winning race j is more likely to be obtained if

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

its chance of winning is regarded as being conditional on the information

available for the other runners in race j. To achieve this, a ‘winningness index’
Wij for horse i in race j is defined as follows:
Xm
ð1Þ W ij ¼ b x ðkÞ þ 1ij
k¼1 k ij

where bk is a coefficient which measures the relative contribution of

information xij(k) to horse i’s chance of winning and 1ij is unperceived
information. If Wij is defined such that the horse with the highest value of the
index wins race j then it can be shown that, if error terms 1ij are independent
and distributed according to the double exponential distribution, the
probability of horse i winning race j is given by the following conditional
logit function (McFadden, 1974):

Pm 
e exp k¼1 bk xij ðkÞ
ð2Þ pij ¼ Pnj
Pm 
i¼1 exp k¼1 bk xij ðkÞ

where the bk are estimated using maximum likelihood procedures. The

conditional logit model has been successfully employed for a range of discrete
choice problems (McFadden, 1974) including a number of studies estimating
the winning probability of racehorses (e.g. Figlewski, 1979; Bolton and
Chapman, 1986; Edelman, 2003).
The one- and two-step procedures differ in terms of when and how
fundamental information concerning the previous performances and
preferences of the horse, its jockey or trainer and race conditions (represented
below in equation (3) as m 2 1 fundamental variables, yij(k)) are combined
with market-generated information (usually employed in the form of
normalised probabilities derived from closing market odds, psij ). In a one-step
procedure the two types of information are combined in a single conditional
logit model of the following form:
s
Pm21
expb bm ln pij þ k¼1 bk yij ðkÞc
ð3Þ peij ¼ Pn h Pm21 i
j s
i¼1 exp bm ln pij þ b
k¼1 k ij y ðkÞ

Benter (1994) was the first to develop a computer model for predicting
winning probabilities of horses in two-steps. The two-step procedure involves
first developing a conditional logit function of the form given in equation (2),
and simply employing the m 2 1 fundamental variables yij(k) (Benter, 1994).
This provides an estimate of the probability of horse i winning race j, pijf ,
which is based solely on the fundamental variables. However, according to
Benter (1994), the fundamental probability consistently diverges from the
observed win percentage for each odds category. As a result, an adjustment
from the fundamental probability to the unobserved true winning chance of a
horse is necessary. Consequently, a second-step is required, incorporating the
natural logarithm of the fundamental model probability, ln ðpijf Þ, as well as

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

the natural logarithm of the normalised closing odds probability, ln ðpsij Þ, based
on a second set of races. Consequently, the final estimated model probability
for horse i in race j is obtained as follows:
    
exp a ln psij þ g ln pfij
ð4Þ peij ¼ Pn     
j
i¼1 exp a ln ps
ij þ g ln pfij

where a and g are parameters to be estimated using maximum likelihood

procedures. The second-step model is designed to capture the subtle
relationship between these two explanatory variables and the outcome of a
race.

(b) Strengths and Weaknesses of One- and Two-step Conditional Logit

Modelling
In order to test the hypothesis that racetrack betting market is semi-strong
form efficient, it is clearly necessary to develop a model which combines both
fundamental and market-generated information. The merits of combining this
information in a one-step model are not only its simplicity but also the
opportunity it affords for examining the significance of each individual
fundamental variable in an explicit manner; since probabilities derived from
the final market odds also appear in the model. This facilitates understanding
of how bettors utilise publicly available information. In particular, the
fundamental information which has been ignored or under-weighted by the
betting public can be identified. Another technical advantage of the one-step
modelling process is that it permits the use of a larger training sample and this
can improve model accuracy. This is particularly true for a conditional logit
model as it treats one race rather than one horse as an observation during
estimation. The two-step modelling procedure, on the other hand, requires that
the training sample is split in two, one for each step; this is required in order to
overcome the potential problem of over-fitting (Benter, 1994). The one-step
model might therefore have particular merit when only a limited sample of
races is available for analysis.
Despite the potential benefits of the one-step procedure indicated above, it
could also be argued that combining all the fundamental variables with the
odds variable may result in counterintuitive signs for the model parameters
due to the variables being highly correlated. This may increase the difficulty
of interpreting the results (Benter, 1994). A second important practical
difficulty associated with the one-step process is that in order to use the
probabilities generated by this process to bet it is necessary to have a good
estimate of the final market odds. Benter (1994) suggests that these can be
obtained from market odds prevailing one or two minutes prior to the start of
the race. However, this would not allow sufficient time in which to run a
complete model incorporating all fundamental variables and a variable
derived from final odds estimates, and then place the bet. It can, therefore, be

