5. TESTS OF MARKET EFFICIENCY
5.2 Efficiency for whole sample home win odds
First we investigate the B365 home win odds for the whole sample. Keep in mind that the spread is measured as the realized probabilities minus the implied probabilities of these home win odds. The graph of this spread is portrayed below.
Figure 2. B365 home win spread for whole sample.
Spread
-100,00%
-80,00%
-60,00%
-40,00%
-20,00%
0,00%
20,00%
40,00%
60,00%
80,00%
100,00%
1,1 1,2 1,33 1,5 1,66 1,83 2,25 2,6 2,87 3,5 4,75 8
Odds
Percentage
Spread
The figure above depicts the spread of 365 home win odds portrayed against their respective odds. The first point of the graph shows the spread of the B365 home win odd of 1,1. This is the odd given to a Manchester United home win over Sunderland played on the 14th of April 2006. This home win odd of 1,1 occurred only once in the sample. As the match between Manchester United and Sunderland ended in a draw the realized probability for this odd is 0%. The implied probability of this odd is approximately 82,88%. This consequently results in a negative spread of -82,88%, which can be seen in the graph by the first dot. Similarly all the other dots in the graph are created. Due to space limitations it is impossible to portray all the accompanying odds on the X-axis. Similar to the example of the 1,1 odd stipulated above, many of the calculated spread values in the graph are based on too little observations and may
therefore form not a good indication of the efficiency of the odds. The current study, for that reason, bundles several odds into odd ranges, which consequently leads to more observations per ´dotspread´ and thus leads to more trustworthy spreads. The bundles are created in such a manner that there are sufficient observations per range and the ranges do not become too width. Especially in the lower ranges bundles are kept deliberately small. This approach is further used throughout the remainder of the sub samples. To give an indication of the bundling process, table 7 is depicted below. Table 7 portrays the spreads of bundles or ranges of odds. The spread table of the whole sample with spreads calculated per odd separately is portrayed in appendix A. This table is thus used to create figure 2 above.
Table 7. Home win spreads calculated per bundle for the whole sample
B365H Himpl # of times odd Really won Realized Spread from-until Prob occurred in sample with the odd Prob
1,1-1,143 80,62% 6 5 83,33% 2,71%
1,16-1,2 76,57% 19 16 84,21% 7,64%
1,22-1,25 73,48% 32 24 75,00% 1,52%
1,28-1,3 70,66% 41 39 95,12% 24,46%
1,33-1,364 67,74% 28 22 78,57% 10,83%
1,4-1,444 64,24% 54 38 70,37% 6,13%
1,5-1,533 60,34% 70 44 62,86% 2,52%
1,57-1,571 58,06% 51 37 72,55% 14,49%
1,61-1,615 56,57% 63 34 53,97% -2,60%
1,66-1,67 54,83% 74 46 62,16% 7,33%
1,72-1,75 52,89% 81 50 61,73% 8,84%
1,8-1,833 50,28% 146 73 50,00% -0,28%
1,9-1,909 47,92% 95 51 53,68% 5,76%
2 45,58% 101 44 43,56% -2,02%
2,1 43,41% 107 47 43,93% 0,51%
2,2-2,25 40,97% 169 76 44,97% 4,00%
2,3-2,38 39,01% 154 59 38,31% -0,70%
2,4-2,5 36,92% 146 55 37,67% 0,75%
2,6-2,63 34,88% 87 26 29,89% -4,99%
2,7-2,75 33,33% 64 19 29,69% -3,64%
2,8-2,875 32,13% 35 10 28,57% -3,56%
3-3,25 29,34% 60 16 26,67% -2,67%
3,4-3,75 25,51% 30 7 23,33% -2,18%
4-4,75 21,23% 53 17 32,08% 10,85%
5-6,00 17,05% 48 4 8,33% -8,72%
6,5-17 12,13% 34 5 14,71% 2,58%
The home implied probabilities in this table are calculated as the sum of the individual implied probabilities multiplied by the number of times each of these implied probabilities occurred in the bundle divided by the total number of times the odds occurred for that specific odd bundle. This leads to a new spread graph, which is shown below.
