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Here's how you can make a PDF with the To obtain an effective betting strategy, the winning probabilities should account for the favourite-longshot bias. Clarke et al. According to Cain et al. If the bookmakers create a bias in their odds because of the presence of insiders, this would suggest that there is no favourite-longshot bias when the Shin model is used.

However, Cain et al. Basic normalization A simple way to obtain winning probabilities that add up to 1 is to divide the implied probabilities by their sum. The winning probabilities determined from betting odds using basic normalization are referred to as scaled probabilities. This method is also known as simple standardization. Suppose one assumes a hypothetical group of insider traders with perfect knowledge of the outcome. Nowadays, more advanced approaches still suggest the existence of insider trading in betting markets and other financial markets, see, for example Coleman , Schnytzer et al.

In the model, bookmakers want to limit their exposure to insiders, especially in the case of low probability-high payout events. They do so by reducing the odds offered for such events. The insiders are a fraction z of the population, which is the measure of the incidence of insider trading. Shin and Clarke et al. Test strategies To test whether standardized probabilities or Shin probabilities provide a better estimate for the actual probability of the outcome, we extend the test strategy in Koning and Boot For simplicity, we take a home win in a soccer match as the leading example.

In both cases, the implied probabilities depend on the stated decimal betting odds. This predictor could also be derived from an extensive regression model see for example, Goddard and Asimakopoulos ; Goddard , or from subjective beliefs of a particular fan.

Some examples of this approach are Blochwitz et al. This binning approach has two major disadvantages: it does not allow for observation specific covariates that may cause Eq. Considering the literature on the existence of the favourite-longshot bias, we would like to allow for deviations from Eq. We take model 8 as our point of departure, and extend it to address these two disadvantages.

Instead of estimating Eq. In the context of informational efficiency of betting markets, we need to test whether the implied probability is an unbiased estimator for the actual probability of outcome. Also, other covariates should not determine the probability of the outcome. In other words, the coefficients of any additional covariates in Eq. Adding covariates will improve the power of the test see Sauer et al.

We take it to be a restricted cubic spline see for example Harrell These splines are characterized by a number of knots, where the curvature of the relation may change. The restricted cubic splines are defined by 1 being cubic splines piecewise third-order polynomials between the knots in the range of the independent variable, 2 being a linear function beyond the boundary knots and 3 being continuous and forcing the first and second derivative of the function to agree at the knots.

The model that is actually estimated is based on a linear transformation of the matrix with all covariates in Eq. With a slight abuse of notation, we only present the estimation results of the transformed model, as that model is exactly identified. Moreover, any information not incorporated in the standardized probabilities or Shin probabilities can be added as the vector with additional covariates w.

Related to our approach is the one in Angelini and De Angelis , Elaad et al. These papers follow the Mincer—Zarnowitz Mincer and Zarnowitz regression based forecast evaluation framework. The model is estimated using weighted least squares, reflecting heteroscedasticity of the dependent variable. For example, our approach should be able to distinguish between a favourite bias and a favourite-longshot bias separately. Also, our estimation approach a logit model reflects the discrete outcome of a match.

This, and the fact that we allow for a nonlinear relationship between the outcome and implied probability, should give our test higher power to the one based on the linear probabilty model in Eq. Note that our methodology to assess the precision of predictions is very general. Empirical results We illustrate our approach to testing informational efficiency of betting odds using data from football-data.

Both leagues represent the highest level of professional soccer in these countries. Both samples of odds were offered by this bookmaker in the UK and not in Spain in the case of La Liga. During most of the time period considered there was no open European market market for offering betting opportunities on sports matches. Even though Spanish bettors could have arranged access to these betting opportunities, such access would require technical proficiency and knowledge of English language.

For them, local betting alternatives will be easier to access. Implied home win probabilities according to either basic normalization Eq. All calculations are done in R R Core Team We start with an analysis of the English Premier League. Results for both models to derive implied home win probabilities from betting odds are presented in Table 1.

Because of the size of the dataset, we take a 0. We conclude that in both cases the hypothesis that betting odds implied probabilities are unbiased for the probability of outcome cannot be rejected. Table 1 Estimation results, basic logit model English Premier League Full size table In a second step, we extend the basic logit model with nonlinear terms, so that a potential favourite-longshot bias can be accommodated.

For this reason, we proceed with the semi-parametric approach by adding restricted cubic spline terms to the basic specification. We choose knots to be at the 0. This choice of quantiles differs from the one suggested in Harrell , since we have moved the first and last knot a bit more to the beginning and end of the distribution. Since the restricted cubic splines regression assumes a linear function beyond the boundary knots, a favourite-longshot bias can be better accounted for using these fixed quantiles.

In Table 2 , we present the results of the extended logit model. More interesting are the last two lines of Table 2. The first of those gives the Wald statistics corresponding to the null hypothesis that all coefficients of the nonlinear terms are jointly 0. This hypothesis cannot be rejected in both cases, judging from the p values that are reported 0. Consequently, adding nonlinear terms has not improved the fit of the model. Also in this case, the null hypotheses cannot be rejected in both models: there is no evidence of a favourite-longshot bias in the market for home wins of the English Premier League.

Table 2 Estimation results, model with restricted cubic spline English Premier League Full size table The first and second step models are shown graphically in Fig. We graph the relation between the scaled and Shin probabilities of outcome horizontal and the probability of outcome vertical. The red line in the left two panels is the estimated logit model from Table 1 , and in the right two panels the models with the nonlinear terms are drawn.

The knots are indicated by the dashed vertical lines. Even though the red line in the right two panels suggest some nonlinearity, this nonlinearity is statistically not significant. Since nonlinearity is not significant, we do not incorporate the restricted cubic splines terms. Even though one can think of many different covariates that are not priced into the betting odds, we consider only two, as an example of the method.

This measure will vary between 0 last three matches are lost and 3 last three matches have been won. Results of this specification are given in Table 3. Again, the Wald tests are most interesting, and we find that the additional covariates are not jointly significant. This is our best assessment of the unbiasedness of standardized and Shin probabilities, as the last test allows both for a potential favourite-longshot bias, and influence of other covariates.

Table 3 Estimation results, model with additional covariates English Premier League Full size table In a second example, we have applied the same approach to betting data for matches of the Spanish La Liga. The results of the logit model without linear terms are shown in Table 4.

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