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

argued that the one-step model would be more difficult to implement in real time.
Step one of the two-step model procedure involves estimating the probability
derived from fundamental variables; these data are available several hours before
the race begins. Step two, which combines probability estimates derived from the
fundamental model and from the final market odds (or those prevailing one or two
minutes before the race start) could be developed in a few seconds, which would
permit the model probabilities to be used to bet. This may therefore represent a
more practical method for implementing a betting strategy in real time.
Therefore, it could be argued that the two-step model is the one which really tests
whether a market is inefficient.
The two-step modelling procedure clearly offers some important
advantages over a one-step procedure but it can only be used to examine
the collective significance of all the fundamental information (by observing
the significance level of the coefficient of the fundamental probability term in
the second-stage model); the marginal significance of each of the individual
fundamental variables cannot be discerned.
As discussed, both the one- and two-step modelling procedures have their
own strengths and weaknesses. However, there has been no investigation
which compares the predictive power of these modelling approaches. The next
section outlines the data and procedures employed here to undertake such an
investigation.

III. DATA AND PROCEDURES

(a) Data
One of the distinguishing features of a good model for making probability
predictions is that it is able to extract the full value from underlying
information. In particular, to make an accurate assessment of horserace
betting market efficiency, it is vital that the model fully utilises the
fundamental and market-generated information. In this regard, in order to set a
difficult test for the one- and two-step modelling procedures evaluated here, it
was decided to limit the set of fundamental variables included in the models to
those included in Bolton and Chapman’s (1986) seminal paper on multi-
covariate modelling in a horseracing context. This paper inspired several other
researchers and practitioners to develop more sophisticated models (the most
notable and successful example being Benter (1994)). It is clear, therefore,
that the fundamental variable set which Bolton and Chapman (1986)
employed has been in the public domain for a considerable period and
according to the efficient market hypothesis it is likely that the betting public
now discounts much of this information in market odds. Consequently, a
modelling procedure which is able to use information extracted from this
fundamental variable set to make predictions which yield abnormal returns
might be regarded as of particular merit. A full list of the variable set used in
the current study is given in Table 1.

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

Bolton and Chapman’s (1986) study predicted winning probabilities at US

racetracks whose topography, configuration and surface are highly
standardised, whereas the current study employs data from the UK where
racetracks are far less uniform. Consequently, to minimise this discrepancy,
data is drawn from races run at one racetrack, Wolverhampton, whose
configuration, topology and surface are similar to that of the US tracks. The
dataset contained details of 16431 horses which ran in 1675 flat races during
the period January 1995 to August 2000. The number of horses in each race
varies from 2 to 13, with a mode of 12. The final market odds for horses in the
sample range from 0.17/1 to 100/1 with a mean value of 13.64/1.
The dataset is split into two parts. The first part, involving 1110 races
(10856 horses) run between January 1995 and December 1998, is used to
develop the conditional logit models. The second part, involving 565 races
(5575 horses) run between December 1998 and August 2000, is preserved for
out-of-sample testing. The one-step model is estimated using all the
observations in the training dataset, but the two-step model requires the
training dataset to be further divided: the fundamental model being estimated
using races run between January 1995 and December 1996 (555 races, 5524
horses) and the model combining fundamental model probabilities and
market-generated probabilities being estimated using races run between
December 1996 and December 1998 (555 races, 5332 horses).
The UK betting market consists of parallel bookmaker and pari-mutuel
markets with odds being generated in both markets. However, the bookmaker

