Judging by its significant potential to affect the outcome of a game in one single action, the penalty kick is arguably the most important set piece in football. Scientific studies on how the ability to convert a penalty kick is distributed among professional football players are scarce. In this paper, we consider how to rank penalty takers in the German Bundesliga based on historical data from 1963 to 2021. We use Bayesian models that improve inference on ability measures of individual players by imposing structural assumptions on an associated high-dimensional parameter space. These methods prove useful for our application, coping with the inherent difficulty that many players only take few penalties, making purely frequentist inference rather unreliable.

Volatility break robust panel unit root tests (PURTs) recently proposed by Herwartz and Siedenburg (Computational Statistics & Data Analysis 2008, 53, 137–150) and Demetrescu and Hanck (Econometrics Letters 2012, 117, 10–13) have different performances under both the null and local alternatives. Common practice in empirical research is to apply multiple tests if none is uniformly superior. We show that this approach tends to produce contradictory evidence for the tests considered, making it unclear whether to reject the null. To address this problem, we advocate a combined testing procedure. Simulation evidence shows that the combined test has good size control and closely tracks the more powerful test. An empirical application reinvestigates whether there is a unit root in OECD inflation rates. We find evidence that inflation is stationary for long observation periods, but we cannot reject nonstationarity in most subsets of countries for the last three decades.