Figure 3. Spread for bundles of the B365 home win whole sample odds.
Spread
-15,00%
-10,00%
-5,00%
0,00%
5,00%
10,00%
15,00%
20,00%
25,00%
30,00%
1,1-1,143 1,22-1,25 1,33-1,364 1,5-1,533 1,61-1,615 1,72-1,75 1,9-1,909 2,1 2,3-2,38 2,6-2,63 2,8-2,875 3,4-3,75 5-6,00
Odds
Percentage
Spread
Although there are some outliers, the graph shows a downward trend. To further investigate the meaning of this graph, statistics are used to investigate the significance of the spread. If appropriate, the statistical tests of all the studies set the significance level at 5%. First an independent sample t-test is used. An independent sample t-test investigates how the mean of a quantitative variable differs between two populations or two subpopulations. The current study prefers this test over a paired sample t-test because we have one quantitative variable, which is in this case the percentage change of a home win or the probability of a home win, and we have two sub populations. Furthermore, the essence of the spread measure is to investigate whether the probabilities or percentage change of home wins differ between the probabilities given by the implied probabilities and the probabilities given by the realized probabilities. A paired sample t-test would in this case assume that the realized and implied probabilities are two quantitative variables, which we think is less logical as treating it as two subpopulations. The basic null hypothesis and alternative hypothesis of this test are:
Null hypothesis: H0 :µ1 =µ2 (i.e.D0 =0) Alternative hypothesis: Ha :µ1 ≠µ2 (two−sided)
Where µ1 is the population mean of the realized probability and µ2 is the population mean of the implied probabilities. The hypotheses thus test whether in the population, the average of realized probabilities and implied probabilities is the same, versus whether the average of realized probabilities differs from the average of implied probabilities. In order to test this we let Excel and Spss run the following test statistic:
2
Where x is a point estimate of the mean of the realized and implied probabilities based on the sample used. Similarly S is an estimated standard deviation and the n is simply the number of the samples used. This test assumes that we use independent samples from both subpopulations and that the variables are normally distributed or that both samples are large, i.e. n1, n2 > 30. This is in line with the central limit theorem, which states that if the sample size n is sufficiently large, then the population of all possible sample means is approximately normally distributed, with mean µ x = µ and standard deviation σ x = σ/ n , no matter what probability distribution describes the sampled population. Although the central limit theory ideally argues that the samples should be equal or larger than 30, Bowerman, O’Connell and Hand (2001) argue that these formulas hold exactly if the sampled population is infinite and hold approximately if the sampled population is finite and much larger than 20 times the size of the sample. This is the case concerning our sample. Furthermore, the B365 home win odd whole sample consists of 26 odd bundles. These bundles thus result in 26 implied probabilities and 26 realized probabilities. Although our n’s are not 30 or more they are fairly close to this number, which will be beneficial for the accuracy of our independent sample t-test results. Bowerman et al. also claim that the population of all possible sample means has a normal distribution, if the sampled population has a normal distribution. To be completely sure the current study tests besides the normality of the variable the normality of the subpopulations as well, i.e. to investigate whether they are close enough to the central limit theorem to assume a normal distribution. In other sub samples where the samples are maybe much smaller than 30, the study simply investigates the normality of the variable.