TABLE 1
DEFINITIONS OF THE INDEPENDENT VARIABLES EMPLOYED IN THE ONE- AND
TWO- STEP MODELS

Independent variable Variable definitions

Market-generated variable
ln ðpsij Þ The natural logarithm of the normalised final odds probability
Fundamental variables
pre_s_ra Speed rating for the previous race in which the horse ran
avgsr4 The average of a horse’s speed rating in its last 4 races; zero when
there is no past run
draw Post-position in current race
eps Total prize money earnings (finishing first, second or third) to
date/Number of races entered
newdis 1 indicates a horse that ran three or four of its last four races at a
distance of 80% less than current distance, and 0 otherwise
weight Weight carried by the horse in current race
win_run The percentage of the races won by the horse in its career
jnowin The number of wins by the jockey in career to date of race
jwinper The winning percentage of the jockey in career to date of race
jst1miss 1 indicates when the other jockey variables are missing; 0 otherwise

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

market is by far the larger and it has been suggested that informed bettors are
more likely to bet in this market (e.g. Bruce and Johnson, 2005).
Consequently, in this study, final market odds in the bookmaker market are
used for model development.
(b) Procedures
The sample involves a choice set of observations, whereby ‘nature’
chooses a winner of each race but the sample size (555 races) is relatively
small. In traditional conditional logit modelling only information concerning
which horse wins the race is employed but Chapman and Staelin (1982),
describe an ‘explosion process’ which can be used to exploit extra information
from the original ranked choice sets (i.e. from horses in a race finished 2nd, 3rd
etc.) without adding too much random noise. This method involves
considering the finishing position of each horse in a given race as a set of
mutually independent choices. Consequently, it is assumed that the horse
which finished second would have won the race if the horse finishing first had
not participated in the race. For example, an explosion from depth one (the
original race) to three can produce two ‘extra races’ by sequentially
eliminating the ‘winner’ from the pared down races (i.e. a race where the
original winner is eliminated and a race where the original winner and second
are eliminated). This is clearly a valuable process as it increases the number of
independent choice sets, which results in more precise parameter estimates.
However, there is a limit to the depth to which races can be exploded since
the latter finishing positions may not truly reflect the competitiveness among
the remaining horses. This arises since it may become obvious to a jockey that
his/her mount will not finish in the first three (where prize money is awarded);
the jockey then has little incentive to ensure that the horse achieves its best
possible finish position. In fact, there may be positive incentives not to do this,
as it helps to conceal the horse’s true ability, which increases the value of the
horse’s connections’ (owners, trainers etc) private information (which they
can exploit in the betting ring in subsequent races). As a result, the maximum
depth of explosion to which a race is exploded is restricted to three in this
study (Bolton and Chapman, 1986).
For certain sets of races it may not even be appropriate to explode to level
three (since this process may introduce too much random noise) and a
statistical measure which can be used to determine the appropriate depth of
explosion is suggested by Watson and Westin (1975). This method involves
iteratively testing the hypothesis that the maximum likelihood estimates for
each individual subgroup of races are equal; the explosion process is
continued until the hypothesis is rejected. This procedure involves
determining the log-likelihood (LL) values for models estimated on the
following separate subgroups of races: (i) all runners included: (E ¼ 1);
(ii) runners which finished first excluded: (E ¼ 2) – (E ¼ 1); (iii) runners
which finished first and second excluded: (E ¼ 3) – (E ¼ 2); (iv) races
falling into categories (i) and (ii) pooled: (E ¼ 2); (v) races falling into

