Analysis of football prediction methods

As we have already seen in the literature review, Karlis and Ntzoufras (2008) were the first ones in this context who focused the attention on the difference of goals, instead of modelling directly the number of the home team and the away team goals.
Modelling the differences instead of the scores themselves has two major advantages: the removal of the correlation imposed by the fact that the two opponent teams compete each other, and the absence of the assumption that the goals scored by each team are marginally Poisson-distributed.
Based on the fact that the distribution of goals is well-fitted by the known Poisson distribution, Karlis and Ntzoufras (2008) stated that the difference of goals at the end of a match can be described by another particular discrete probability distribution, that is in fact based on the difference between two Poisson random variables, i.e. the Skellam distribution (see Appendix A for its brief description).
In order to give more support to this aspect, Karlis and Ntzoufras (2008) pointed out that, for the range of mean values observed in football games, the distribution of the difference of goals cannot be sufficiently approximated by another distribution that seems to work well in this context, i.e. the Normal distribution, and therefore inference based on a simple Normal regression can be misleading.

From the Central Limit Theorem we know though that if goals had a Poisson distribution with a large mean, they would be approximately described by a Normal, and then the difference would be a difference of Normals, which is a Normal itself. However, the problem is that with very small means, as occurred in the distribution of goals, this argument does not apply. Although this does not automatically mean that the difference of goals cannot be approximated by a Normal, here we cannot use the Central Limit Theorem to say that a Normal approximation might work well in this context.
However we are still unconvinced that the Skellam distribution can fit the difference of goals so much better than the Normal. For this reason we want to compare in our analysis these two distributions. In order to do it, we are going to fit the final outcome of all matches from our dataset using four different basic models: a model based on the Skellam distribution, as Karlis and Ntzoufras (2008) proposed, a second model based on the Skellam as well but slightly different from the first one, and then two new models both based on the Normal distribution.
Firstly we are going to explain in detail what advantages can be taken by using the difference of goals. Then we will test the effectiveness of the models based on the Skellam distribution against those based on the Normal, in order to understand if the Skellam distribution is absolutely better than the Normal in approximating the difference of goals, as Karlis and Ntzoufras (2008) showed, or, with appropriate formulations, if the Normal distribution can give results similar to the Skellam's or even better.