Comparing two methods for testing the efficiency of sports betting markets

Sports betting markets can be considered strongly efficient if expected returns on all possible bets on an event are equal. If this form of efficiency holds, then there is a direct mapping from betting odds into probabilities of outcomes of sporting events. We compare two regression-based methods for testing this form of efficiency that have been used in previous research: One that uses normalized probabilities as the explanatory variable for event outcomes and one that uses the inverse of the decimal odds. We show that the normalized probability method produces good tests of the null hypothesis of strong market efficiency but that the inverse odds method does not, with results biased against finding favorite-longshot bias. We illustrate this finding using large datasets of bets and outcomes for tennis and soccer and also with realistic simulations.