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

categories (i), (ii) and (iii) pooled: (E ¼ 3). The statistic used to test the
hypothesis that the maximum likelihood estimates for each individual
subgroup of races are equal compares, for example, the LL of explosion depth
two (E ¼ 2) with that from the subgroups [(E ¼ 1) and (E ¼ 2) – (E ¼ 1)]
combined. This statistic is, therefore, defined as 2 2 {LL(E ¼ 2) – [LL
(E ¼ 1) þ LL((E ¼ 2) 2 (E ¼ 1))]} (Chapman and Staelin, 1982), and
follows the chi-square distribution with the degrees of freedom equal to the
number of parameters in the conditional logit model (Wald, 1943). A
particular subset of races is only exploded to a depth where the hypothesis that
the maximum log-likelihood estimates for all the subgroups of races are equal
is not rejected.
The aim of the paper is to compare the predictive ability of one- and two-
step conditional logit models. The approach is, therefore, to use the exploded
data sets to build both types of model. For the one-step model, the appropriate
depth of explosion is determined by calculating the test statistic discussed
above for the whole test sample of 1110 races (January 1995- December
1998). For the two-step model, the subset of 555 races (January 1995–
December 1996) used to estimate the fundamental model and the subset of
555 races (December 1996 – December 1998) used to estimate the model
incorporating fundamental and market-generated probabilities are both tested
separately to determine the appropriate depth of explosion.

(c) Model Evaluation

The predictive ability of the two models is compared by developing a
betting strategy based on model predictions. In particular, a Kelly wagering
strategy (Kelly, 1956) is employed, based on probabilities derived from the
one and two-step models. The Kelly strategy ensures that total wealth grows
optimally at an exponential rate in the long run with zero possibility of ruin.
A Kelly betting strategy involves betting a proportion of total wealth on a given
runner. As wealth levels increase later in the sequence of bets, very large absolute
value bets can be recommended. To ensure that the success of one of the
modelling procedures is not biased by one or two large wins later in the sequence
of bets, a Kelly strategy without re-investment is employed; the wealth level is
therefore returned to unity after each bet, whatever the outcome of the previous
bet. The accuracy of the winning probabilities estimated by the one- and two-step
modelling procedures are assessed by comparing the rates of return obtained by
using these probabilities in a Kelly strategy to bet on the out-of-sample races.

IV. RESULTS AND DISCUSSION

(a) Model Estimates and Model Fit: Two-step Model
1. Step One: Fundamental Model
The coefficients for the ten fundamental variables in the conditional logit
model were estimated for depths of explosion one, two and three, respectively,

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 TABLE 2
COEFFICIENTS AND TEST STATISTICS OF CONDITIONAL LOGIT MODELS INCORPORATING FUNDAMENTAL VARIABLES FOR EXPLOSION DEPTHS OF 1, 2, AND 3
( STEP- ONE OF A TWO- STEP PROCEDURE)

Explosion Strategy
Variable1 t-ratio p-val.2
E¼1 E¼2 E¼3
Coef. Std. error Coef. Std. error Coef. Std. error
pre_s_ra** 0.1968 0.0684 0.2003 0.0490 0.2005 0.0405 4.96 0.000
avgsr4** 0.3288 0.0858 0.3751 0.0612 0.3423 0.0505 6.78 0.000
draw** 0.2481 0.0488 0.2462 0.0347 0.2117 0.0282 7.51 0.000
eps 0.1244 0.0907 0.1143 0.0654 0.0881 0.0550 1.60 0.109
newdis** 20.2191 0.0617 2 0.1648 0.0435 20.1968 0.0358 25.50 0.000

51
weight** 0.1631 0.0641 0.1644 0.0456 0.1549 0.0371 4.17 0.000
win_run 0.0237 0.0722 0.0018 0.0523 0.0117 0.0444 0.26 0.792
jnowin** 0.2433 0.1278 0.3116 0.0916 0.2454 0.0775 3.17 0.002
jwinper** 0.0776 0.0308 0.0631 0.0226 0.0602 0.0190 3.17 0.002
All rights reserved

jst1miss 2 0.0496 0.0577 2 0.0799 0.0443 20.0175 0.0272 20.64 0.521

Summary Statistics
No. races 555 1,109 1,659
L(u ¼ _0) 2 1,254 22,444 2 3,558
Lðu ¼ u_Þ 2 1,134 22,210 2 3,263
2
PseudoR 0.0958 0.0957 0.0829
PREDICTING OUTCOMES IN A SPECULATIVE MARKET