In order to be completely sure of the above, the current study now tests the normality of the whole finite population, i.e. in this case the unbundled odds, and the samples, i.e. in this case the bundled odds and the normality of the variable. Consequently, the independent sample
t-test results are given and finally a non-parametric t-test is used when there is any sign that the rather small sample may endanger the normality assumptions and thus the ability to generalize the test results. For the independent sample t-test we have two samples, the realized probability and the implied probability. For the population descriptive statistics are presented in appendix A for these two finite populations and the samples thereof. The normal distribution histograms of the two populations are presented below. Notice that there are more observations for the home implied probabilities than for the realized probabilities. This is due to the bundling process described earlier. Furthermore in order to calculate the realized probabilities correctly, we have to group together per bundle the amount of times the odds for that specific bundle really occurred. For the realized probability the numbers presented in table 7 are used. For the implied probabilities the whole population is used and not the average numbers presented in table 7.
0,00 0,20 0,40 0,60 0,80 1,00
Realizedprob
0 2 4 6 8 10 12
Frequency
Mean = 0,4619 Std. Dev. = 0,31305 N = 82
Histogram
Figure 4. Histograms and normal distributions of the whole finite population home win 365 odds, for respectively the realized probabilities and the implied probabilities
0,00 0,20 0,40 0,60 0,80 1,00
Himpliedprobunbundled
0 50 100 150 200
Frequency
Mean = 0,4452 Std. Dev. = 0,13503 N = 1.848
Histogram
The histogram of the home win implied probabilities looks more bell shaped than the histogram of the home win realized probabilities. To be completely sure whether the populations are normally distributed descriptive statistics are used. Two important measures that can be applied to check for normality are the skewness and kurtosis of a variable. The skewness and the kurtosis of the realized probabilities are respectively 0,221 and -0,851.
Similarly the skewness and the kurtosis of the implied probabilities are ,058 and ,098. If these outcomes differentiate significantly from zero the populations are probably not normally distributed. One may suspect therefore that the realized probability population is not completely normally distributed. To test whether the populations are skewed to a significant degree the skewness and the kurtosis of each population should not be more than two values of standard error of skewness or of kurtosis. The standard error of skewness and the standard error of kurtosis for the realized probability population are respectively 0,266 and 0,526.
Multiplying these numbers by two results in values of 0,532 and 1,052. Both are bigger than the skewness and kurtosis values reported above, i.e. 0,532>0,221 and 1,052>-0,851. The standard errors for skewness and kurtosis of the implied probability population are 0,057 and 0,114. Multiplying these values by two again indicates that there is no real evidence of positive or negative skewness for the implied probability population. This approach is in line with Tabachnick and Fidell (1996), who estimate the standard error of skewness, ses, and standard error of kurtosis, sek, by the following two equations:
ses N6
=
ses 24N
=
The sampled populations therefore seem to be normally distributed. According to Bowerman, O’Connell and Hand (2001) who claim that the population of all possible sample means has a normal distribution, if the sampled population has a normal distribution, the results of the independent sample t-tests that follow should be accurate. For the independent sample t-test we use the implied probability column and the realized probability column of table 7. This are thus the sample populations. To be complete the descriptive statistics, including tests for normality of these samples are given in appendix A. The Shapiro-Wilk tests of the realized probability sample and the implied probability sample of the sample are respectively 0,775 and 0,926. Both are bigger than 0,05, which according to this test implies that both distributions of the samples are normally distributed, which probably is due to the fact that samples approximate the 30 observations ideally necessary for the central limit theorem to hold. Overall there is no evidence in favor of positive or negative skewness, which should underline the accuracy of the independent sample t-test results.
The descriptive statistics of the quantitative variable, the percentages or probabilities of the odd outcome a home win for the sample, are given in appendix A as well. The descriptive statistics indicate that the Shapiro-Wilk test’s p-value is 0,796. This value is significantly above 0,05, which implies that the home win odd variable is normally distributed. Something that is further underlined by the histogram of the variable below that looks pretty bell-shaped.
0,00 0,20 0,40 0,60 0,80 1,00
variableproboddoutcome
0 2 4 6 8
Frequency
Mean = 0,4808 Std. Dev. = 0,20639 N = 51
Histogram
Figure 5. Histogram and normal distribution of the variable that describes the odd outcome of a home win
Concerning the independency there is no grounded reason to assume that the two subpopulations are not independent, i.e. none of the sample’s matches are played in the Italian League. Furthermore, the following equation holds for the samples.
