Modelling Time-Varying Rankings with Autoregressive and Score-Driven Dynamics

We develop a new statistical model to analyse time-varying ranking data. The model can be used with a large number of ranked items, accommodates exogenous time-varying covariates and partial rankings, and is estimated via maximum likelihood in a straightforward manner. Rankings are modelled using the Plackett-Luce distribution with time-varying worth parameters that follow a mean-reverting time series process. To capture the dependence of the worth parameters on past rankings, we utilize the conditional score in the fashion of the generalized autoregressive score (GAS) models. Simulation experiments show that small-sample properties of the maximum-likelihood estimator improve rapidly with the length of the time series and suggest that statistical inference relying on conventional Hessian-based standard errors is usable even for medium-sized samples. As an illustration, we apply the model to the results of the Ice Hockey World Championships. We also discuss applications to rankings based on underlying indices, repeated surveys, and non-parametric efficiency analysis.