1
**Significant at the 5% level;
2
*The test statistics are taken from the data for an explosion depth of three.
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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

based on the first subset of 555 races. These results are presented in Table 2.
To evaluate which rank ordered explosion is appropriate, sequential tests of
the hypothesis that the maximum likelihood estimates for individual
subgroups of races are equal are undertaken. These results are reported in
Table 3. The chi-square test statistics for explosion depth two (4.97) and three
(17.71) are both less than the 5% critical value (18.31), suggesting that the
hypotheses that the exploded rank ordered samples follow the same
distribution as the population cannot be rejected at the 5% level. It is,
therefore, valid to explode the choice set to a depth of 3 for the purpose of
model estimation. The value of increasing the number of observations using
the exploding procedure is demonstrated by an increase in model precision;
the estimated standard errors of the model coefficients decrease on average as
the depth of explosion increases by 28 percent (from depth explosion one to
two), and 19% (from depth two to three).
A LL ratio test comparing the fundamental variable model estimated for
explosion depth three (shown in Table 2) with one where no explanatory
predictor is incorporated demonstrates that the ten fundamental variables,
collectively, have a significant amount of explanatory power (LL ratio ¼ 590,
x210 ð0:05Þ ¼ 18:31). Of the ten variables, seven are significant at the 5% level.
Two of these are associated with the situation in the current race (i.e. post-
position and the weight carried by the horse), one with the horse’s preferences
(i.e. whether the horse is running at a new distance), two with the historical
performances of the horse (variables involving past speed ratings), and two
with jockey-related variables. The model clearly demonstrates that these
variables have an impact on which horse wins a given race. In addition, the
model appears sensible, since all of the significant variables have coefficients
with the anticipated signs.
2. Step Two: Combining Fundamental and Market-Generated Information
A second-step conditional logit model, including the natural logarithm of
(i) the estimated fundamental probability from the first-step model and (ii) the
normalised probability implied by the closing bookmaker market odds is
developed, based on the second subset of 555 races. The second-step model is

TABLE 3
LOG- LIKELIHOOD VALUES AND TEST STATISTICS FOR DETERMINING THE OPTIMAL EXPLOSION
DEPTH FOR FIRST- STEP MODEL ESTIMATES

x210 (.05)
Subgroup of races No. of Races LL Value LL ratio test statistic critical value
(E ¼ 1) 555 2 1,134
(E ¼ 2) 2 (E ¼ 1) 554 2 1,073
(E ¼ 3) 2 (E ¼ 2) 550 2 1,044
(E ¼ 2) 1,109 2 2,210 4.97 18.31
(E ¼ 3) 1,659 2 3,263 17.71 18.31

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 TABLE 4
COEFFICIENTS AND TEST STATISTICS OF CONDITIONAL LOGIT MODELS INCORPORATING FUNDAMENTAL AND MARKET- GENERATED VARIABLES FOR EXPLOSION
DEPTHS OF 1, 2, AND 3 ( STEP- TWO OF A TWO- STEP PROCEDURE)

Explosion Strategy
Variable t-ratio p-val.1
E¼1 E¼2 E¼3
Coef. Std. error Coef. Std. error Coef. Std. error

53
ln ðpsij Þ** 0.7977 0.0632 0.7798 0.0454 0.6951 0.0372 17.19 0.0000
ln ðpijf Þ** 0.1658 0.0625 0.1386 0.0446 0.1742 0.0366 3.11 0.0020
Summary Statistics
Number of Races 555 1,110 1,663
All rights reserved

0)
L(u ¼ _ 2 1,234 2 2,400 23,488
Lðu ¼
_2
uÞ 2 1,060 2 2,090 23,090
Adj R 0.1408 0.1293 0.1141