( ) ( )
(
impliedprobabilities realizedprobabilities)
P(
impliedprobabilities)
P
ly equivalent or
s obabilitie realizedpr
P babilities impliedpro
s obabilitie realizedpr
P
=
= , ,
The former equation indicates that the probability of the events in the realized probability sample, given the condition that the events in the implied probability have occurred, is simply the probability of the realized probabilities. Similarly the second equation can be explained (Bowerman, O’Connell & Hand, 2001).
Overall checking for the assumptions indicates that the results of the independent sample t-test that follow will be pretty accurate. The samples are independent from both subpopulations, both samples approximate the central limit theorem and are normally distributed and finally the quantitative variable is normally distributed. These results thus do not indicate the necessity of applying a non-parametric test.
The results of the independent samples t-test are given below. First the group statistics are given and thereafter the independent samples test results are given.
Table 8. Results of the independent samples t-test for the home win odd samples.
Group Statistics
26 ,5027 ,23108 ,04532
26 ,4715 ,18852 ,03697
RealizedandImplied Realized
Implied variableproboddoutcome
N Mean Std. Deviation
Std. Error Mean
Independent Samples Test
1,456 ,233 ,533 50 ,597 ,03115 ,05849 -,08632 ,14863
,533 48,063 ,597 ,03115 ,05849 -,08644 ,14875
Equal variances assumed Equal variances not assumed variableproboddoutcome
F Sig.
Levene's Test for Equality of Variances
t df Sig. (2-tailed) Mean Difference
Std. Error
Difference Lower Upper 95% Confidence
Interval of the Difference t-test for Equality of Means
The group statistics show that the means between the realized probabilities and the implied probabilities do not differ enormously, i.e. 50,27% versus 47,15%. Thereafter the independent samples test box indicates a p value for the Levene’s test of 0,233, which means that there is no reason to assume that the variances differ. Consequently we focus on the equal variances row and find a p value of 0,597. This high p value thus implies that there is no reason to reject the H0 that the means differ between the realized probabilities and the implied probabilities.
Overall, one may therefore conclude that for the finite population of home win odds, which the current study has selected, there is no real prove of inefficiency in the odds set by bookmakers. Whether this result may hold for the whole population, i.e. all home win odds, is question to debate.
Relating these findings to the spread graph depicted in figure 10, one may argue that some of the positive spreads in the lower odd ranges or bundles are neutralized by some of the negative spreads in the higher odd ranges. This is something that will be further scrutinized in chapter 6. However, it may be interesting now to see whether the independent sample t-test finds significant differences in means between the realized and implied probabilities if we split the sample in two parts. One group for the lower odd ranges and one group for the higher odd ranges. Here we present the results for the lower odd bundles, as the literature especially indicates that spreads will be most significant in these lower odd bundles. The lower group or sample consists of odds until 1,75 and the higher group or sample consists of odds higher than 1,75. Checking the assumption of normality for the variable for the lower half leads to a Shapiro-Wilk test with a p value of 0,502, which indicates that the variable is normally distributed and independent sample t-test outcomes are accurate. The findings of this test for the lower half are portrayed below.
Table 9. Results of the independent samples t-test for the home win odd lower half samples.
Group Statistics
11 ,7273 ,12067 ,03638
11 ,6518 ,09443 ,02847
Realizedand Impliedlowerhalf ,00
1,00 variableproboddo utcomelowerhalf
N Mean Std. Deviation
Std. Error Mean
Independent Samples Test
,419 ,525 1,633 20 ,118 ,07545 ,04620 -,02092 ,17182
1,633 18,907 ,119 ,07545 ,04620 -,02127 ,17218
Equal variances assumed Equal variances not assumed variableproboddo
utcomelowerhalf
F Sig.