1
The test statistics are taken from the data for an explosion depth of two.
PREDICTING OUTCOMES IN A SPECULATIVE MARKET
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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

estimated by exploding the 555-race sub-sample to depths of one, two and

three (see Table 4 for detailed results). The resulting log-likelihood values and
the number of races for each depth explosion are displayed in Table 5. The
value of the Watson and Westin (1975) sequential pooling test statistic for
E ¼ 3 is 11.60, which is larger than the chi-square critical value (with 2
degrees of freedom: 5.99) at the 5% level of statistical significance. The
hypothesis that the exploded rank ordered sample at explosion depth 3 follows
the same distribution as the population is therefore rejected. Consequently, it
is only appropriate to explode the 555-race sub-sample to a depth of two (1110
races).
As in step one, the standard errors of the parameter estimates of the
conditional logit model improve as more observations are added (i.e. at
explosion depth 2: see Table 4). In addition, a LL ratio test comparing the
likelihood of the data under the alternative hypothesis that all the parameters
in the model with depth explosion two are not equal to zero, against the
likelihood of the data under the null hypothesis that all the parameters in the
model are equal to zero, indicates that the combination of the two variables
offers significant predictive power (LL ratio ¼ 620, x22 ð0:05Þ ¼ 5:99). In
addition, the coefficients of both the log of the normalised closing bookmaker
market odds probability and the log of the fundamental model probability are
both significant at the 5% level (t-ratio ¼ 17.19 and 3.11 respectively). This
suggests that both these variables provide valuable information for estimating
winning probabilities. This is confirmed by a LL ratio test comparing the log-
likelihood of the combined model (LL ¼ 2 2,090) with the log-likelihood of
a model simply incorporating the log of the probability derived from the final
market odds (LL ¼ 2 2,095); (LL ratio ¼ 10, which is significant at the 1
per cent level (x21 ð0:01Þ ¼ 6:64)).
(b) Model Estimates and Model Fit: Ono-step Model
The one-step model is estimated by exploding the 1110 races run between
January 1995 and December 1998 to depths of one, two and three. The log-
likelihood values of each explosion depth are summarised in Table 6. The test
statistic for depth explosion two is less than the chi-square critical value but
the test statistic for depth explosion three is greater than this critical value,

TABLE 5
LOG- LIKELIHOOD VALUES AND TEST STATISTICS FOR DETERMINING THE OPTIMAL EXPLOSION
DEPTH FOR SECOND- STEP MODEL ESTIMATES

Choice Group No. of Races LL Value LL ratio x22 (.05) critical value
(E ¼ 1) 555 2 1,060
(E ¼ 2) 2 (E ¼ 1) 555 2 1,029
(E ¼ 3) 2 (E ¼ 2) 553 2994
(E ¼ 2) 1,110 2 2,090 1.20 5.99
(E ¼ 3) 1,663 2 3,090 11.60 5.99

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

indicating that it is only appropriate to explode races to a depth of two for

model estimation; expanding the in-sample size from 1110 to 2219 races
(20,601 horses).
Table 7 reports the results of estimating a one-step conditional logit model
(referred to as model A) incorporating the 10 fundamental variables together
with log of the normalised probability derived from final bookmaker market
odds. This model is estimated using data exploded to a depth of two.
A comparison of model A with the model including only fundamental
variables, which is developed at step-one of the two-step procedure (results
presented in Table 2), reveals several interesting issues relevant to betting
market efficiency. Coefficients of two variables (i.e. speed rating for the
previous race in which the horse ran and average career earnings) change to
counter-intuitive signs in model A (suggesting that the public over-emphasise
this information in their assessment of winning probabilities). In addition, four
variables which were significant in the fundamental variable model
(developed at step one of the two-step procedure) are not significant at the
5% level when combined with log of the normalised odds probability (i.e.
weight carried by the horse in current race, speed rating for the previous race
in which the horse ran, the number of wins and the winning percentage of the
jockey throughout career). This suggests that, whilst these factors influence
the winning probabilities, they are fully discounted in odds. Three variables,
which were significant in the model including only fundamental variables
(which is developed at step-one of the two-step procedure), remain significant
in model A. These include post-position, average speed rating of the horse’s
last four runs and whether the horse is running at a significantly longer
distance than in its last four races. This finding implies that post-position is not
fully incorporated into the closing market prices and is consistent with
existing studies which explore the role of post-position (e.g. Quirin, 1979;
Canfield, Fauman and Ziemba, 1987; Betton, 1994). The other two variables
which are significant in both models involve information derived from several
past performances of a horse; these might, therefore, be regarded as relatively
opaque variables, and the fact that the betting public does not appear to fully