Levene's Test for Equality of Variances
t df Sig. (2-tailed)
Mean Difference
Std. Error
Difference Lower Upper 95% Confidence
Interval of the Difference t-test for Equality of Means
The tables clearly indicate that compared to results of the whole sample the p value dropped enormously, which indicates that the means of the two samples differentiate more in these lower bundle ranges. However, the means still do not significantly differ. According to these findings there is thus no real evidence in favor of inefficiently set odds in the home win odd population.
To check the robustness of these findings, the current study also models several regressions and singles out the regression with the highest adjusted R squared. The basic regression models the spread as the dependent variable and the odds as the independent variable. Before the current study starts modeling other multiple regressions it is wise to portray the scatter plot of this basic regression to find out whether there is a positive or negative relation between the spread and odds and whether any other factor may explain the spread (Kerckhoffs, personal communication, October 2002).
Figure 6. Scatter graph of the spread tabulated against the odds. Notice that spread scale should be multiplied by 100 to obtain the true spread percentages.
1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00
Odds
-0,10 0,00 0,10 0,20 0,30
Spread
The figure clearly indicates that there is a downward trend, i.e. if the odds increase the spread decreases, but that there is not a one for one relationship between the two, which probably indicates that other factors influence the relationship as well. It may therefore be wise to model several regressions. Based on this scatter plot the current study creates the basic regression in which the spread is the dependent variable and in which the odds are the independent variable. The second regression contains a dummy variable for the odd ranges.
Similar to the independent sample t-test the sample will be split for odds lower than 1,75, which are assigned a dummy value of 1, and for odds higher than 1,75, which are assigned a dummy value of 0. The third model also includes an interaction variable to test whether the
odds effect depends on the range of odds, i.e. lower or higher odds. This results in the following three regressions:
1) spread =β0 +β1odds+ε
2) spread =β0 +β1odds+β2lowerrange+ε
3) spread =β0 +β1odds+β2lowerrange+β3odds*lowerrange+ε
The current study models all of the above regressions. The full regression results are given in appendix A, because these regressions still relate to the whole sample B365 home win odds.
Plugging the numbers into the regressions leads to the following equations for respectively regression 1, 2 and 3.
1) spreaˆd =0,068−0,016odds
2) spreaˆd =−0,003−0,0000398odds+0,079lowerrange
3) spreaˆd =−0,004+0,000odds+0,114lowerrange−0,24odds*lowerrange
Important to keep in mind is that spread is measured in percentages. Thus that would mean for the first regression that an increase of 1 in odds would lead to a 1,6% decrease in spread.
Before further conclusions are drawn from these regressions it is important to check the adjusted R squared of the regressions and some regression assumptions. The adjusted R squared of regression one, two and three are respectively 0,070; 0,258 and 0,227. The adjusted R square measures how much of the variation in spread is captured by the independent variables and thus tests the utility of the model. According to these findings other factors influence the dependent variable spread as well. Some of the assumptions that the regressions must satisfy are the assumption of correct functional form, the assumption of constant variance, the assumption of normality and the assumption of independence. These assumptions are all properties of the error term ((Kerckhoffs, personal communication, October 2002). Furthermore, regression coefficients have nice properties, provided the error regression assumptions are roughly satisfied. The former assumption indicates that the error is on average zero for any value of the independent variable. The second suggests that for any value of the independent variable the error has the same variance. The third assumptions holds that the error is drawn from a normal distribution. The latter implies that each error is
independent from any other error. To check these assumptions the residuals are plotted against the independent variable odds and against the predicted dependent variable spreads.
Figure 7. Normal distribution of residuals and residuals plotted against odds and spread.
-0,10000 0,00000 0,10000 0,20000
Unstandardized Residual
10
Mean = 2,6020852E-18
Mean = 2,6020852E-18