TABLE 6
LOG- LIKELIHOOD VALUES FOR DETERMINING THE OPTIMAL EXPLOSION DEPTH FOR MODEL
ESTIMATES, WHICH INCLUDES THE TEN FUNDAMENTAL VARIABLES AND THE LOG OF THE
NORMALISED ODDS PROBABILITY

Choice Group No. of Races LL Value LL ratio x211 (.05) critical value
(E ¼ 1) 1,110 22,106
(E ¼ 2) 2 (E ¼ 1) 1,109 22,038
(E ¼ 3) 2 (E ¼ 2) 1,103 21,978
(E ¼ 2) 2,219 24,149 9.67 19.68
(E ¼ 3) 3,322 26,141 29.98 19.68

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

TABLE 7
COEFFICIENTS AND TEST STATISTICS OF CONDITIONAL LOGIT MODELS INCORPORATING
FUNDAMENTAL AND MARKET- GENERATED VARIABLES FOR EXPLOSION DEPTHS OF 1, 2, AND 3
(IN A ONE- STEP PROCEDURE)

Model A
Variables
Coefficients Std. error t-ratio
ln ðpsij Þ **0.8091 0.0337 23.99
pre_s_ra 2 0.0123 0.0362 20.34
avgsr4 **0.1413 0.0449 3.15
draw **0.1367 0.0251 5.44
eps 2 0.0506 0.0464 21.09
newdis ** 2 0.0960 0.0332 22.90
weight *0.0554 0.0330 1.68
win_run 0.0117 0.0394 0.30
jnowin *0.0620 0.0369 1.68
jwinper 0.0350 0.0225 1.56
jst1miss * 2 0.0708 0.0376 21.88

Summary statistics
No. of races 2219 (20601 runners)
Lð
u ¼ 0
_
Þ 24,844
L u ¼ u_ 24,149
2
PseudoR 0.1436

**Significant at the 5% level.

*Significant at the 10% level.

discount this information in odds is in line with laboratory based research

which indicates that individuals take less account of data which is less readily
discernable when forming their judgements (Tversky and Kahneman, 1974).
Finally, one of the most striking findings associated with model A and the
model incorporating fundamental probabilities and market-generated
probabilities (at step-two of the two-step procedure) is that the log of the
normalised odds probability appears to be highly significant, implying it has a
dominant influence in predicting the race winner.

(c) Comparison of models’ predictive ability based on a Kelly wagering

strategy
The accuracy of winning probabilities predicted by models developed by
the one and two-step procedures (both of which incorporate market-generated
and fundamental variables) are tested by a simulated betting exercise. Both
models are used to predict probabilities for the 565 out-of-sample races run
between December 1998 and August 2000, and these are used as inputs to
implement Kelly wagering strategies without re-investment of profits.
The rates of return obtained for the models developed by the one- and two-step
procedures over this period were 0.96 percent and 17.53 percent, respectively.

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

Clearly both models suggest that the market is semi-strong form inefficient,
but the two-step model appears to capture significantly more information
relevant for winning probability prediction than the one-step model.

V. CONCLUSION
The paper set out to compare the accuracy of probability estimates based on
one- and two-step conditional logit analysis. In particular, the paper assessed the
ability of these models to make accurate assessments of the winning probability
of horses running in flat races in the UK. The models were set a difficult task since
the only independent variables which were employed were those which have
been widely publicised as having an influence on winning probability (i.e.
variables employed in Bolton and Chapman, 1986).
The results suggest a number of important conclusions. Most importantly,
in relation to the central objective of this paper, the analysis conducted here
suggests that the two-step model captures more information contained in the
independent variables; as significantly larger profits were obtained in the out-
of-sample period using a betting strategy based on the predicted probabilities
from this model. These results imply that tests of market efficiency which
employ the one-step model may over-estimate the degree to which market
odds discount fundamental information. One of the reasons for this may be
that the two-step model reduces the impact of multicollinearity. Odds clearly
play a dominant role in predicting the outcome of a race and under these
conditions the correlations between the odds and other fundamental variables
are likely to be high. A model which incorporates odds and fundamental
variables in one step is, therefore, likely to produce more unstable predictions
due to muticollinearity. In addition, the coefficients of the fundamental
variables do not aid understanding of the relationship between winning
probability and the fundamental variables since these coefficients will be
affected by the degree to which bettors account for these variables in odds.
A two-step modelling process separates the odds-related variable from the
fundamental variables and allows these fundamental variables to compete for
importance in one model. This may reduce the problems resulting from
multicollinearity. An additional benefit of the two-step model, as discussed
earlier, is that it can be used in practice to capitalise on any market
inefficiency identified, since step one (the development of a fundamental
variable model) can be undertaken well before the betting period starts. This
enables bettors to complete step two in the last two minutes of the betting
period, allowing for market odds close to the start of the race. The one-step
modelling procedure would take far too long to complete, even with modern
computers, to enable predictions of winning probabilities to be generated in
the last one or two minutes before the race starts. Consequently, it can be
argued that the two-step modelling procedure is the only one of these
approaches which allows for practical application of the model in real time;

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2007, 1 1 THE JOURNAL OF PREDICTION MARKETS

and is therefore the only one which can be applied to test for true market
efficiency.
A further interesting finding reported here is that the variables which are
significant in the one-step model are, to some degree, different from the
variables which are significant in the fundamental model of the two-step
model. For example, jockey-related variables are significant in the
fundamental model of the two-step procedure but are no longer significant
when they are included alongside the odds variable in the one-step model.
This implies that the odds variable incorporates information in relation to the
past performances of jockeys. On the other hand, if the results of the different
modelling procedures had not been compared, the importance of each variable
with and without the odds variable included would not be revealed. In other
words, the empirical comparison between the modelling methods aids our
understanding the important factors affecting the results of horse races and the
extent to which this information is used efficiently.
Finally, the study offers important conclusions concerning the degree of
semi-strong form efficiency in the UK betting market. The results demonstrate
that certain types of information are not accounted for in market odds in the UK.
The one- and two-step models both identified a number of significant explanatory
variables derived from publicly available information. For example, the position
of a horse in the starting stalls (post-position) appears to be significant at the 5%
level. This is a surprising finding for two reasons: (i) post-position is normally
made public the day before the race and this should provide sufficient time for the
public to take this information into account, (ii) Bolton and Chapman (1986)
reported post-position to have non-trivial effects on winning probabilities. In
addition, it is interesting to note that the average speed rating for a horse in its last
four races plays a significant role in forecasting the outcomes of unseen races. To
take this variable into account the betting public need to transform the underlying
data. The fact that they do not appear to account for this variable in their betting
decisions suggests that data which is not readily available (i.e. requires some prior
analysis) may not be acted on by the betting public. The study also confirms the
strongly positive relationship between odds and the likelihood of a horse winning
a race, confirming that the odds variable contains a considerable amount of
information associated with a horse’s relative competitiveness in a race. The
return of 17.53% over the holdout sample period for a betting strategy based on
probabilities predicted by the two-step model suggests that the UK bookmaker-
based betting market is not semi-strong form efficient. This is surprising since the
variables employed have been in the public domain since Bolton and Chapman
published their article in 1986. This finding runs counter to the efficient market
hypothesis which would predict that markets react to the publication of
information and discount it in market prices.
In summary, the results reported here further our understanding of modelling
of outcome probabilities in a speculative market and confirm that levels of
efficiency identified in these markets are highly dependent on the modelling
technique employed.

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PREDICTING OUTCOMES IN A SPECULATIVE MARKET

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